Liang Huang,, Ying-Cheng Lai1School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China 2School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, United States of America 3Department of Physics, Arizona State University, Tempe, AZ 85287, United States of America
Abstract Quantum Chaos has been investigated for about a half century. It is an old yet vigorous interdisciplinary field with new concepts and interesting topics emerging constantly. Recent years have witnessed a growing interest in quantum chaos in relativistic quantum systems, leading to the still developing field of relativistic quantum chaos. The purpose of this paper is not to provide a thorough review of this area, but rather to outline the basics and introduce the key concepts and methods in a concise way. A few representative topics are discussed, which may help the readers to quickly grasp the essentials of relativistic quantum chaos. A brief overview of the general topics in quantum chaos has also been provided with rich references. Keywords：quantum chaos;relativistic quantum systems;Dirac billiards
Quantum chaos is a branch of fundamental physics investigating the intercapillary field of quantum mechanics, statistical physics, and nonlinear dynamics [1–8]. Even before the establishment of quantum mechanics, in 1913, Bohr proposed quantization rule and used it to successfully predict the energy spectrum of hydrogen atom, which explained the Balmer formula obtained from experimental observations well. Later in 1917, Einstein extended Bohr’s quantization rule to integrable systems with global torus structure in phase space [9]. Then he noticed that these quantization rules are only applicable to integrable systems, and would fail for more general, non-integrable systems [9, 10]. About a half century later, in 1970s, inspired by extensive investigations of nonlinear dynamics and chaos, the issue of how to extend the semiclassical quantization rule to non-integrable systems was perceived again by the community, leading to the development of Gutzwiller’s trace formula that although being measure zero, the unstable periodic orbits play a crucial rule in shaping the quantum spectral fluctuation behaviors [5, 11–23]. There are quantum systems, e.g. quantum billiards, whose classical counterpart can be chaotic. It is thus mystical that since the Schrödinger equation is linear and thus there is no real chaos in the quantum system, how does it emerge in the semiclassical limit? Note that here ‘quantum system’ is specifically for the single particle system described by the Schrödinger equation, where many-body effects are excluded. Alternatively, how does the nonlinear and chaotic dynamics that are ubiquitous in the classical world affect the behavior of the corresponding quantum systems? Are there any indicators in the quantum system that can be used to tell whether its semiclassical limit is integrable or chaotic? The efforts to understand these questions and the results constitute the field of quantum chaos, which has attracted extensive attention during the past half century, and has led to profound understandings to the principles of the classical-quantum correspondence [3].
An intriguing phenomenon regarding classical orbits in quantum systems is quantum scar [15, 24–33], where chaotic systems leave behind scars of paths that seem to be retraced in the quantum world [34–40]. Classically, a chaotic system has periodic orbits, but the chaotic nature renders all the orbits being unstable, i.e. an arbitrarily small perturbation to the particle moving along such an orbit could push its motion out of the orbit completely. A paradigmatic model is the two dimensional billiard system, where a particle moves freely inside the billiard, and reflects specularly at the boundary. Thus the shape of the boundary determines the dynamics of the billiard system [8, 41, 42]. A quantum billiard can be constructed similarly, i.e. a two dimensional infinite potential well whose boundary has the same shape of the corresponding classical billiard. Thus in the short wavelength limit, wave dynamics become ray dynamics, and the quantum billiard degenerates to the classical billiard. Classically, the probability to find a particle moving exactly on an unstable periodic orbit is zero, as the measure of these orbits in the phase space vanishes. This leads to the ergodicity of the dynamics. Semiclassically, the averaged Wigner function can be assumed to take the ‘microcanonical’ form [43], resulting in a Gaussian random function in the coordinate space. Surprisingly, for the quantum counterpart, some of the eigenwavefunctions would concentrate especially around these orbits, forming the quantum scars. The density of a scar along the orbit is a constant and does not depend on ℏ sensitively. But the width of the scars is typically of the order of the wavelength, as ℏ or the wavelength goes to zero, the scar will finally disappear in the random background. ‘In this way, the scar ‘heals’ as ${\hslash }\to 0$’ [24]. Scars have been searched and analyzed in mesoscopic systems [44–54]. Due to the similarity of the equations for different types of waves, scars have been observed in microwave [55–61], optical fiber and microcavity [62–66], acoustic and liquid surface wave systems [67–70]. Quantum scars in phase space could reveal more information about the classical orbit [24] and have been discussed in [35, 71–77], with numerical evidence of antiscars provided in [76]. The statistical properties of scars has been discussed in [78]. Many-body effects in billiard models have been investigated using the Kohn and Sham (KS) equations in the mean-field approximation, i.e. noninteracting particles moving in some fictitious effective field, and scarring could take place when the disorder is weak and the electron density is sufficiently high [30]. For a comprehensive analysis of quantum scars, please see [79, 80]. Besides the conventional situations, scars on quantum networks are found to be insensitive to the Lyapunov exponents [81]. Quantum scars have also been identified by accumulation of atomic density for certain energies in spin–orbit-coupled atomic gases [82], and observed in the two-dimensional harmonic oscillators due to local impurities [83–85]. Quantum many-body scars become a hot topic recently due to the weak ergodicity breaking caused by these states [86–95], which provides a new route to the departure from the eigenstate thermalization hypothesis (ETH) scenario other than many-body localization (MBL). Their analogy in a driven fracton system, namely, dynamical scar, has also been observed [96]. Note that in these investigations, scar states do not relate to classical periodic orbits as in its original setup, but rather a small number of localized states in an otherwise thermalizing spectrum, while in contrast to both ETH where most of the states behave like thermal states, and MBL in which essentially all eigenstates are athermal.
Another cornerstone of quantum chaos is the random matrix theory (RMT), which was mostly developed in 1950s by Wigner [97–99] when dealing with the energy spectrum of complex quantum systems such as the complex nuclei and later in 1960s by Dyson [100–104]. These works form the basis for random matrix theory (see [105–111] for an overview and recent developments). The idea is such that since the interactions are so complex, it could be efficient to approximate the Hamiltonian by a random matrix with elements following certain statistical properties imposed by the symmetry of the system [105–112]. Density distributions of the energy levels are given, and the distributions of the spacings between nearest neighboring levels are investigated extensively due to the findings that they follow different universal functional forms, namely, Poisson [113] or Wigner–Dyson statistics [114], if the corresponding classical dynamics are integrable or chaotic, respectively. In particular, for systems corresponding to classically chaotic dynamics with no additional geometric symmetry, if time reversal symmetry is preserved, the level spacing statistics would follow RMT with Gassian orthogonal ensembles (GOE). If time reversal symmetry is broken, then Gassian unitary ensembles (GUE) would apply. Gassian symplectic ensembles (GSE) would also appear if the system possesses symplectic symmetry. Long range correlations, i.e. the number variance Σ2 and spectral rigidity Δ3, and higher order correlations in spectra are also found to have distinct behavior for quantum systems with integrable or chaotic classical dynamics [106, 115, 116]. Various numerical and experimental evidences are provided [117–121]. Through a series of works [115, 122–125], a connection between classical periodic orbits in chaotic systems and the spectral correlation of the corresponding quantum system represented by the form factor was established, laid the foundation of the universality of the spectral statistics for classically chaotic systems in RMT, which has also been extended into spin 1/2 [126] and many-body situations [127]. These findings are quite prominent as they could serve as the quantum indications of their classical dynamics and symmetry properties. Note that these statements are for generic systems. Non-generic systems, however, may violate such rules [128]. Level spacing statistics of quantum quasidengeneracy has been investigated and Shnirelman peak was identified [129]. Model for experimental level spacing distributions with missing and spurious levels has also been considered [130, 131]. Generalization of the nearest level spacing statistics to open chaotic wave systems with non-Hermitian Hamiltonian was demonstrated in [132]. Between integrable and chaotic systems, there are pseudo-integrable systems [133], and also mixed dynamical systems with both Kolmogorov–Arnold–Moser (KAM) islands and chaotic sea in the phase space [134]. Non-universal behaviors have been noticed [133–138]. In particular, for singular quantum billiards with a point-like scatterer inside an integrable billiard [137], the level spacing statistics may exhibit intermediate statistics, namely, semi-Poisson [139], that exhibits both level repulsion (in the chaotic case) so the probability to find closeby levels are small, and exponential decay for large spacings as in the Poisson distribution (the integrable case) [140–146]. This feature also appears in quantum systems with parameters close to the metal-insulator transition [147]. While for another class of pseudo-integrable systems, namely, polygonal (particularly triangular) quantum billiards that introduce dividing scattering only at the corners [133, 135], although there are conjectures and numerical discussions [139], the spectral statistic is rather complex and is still an open issue for general cases.
There are many other important topics involving quantum billiards, such as nodal line structures and wavefunction statistics [148–157], quantum chaotic scattering [158–166] with experimental demonstrations on two dimensional electron gas (2DEG) [50, 167–170] where the electrons are described by the two dimensional Schrödinger equation, quantum pointer (preferred) states and decoherence [52, 171–175], universal conductance fluctuations [176–180], chaos-assisted quantum tunneling [181–195], effects of electron–electron interaction [30, 195–199], Loschmidt echo and fidelity [200–236], etc. Being described by the same Helmholtz equation for the spatial wavefunction, e.g. $({{\rm{\nabla }}}^{2}+{k}^{2})\psi =0$ or its extensions, where ∇2 is the Laplacian operator and k is the wave number, the quantum billiard can be simulated with other wave systems, such as microwave [55, 59–61, 182, 237–254], light in optical fibers [62, 66] and optical microcavities [255–264], acoustic waves and plate vibrations [68, 265–270], liquid surface waves [67, 69, 70, 271–273], etc.
Other prototypical models that have been investigated extensively in the development of the field of quantum chaos include kicked rotors in terms of diffusion [274–288], entanglement as signatures of referring classical chaos [289–291], experimental realizations [292–295], and other related topics [285, 296–299], and the Dicke model [300–306] and the Lipkin–Meshkov–Glick model [307–310] to account for the many-body effects. In particular, in the phenomena of dynamical localization of kicked rotors with parameters in the classically chaotic region, the momentum localization length has an integer scaling property versus the reduced Planck constant ${\hslash };$ while in the vicinity of the golden cantori, a fractional ℏ scaling is observed, which was argued as the quantum signature of the golden cantori [311–316]. However, in a following work with a random-pair-kicked particle model, it is found that the fractional ℏ scaling can emerge in systems even without the golden cantori structure at all [296], thus it is not a quantum signature of the classical cantori, but has an origin of inherent quantum nature. Abnormal diffusion in one-dimensional tight-binding lattices [317–322] is another interesting subject which is related to the kicked rotor model, as if the diagonal potential is periodic, it can be mapped into a periodically driven time-dependent quantum problem [276]. Ionization of Rydberg atoms [323–344] and dynamics of Bose–Einstein condensation [197, 345–353] have also been investigated extensively in quantum chaos. Due to the interdisciplinary nature, phenomena and effects in quantum chaos have broad applications in nuclei physics [110, 111, 354, 355], cold atom physics [356–361], controlled laser emission [64, 256, 257, 259, 362–364], quantum information [365–370], etc.
Although being an old field, there are still hot topics and astonishing findings emerging recently due to deeper understandings of the theory, the advances of the computation power and the experimental techniques, such as quantum graphs and their microwave network simulations [131, 371–389], universal quantum manifestations for different classical dynamics [125, 284, 380, 390–396], many-body localization [397–404], quantum thermalization [309, 405–417], quantum thermaldynamics [304, 418–426], out-of-time-ordered correlator (OTOC) [427–436], and other related topics [437, 438]. In particular, there are a number of active groups in China publishing recent works in Communications in Theoretical Physics [166, 207, 236, 285, 340, 359, 439, 440], Chinese Physics B [254, 320, 341, 343, 369, 398, 403, 441–443], Chinese Physics Letters [175, 344, 416, 444, 445], and Science China Physics, Mechanics & Astronomy [288, 446–449].
Among the new developments, one interesting field is to expand the broadly investigated quantum chaos into relativistic quantum systems, and see what happens when the relativistic effect cannot be neglected, e.g. what classical chaos can bring to the relativistic quantum systems. This resulted in the still developing field of relativistic quantum chaos. The first study of relativistic quantum chaos was carried out in 1987, by Berry and Mondragon [450], who invented a two dimensional neutrino billiard by imposing infinite mass confinement on the boundary, i.e. the MIT bag model [451], where the neutrino, with spin 1/2, at that time was believed to be massless, and was described by the massless Dirac equation. The reason for not choosing the conventional electric potential for confinement is that the Klein tunneling for relativistic particles will invalidate such a confinement. We shall denote such system as massless Dirac billiards. The intriguing point is that, with the infinite mass potential, the Hamiltonian breaks the time reversal symmetry, leading to GUE spectral statistics. In a following work by Antoine et al [452], a 2D fermionic billiard in a curved space coupled with a magnetic field is considered. Results were obtained under a generalized boundary condition, which confirmed the results by Berry and Mondragon when the boundary condition reduces to the same one. The generalization of the neutrino billiard in a three dimensional cavity has been investigated in [453], with the finding that the orbital lengths seem to be the same as in the scalar spinless case. After Berry and Mondragon’s seminal investigation of neutrino (or massless Dirac) billiards [450], there are only a few works in this topic [452, 453]. Only when graphene and other 2D Dirac materials emerged in 2000s [454–460] rendering experimental observation of the effects possible, the field became prosperous and many different aspects have been investigated extensively, e.g. graphene/Dirac billiards were proposed [461, 462] and has started the enthusiasm in this field with either graphene, or microwave artificial graphene (microwave photonic crystal with honeycomb lattice), or by directly solving the massless Dirac equation in a confined region [463, 464].
Since there are different approaches for relativistic quantum chaos, before we proceed further, we shall define the boundary of the our discussions clearly to avoid confusion. First, it should be distinguished from relativistic chaos, where the motion of the particle is in relativistic regime, i.e. its speed is comparable to the speed of light, but is described by the classical dynamics, not quantum mechanics. There are many interesting results in this topic [465–479], but it is not considered as relativistic quantum chaos. Secondly, quantum chaos has a great motivation regarding classical-quantum correspondence. While for relativistic quantum systems, there are quantities such as spin that does not have a classical correspondence. However, for a relativistic quantum systems and its ‘corresponding’ classical counterpart by just considering the ‘trajectory’ of the particle, the properties of the former can be affected significantly by the classical dynamics, i.e. whether chaotic or integrable, of the latter. Therefore, studies of this field are to reveal how classical dynamics may have influence to the ‘corresponding’ relativistic quantum systems, not to demonstrate the one-to-one correspondence of the two limiting cases. In this sense, there are studies of the relation between entanglement and classical dynamics [289–291, 353, 443, 445, 480–483], spin transport versus classical dynamics [484–489], that although there are no direct one-to-one correspondence, there are significant influence to the behavior of entanglement and spin transport from classical dynamics. Thirdly, the term relativistic quantum chaos is actually not new, but was proposed explicitly about three decades ago in a paper by Tomaschitz entitled ‘Relativistic quantum chaos in Robertson–Walker cosmologies’ [490]. In this work, Tomaschitz found localized wave fields, which are solutions of the Klein–Gordon equation, quantized on the bounded trajectories in the classical geodesic motion. Actually there is a series of works on this line in quantum cosmology [490–498] concerning chaotic quantum billiards in the vicinity of a cosmological singularity in quantum cosmology, where the local behavior of a part of the metric functions can be described by a billiard on a space of constant negative curvature, leading to the formation of spatial chaos. These results could be helpful to understand the early stages of the Universe. Another line is to examine the spectral properties of the quantum chromodynamics (QCD) lattice Dirac operator [499–507] and Dirac operator on quantum graphs [508, 509], where agreement with chiral random matrix theory has been confirmed.
While our focus has been on billiard systems, kicked rotor was an important model for this field and is still an active research topic [442, 510–512]. As a side note, looking for semiclassical treatment of quantum spinor particles has been a persistent effort from 1930s by Pauli [513] to early of this century [514–531]. These semiclassical results provide insights in understanding the spectral fluctuations in graphene nano-structures [532, 533].
2. Spectral statistics
The main results in level spacing statistics are as follows. For a system with energy levels {En, n=1, 2, ⋯}, let $\widetilde{N}(E)$ be the number of levels below E. Generally, the density of the spectra is not uniform, therefore, to make comparison of level spacings meaningful, the spectra needs to be unfolded: ${x}_{n}\equiv \langle \widetilde{N}({E}_{n})\rangle $, where $\langle \widetilde{N}(E)\rangle $ is the smooth part of $\widetilde{N}(E)$. Then in general the statistics of xn follow universal rules depending only on the symmetry of the original quantum system and the corresponding classical dynamics [113, 114], not on the details of the systems. One important quantity is the level spacing distribution P(S), which is the distribution function of the nearest-neighbor spacing, e.g. ${S}_{n}={x}_{n+1}-{x}_{n}$, of the unfolded spectrum {xn}. Another quantity is the spectral rigidity Δ3(L), for detailed calculations please refer to page 5 of [534] and references therein.
2.1. Berry and Mondragon’s result revisited
Berry and Mondragon investigated two dimensional billiard with the African shape (guaranteeing chaotic dynamics with no geometric symmetry) of confined massless spin-1/2 particles, and found GUE statistics [450]. This result is quite surprising as there is no magnetic field or magnetic flux in the system which are typically required to break the T-symmetry. Here the time reversal symmetry is broken by the confinement boundary. The Hamiltonian is given by$\begin{eqnarray}\hat{H}=-{\rm{i}}{\hslash }v\hat{{\boldsymbol{\sigma }}}\cdot {\boldsymbol{\nabla }}+V({\boldsymbol{r}}){\hat{\sigma }}_{z},\end{eqnarray}$where v is the Fermi velocity for quasiparticles or the speed of light for a true massless relativistic particle, $\hat{{\boldsymbol{\sigma }}}=({\hat{\sigma }}_{x},{\hat{\sigma }}_{y})$ and ${\hat{\sigma }}_{z}$ are Pauli matrices, and $V({\boldsymbol{r}})$ is the infinite-mass confinement potential, i.e. $V({\boldsymbol{r}})=0$ for ${\boldsymbol{r}}$ inside the billiard region D, while $V({\boldsymbol{r}})=\infty $ otherwise. The time-reversal operator is given by $\hat{T}={\rm{i}}{\sigma }_{y}\hat{K}$, where $\hat{K}$ denotes complex conjugate. It can be readily verified that $\hat{T}\hat{H}{\hat{T}}^{-1}=-{\rm{i}}{\hslash }v\hat{{\boldsymbol{\sigma }}}\cdot {\boldsymbol{\nabla }}\,-V({\boldsymbol{r}}){\hat{\sigma }}_{z}\ne \hat{H}$, i.e. the free motion of the particle is unchanged, but the confinement potential changes sign and breaks T-symmetry. Microscopically, since the spin is locked with the momentum, at each reflection there will be an extra phase due to the rotation of the spin. While for periodic orbits, if the period, or the number of bouncings at the boundary N, is even, then the accumulated phase along the orbit counterclockwisely and clockwisely are the same modulo 2π, thus both orientations will satisfy the quantization rule simultaneously, i.e. if one orientation is a solution of the system, the other orientation (time-reversed) will also be a solution. This is the same for non-relativistic quantum billiards without magnetic field. However, if N is odd, then the accumulated phase difference for two opposite orientations will be π modulo 2π. Thus if one orientation satisfies the quantization condition, i.e. the overall accumulated phase along the complete orbit is integer multiples of 2π, and is thus a solution of the system, the reversed orientation will have an extra π phase and will not satisfy the quantization condition, thus will not be a solution. This breaks the T-symmetry as it requires that the two orientations must be or be not solutions of the system simultaneously.
Although this effect is quite subtle, it can result in GUE, instead of GOE, spectral statistics [450], which has also been verified by solving the system using other numerical techniques such as direct discretization [536], conformal mapping [535], and extended boundary integral method [537] (see also figure 1). Figure 1 shows the results for three billiards: the circular, the African, and the heart-shaped billiards. The circular billiard is integrable, leads to Poisson statistics. The African billiard is chaotic, leads to GUE. The heart-shaped billiard has a mirror symmetry, although it is not symmetric under the time-reversal operation for the corresponding Dirac billiard, it is symmetric under the joint parity and time-reversal operations. Thus again the GOE statistics are recovered. Since the pseudoparticles in graphene follow the same 2D massless Dirac equation as in [450], it is quite natural to ask whether the graphene billiard follow the same GUE statistics. In this regard, the experimental work [462] by counting the resonance peaks in the transport measurement as approximations of intrinsic energy levels, obtained GUE statistics. However, subsequent numerical calculations provide concrete results of GOE statistics in chaotic graphene billiards in the absence of magnetic fields [534, 538, 539], see figure 2, which lead to further experimental investigations using artificial graphene with much higher accuracy and confirmed GOE statistics [540].
Figure 1.
New window|Download| PPT slide Figure 1.Level spacing statistics of the massless Dirac billiards with the boundary being a circle, the Africa shape, the heart shape for left, middle, and right, respectively. The first row shows the unfolded level-spacing distribution P(S), and the second row shows the spectral rigidity Δ3(L). The green dashed–dotted line, cyan dashed line, and blue solid line are for Poisson, GOE, and GUE. Red staircase curves and symbols are numerical results from 13000 energy levels for each shape by diagonalizing the operator $\hat{H}$ given by equation (1). Adapted from [535] with permission.
Figure 2.
New window|Download| PPT slide Figure 2.(a) Chaotic graphene billiard with Africa shape cut from a graphene sheet. The system has 42 505 carbon atoms. The outline is determined by the equation $x+{\rm{i}}{y}=70a$ $(z+0.2{z}^{2}+0.2{z}^{3}{{\rm{e}}}^{{\rm{i}}\pi /3})$, where z is the unit circle in the complex plane, a=2.46 Å is the lattice constant for graphene. The area is A=1117 nm2. (b)–(d) are the level spacing distribution, integrated level spacing distribution, and the spectral rigidity, respectively, for 664 energy levels in the range $0.02\lt {E}_{n}/t\lt 0.4$, where t is the hopping energy between nearest neighboring atoms. Dashed line is Poisson, solid line is GOE, and dotted line is GUE. The results show clear evidence of GOE. Adopted from [539] with permission.
2.2. Chaotic graphene billiard
Numerically, the graphene billiard is a graphene sheet where the boundary is cut following a specific shape that carries desired classical dynamics. This is effectively an infinite potential well on the graphene sheet: the potential on the boundary is infinite, and the probability to find an electron on the boundary is zero. The general tight-binding Hamiltonian is given by$\begin{eqnarray}\hat{H}=\sum (-{\varepsilon }_{i})| i\rangle \langle i| +\sum (-{t}_{{ij}})| i\rangle \langle j| ,\end{eqnarray}$where i and j are the indices of the atoms (or lattice sites), the first summation is over all the atoms within the billiard and the second summation is over pairs of all necessary neighboring atoms, which could be the nearest neighboring pairs, or the next or next–next nearest neighboring pairs, with their respective hopping energy tij's. Note that hopping energies between atoms close to the boundary may be different from those far from the boundaries. For clean graphene the onsite energy ϵi is identical for all atoms, thus it is convenient to set it to zero. If there are static electric disorders, ϵi will be position dependent. In the atomic (or lattice site) basis $| j\rangle $, the Hamiltonian matrix element can be calculated as ${H}_{{ij}}=\langle i| \hat{H}| j\rangle $, which is given by $(-{\varepsilon }_{i})$ for the diagonal element Hii and $(-{t}_{{ij}})$ for element Hij. Once the Hamiltonian matrix is obtained, it can be diagonalized to yield the eigenenergies and the eigenstates. The results of spectral statistics for the African shaped graphene billiard is shown in figure 2, which assumes ϵi=0 and uniform hopping energies tij=t between only the nearest neighboring atoms. Thus the energy is in units of t, and it is convenient to use En/t for the values of the eigenenergies. It is clear that they follow GOE statistics. Non-idealities such as interactions beyond the nearest neighbors, lattice orientation, effect of boundary bonds and staggered potentials caused by substrates, etc. may have influence to the details of the system, but the GOE statistics are robust and persistent in these non-ideal situations [534].
This might be counterintuitive as one would expect that the graphene chaotic billiards should exhibit the same GUE level-spacing distribution as the massless Dirac billiard [450], since they obey the same equation. The reasoning is as follows. Graphene has two non-equivalent Dirac points (valleys). Quasiparticles in the vicinity of a Dirac point obey the same massless Dirac equation, but the abrupt edge termination in graphene billiard couples the two valleys. As a result, a full set of equations taking into account the effects of both the two nonequivalent Dirac points and the boundary conditions are thus necessary to describe the motion of the relativistic particle. The time-reversal operation for the massless Dirac particle interchanges the two valleys. Thus as a whole, the time-reversal symmetry is preserved [456], resulting in GOE statistics.
Spectral statistics of disordered graphene sheets has also been investigated extensively with both experiments [541] and numerical simulations [542–544], where GOE statistics have been identified in general. Reference [545] examined the level spacing statistics for the edge states only for energies close to the Dirac point. Since these states are localized, it was expected that the statistics may follow that of Poisson, but it turned out that the level spacing statistics was GOE, which can be attributed to the chiral symmetry that introduced long-range correlation between the edge states on different sides, and thus level repulsion. Indeed, when the symmetry is broken by non-zero next nearest neighbor hopping energies, the level spacing statistics becomes Poisson.
Much effort has been devoted to searching for GUE in graphene billiards, e.g. by decoupling the two valleys. A smooth varying mass term was added in [538], however, GUE statistics were not found, which was attributed to the residual inter-valley scatterings. Indications of GUE statistics were found in triangular graphene billiards with zigzag edges and smooth impurity potentials [544], and with an asymmetric strain [546] due to the induced pseudomagnetic field [547, 548].
In the spectral statistics, there is a series of works employing microwave artificial graphene, e.g. a manmade honeycomb lattice not for electrons, but for microwaves [253, 540, 549–563]. Due to the inherent similarity of the wave equations, the quasiparticles follow the same massless Dirac equation and behave similarly as those in graphene. Especially, the Darmstadt group of A. Richter used super-conducting microwave cavities filling in photonic crystals, obtained spectra with unprecedentedly high accuracy, yielding convincing statistics [240, 253, 540, 562, 563].
2.3. Beyond Berry and Tabor’s conjecture
Berry and Tabor [113] proposed that for generically integrable systems, the energy levels are uncorrelated and the resulting statistics would be Poisson. This has been verified by extensive numerical and experimental studies. However, it is found that for graphene billiard with a sector shape where the corresponding classical dynamics are generically integrable, for energy levels close to the Dirac point, the spectral statistics are in general GOE, not Poisson [564]. Only close to the band edge (E/t=±3) where the pseudoparticles follow the Schrödinger equation, the statistics become Poisson. The reason for this abnormal phenomenon is that when the energy is close to the Dirac point, the edges play an important role. Even for an ideal situation, say, 60° sector with both straight edges being armchair, the level spacing statistics could be Poisson, as figure 3(a) shows, but changing a few atoms around the tip, or adding or removing one line of atoms along one edge, as demonstrated in the insets, the level spacing statistic becomes that of the GOE (figure 3(b)). Thus the system is extremely sensitive to the imperfections of the boundary, and for sectors with arbitrary angles, the results are generally GOE [564]. An interesting question is that, is this result due to the particular lattice structure of graphene, or due to relativistic nature? A preliminary examination reveals that this might be caused by the complex boundary condition provoked by the multi-component spinor wavefunction. Concrete conclusion may require further investigation.
Figure 3.
New window|Download| PPT slide Figure 3.Level spacing statistics of graphene billiard with the shape of a 60° sector with armchair edges. (a) With perfect edges and 227 254 atoms. (b) With one row of atoms removed along one edge so the structure is no longer symmetric (as indicated by the red dots in the inset) and 226 315 atoms. The energy range is 0.02<En/t < 0.2. Dashed line is Poisson, solid line is GOE. Insets show the magnified view of the lattice structure close to the tip of the sector to illustrate the differences. Adopted from [564] with permission.
3. Quantum scars
Quantum scar has been an important pillar for quantum chaos. In the development of relativistic quantum chaos, one natural question is whether scars exist in relativistic quantum systems, and if so, are there any unique features that can distinguish them from the conventional quantum scars?
3.1. Relativistic quantum scars
The existence of scars in relativistic quantum systems was confirmed with a stadium shaped graphene billiard [565]. Scars in the Wimmer system (distorted circular) filled with graphene was also observed [566]. Employing the tight-binding Hamiltonian equation (2), the eigenstates ψn can be calculated. By examining the spatial distribution of $| {\psi }_{n}{| }^{2}$ for eigenstates close to Dirac point, unequivocal scars on periodic orbits are observed (figure 4). The scarring state can be formed when the particle, after traveling the orbit for a complete cycle, gains a global phase that is an integer multiple of 2π. Thus when there is a scar occurring at the wavenumber k0, as the wavenumber (or energy) is changed, there will be a scar again at (or close to) wavenumber k if ${\rm{\Delta }}k\cdot L=2n\pi $, where ${\rm{\Delta }}k=k-{k}_{0},L$ is the length of the orbit and n is an integer. For two adjacent scarring states, one has ${\rm{\Delta }}k=2\pi /L$. This holds for both massless relativistic and non-relativistic quantum systems. Note that this does not hold for massive relativistic quantum billiard systems, as when varying k (or energy E), besides the Δk·L term, there will be an additional term that would lead to an extra phase depending on k, which would also need to be taken into account in the quantization formula. For massless relativistic and non-relativistic quantum systems, the key difference lies in the dispersion relation, with E∝k for the former and E∝k2 for the latter. Therefore, in terms of E, it will be either E or $\sqrt{E}$ that will be equally spaced for recurrent scars, corresponding to massless relativistic (graphene) or nonrelativistic quantum cases. Figure 5 shows that for two representative scars as shown in the insets, the energy values where they occur versus the relative index. Despite small fluctuations, the linear relation is apparent, corroborating the massless relativistic predictions. In particular, for graphene, since $E={\hslash }{v}_{F}k$, where ${v}_{F}=\sqrt{3}{ta}/(2{\hslash })$ is the Fermi velocity, t is the hopping energy between the nearest neighbors, a=2.46 Å is the lattice constant, one has ${\rm{\Delta }}E={\hslash }{v}_{F}{\rm{\Delta }}k={{hv}}_{F}/L$. For the scar shown in the left inset, the length of the orbit is 263a, yielding ΔE=0.0207t; while from figure 5, the average ΔE equals to 0.0203t, which agrees well. For the other scar, the length of the orbit is 275a, leading to ΔE=0.0198t, agrees well with 0.0195t from figure 5.
Figure 4.
New window|Download| PPT slide Figure 4.(a) A stadium shaped graphene billiard with 11814 atoms. (b) and (c) show $| {\psi }_{n}{| }^{2}$ with ${E}_{n}/t=0.363\,58$ and 0.576 65, respectively. Adopted from [565] with permission.
Figure 5.
New window|Download| PPT slide Figure 5.For two representative scars with orbit length 263a (left) and 275a (right), the energy values where it appears. Vertical axis shows the relative index of these scars. From [464] with permission.
Note that only when the energy is small, the dispersion relation is homogeneous and the pseudoparticles follow the massless Dirac equation. When the energy is large, the dispersion relation is no longer homogeneous but direction dependent, and the group velocity ${{\rm{\nabla }}}_{{\boldsymbol{k}}}E$ is concentrated in only three directions according to the symmetry of the honeycomb lattice. In this case, the motion of the pseudoparticles deviates from the massless Dirac equation. However, along these three directions, E is still approximately linear to $| {\boldsymbol{k}}| $ even for E close to t. Since the scars are also constrained on orbits that are composed by straight lines only in these three directions, the relation ${\rm{\Delta }}E={{hv}}_{F}/L$ holds for almost the whole range from 0 to t. This makes it easier to examine the scars and to verify this relation in experiments. Indeed, this feature of equal spacing of E in the recurring scarring states has been confirmed experimentally in a mesoscopic graphene ring system [567].
3.2. Chiral scars
Although pseudoparticles close to one Dirac point in graphene and Berry and Mondragon’s ‘neutrino’ follow the same 2D massless Dirac equation, due to the coupling of the two Dirac points by the boundary, a complete description for the pseudoparticles in graphene will be different. Thus it is still intriguing to examine the scars in the ‘neutrino’ billiard and see how the time-reversal symmetry broken by the infinite mass boundary condition is revealed in scars. A direct discretization method was developed to solve the massless Dirac billiard in a confined region, where scars in an African billiard and a bow-tie shaped billiard were identified [536], but due to limited spatial resolution, recurrent rhythm can not be determined. Later, a conformal mapping method was developed where a huge number of eigenstates with extremely high spatial resolution can be obtained [568]. By solving the eigenproblem of a heart-shaped 2D massless Dirac billiard, scars on periodic orbits are identified. Furthermore, it is found that the properties of the scars depend on whether the orbit has even or odd bounces at the boundary, and the relation $k-{k}_{0}=2\pi n/L$ for recurring scars is no longer fulfilled for the odd orbits. Particularly, for a given reference point k0 with pronounced scarring patterns, let $\delta k=2\pi /L$ and define $\eta (m)=({k}_{m}-{k}_{0})/\delta k-[({k}_{m}-{k}_{0})/\delta k]$, where [x] denotes the integer part of x and km is the eigenwavenumber of the mth identified scarring state on the same orbit, if the above relation is satisfied, then numerically, η(m) will take values that are either close to zero or close to one. As shown in figure 6, this is indeed the case for period-4 orbits. But for period-3 orbits, η takes an extra value close to 1/2 [568].
Figure 6.
New window|Download| PPT slide Figure 6.For scars on two representative orbits, period-4 for the left panels, and period-3 for the right panels, the upper panels show the corresponding eigenenergies of the scars, and the lower panels show η (see text) for these sates. Adopted from [568] with permission.
A complete understanding would involve many more details [569]. Here we would only provide the main arguments. The quantization condition is such that, following the orbit, after a complete cycle, the total phase accumulation should be integer multiples of 2π. For the massless Dirac billiard with a magnetic flux $\alpha {{\rm{\Phi }}}_{0}/2\pi $ (${{\rm{\Phi }}}_{0}\equiv h/e$ is the magnetic flux quanta) at the center of the billiard, the total accumulated phase after one complete cycle is$\begin{eqnarray*}{{\rm{\Phi }}}^{\pm }=\displaystyle \frac{1}{{\hslash }}S-\displaystyle \frac{\sigma \pi }{2}+{\beta }^{\pm },\end{eqnarray*}$where ‘±’ indicates whether the flow of the orbit is counterclockwise or clockwise. ${\beta }^{\pm }={\sum }_{i}{\delta }_{i}^{\pm }$ is the extra phase due to spin rotation imposed by reflections at the boundary, e.g. see figure 7. For a given periodic orbit, at each reflection point, the angle ${\delta }_{i}^{\pm }$ can be calculated explicitly, which determines ${\beta }^{\pm }$ unsuspiciously [569]. The Maslov index σ is the number of conjugate points along the orbit and is canonically invariant [17]. For the heart-shaped chaotic billiard, the value of σ is nothing but the number of reflections along a complete orbit [19]. The action is$\begin{eqnarray*}S=\oint {\boldsymbol{p}}\cdot {\rm{d}}{\boldsymbol{q}}={\hslash }\oint {\boldsymbol{k}}\cdot {\rm{d}}{\boldsymbol{q}}+e\oint {\boldsymbol{A}}\cdot {\rm{d}}{\boldsymbol{q}}=k\cdot L\pm W\alpha ,\end{eqnarray*}$where W is the winding number of the orbit with respective to the flux, i.e. how many times it circulates the flux. One thus has$\begin{eqnarray}{{\rm{\Phi }}}^{\pm }=k\cdot L\pm W\alpha -\displaystyle \frac{\sigma \pi }{2}+{\beta }^{\pm }.\end{eqnarray}$
Figure 7.
New window|Download| PPT slide Figure 7.Definition of the angles. ${\delta }_{i}^{+}=({\theta }_{i}-{\theta }_{i-1})/2$ is the extra phase due to the rotation of the spin, where ‘+’ indicates counterclockwise orientation. Typically, the phase associated with the time reversed reflection, e.g. ${\delta }_{i}^{-}$, from $-{{\boldsymbol{k}}}_{i}$ to $-{{\boldsymbol{k}}}_{i-1}$, would be different from ${\delta }_{i}^{+}$.
For semiclassically allowed states, the phase accumulation around one cycle should be an integer multiple of 2π, i.e. ${{\rm{\Phi }}}^{\pm }=2\pi n$ (n=1, 2, ⋯) so as to ensure that the wavefunction is single-valued. One thus has$\begin{eqnarray}{k}^{\pm }=(2\pi n\mp W\alpha +\displaystyle \frac{\sigma \pi }{2}-{\beta }^{\pm })/L.\end{eqnarray}$This is the quantization rule for a scarring state on periodic orbit with length L, which tells at (or close to) which value of wavenumber k (or E) a scar can form. A comparison between this formula and numerical results for two representative orbits are shown in figure 8, which shows good agreement. The time-reversal symmetry is then imprinted with whether there are integers for both counterclockwise and clockwise orientations that could satisfy equation (4) simultaneously for the same value of k, and thus both states with counterclockwise and clockwise local current flows are solutions of the system. This requires ${\rm{\Delta }}{\rm{\Phi }}\equiv {{\rm{\Phi }}}^{+}-{{\rm{\Phi }}}^{-}=0$ modulo 2π, or $2W\alpha +{\rm{\Delta }}\beta =0$ modulo 2π, where ${\rm{\Delta }}\beta ={\beta }^{+}-{\beta }^{-}$. For systems without a magnetic flux, α=0, the condition becomes Δβ=0. It is surprising that Δβ only depends on whether the periodic orbit has even or odd number of bounces at the boundary: Δβ=0 modulo 2π for even orbits—orbits with even number of bounces, and Δβ=π modulo 2π for odd orbits. Thus although each reflection breaks the time-reversal symmetry due to the polarization of the spin and the tangential current at each reflection point [569], even orbits, when considering the overall accumulated phase, preserve the T-symmetry, thus only odd orbits lead to T-symmetry broken. This provides an understanding of the behaviors of η in figure 6. For the period-4 orbit, at α=0, the values of k for scars with different orientation coincide with each other, as shown in figure 8(a) thus they are both allowed when k satisfies the quantization rule. The difference in neighboring k is then 2π/L, leading to η to be either close to 1 or close to 0. However, at α=0, for the period-3 orbit, the values of k for scars with different orientation are interlaced, e.g. clockwise, counterclockwise, clockwise, and so on, as shown in figure 8(b), while for each orientation, the space between neighboring k is 2π/L, but if one does not differentiate the orientations, the difference becomes π/L for the two neighboring k values corresponding to different orientations, leading to η=1/2. Note that as the magnetic flux is varied, the system is periodic with α=2π. Furthermore, since ${\rm{\Delta }}{\rm{\Phi }}=2W\alpha +{\rm{\Delta }}\beta $, for period-4 orbit, Δβ=0, W=1, thus when α=π/2, as indicated by the vertical lines in figure 8, Δ Φ will become π, which will be similar as that for period-3 at α=0. On the other hand, for period-3 orbit, Δβ=π, thus when α=π/2, Δ Φ=2π, or 0 modulo 2π, which is similar to the case of periodic-4 at α=0. Thus by applying a magnetic flux of α=π/2, the chirality interchanges for these two orbits.
Figure 8.
New window|Download| PPT slide Figure 8.Validation of the quantization rule equation (4). Shown are the relations between wavenumber k and magnetic flux α, for (a) the period-4 scar in figure 6(a), and (b) the period-3 scar in figure 6(c). The orange up-triangles indicate scars with a counterclockwise flow, and the blue down-triangles are those with a clockwise flow. The gray squares mark the scars whose flow orientations cannot be identified, which typically occur close to the cross points of the two orientation cases. The solid lines are theoretical predictions of equation (4). Vertical lines indicate the position of α=π/2. The step in the variation of α is 0.01. Adapted from [569] with permission.
3.3. Unification of chiral scar and nonrelativistic quantum scars
Recently we have developed quantization rule for scars in massive 2D Dirac billiards with infinite mass confinement [570]. Compare to the massless case, there is a new phase emerging during each reflection j. For the massless case, the reflection coefficient Rj is 1. While for the massive case, although the module of Rj is still 1, it has a non-trivial phase, i.e. ${R}_{j}^{\pm }={{\rm{e}}}^{{\rm{i}}({\delta }_{j}^{\pm }+2{\omega }_{j}^{\pm })}$, where ${\delta }_{j}^{\pm }=({\theta }_{j}^{\pm }-{\theta }_{j-1}^{\pm })/2$ is the same as in the massless case (figure 7), but $2{\omega }_{j}^{\pm }$ is a complicated function of the angles (${\theta }_{i-1},{\theta }_{j}$), the mass m, and the wavenumber k (or energy E) [570]. Let ${\gamma }^{\pm }={\sum }_{j}2{\omega }_{j}^{\pm }$ and in the absence of magnetic flux, i.e. α=0, the total phase accumulation around one complete cycle is then$\begin{eqnarray}{{\rm{\Phi }}}^{\pm }=k\cdot L-\displaystyle \frac{\sigma \pi }{2}+2{\beta }^{\pm }+{\gamma }^{\pm }.\end{eqnarray}$Since mod(Δ 2β,2π)=0, where ${\beta }^{\pm }={\sum }_{j}{\delta }^{\pm }$, we then have ${\rm{\Delta }}{\rm{\Phi }}={\rm{\Delta }}\gamma \equiv {\gamma }^{+}-{\gamma }^{-}$. Thus for massive Dirac billiards, the complex behavior can be all attributed to Δγ. We have found that when m goes to zero, Δγgoes to 2π or π for even or odd orbits, respectively (see figure 9 when $m\to 0$), degenerating to the massless cases. When the mass m goes to infinity, Δγgoes to zero for both even and odd orbits (see figure 9 when $k\to 0$): hence the difference between even and odd orbits diminish and the system becomes effectively a nonrelativistic quantum billiard. Thus through the modulation of the extra phase in the reflection coeffecients, the relativistic chiral scar and the nonrelativistic quantum scar can be unified as the two limiting cases of the massive Dirac billiards.
Figure 9.
New window|Download| PPT slide Figure 9.Δγ between counterclockwise and clockwise scaring states on a period-3 orbit (a) and a period-4 orbit (b). The massless Dirac regime is $m\to 0$, while $k\to 0$ is effectively $m\to \infty $ and is the Schrödinger limit. Adapted from [570] with permission.
4. Scattering and tunneling
For open quantum systems, an important topic of quantum chaos is quantum chaotic scattering [50, 158–170]. In nonrelativistic systems, a general observation is that, for classically mixed systems, the transmission (or conductance) of the corresponding quantum system exhibit many sharp resonances caused by the strongly localized states around the classically stable periodic orbits, while for classically chaotic system, the peaks are either broadened or removed. That is, chaos regularizes the quantum transport and makes the transmission curve smoother. Note that, for closed systems, such as a quantum billiard, although there are localized scarring state on the unstable periodic orbits for classically chaotic system, these states are unstable that once the system is opened up, due to the spanning chaotic sea in the phase space, they are typically washed out, leaving few or no localized states.
Similar investigation has been carried out for graphene/Dirac quantum dots with different classical dynamics [538, 571–577]. It has been found that, classical dynamics can indeed influence the quantum transport, e.g. by varying the boundary of the quantum dot to change the corresponding classical dynamics from mixed to chaotic, most of the sharp resonances are broadened or removed, however, there are residual sharp resonances, with still strong localized states on classically unstable periodic orbits that would not exist for nonrelativistic systems [573]. In particular, a cosine billiard [163] is adopted to demonstrate this phenomenon. The boundary is given by two hard walls at y=0 and $y=W+(M/2)[1-\cos (2\pi x/L)]$ for $0\leqslant x\leqslant L$, with two semi-infinite leads of width W attached at the two openings of the billiard, whose length is L and the widest part is (W+M). By changing the geometric parameters M, W, and L, the classical dynamics can be either mixed, e.g. for W/L= 0.18 and M/L=0.11, or chaotic, e.g. W/L=0.36 and M/L=0.22. A tight-binding approach is employed, and Green’s function formalism is used to calculate the transmission and the local density of states (LDS) [578–581].
Assume the isolated dot region (0≤x≤L) has Hamiltonian Hc with a set of eigenenergies and eigenfuctions $\{{E}_{0\alpha },{\psi }_{0\alpha }| \alpha =1,2,\cdots \}$. The effects of the semi-infinite leads can be incorporated into the retarded self-energy matrices, ${{\rm{\Sigma }}}^{R}={{\rm{\Sigma }}}_{L}^{R}+{{\rm{\Sigma }}}_{R}^{R}$, with the lower indices indicate whether it is due to the left or right leads. Then the whole Hamiltonian with the effects of the leads is ${H}_{c}+{{\rm{\Sigma }}}^{R}$. Since ΣR in general can be complex, and it is small that it can be regarded as a perturbation, the new set of eigenenergies becomes ${E}_{\alpha }={E}_{0\alpha }-{{\rm{\Delta }}}_{\alpha }-{\rm{i}}{\gamma }_{\alpha }$, where Δα and γα are generally small. Δα represents a shift in ${E}_{0\alpha }$, and γα is the width of the resonance for the α's state. 1/γα can be regarded as the life-time of the state [578]. For detailed formulas of calculating γ and the determining factors, please refer to [582].
The values of γα for four cases with mixed or chaotic dynamics and 2-dimensional electron gas (2DEG) or graphene quantum dots are shown in figure 10. Note that smaller γα will result in sharper transmission resonances. For 2DEG quantum dots with mixed dynamics (figure 10(a)), there are many cases that γα takes very small values, in the order of 10−4, indicating extremely sharp resonances. When the classical dynamics change from mixed to chaotic, beside the envelope, the small values in γα are almost all removed (figure 10(b)). For graphene quantum dots, when the classical dynamics is mixed, beside the smooth envelope for γα∼10−2, there is a cluster of points for small γα values (figure 10(c)). When the classical dynamics becomes chaotic (figure 10(d)), although the overall trend is that the small values are shifted upwards, there are still a big cluster of points take apparently smaller values than the envelope, indicating the persistence of the sharp resonances.
Figure 10.
New window|Download| PPT slide Figure 10.The imaginary part γ of the eigenenergies due to coupling between the dot and the leads, which is an effective indicator of the resonance width. The left panels are for 2DEG quantum dots, and the right panels are for graphene quantum dots. Upper panels are for the cases with mixed dynamics, and lower panels are for classically chaotic dynamics. Adapted from [573] with permission.
In addition, figure 11 shows the LDS for the most pronounced patterns in both the 2DEG and graphene quantum dots. (a) and (d) are for classically mixed dynamics, which show strong localizations on the stable periodic orbits in both cases. (b), (c) and (e), (f) are for classically chaotic dynamics. It is clear that for 2DEG cases, chaos ruined the localized states on the unstable periodic orbits that are present in the closed case, i.e. the scars; but for graphene quantum dot, localization on unstable periodic orbits still persists. Actually, there are many such states, corroborating the results in figure 10. Although weaker, the effect that classical chaos can make the conductance fluctuation becomes smoother can be exploited to articulate a controlling scheme to modulate the conductance fluctuations in quantum transport through a quantum dot, by changing the underlying classical dynamics [583, 584]. In the presence of a strong magnetic field, the difference caused by the classical dynamics can be suppressed further [585].
Figure 11.
New window|Download| PPT slide Figure 11.Typical local density of states for quantum dots with classically mixed (a), (d) and chaotic (b) ,(c), (e), (f) dynamics. The upper panels are for 2DEG quantum dots, and lower panels are for graphene quantum dots.
The same phenomenon has also been observed in bilayer graphene [586]. The pseudoparticles in bilayer graphene follow the 2D massive Dirac equation. Thus this indicates that the suppression of the effect of eliminating sharp resonances by chaos also persists for massive Dirac systems. In addition, when the pseudoparticle is traveling along the classical ballistic orbit, it tends to hop back and forth between the two layers, exhibiting a Zitterbewegung-like effect.
Besides scattering, there are other phenomena that the effect of chaos has been suppressed. For example, for regulation of tunneling rates by chaos [191, 192], it has been found that if 2DEG is replaced by graphene or the massless Dirac fermion, although the regularization effect persist, it is much weaker than the 2DEG case [193–195]. The same hold for persistent currents [587, 588] in Aharonov–Bohm (AB) rings [589]. Conventional metallic [590–593] or semiconductor [594] ring systems with a central AB magnetic flux may exhibit dissipationless currents, e.g. the persistent or permanent current. However, the current is quite sensitive that small non-idealities such as boundary deformation or disorders may destroy the persistent current drastically [595–598]. While in a ring of massless Dirac fermions, due to the Dirac whispering gallery modes [599–601], the persistent current is quite robust against boundary deformations that even in the case where the classical dynamics become chaotic there is still a quite large amount of persistent currents [602–604]. Furthermore, recently, Han et al [605] investigated out-of-time-order correlator in relativistic quantum billiard systems and found that the signatures of classical chaos are less pronounced than in the nonrelativistic case. Here again the effect of chaos is suppressed.
5. Quantum chaos in pseudospin-1 Dirac materials
Dirac materials hosting pseudospin-1 quasiparticles with a conical intersection of triple degeneracy in the underlying energy band have attracted a great deal of attention [606–633]. The physics of these 2D Dirac materials is described by the generalized Dirac-Weyl equation for massless spin-1 particles [607, 608, 626]. Pseudospin-1 quasiparticles are different from Dirac, Weyl and Majorana fermions, and are of particular interest to the broad research community with diverse experimental realization schemes such as artificial photonic lattices [612, 616, 620, 621, 624], optical [622] and electronic Lieb lattices [631, 632], as well as superconducting qutrits [633]. A striking relativistic quantum hallmark of pseudospin-1 particles is super-Klein tunneling through a scalar potential barrier [608, 610, 623, 634, 635], where omnidirectional and perfect transmission of probability one occurs when the incident energy is about one half of the potential height. Generally, Klein tunneling defines optical-like, negatively refracted ray paths through the barrier interface via angularly resolved transmittance in the short wavelength limit [636–638].
A recent study [639] addressed the issue of confinement of quasiparticles in pseudospin-1 materials. When both super-Klein tunneling and chaos are present, one may intuitively expect severe leakage to predominantly occur so that trapping would be impossible. However, quite counterintuitively, an energy range was found in which robust wave confinement occurs in spite of chaos and super-Klein tunneling. Especially, the three-component spinor wave concentrates in a particular region of the boundary through strongly squeezed local current vortices generated there, whose pattern in physical space can be manipulated in a reconfigurable manner, e.g. by deforming the boundary shape or setting the direction of excitation wave. While these modes are distributed unevenly in physical space because of the irregular deformations, even fully developed chaos and super-Klein tunneling are not able to reduce their trapping lifetime. That is, these modes contradict the intuitive expectation that electrostatically confining relativistic type of carriers/particles to a finite chaotic domain is impossible due to the simultaneous presence of two leaking (Q-spoiling) mechanisms: chaos assisted tunneling and Klein tunneling. This phenomenon has no counterpart in nonrelativistic quantum or even in pseudospin-1/2 systems. The resulting narrow resonances are also characteristically different from those due to scarring modes concentrating on periodic orbits in conventional wave chaotic scattering, in quantum dots [573, 640–645] or in open optical microcavities [646–648].
6. Discussions
Beside the above discussed few topics in relativistic quantum chaos, there are many other interesting topics that have been investigated in depth, such as quantum tunneling without [193, 194] and with electron–electron interactions [195], super-persistent currents that are robust to boundary deformations [602, 603] and the presence of disorders [604], relativistic quantum chimera states that electrons with different spins exhibit distinct scattering behaviors as they follow different classical dynamics [649], OTOC for relativistic quantum systems [605], anomalous entanglement in chaotic Dirac billiards [650], relativistic quantum kicked rotors [442, 510–512], kicked relativistic particle in a box [651], etc. More efforts are needed to gain deeper understandings of these interesting subjects. In addition, electron–electron interaction effects [652] in a chaotic graphene quantum billiard have also been considered and compared with scanning tunneling microscopy (STM) experiments, which could explain both the measured density of state values and the experimentally observed topography patterns [198]. Most of the understandings achieved so far for relativistic quantum chaos are for massless cases. Massive Dirac billiards have been considered only recently, where quantization formula for scarred states in confined 2D massive Dirac billiard has been proposed and validated numerically, and restoration of time-reversal symmetry in the infinite mass limit has been unveiled [570]. However, there are many other issues to be understood in the massive Dirac billiards, e.g. to what extent the intriguing observations for massless Dirac billiard persist in the massive case? Pseudo-spin one systems [607–609] have attracted much attention recently. Due to the flat band, it has many interesting properties regarding quantum chaotic scattering, such as superscattering that could even defy chaos Q-spoiling and Klein tunneling [635, 639, 653, 654]. There are still many open questions concerning pseudo-spin one system and quantum chaos.
Retrospecting the half-century development of quantum chaos, there are many subjects that would be interesting to extend into the relativistic quantum realm, such as the validity of the proposed indicators of universality corresponding to different classical dynamics, Loschmidt echo, many-body effects, quantum thermalization, etc. that have been discussed in section 1, as it is not straightforward to speculate what will happen when stepping into the relativistic regime. Efforts in trying to understand the behaviors of these subjects in relativistic quantum systems may not only advance the knowledge on the fundamental physics of relativistic quantum chaos, but may also bring new concepts of applications base on the state-of-art Dirac material technologies.
Acknowledgments
We thank Professor Kai Chang for the invitation to contribute a mini-review for Commun. Theor. Phys., and Mr Chenrong Liu for the help with organizing part of the references. We also thank Professor Wenge Wang and Professor Jiao Wang for carefully reading of the paper and detailed constructive suggestions. This work was supported by NSFC under Grants No.11775101 and No.11422541. YCL is supported by ONR through Grant No. N00014-16-1-2828.
MillerW H1975 Semiclassical quantization of nonseparable systems: a new look at periodic orbit theory J. Chem. Phys. 63 996999 DOI:10.1063/1.431410
BerryMTaborM1977 Calculating the bound spectrum by path summation in action-angle variables J. Phys. A: Math. Gen. Math. Theor. 10 371379 DOI:10.1088/0305-4470/10/3/009
DuM LDelosJ B1987 Effect of closed classical orbits on quantum spectra: ionization of atoms in a magnetic field Phys. Rev. Lett. 58 17311733 DOI:10.1103/PhysRevLett.58.1731
CreaghS CRobbinsJ MLittlejohnR G1990 Geometrical properties of Maslov indices in the semiclassical trace formula for the density of states Phys. Rev. A 42 1907 DOI:10.1103/PhysRevA.42.1907 [Cited within: 1]
PrimackHSmilanskyU1998 On the accuracy of the semiclassical trace formula J. Phys. A: Math. Gen. 31 62536277 DOI:10.1088/0305-4470/31/29/016
CohenDPrimackHSmilanskyU1998 Quantal-classical duality and the semiclassical trace formula Ann. Phys. 264 108170 DOI:10.1006/aphy.1997.5776
YangSKellmanM E2002 Perspective on semiclassical quantization: how periodic orbits converge to quantizing tori Phys. Rev. A 66 052113 DOI:10.1103/PhysRevA.66.052113
HellerE J1984 Bound-state eigenfunctions of classically chaotic Hamiltonian systems—scars of periodic-orbits Phys. Rev. Lett. 53 15151518 DOI:10.1103/PhysRevLett.53.1515 [Cited within: 3]
McdonaldS WKaufmanA N1988 Wave chaos in the stadium—statistical properties of short-wave solutions of the Helmholtz equation Phys. Rev. A 37 30673086 DOI:10.1103/PhysRevA.37.3067
WintgenDHönigA1989 Irregular wave functions of a hydrogen atom in a uniform magnetic field Phys. Rev. Lett. 63 14671470 DOI:10.1103/PhysRevLett.63.1467
KusMZakrzewskiJZyczkowskiK1991 Quantum scars on a sphere Phys. Rev. A 43 42444248 DOI:10.1103/PhysRevA.43.4244
MaltaC PDeaguiarM A MDealmeidaA M O1993 Quantum signature of a period-doubling bifurcation and scars of periodic orbits Phys. Rev. A 47 16251632 DOI:10.1103/PhysRevA.47.1625
BellomoPUzerT1995 Quantum scars and classical ghosts Phys. Rev. A 51 16691672 DOI:10.1103/PhysRevA.51.1669
SimonottiF PVerginiESaracenoM1997 Quantitative study of scars in the boundary section of the stadium billiard Phys. Rev. E 56 38593867 DOI:10.1103/PhysRevE.56.3859
LiB1997 Numerical study of scars in a chaotic billiard Phys. Rev. E 55 53765379 DOI:10.1103/PhysRevE.55.5376
AgamOFishmanS1993 Quantum eigenfunctions in terms of periodic orbits of chaotic systems J. Phys. A: Math. Gen. 26 21132137 DOI:10.1088/0305-4470/26/9/010
AgamOFishmanS1994 Semiclassical criterion for scars in wave functions of chaotic systems Phys. Rev. Lett. 73 806809 DOI:10.1103/PhysRevLett.73.806
FishmanSGeorgeotBPrangeR E1996 Fredholm method for scars J. Phys. A: Math. Gen. 29 919937 DOI:10.1088/0305-4470/29/4/019
EckhardtBFishmanSKeatingJAgamOMainJMüllerK1995 Approach to ergodicity in quantum wave functions Phys. Rev. E 52 58935903 DOI:10.1103/PhysRevE.52.5893
FromholdT MWilkinsonP BSheardF WEavesLMiaoJEdwardsG1995 Manifestations of classical chaos in the energy level spectrum of a quantum well Phys. Rev. Lett. 75 11421145 DOI:10.1103/PhysRevLett.75.1142 [Cited within: 1]
WilkinsonP BFromholdT MEavesLSheardF WMiuraNTakamasuT1996 Observation of ‘scarred’ wavefunctions in a quantum well with chaotic electron dynamics Nature 380 608610 DOI:10.1038/380608a0
MonteiroT SDelandeDConneradeJ P1997 Have quantum scars been observed? Nature 387 863864 DOI:10.1038/43096
AkisRFerryD KBirdJ P1997 Wave function scarring effects in open stadium shaped quantum dots Phys. Rev. Lett. 79 123126 DOI:10.1103/PhysRevLett.79.123
NarimanovE EStoneA D1998 Origin of strong scarring of wave functions in quantum wells in a tilted magnetic field Phys. Rev. Lett. 80 4952 DOI:10.1103/PhysRevLett.80.49
BirdJ PAkisRFerryD KVasileskaDCooperJAoyagiYSuganoT1999 Lead-orientation-dependent wave function scarring in open quantum dots Phys. Rev. Lett. 82 46914694 DOI:10.1103/PhysRevLett.82.4691
CrookRSmithC GGrahamA CFarrerIBeereH ERitchieD A2003 Imaging fractal conductance fluctuations and scarred wave functions in a quantum billiard Phys. Rev. Lett. 91 246803 DOI:10.1103/PhysRevLett.91.246803 [Cited within: 2]
LeRoyB JBleszynskiA CAidalaK EWesterveltR MKalbenAHellerE JShawS E JMaranowskiK DGossardA C2005 Imaging electron interferometer Phys. Rev. Lett. 94 126801 DOI:10.1103/PhysRevLett.94.126801
BrunnerRAkisRFerryD KKucharFMeiselsR2008 Coupling-induced bipartite pointer states in arrays of electron billiards: Quantum Darwinism in action? Phys. Rev. Lett. 101 024102 DOI:10.1103/PhysRevLett.101.024102 [Cited within: 1]
BurkeA MAkisRDayT ESpeyerGFerryD KBennettB R2010 Periodic scarred states in open quantum dots as evidence of quantum Darwinism Phys. Rev. Lett. 104 176801 DOI:10.1103/PhysRevLett.104.176801
SridharSHellerE J1992 Physical and numerical experiments on the wave mechanics of classically chaotic systems Phys. Rev. A 46 R1728R1731 DOI:10.1103/PhysRevA.46.R1728
KudrolliAKidambiVSridharS1995 Experimental studies of chaos and localization in quantum wave functions Phys. Rev. Lett. 75 822825 DOI:10.1103/PhysRevLett.75.822 [Cited within: 2]
WaterlandR LYuanJ-MMartensC CGillilanR EReinhardtW P1988 Classical-quantum correspondence in the presence of global chaos Phys. Rev. Lett. 61 27332736 DOI:10.1103/PhysRevLett.61.2733 [Cited within: 1]
JensenR VSandersM MSaracenoMSundaramB1989 Inhibition of quantum transport due to scars of unstable periodic-orbits Phys. Rev. Lett. 63 27712775 DOI:10.1103/PhysRevLett.63.2771
FeingoldMLittlejohnR GSolinaS BPehlingJPiroO1990 Scars in billiards: The phase space approach Phys. Lett. A 146 199203 DOI:10.1016/0375-9601(90)90165-K
DepolaviejaG GBorondoFBenitoR M1994 Scars in groups of eigenstates in a classically chaotic system Phys. Rev. Lett. 73 16131616 DOI:10.1103/PhysRevLett.73.1613
ArranzF JBorondoFBenitoR M1998 Scar formation at the edge of the chaotic region Phys. Rev. Lett. 80 944947 DOI:10.1103/PhysRevLett.80.944
TurnerC JMichailidisA AAbaninD ASerbynMPapićZ2018 Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations Phys. Rev. B 98 155134 DOI:10.1103/PhysRevB.98.155134
MoudgalyaSRegnaultNBernevigB A2018 Entanglement of exact excited states of Affleck–Kennedy–Lieb–Tasaki models: exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis Phys. Rev. B 98 235156 DOI:10.1103/PhysRevB.98.235156
HoW WChoiSPichlerHLukinM D2019 Periodic orbits, entanglement, and quantum many-body scars in constrained models: matrix product state approach Phys. Rev. Lett. 122 040603 DOI:10.1103/PhysRevLett.122.040603
ChoiSTurnerC JPichlerHHoW WMichailidisA APapićZSerbynMLukinM DAbaninD A2019 Emergent SU(2) dynamics and perfect quantum many-body scars Phys. Rev. Lett. 122 220603 DOI:10.1103/PhysRevLett.122.220603
LinC-JMotrunichO I2019 Exact quantum many-body scar states in the Rydberg-blockaded atom chain Phys. Rev. Lett. 122 173401 DOI:10.1103/PhysRevLett.122.173401
MamaevMKimchiIPerlinM ANandkishoreR MReyA M2019 Quantum entropic self-localization with ultracold fermions Phys. Rev. Lett. 123 130402 DOI:10.1103/PhysRevLett.123.130402
SchecterMIadecolaT2019 Weak ergodicity breaking and quantum many-body scars in spin-1 xy magnets Phys. Rev. Lett. 123 147201 DOI:10.1103/PhysRevLett.123.147201
WilmingHGoihlMRothIEisertJ2019 Entanglement-ergodic quantum systems equilibrate exponentially well Phys. Rev. Lett. 123 200604 DOI:10.1103/PhysRevLett.123.200604
TerasakaTMatsushitaT1985 Statistical properties of the quantized energy spectrum of a Hamiltonian system with classically regular and chaotic trajectories: a numerical study of level-spacing distributions for two-dimensional coupled Morse-oscillator systems Phys. Rev. A 32 538551 DOI:10.1103/PhysRevA.32.538 [Cited within: 1]
JiangY1988 Level statistics of doublet spectrum of Sinai’s billiard Chin. Phys. Lett. 5 541544 DOI:10.1088/0256-307X/5/12/004
DelandeDGayJ C1986 Quantum chaos and statistical properties of energy levels: numerical study of the hydrogen atom in a magnetic field Phys. Rev. Lett. 57 20062009 DOI:10.1103/PhysRevLett.57.2006
FromholdT MWilkinsonP BSheardF WEavesLMiaoJEdwardsG1995 Manifestations of classical chaos in the energy-level spectrum of a quantum-well Phys. Rev. Lett. 75 11421145 DOI:10.1103/PhysRevLett.75.1142
ZhouWChenZZhangBYuC HLuWShenS C2010 Magnetic field control of the quantum chaotic dynamics of hydrogen analogs in an anisotropic crystal field Phys. Rev. Lett. 105 024101 DOI:10.1103/PhysRevLett.105.024101 [Cited within: 1]
SieberMRichterK2001 Correlations between periodic orbits and their role in spectral statistics Phys. Scr. T90 128 DOI:10.1238/Physica.Topical.090a00128
HeuslerSMüllerSBraunPHaakeF2004 universal spectral form factor for chaotic dynamics J. Phys. A: Math. Gen. 37 L31L37 DOI:10.1088/0305-4470/37/3/L02
LehmannK KCoyS L1987 The Gaussian orthogonal ensemble with missing and spurious levels: A model for experimental level spacing distributions J. Chem. Phys. 87 54155418 DOI:10.1063/1.453660 [Cited within: 1]
BiałousMYunkoVBauchSŁawniczakMDietzBSirkoL2016 Power spectrum analysis and missing level statistics of microwave graphs with violated time reversal invariance Phys. Rev. Lett. 117 144101 DOI:10.1103/PhysRevLett.117.144101 [Cited within: 2]
PoliCLuna-AcostaG AStöckmannH-J2012 Nearest level spacing statistics in open chaotic systems: Generalization of the Wigner surmise Phys. Rev. Lett. 108 174101 DOI:10.1103/PhysRevLett.108.174101 [Cited within: 1]
IzrailevF M1989 Intermediate statistics of the quasi-energy spectrum and quantum localisation of classical chaos J. Phys. A: Math. Gen. 22 865878 DOI:10.1088/0305-4470/22/7/017
ProsenTRobnikM1994 Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry–Robnik behaviour J. Phys. A: Math. Gen. 27 80598077 DOI:10.1088/0305-4470/27/24/017 [Cited within: 1]
ShigeharaT1994 Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer Phys. Rev. E 50 43574370 DOI:10.1103/PhysRevE.50.4357
BogomolnyEGiraudOSchmitC2002 Nearest-neighbor distribution for singular billiards Phys. Rev. E 65 056214 DOI:10.1103/PhysRevE.65.056214
García-GarcíaA MWangJ2006 Semi-Poisson statistics in quantum chaos Phys. Rev. E 73 036210 DOI:10.1103/PhysRevE.73.036210
TuanP HLiangH CTungJ CChiangP YHuangK FChenY F2015 Manifesting the evolution of eigenstates from quantum billiards to singular billiards in the strongly coupled limit with a truncated basis by using RLC networks Phys. Rev. E 92 062906 DOI:10.1103/PhysRevE.92.062906
BiałousMYunkoVBauchSŁawniczakMDietzBSirkoL2016 Long-range correlations in rectangular cavities containing pointlike perturbations Phys. Rev. E 94 042211 DOI:10.1103/PhysRevE.94.042211 [Cited within: 1]
ShklovskiiB IShapiroBSearsB RLambrianidesPShoreH B1993 Statistics of spectra of disordered systems near the metal-insulator transition Phys. Rev. B 47 1148711490 DOI:10.1103/PhysRevB.47.11487 [Cited within: 1]
SrivastaNMullerG1990 Quantum images of Hamiltonian chaos Phys. Lett. A 147 282286 DOI:10.1016/0375-9601(90)90448-W
LanB LWardlawD M1993 Signatures of chaos in the modulus and phase of time-dependent wave functions Phys. Rev. E 47 21762179 DOI:10.1103/PhysRevE.47.2176
WangWXuGFuD1994 Manifestation of destruction of quantum integrability with expectation and uncertainty values of quantum observables Phys. Lett. A 190 377381 DOI:10.1016/0375-9601(94)90718-8
BiesW EKaplanLHaggertyM RHellerE J2001 Localization of eigenfunctions in the stadium billiard Phys. Rev. E 63 066214 DOI:10.1103/PhysRevE.63.066214
BlumGGnutzmannSSmilanskyU2002 Nodal domains statistics: a criterion for quantum chaos Phys. Rev. Lett. 88 114101 DOI:10.1103/PhysRevLett.88.114101
BerkolaikoGKeatingJ PWinnB2003 Intermediate wave function statistics Phys. Rev. Lett. 91 134103 DOI:10.1103/PhysRevLett.91.134103
KotimäkiVRäsänenEHennigHHellerE J2013 Fractal dynamics in chaotic quantum transport Phys. Rev. E 88 022913 DOI:10.1103/PhysRevE.88.022913
MasonD JBorundaM FHellerE J2015 Revealing the flux: using processed Husimi maps to visualize dynamics of bound systems and mesoscopic transport Phys. Rev. B 91 165405 DOI:10.1103/PhysRevB.91.165405
KetzmerickR1996 Fractal conductance fluctuations in generic chaotic cavities Phys. Rev. B 54 1084110844 DOI:10.1103/PhysRevB.54.10841
HuckesteinBKetzmerickRLewenkopfC H2000 Quantum transport through ballistic cavities: soft vs hard quantum chaos Phys. Rev. Lett. 84 55045507 DOI:10.1103/PhysRevLett.84.5504 [Cited within: 1]
CasatiGGuarneriIMasperoG2000 Fractal survival probability fluctuations Phys. Rev. Lett. 84 6366 DOI:10.1103/PhysRevLett.84.63
BäckerAManzeAHuckesteinBKetzmerickR2002 Isolated resonances in conductance fluctuations and hierarchical states Phys. Rev. E 66 016211 DOI:10.1103/PhysRevE.66.016211
HuangW-MMouC-YChangC-H2010 Scattering phase correction for semiclassical quantization rules in multi-dimensional quantum systems Commun. Theor. Phys. 53 250 DOI:10.1088/0253-6102/53/2/09 [Cited within: 2]
MarcusC MRimbergA JWesterveltR MHopkinsP FGossardA C1992 Conductance fluctuations and chaotic scattering in ballistic microstructures Phys. Rev. Lett. 69 506509 DOI:10.1103/PhysRevLett.69.506 [Cited within: 1]
MarcusC MWesterveltR MHopkinsP FGossardA C1993 Conductance fluctuations and quantum chaotic scattering in semiconductor microstructures Chaos 3 643653 DOI:10.1063/1.165927
TaylorR Pet al. 1997 Self-similar magnetoresistance of a semiconductor Sinai billiard Phys. Rev. Lett. 78 19521955 DOI:10.1103/PhysRevLett.78.1952
SachrajdaA SKetzmerickRGouldCFengYKellyP JDelageAWasilewskiZ1998 Fractal conductance fluctuations in a soft-wall stadium and a Sinai billiard Phys. Rev. Lett. 80 19481951 DOI:10.1103/PhysRevLett.80.1948 [Cited within: 2]
FerryD KAkisRBirdJ P2004 Einselection in action: Decoherence and pointer states in open quantum dots Phys. Rev. Lett. 93 026803 DOI:10.1103/PhysRevLett.93.026803
FerryD KHuangLYangRLaiY-CAkisR2010 Open quantum dots in graphene: scaling relativistic pointer states J. Phys.: Conf. Ser. 220 012015 DOI:10.1088/1742-6596/220/1/012015
WangW-GHeLGongJ2012 Preferred states of decoherence under intermediate system-environment coupling Phys. Rev. Lett. 108 070403 DOI:10.1103/PhysRevLett.108.070403
KaplanS BHartsteinA1986 universal conductance fluctuations in narrow Si accumulation layers Phys. Rev. Lett. 56 24032406 DOI:10.1103/PhysRevLett.56.2403
SkocpolW JMankiewichP MHowardR EJackelL DTennantD MStoneA D1986 universal conductance fluctuations in silicon inversion-layer nanostructures Phys. Rev. Lett. 56 28652868 DOI:10.1103/PhysRevLett.56.2865
IidaSWeidenmüllerH AZukJ A1990 Wave propagation through disordered media and universal conductance fluctuations Phys. Rev. Lett. 64 583586 DOI:10.1103/PhysRevLett.64.583
DembowskiCGräfH-DHeineAHofferbertRRehfeldHRichterA2000 First experimental evidence for chaos-assisted tunneling in a microwave annular billiard Phys. Rev. Lett. 84 867870 DOI:10.1103/PhysRevLett.84.867 [Cited within: 1]
SteckD AOskayW HRaizenM G2001 Observation of chaos-assisted tunneling between islands of stability Science 293 274278 DOI:10.1126/science.1061569
de MouraA P SLaiY-CAkisRBirdJFerryD K2002 Tunneling and nonhyperbolicity in quantum dots Phys. Rev. Lett. 88 236804 DOI:10.1103/PhysRevLett.88.236804
BäckerAKetzmerickRMonastraA G2005 Flooding of chaotic eigenstates into regular phase space islands Phys. Rev. Lett. 94 054102 DOI:10.1103/PhysRevLett.94.054102
BäckerAKetzmerickRLöckSRobnikMVidmarGHöhmannRKuhlUStöckmannH-J2008 Dynamical tunneling in mushroom billiards Phys. Rev. Lett. 100 174103 DOI:10.1103/PhysRevLett.100.174103
BäckerAKetzmerickRLöckSSchillingL2008 Regular-to-chaotic tunneling rates using a fictitious integrable system Phys. Rev. Lett. 100 104101 DOI:10.1103/PhysRevLett.100.104101
RongSHaiWXieQZhuQ2009 Chaos enhancing tunneling in a coupled Bose–Einstein condensate with a double driving Chaos 19 033129 DOI:10.1063/1.3215764
LöckSBäckerAKetzmerickRSchlagheckP2010 Regular-to-chaotic tunneling rates: from the quantum to the semiclassical regime Phys. Rev. Lett. 104 114101 DOI:10.1103/PhysRevLett.104.114101
NiXHuangLYingLLaiY-C2013 Relativistic quantum tunneling of a Dirac fermion in nonhyperbolic chaotic systems Phys. Rev. B 87 224304 DOI:10.1103/PhysRevB.87.224304 [Cited within: 1]
YingLWangGHuangLLaiY-C2014 Quantum chaotic tunneling in graphene systems with electron-electron interactions Phys. Rev. B 90 224301 DOI:10.1103/PhysRevB.90.224301 [Cited within: 4]
UgajinR1997 Spectral statistics of correlated electrons in a square-well quantum dot Physica A 237 220228 DOI:10.1016/S0378-4371(96)00381-0
ZhangCLiuJRaizenM GNiuQ2004 Quantum chaos of Bogoliubov waves for a Bose–Einstein condensate in stadium billiards Phys. Rev. Lett. 93 074101 DOI:10.1103/PhysRevLett.93.074101 [Cited within: 1]
BychekA AMuraevP SKolovskyA R2019 Probing quantum chaos in many-body quantum systems by the induced dissipation Phys. Rev. A 100 013610 DOI:10.1103/PhysRevA.100.013610 [Cited within: 1]
BonciLRoncagliaRWestB JGrigoliniP1991 Quantum irreversibility and chaos Phys. Rev. Lett. 67 25932596 DOI:10.1103/PhysRevLett.67.2593
TomsovicSHellerE J1991 Semiclassical dynamics of chaotic motion: Unexpected long-time accuracy Phys. Rev. Lett. 67 664667 DOI:10.1103/PhysRevLett.67.664
TomsovicSHellerE J1993 Long-time semiclassical dynamics of chaos: the stadium billiard Phys. Rev. E 47 282299 DOI:10.1103/PhysRevE.47.282
JieQXuG1995 Quantum signature of classical chaos in the temporal mean of expectation values of observables Chin. Phys. Lett. 12 577580 DOI:10.1088/0256-307X/12/10/001
XieR-HXuG-O1996 Quantum signature of classical chaos in a Lipkin model: Sensitivity of eigenfunctions to parameter perturbations Chin. Phys. Lett. 13 329332 DOI:10.1088/0256-307X/13/5/003
JieQ-LXuG-O1996 Numerical evidence of quantum correspondence to the classical ergodicity Commun. Theor. Phys. 26 191196 DOI:10.1088/0253-6102/26/2/191
XingY-ZXuG-OLiJ-Q2001 The relation between one-to-one correspondent orthonormal eigenstates of h0 and $h(\lambda )={h}_{0}+\lambda v$ Commun. Theor. Phys. 35 1114 DOI:10.1088/0253-6102/35/1/11 [Cited within: 1]
JacquodPSilvestrovPBeenakkerC2001 Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo Phys. Rev. E 64 055203 DOI:10.1103/PhysRevE.64.055203
JalabertR APastawskiH M2001 Environment-independent decoherence rate in classically chaotic systems Phys. Rev. Lett. 86 24902493 DOI:10.1103/PhysRevLett.86.2490
LiJ-QLiuFXingY-ZZuoW2002 Quantitative measurement of the exponential growth of spreading width of a quantum wave packet in chaotic systems Commun. Theor. Phys. 37 671674 DOI:10.1088/0253-6102/37/6/671
KarkuszewskiZ PJarzynskiCZurekW H2002 Quantum chaotic environments, the butterfly effect, and decoherence Phys. Rev. Lett. 89 170405 DOI:10.1103/PhysRevLett.89.170405
CerrutiN RTomsovicS2002 Sensitivity of wave field evolution and manifold stability in chaotic systems Phys. Rev. Lett. 88 054103 DOI:10.1103/PhysRevLett.88.054103
CucchiettiF MLewenkopfC HMuccioloE RPastawskiH MVallejosR O2002 Measuring the Lyapunov exponent using quantum mechanics Phys. Rev. E 65 046209 DOI:10.1103/PhysRevE.65.046209
CucchiettiF MDalvitD A RPazJ PZurekW H2003 Decoherence and the Loschmidt echo Phys. Rev. Lett. 91 210403 DOI:10.1103/PhysRevLett.91.210403
JacquodPAdagideliİBeenakkerC W J2003 Anomalous power law of quantum reversibility for classically regular dynamics Europhys. Lett. 61 729735 DOI:10.1209/epl/i2003-00289-y
AdamovYGornyiI VMirlinA D2003 Loschmidt echo and Lyapunov exponent in a quantum disordered system Phys. Rev. E 67 056217 DOI:10.1103/PhysRevE.67.056217
VaníčekJHellerE J2003 Semiclassical evaluation of quantum fidelity Phys. Rev. E 68 056208 DOI:10.1103/PhysRevE.68.056208
SankaranarayananRLakshminarayanA2003 Recurrence of fidelity in nearly integrable systems Phys. Rev. E 68 036216 DOI:10.1103/PhysRevE.68.036216
GorinTProsenTSeligmanT HStrunzW T2004 Connection between decoherence and fidelity decay in echo dynamics Phys. Rev. A 70 042105 DOI:10.1103/PhysRevA.70.042105
LiuJWangWZhangCNiuQLiB2005 Fidelity for the quantum evolution of a Bose–Einstein condensate Phys. Rev. A 72 063623 DOI:10.1103/PhysRevA.72.063623
WeinsteinY SHellbergC S2005 Quantum fidelity decay in quasi-integrable systems Phys. Rev. E 71 016209 DOI:10.1103/PhysRevE.71.016209
ProsenTŽnidaričM2005 Quantum freeze of fidelity decay for chaotic dynamics Phys. Rev. Lett. 94 044101 DOI:10.1103/PhysRevLett.94.044101
WangW-GLiB2005 Uniform semiclassical approach to fidelity decay: from weak to strong perturbation Phys. Rev. E 71 066203 DOI:10.1103/PhysRevE.71.066203
GorinTProsenTSeligmanT HZ˘nidaricM2006 Dynamics of Loschmidt echoes and fidelity decay Phys. Rep. 435 33156 DOI:10.1016/j.physrep.2006.09.003
QuanH TSongZLiuX FZanardiPSunC P2006 Decay of Loschmidt echo enhanced by quantum criticality Phys. Rev. Lett. 96 140604 DOI:10.1103/PhysRevLett.96.140604
PellegriniFMontangeroS2007 Fractal fidelity as a signature of quantum chaos Phys. Rev. A 76 052327 DOI:10.1103/PhysRevA.76.052327
ZanardiPQuanH TWangXSunC P2007 Mixed-state fidelity and quantum criticality at finite temperature Phys. Rev. A 75 032109 DOI:10.1103/PhysRevA.75.032109
IanHGongZ RLiuY-XSunC PNoriF2008 Cavity optomechanical coupling assisted by an atomic gas Phys. Rev. A 78 013824 DOI:10.1103/PhysRevA.78.013824
HöhmannRKuhlUStöckmannH-J2008 Algebraic fidelity decay for local perturbations Phys. Rev. Lett. 100 124101 DOI:10.1103/PhysRevLett.100.124101
HuangJ-FLiYLiaoJ-QKuangL-MSunC P2009 Dynamic sensitivity of photon-dressed atomic ensemble with quantum criticality Phys. Rev. A 80 063829 DOI:10.1103/PhysRevA.80.063829
QuanH TCucchiettiF M2009 Quantum fidelity and thermal phase transitions Phys. Rev. E 79 031101 DOI:10.1103/PhysRevE.79.031101
GutkinBWaltnerDGutiérrezMKuipersJRichterK2010 Quantum corrections to fidelity decay in chaotic systems Phys. Rev. E 81 036222 DOI:10.1103/PhysRevE.81.036222
KohlerHSommersH-JÅbergSGuhrT2010 Exact fidelity and full fidelity statistics in regular and chaotic surroundings Phys. Rev. E 81 050103 DOI:10.1103/PhysRevE.81.050103
RamsM MDamskiB2011 Quantum fidelity in the thermodynamic limit Phys. Rev. Lett. 106 055701 DOI:10.1103/PhysRevLett.106.055701
CaiC YAiQQuanH TSunC P2012 Sensitive chemical compass assisted by quantum criticality Phys. Rev. A 85 022315 DOI:10.1103/PhysRevA.85.022315
DoronESmilanskyUFrenkelA1990 Experimental demonstration of chaotic scattering of microwaves Phys. Rev. Lett. 65 30723075 DOI:10.1103/PhysRevLett.65.3072
HaakeFLenzGSebaPSteinJStöckmannH-JŻyczkowskiK1991 Manifestation of wave chaos in pseudo-integrable microwave resonators Phys. Rev. A 44 R6161R6164 DOI:10.1103/PhysRevA.44.R6161
GräfH-DHarneyH LLengelerHLewenkopfC HRangacharyuluCRichterASchardtPWeidenmüllerH A1992 Distribution of eigenmodes in a superconducting stadium billiard with chaotic dynamics Phys. Rev. Lett. 69 12961299 DOI:10.1103/PhysRevLett.69.1296 [Cited within: 1]
KudrolliASridharSPandeyARamaswamyR1994 Signatures of chaos in quantum billiards: microwave experiments Phys. Rev. E 49 R11R14 DOI:10.1103/PhysRevE.49.R11
SoPAnlageS MOttEOerterR N1995 Wave chaos experiments with and without time reversal symmetry: GUE and GOE statistics Phys. Rev. Lett. 74 26622665 DOI:10.1103/PhysRevLett.74.2662
DeusSKochP MSirkoL1995 Statistical properties of the eigenfrequency distribution of three-dimensional microwave cavities Phys. Rev. E 52 11461155 DOI:10.1103/PhysRevE.52.1146
StoffregenUSteinJStöckmannH-JKuśMHaakeF1995 Microwave billiards with broken time reversal symmetry Phys. Rev. Lett. 74 26662669 DOI:10.1103/PhysRevLett.74.2666
AltHGräfH-DHofferbertRRangacharyuluCRehfeldHRichterASchardtPWirzbaA1996 Chaotic dynamics in a three-dimensional superconducting microwave billiard Phys. Rev. E 54 23032312 DOI:10.1103/PhysRevE.54.2303
AltHGräfH-DGuhrTHarneyH LHofferbertRRehfeldHRichterASchardtP1997 Correlation-hole method for the spectra of superconducting microwave billiards Phys. Rev. E 55 66746683 DOI:10.1103/PhysRevE.55.6674
SirkoLKochP MBlümelR1997 Experimental identification of non-newtonian orbits produced by ray splitting in a dielectric-loaded microwave cavity Phys. Rev. Lett. 78 29402943 DOI:10.1103/PhysRevLett.78.2940
KottosTSmilanskyUFortunyJNestiG1999 Chaotic scattering of microwaves Radio Sci. 34 747758 DOI:10.1029/1999RS900037
SirkoLBauchSHlushchukYKochPBlümelRBarthMKuhlUStöckmannH-J2000 Observation of dynamical localization in a rough microwave cavity Phys. Lett. A 266 331335 DOI:10.1016/S0375-9601(00)00052-9
StöckmannH-JBarthMDörrUKuhlUSchanzeH2001 Microwave studies of chaotic billiards and disordered systems Physica E 9 571577 DOI:10.1016/S1386-9477(00)00264-2
DembowskiCDietzBGräfH-DHeineALeyvrazFMiski-OgluMRichterASeligmanT H2003 Phase shift experiments identifying Kramers doublets in a chaotic superconducting microwave billiard of threefold symmetry Phys. Rev. Lett. 90 014102 DOI:10.1103/PhysRevLett.90.014102
KuhlUStöckmannH-JWeaverR2005 Classical wave experiments on chaotic scattering J. Phys. A: Math. Gen. 38 1043310463 DOI:10.1088/0305-4470/38/49/001
DietzBRichterA2015 Quantum and wave dynamical chaos in superconducting microwave billiards Chaos 25 097601 DOI:10.1063/1.4915527 [Cited within: 2]
ZhangRZhangWDietzBChaiGHuangL2019 Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards Chin. Phys. B 28 100502 DOI:10.1088/1674-1056/ab3f96 [Cited within: 2]
LeeS-BYangJMoonSLeeS-YShimJ-BKimS WLeeJ-HAnK2009 Observation of an exceptional point in a chaotic optical microcavity Phys. Rev. Lett. 103 134101 DOI:10.1103/PhysRevLett.103.134101
SongQFangWLiuBHoS-TSolomonG SCaoH2009 Chaotic microcavity laser with high quality factor and unidirectional output Phys. Rev. A 80 041807 DOI:10.1103/PhysRevA.80.041807 [Cited within: 1]
MortessagneFLegrandOSornetteD1993 Transient chaos in room acoustics Chaos 3 529541 DOI:10.1063/1.165958
EllegaardCGuhrTLindemannKLorensenH QNygårdJOxborrowM1995 Spectral statistics of acoustic resonances in aluminum blocks Phys. Rev. Lett. 75 15461549 DOI:10.1103/PhysRevLett.75.1546
EllegaardCGuhrTLindemannKNygårdJOxborrowM1996 Symmetry breaking and spectral statistics of acoustic resonances in quartz blocks Phys. Rev. Lett. 77 49184921 DOI:10.1103/PhysRevLett.77.4918
LeitnerD M1997 Effects of symmetry breaking on statistical properties of near-lying acoustic resonances Phys. Rev. E 56 48904891 DOI:10.1103/PhysRevE.56.4890
LindelofP ENorregaardJHanbergJ1986 New light on the scattering mechanisms in Si inversion layers by weak localization experiments Phys. Scr. 1986 1726 DOI:10.1088/0031-8949/1986/T14/003 [Cited within: 1]
TangYShenYYangJLiuXZiJLiB2008 Experimental evidence of wave chaos from a double slit experiment with water surface waves Phys. Rev. E 78 047201 DOI:10.1103/PhysRevE.78.047201 [Cited within: 1]
CasatiGChirikovB VIzraelevF MFordJ1979 Stochastic behavior of a quantum pendulum under a periodic perturbation Stochastic Behavior in Classical and Quantum Hamiltonian SystemsCasatiGFordJ BerlinSpringer334352 [Cited within: 1]
FishmanSGrempelD RPrangeR E1982 Chaos, quantum recurrences, and Anderson localization Phys. Rev. Lett. 49 509512 DOI:10.1103/PhysRevLett.49.509
GrempelD RFishmanSPrangeR E1982 Localization in an incommensurate potential: an exactly solvable model Phys. Rev. Lett. 49 833836 DOI:10.1103/PhysRevLett.49.833 [Cited within: 1]
PrangeR EGrempelD RFishmanS1984 Solvable model of quantum motion in an incommensurate potential Phys. Rev. B 29 65006512 DOI:10.1103/PhysRevB.29.6500
BerryM V1984 Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator Physica D 10 369378 DOI:10.1016/0167-2789(84)90185-4
FishmanSPrangeR EGriniastyM1989 Scaling theory for the localization length of the kicked rotor Phys. Rev. A 39 16281633 DOI:10.1103/PhysRevA.39.1628
IzrailevF M1990 Simple models of quantum chaos: spectrum and eigenfunctions Phys. Rep. 196 299392 DOI:10.1016/0370-1573(90)90067-C
ChirikovB V1991 A theory of quantum diffusion localization Chaos 1 95100 DOI:10.1063/1.165820
IominAFishmanSZaslavskyG M2002 Quantum localization for a kicked rotor with accelerator mode islands Phys. Rev. E 65 036215 DOI:10.1103/PhysRevE.65.036215
GongJWangJ2007 Quantum diffusion dynamics in nonlinear systems: a modified kicked-rotor model Phys. Rev. E 76 036217 DOI:10.1103/PhysRevE.76.036217
WangHWangJGuarneriICasatiGGongJ2013 Exponential quantum spreading in a class of kicked rotor systems near high-order resonances Phys. Rev. E 88 052919 DOI:10.1103/PhysRevE.88.052919
WangJTianCAltlandA2014 Unconventional quantum criticality in the kicked rotor Phys. Rev. B 89 195105 DOI:10.1103/PhysRevB.89.195105
GhoseSStockRJessenPLalRSilberfarbA2008 Chaos, entanglement, and decoherence in the quantum kicked top Phys. Rev. A 78 042318 DOI:10.1103/PhysRevA.78.042318
AmmannHGrayRShvarchuckIChristensenN1998 Quantum delta-kicked rotor: experimental observation of decoherence Phys. Rev. Lett. 80 41114115 DOI:10.1103/PhysRevLett.80.4111
ChaudhurySSmithAAndersonBGhoseSJessenP S2009 Quantum signatures of chaos in a kicked top Nature 461 768 DOI:10.1038/nature08396
HainautCFangPRançonAClémentJ-F M CSzriftgiserPGarreauJ-CTianCChicireanuR2018 Experimental observation of a time-driven phase transition in quantum chaos Phys. Rev. Lett. 121 134101 DOI:10.1103/PhysRevLett.121.134101 [Cited within: 1]
WangJMonteiroT SFishmanSKeatingJ PSchubertR2007 Fractional ${\hslash }$ scaling for quantum kicked rotors without cantori Phys. Rev. Lett. 99 234101 DOI:10.1103/PhysRevLett.99.234101 [Cited within: 2]
WangJGarcía-GarcíaA M2009 Anderson transition in a three-dimensional kicked rotor Phys. Rev. E 79 036206 DOI:10.1103/PhysRevE.79.036206
WangJGongJ2009 Butterfly floquet spectrum in driven SU(2) systems Phys. Rev. Lett. 102 244102 DOI:10.1103/PhysRevLett.102.244102
LewenkopfC HNemesM CMarvulleVPatoM PWreszinskiW F1991 Level statistics transitions in the spin-boson model Phys. Lett. A 155 113116 DOI:10.1016/0375-9601(91)90575-S
CibilsM BCucheYMarvulleVWreszinskiW FAmietJ PBeckH1991 The semiclassical limit of the spin boson model J. Phys. A: Math. Gen. 24 16611675 DOI:10.1088/0305-4470/24/7/035
EmaryCBrandesT2003 Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model Phys. Rev. Lett. 90 044101 DOI:10.1103/PhysRevLett.90.044101
Chávez-CarlosJLópez-del CarpioBBastarrachea-MagnaniM AStránskýPLerma-HernándezSSantosL FHirschJ G2019 Quantum and classical Lyapunov exponents in atom-field interaction systems Phys. Rev. Lett. 122 024101 DOI:10.1103/PhysRevLett.122.024101
LewisswanR JSafavinainiABollingerJ JReyA M2019 Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model Nat. Commun. 10 1581 DOI:10.1038/s41467-019-09436-y [Cited within: 1]
LipkinH JMeshkovNGlickA J1965 Validity of many-body approximation methods for a solvable model. i. exact solutions and perturbation theory Nucl. Phys. 62 188198 DOI:10.1016/0029-5582(65)90862-X [Cited within: 1]
MeredithD CKooninS EZirnbauerM R1988 Quantum chaos in a schematic shell model Phys. Rev. A 37 34993513 DOI:10.1103/PhysRevA.37.3499
GeiselTRadonsGRubnerJ1986 Kolmogorov-Arnol’d-Moser barriers in the quantum dynamics of chaotic systems Phys. Rev. Lett. 57 28832886 DOI:10.1103/PhysRevLett.57.2883 [Cited within: 1]
FishmanSGrempelD RPrangeR E1987 Temporal crossover from classical to quantal behavior near dynamical critical points Phys. Rev. A 36 289305 DOI:10.1103/PhysRevA.36.289
RadonsGPrangeR E1988 Wave functions at the critical Kolmogorov-Arnol’d-Moser surface Phys. Rev. Lett. 61 16911694 DOI:10.1103/PhysRevLett.61.1691
MaitraN THellerE J2000 Quantum transport through cantori Phys. Rev. E 61 36203631 DOI:10.1103/PhysRevE.61.3620
CreffieldC EHurGMonteiroT S2006 Localization-delocalization transition in a system of quantum kicked rotors Phys. Rev. Lett. 96 024103 DOI:10.1103/PhysRevLett.96.024103
ZhangZTongPGongJLiB2012 Quantum hyperdiffusion in one-dimensional tight-binding lattices Phys. Rev. Lett. 108 070603 DOI:10.1103/PhysRevLett.108.070603
QinPYinCChenS2014 Dynamical Anderson transition in one-dimensional periodically kicked incommensurate lattices Phys. Rev. B 90 054303 DOI:10.1103/PhysRevB.90.054303
LikhachevV NShevaleevskiiO IVinogradovG A2016 Quantum dynamics of charge transfer on the one-dimensional lattice: wave packet spreading and recurrence Chin. Phys. B 25 018708 DOI:10.1088/1674-1056/25/1/018708 [Cited within: 1]
ZhaoW-LGongJWangW-GCasatiGLiuJFuL-B2016 Exponential wave-packet spreading via self-interaction time modulation Phys. Rev. A 94 053631 DOI:10.1103/PhysRevA.94.053631
GholamiELashkamiZ M2017 Noise, delocalization, and quantum diffusion in one-dimensional tight-binding models Phys. Rev. E 95 022216 DOI:10.1103/PhysRevE.95.022216 [Cited within: 1]
LeopoldJ GPercivalI C1978 Microwave ionization and excitation of Rydberg atoms Phys. Rev. Lett. 41 944947 DOI:10.1103/PhysRevLett.41.944
CasatiGChirikovB VShepelyanskyD L1984 Quantum limitations for chaotic excitation of the hydrogen atom in a monochromatic field Phys. Rev. Lett. 53 25252528 DOI:10.1103/PhysRevLett.53.2525
van LeeuwenK A HOppenG vRenwickSBowlinJ BKochP MJensenR VRathORichardsDLeopoldJ G1985 Microwave ionization of hydrogen atoms: experiment versus classical dynamics Phys. Rev. Lett. 55 22312234 DOI:10.1103/PhysRevLett.55.2231
CasatiGChirikovB VGuarneriIShepelyanskyD L1986 Dynamical stability of quantum ‘chaotic’ motion in a hydrogen atom Phys. Rev. Lett. 56 24372440 DOI:10.1103/PhysRevLett.56.2437
CasatiGChirikovB VShepelyanskyD LGuarneriI1986 New photoelectric ionization peak in the hydrogen atom Phys. Rev. Lett. 57 823826 DOI:10.1103/PhysRevLett.57.823
CasatiGChirikovB VGuarneriIShepelyanskyD L1987 Localization of diffusive excitation in the two-dimensional hydrogen atom in a monochromatic field Phys. Rev. Lett. 59 29272930 DOI:10.1103/PhysRevLett.59.2927
CasatiGChirikovB VShepelyanskyD LGuarneriI1987 Relevance of classical chaos in quantum mechanics: the hydrogen atom in a monochromatic field Phys. Rep. 154 77123 DOI:10.1016/0370-1573(87)90009-3
CasatiGChirikovB VShepelyanskyD LGuarneriI1987 Relevance of classical chaos in quantum mechanics: The hydrogen atom in a monochromatic field Phys. Rep. 154 77123 DOI:10.1016/0370-1573(87)90009-3
ChirikovB VIzrailevF MShepelyanskyD L1988 Quantum chaos: localization versus ergodicity Physica D 33 7788 DOI:10.1016/S0167-2789(98)90011-2
BlümelRSmilanskyU1989 Ionization of excited hydrogen atoms by microwave fields: a test case for quantum chaos Phys. Scr. 40 386393 DOI:10.1088/0031-8949/40/3/022
BayfieldJ ECasatiGGuarneriISokolD W1989 Localization of classically chaotic diffusion for hydrogen atoms in microwave fields Phys. Rev. Lett. 63 364367 DOI:10.1103/PhysRevLett.63.364
BlümelRGrahamRSirkoLSmilanskyUWaltherHYamadaK1989 Microwave excitation of Rydberg atoms in the presence of noise Phys. Rev. Lett. 62 341344 DOI:10.1103/PhysRevLett.62.341
JensenRSusskindSSandersM1991 Chaotic ionization of highly excited hydrogen atoms: Comparison of classical and quantum theory with experiment Phys. Rep. 201 156 DOI:10.1016/0370-1573(91)90113-Z
YoakumSSirkoLKochP M1992 Stueckelberg oscillations in the multiphoton excitation of helium Rydberg atoms: Observation with a pulse of coherent field and suppression by additive noise Phys. Rev. Lett. 69 19191922 DOI:10.1103/PhysRevLett.69.1919
HaffmansABlümelRKochP MSirkoL1994 Prediction of a new peak in two-frequency microwave ‘ionization’ of excited hydrogen atoms Phys. Rev. Lett. 73 248251 DOI:10.1103/PhysRevLett.73.248
KochP Mvan LeeuwenK H A1995 The importance of resonances in microwave ‘ionization’ of excited hydrogen atoms Phys. Rep. 256 289403 DOI:10.1016/0370-1573(94)00093-I
KangSChenC-Y2010 Statistics and correlation properties of diamagnetic high Rydberg hydrogen atom Commun. Theor. Phys. 53 105 DOI:10.1088/0253-6102/53/1/23 [Cited within: 1]
AßmannMThewesJFröhlichDBayerM2016 Quantum chaos and breaking of all anti-unitary symmetries in Rydberg excitons Nat. Mater. 15 741745 DOI:10.1038/nmat4622
ZhangYXuXKangLCaiXTangX2018 Analysis of the fractal intrinsic quality in the ionization of Rydberg helium and lithium atoms Chin. Phys. B 27 053401 DOI:10.1088/1674-1056/27/5/053401 [Cited within: 1]
XuLFuL-B2019 Understanding tunneling ionization of atoms in laser fields using the principle of multiphoton absorption Chin.. Phys. Lett. 36 043202 DOI:10.1088/0256-307X/36/4/043202 [Cited within: 2]
GardinerS A2002 Quantum chaos in Bose–Einstein condensates J. Mod. Opt. 49 19711977 DOI:10.1080/09500340210140777
FranzosiRPennaV2003 Chaotic behavior, collective modes, and self-trapping in the dynamics of three coupled Bose–Einstein condensates Phys. Rev. E 67 046227 DOI:10.1103/PhysRevE.67.046227
XieQHaiW2005 Quantum entanglement and chaos in kicked two-component Bose–Einstein condensates Euro. Phys. J. D 33 265272 DOI:10.1140/epjd/e2005-00054-4
MahmudK WPerryHReinhardtW P2005 Quantum phase-space picture of Bose–Einstein condensates in a double well Phys. Rev. A 71 023615 DOI:10.1103/PhysRevA.71.023615
JunXWen-HuaHHuiL2007 Generation and control of chaos in a Bose–Einstein condensate Chin. Phys. 16 2244 DOI:10.1088/1009-1963/16/8/015
KronjägerJSengstockKBongsK2008 Chaotic dynamics in spinor Bose–Einstein condensates New J. Phys. 10 045028 DOI:10.1088/1367-2630/10/4/045028
KoberlePCartariusHFabcicTMainJWunnerG2009 Bifurcations, order and chaos in the Bose–Einstein condensation of dipolar gases New J. Phys. 11 023017 DOI:10.1088/1367-2630/11/2/023017
ValdezM AShchedrinGHeimsothMCreffieldC ESolsFCarrL D2018 Many-body quantum chaos and entanglement in a quantum ratchet Phys. Rev. Lett. 120 234101 DOI:10.1103/PhysRevLett.120.234101 [Cited within: 2]
LiHHaiW-HXuJ2008 Quantum signatures of chaos in adiabatic interaction between a trapped ion and a laser standing wave Commun. Theor. Phys. 49 143 DOI:10.1088/0253-6102/49/1/32 [Cited within: 1]
KrivolapovYFishmanSOttEAntonsenT M2011 Quantum chaos of a mixed open system of kicked cold atoms Phys. Rev. E 83 016204 DOI:10.1103/PhysRevE.83.016204
FrischAMarkMAikawaKFerlainoFBohnJ LMakridesCPetrovAKotochigovaS2014 Quantum chaos in ultracold collisions of gas-phase erbium atoms Nature 507 475479 DOI:10.1038/nature13137 [Cited within: 1]
SpillaneS MKippenbergT JVahalaK J2002 Ultralow-threshold Raman laser using a spherical dielectric microcavity Nature 415 621623 DOI:10.1038/415621a [Cited within: 1]
GenstyTBeckerKFischerIElsäßerWDegenCDebernardiPBavaG P2005 Wave chaos in real-world vertical-cavity surface-emitting lasers Phys. Rev. Lett. 94 233901 DOI:10.1103/PhysRevLett.94.233901
FangWCaoHSolomonG S2007 Control of lasing in fully chaotic open microcavities by tailoring the shape factor Appl. Phys. Lett. 90 081108 DOI:10.1063/1.2535692 [Cited within: 1]
BarangerHMelloP1996 Short paths and information theory in quantum chaotic scattering: transport through quantum dots Europhys. Lett. 33 465 DOI:10.1209/epl/i1996-00364-5
ProsenTZnidaricM2001 Can quantum chaos enhance the stability of quantum computation? J. Phys. A: Math. Gen. 34 L681 DOI:10.1088/0305-4470/34/47/103
PoulinDLaflammeRMilburnG JPazJ P2003 Testing integrability with a single bit of quantum information Phys. Rev. A 68 022302 DOI:10.1103/PhysRevA.68.022302
KottosTSmilanskyU1999 Periodic orbit theory and spectral statistics for quantum graphs Ann. Phys. 274 76124 DOI:10.1006/aphy.1999.5904
BarraFGaspardP2000 On the level spacing distribution in quantum graphs J. Stat. Phys. 101 283319 DOI:10.1023/A:1026495012522
KottosTSmilanskyU2000 Chaotic scattering on graphs Phys. Rev. Lett. 85 968971 DOI:10.1103/PhysRevLett.85.968
PakonskiPZyczkowskiKKusM2001 Classical 1d maps, quantum graphs and ensembles of unitary matrices J. Phys. A: Math. Gen. 34 9303 DOI:10.1088/0305-4470/34/43/313
BerkolaikoGBogomolnyE BKeatingJ P2001 Star graphs and Seba billiards J. Phys. A: Math. Gen. 34 335350 DOI:10.1088/0305-4470/34/3/301
BlümelRDabaghianYJensenR2002 Explicitly solvable cases of one-dimensional quantum chaos Phys. Rev. Lett. 88 044101 DOI:10.1103/PhysRevLett.88.044101
PakońskiPTannerGŻyczkowskiK2003 Families of line-graphs and their quantization J. Stat. Phys. 111 13311352 DOI:10.1023/A:1023012502046
GnutzmannSSmilanskyU2006 Quantum graphs: applications to quantum chaos and universal spectral statistics Adv. Phys. 55 527625 DOI:10.1080/00018730600908042
ŁawniczakMBauchSHulOSirkoL2010 Experimental investigation of the enhancement factor for microwave irregular networks with preserved and broken time reversal symmetry in the presence of absorption Phys. Rev. E 81 046204 DOI:10.1103/PhysRevE.81.046204
ŁawniczakMBauchSHulOSirkoL2011 Experimental investigation of the enhancement factor and the cross-correlation function for graphs with and without time-reversal symmetry: the open system case Phys. Scr. 2011 014014 DOI:10.1088/0031-8949/2011/T143/014014
HulOŁawniczakMBauchSSawickiAKuśMSirkoL2012 Are scattering properties of graphs uniquely connected to their shapes? Phys. Rev. Lett. 109 040402 DOI:10.1103/PhysRevLett.109.040402
AllgaierMGehlerSBarkhofenSStöckmannH-JKuhlU2014 Spectral properties of microwave graphs with local absorption Phys. Rev. E 89 022925 DOI:10.1103/PhysRevE.89.022925
RehemanjiangAAllgaierMJoynerC HMüllerSSieberMKuhlUStöckmannH-J2016 Microwave realization of the Gaussian symplectic ensemble Phys. Rev. Lett. 117 064101 DOI:10.1103/PhysRevLett.117.064101
DietzBYunkoVBialousMBauchSŁawniczakMSirkoL2017 Nonuniversality in the spectral properties of time-reversal-invariant microwave networks and quantum graphs Phys. Rev. E 95 052202 DOI:10.1103/PhysRevE.95.052202
ŁawniczakMLipovskýJ C VSirkoL2019 Non-Weyl microwave graphs Phys. Rev. Lett. 122 140503 DOI:10.1103/PhysRevLett.122.140503
BiałousMDietzBSirkoL2019 Experimental investigation of the elastic enhancement factor in a microwave cavity emulating a chaotic scattering system with varying openness Phys. Rev. E 100 012210 DOI:10.1103/PhysRevE.100.012210 [Cited within: 1]
AurichRBolteJSteinerF1994 universal signatures of quantum chaos Phys. Rev. Lett. 73 13561359 DOI:10.1103/PhysRevLett.73.1356
BlumGGnutzmannSSmilanskyU2002 Nodal domains statistics: a criterion for quantum chaos Phys. Rev. Lett. 88 114101 DOI:10.1103/PhysRevLett.88.114101
RelañoAGómezJ M GMolinaR ARetamosaJFaleiroE2002 Quantum chaos and 1/f noise Phys. Rev. Lett. 89 244102 DOI:10.1103/PhysRevLett.89.244102
FaleiroEGómezJ M GMolinaR AMuñozLRelañoARetamosaJ2004 Theoretical derivation of 1/f noise in quantum chaos Phys. Rev. Lett. 93 244101 DOI:10.1103/PhysRevLett.93.244101
HemmadySZhengXOttEAntonsenT MAnlageS M2005 universal impedance fluctuations in wave chaotic systems Phys. Rev. Lett. 94 014102 DOI:10.1103/PhysRevLett.94.014102
NandkishoreRHuseD A2015 Many-body localization and thermalization in quantum statistical mechanics Annu. Rev. Condens. Matter Phys. 6 1538 DOI:10.1146/annurev-conmatphys-031214-014726
SchreiberMHodgmanSBordiaPLuschenH PFischerM HVoskRAltmanESchneiderUBlochI2015 Observation of many-body localization of interacting fermions in a quasirandom optical lattice Science 349 842845 DOI:10.1126/science.aaa7432
ChoiJHildSZeiherJSchausPRubioabadalAYefsahTKhemaniVHuseD ABlochIGrossC2016 Exploring the many-body localization transition in two dimensions Science 352 15471552 DOI:10.1126/science.aaf8834
ImbrieJ Z2016 On many-body localization for quantum spin chains J. Stat. Phys. 163 9981048 DOI:10.1007/s10955-016-1508-x
RigolMDunjkoVOlshaniiM2008 Thermalization and its mechanism for generic isolated quantum systems Nature 452 854 DOI:10.1038/nature06838
RigolMSantosL F2010 Quantum chaos and thermalization in gapped systems Phys. Rev. A 82 011604 DOI:10.1103/PhysRevA.82.011604
AlbaV2015 Eigenstate thermalization hypothesis and integrability in quantum spin chains Phys. Rev. B 91 155123 DOI:10.1103/PhysRevB.91.155123
De PalmaGSerafiniAGiovannettiVCramerM2015 Necessity of eigenstate thermalization Phys. Rev. Lett. 115 220401 DOI:10.1103/PhysRevLett.115.220401
KaufmanA MTaiM ELukinARispoliMSchittkoRPreissP MGreinerM2016 Quantum thermalization through entanglement in an isolated many-body system Science 353 794800 DOI:10.1126/science.aaf6725
D’AlessioLKafriYPolkovnikovARigolM2016 From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics Adv. Phys. 65 239362 DOI:10.1080/00018732.2016.1198134
TianCYangKFangPZhouH-JWangJ2018 Hidden thermal structure in fock space Phys. Rev. E 98 060103 DOI:10.1103/PhysRevE.98.060103
FangPZhaoLTianC2018 Concentration-of-measure theory for structures and fluctuations of waves Phys. Rev. Lett. 121 140603 DOI:10.1103/PhysRevLett.121.140603
AnzaFGogolinCHuberM2018 Eigenstate thermalization for degenerate observables Phys. Rev. Lett. 120 150603 DOI:10.1103/PhysRevLett.120.150603
XuJLiY2019 Eigenstate distribution fluctuation of a quenched disordered bose-hubbard system in thermal-to-localized transitions Chin. Phys. Lett. 36 027201 DOI:10.1088/0256-307X/36/2/027201 [Cited within: 1]
CampisiMGooldJ2017 Thermodynamics of quantum information scrambling Phys. Rev. E 95 062127 DOI:10.1103/PhysRevE.95.062127
BinderFCorreaL AGogolinCAndersJAdessoG2018 Thermodynamics in the quantum regime Fundamental Theories of PhysicsBerlinSpringer [Cited within: 1]
AleinerI LFaoroLIoffeL B2016 Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves Ann. Phys. 375 378406 DOI:10.1016/j.aop.2016.09.006 [Cited within: 1]
RozenbaumE BGalitskiVGaneshanS2017 Lyapunov exponent and out-of-time-ordered correlator’s growth rate in a chaotic system Phys. Rev. Lett. 118 086801 DOI:10.1103/PhysRevLett.118.086801
HalpernN Y2017 Jarzynski-like equality for the out-of-time-ordered correlator Phys. Rev. A 95 012120 DOI:10.1103/PhysRevA.95.012120
HuangYZhangY-LChenX2017 Out-of-time-ordered correlators in many-body localized systems Ann. Phys. 529 1600318 DOI:10.1002/andp.201600318
LiJFanRWangHYeBZengBZhaiHPengXDuJ2017 Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator Phys. Rev. X 7 031011 DOI:10.1103/PhysRevX.7.031011
HalpernN YSwingleBDresselJ2018 Quasiprobability behind the out-of-time-ordered correlator Phys. Rev. A 97 042105 DOI:10.1103/PhysRevA.97.042105
GarciamataIJalabertR ASaracenoMRoncagliaA JWisniackiD A2018 Chaos signatures in the short and long time behavior of the out-of-time ordered correlator Phys. Rev. Lett. 121 210601 DOI:10.1103/PhysRevLett.121.210601
RakovszkyTPollmannFVon KeyserlingkC2018 Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation Phys. Rev. X 8 031058 DOI:10.1103/PhysRevX.8.031058
AlonsoJ R GHalpernN YDresselJ2019 Out-of-time-ordered-correlator quasiprobabilities robustly witness scrambling Phys. Rev. Lett. 122 040404 DOI:10.1103/PhysRevLett.122.040404
LakshminarayanA2019 Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices Phys. Rev. E 99 012201 DOI:10.1103/PhysRevE.99.012201 [Cited within: 1]
ZhangY-HZhangJ-QXuX-YLinS-L2009 The quantum spectral analysis of the two-dimensional annular billiard system Chin. Phys. B 18 3539 DOI:10.1088/1674-1056/18/1/006 [Cited within: 1]
YuHRenZZhangX2019 Dynamical stable-jump-stable-jump picture in a non-periodically driven quantum relativistic kicked rotor system Chin. Phys. B 28 20504 DOI:10.1088/1674-1056/28/2/020504 [Cited within: 2]
LiuC-RYuPChenX-ZXuH-YHuangLLaiY-C2019 Enhancing von neumann entropy by chaos in spin-orbit entanglement Chin. Phys. B 28 100501 DOI:10.1088/1674-1056/ab3dff [Cited within: 2]
TanJ-TLuoY-RZhouZHaiW-H2016 Combined effect of classical chaos and quantum resonance on entanglement dynamics Chin. Phys. Lett. 33 070302 DOI:10.1088/0256-307X/33/7/070302 [Cited within: 2]
GeMZhangYWangDDuMLinS2005 The dynamical properties of Rydberg hydrogen atom near a metal surface Sci. China-Phys. Mech. Astron. 48 667675 DOI:10.1360/142004-85 [Cited within: 1]
LiHGaoSXuXLinS2008 Scattering matrix theory for Cs Rydberg atoms in magnetic field Sci. China-Phys. Mech. Astron. 51 499506 DOI:10.1007/s11433-008-0063-0
XinJLiangJ2014 Exact solutions of a spin–orbit coupling model in two-dimensional central-potentials and quantum–classical correspondence Sci. China-Phys. Mech. Astron. 57 15041510 DOI:10.1007/s11433-014-5531-0
BerryM VMondragonR J1987 Neutrino billiards—time-reversal symmetry-breaking without magnetic-fields Proc. R. Soc. A 412 5374 DOI:10.1098/rspa.1987.0080 [Cited within: 6]
AntoineMComtetAKnechtM1990 Heat kernel expansion for fermionic billiards in an external magnetic field J. Phys. A: Math. Gen. 23 L35 DOI:10.1088/0305-4470/23/1/007 [Cited within: 2]
PhatakS CPalSBiswasD1995 Semiclassical features in the quantum description of a Dirac particle in a cavity Phys. Rev. E 52 13331344 DOI:10.1103/PhysRevE.52.1333 [Cited within: 2]
NovoselovK SGeimA KMorozovS VJiangDZhangYDubonosS VGrigorievaI VFirsovA A2004 Electric field effect in atomically thin carbon films Science 306 666669 DOI:10.1126/science.1102896 [Cited within: 1]
NovoselovK SGeimA KMorozovS VJiangDKatsnelsonM IGrigorievaI VDubonosS VFirsovA A2005 Two-dimensional gas of massless Dirac fermions in graphene Nature 438 197200 DOI:10.1038/nature04233
LockhartC MMisraBPrigogineI1982 Geodesic instability and internal time in relativistic cosmology Phys. Rev. D 25 921929 DOI:10.1103/PhysRevD.25.921
ChernikovA ATélTVattayGZaslavskyG M1989 Chaos in the relativistic generalization of the standard map Phys. Rev. A 40 40724076 DOI:10.1103/PhysRevA.40.4072
NomuraYIchikawaY HHortonW1992 Nonlinear dynamics of the relativistic standard map Phys. Rev. A 45 11031115 DOI:10.1103/PhysRevA.45.1103
KimJ-HLeeH-W1995 Relativistic chaos in the driven harmonic oscillator Phys. Rev. E 51 15791581 DOI:10.1103/PhysRevE.51.1579
DrakeS PDettmannC PFrankelN ECornishN J1996 Chaos in special relativistic dynamics Phys. Rev. E 53 13511361 DOI:10.1103/PhysRevE.53.1351
PodolskýJ VVeselýK1998 Chaos in pp-wave spacetimes Phys. Rev. D 58 081501 DOI:10.1103/PhysRevD.58.081501
BarrowJ DLevinJ1998 Chaos in the Einstein-Yang-Mills equations Phys. Rev. Lett. 80 656659 DOI:10.1103/PhysRevLett.80.656
TomaschitzR2000 Tachyons, Lamb shifts and superluminal chaos Euro. Phys. J. B 17 523536 DOI:10.1007/s100510070130
BurnellFMannR BOhtaT2003 Chaos in a relativistic 3-body self-gravitating system Phys. Rev. Lett. 90 134101 DOI:10.1103/PhysRevLett.90.134101
MotterA E2003 Relativistic chaos is coordinate invariant Phys. Rev. Lett. 91 231101 DOI:10.1103/PhysRevLett.91.231101
MotterA ESaaA2009 Relativistic invariance of Lyapunov exponents in bounded and unbounded systems Phys. Rev. Lett. 102 184101 DOI:10.1103/PhysRevLett.102.184101
BernalJ DSeoaneJ MSanjuánM A F2017 Global relativistic effects in chaotic scattering Phys. Rev. E 95 032205 DOI:10.1103/PhysRevE.95.032205
BernalJ DSeoaneJ MSanjuánM A F2018 Uncertainty dimension and basin entropy in relativistic chaotic scattering Phys. Rev. E 97 042214 DOI:10.1103/PhysRevE.97.042214 [Cited within: 1]
ZaitsevOFrustagliaDRichterK2004 Role of orbital dynamics in spin relaxation and weak antilocalization in quantum dots Phys. Rev. Lett. 94 026809 DOI:10.1103/PhysRevLett.94.026809
ChangC HMal’shukovA GChaoK A2004 Spin relaxation dynamics of quasiclassical electrons in ballistic quantum dots with strong spin–orbit coupling Phys. Rev. B 70 245309 DOI:10.1103/PhysRevB.70.245309
AkgucG BGongJ2008 Spin-dependent electron transport in two-dimensional waveguides of arbitrary geometry Phys. Rev. B 77 205302 DOI:10.1103/PhysRevB.77.205302
YingLLaiY-C2016 Enhancement of spin polarization by chaos in graphene quantum dot systems Phys. Rev. B 93 085408 DOI:10.1103/PhysRevB.93.085408
TomaschitzR1991 Relativistic quantum chaos in Robertson-Walker cosmologies J. Math. Phys. 32 25712579 DOI:10.1063/1.529102 [Cited within: 2]
BergerB K1989 Quantum chaos in the mixmaster universe Phys. Rev. D 39 24262429 DOI:10.1103/PhysRevD.39.2426
KirillovA AMelnikovV N1995 Dynamics of inhomogeneities of the metric in the vicinity of a singularity in multidimensional cosmology Phys. Rev. D 52 723729 DOI:10.1103/PhysRevD.52.723
CalzettaEGonzalezJ J1995 Chaos and semiclassical limit in quantum cosmology Phys. Rev. D 51 68216828 DOI:10.1103/PhysRevD.51.6821
CornishN JShellardE P S1998 Chaos in quantum cosmology Phys. Rev. Lett. 81 35713574 DOI:10.1103/PhysRevLett.81.3571
DamourTHenneauxM2001 ${E}_{10},{\mathrm{BE}}_{10}$ and arithmetical chaos in superstring cosmology Phys. Rev. Lett. 86 47494752 DOI:10.1103/PhysRevLett.86.4749
BojowaldMDateG2004 Quantum suppression of the generic chaotic behavior close to cosmological singularities Phys. Rev. Lett. 92 071302 DOI:10.1103/PhysRevLett.92.071302
ShuryakEVerbaarschotJ J M1993 Random matrix theory and spectral sum rules for the Dirac operator in QCD Nucl. Phys. 560 306320 DOI:10.1016/0375-9474(93)90098-I
VerbaarschotJ J MZahedI1993 Spectral density of the QCD Dirac operator near zero virtuality Phys. Rev. Lett. 70 38523855 DOI:10.1103/PhysRevLett.70.3852
VerbaarschotJ1994 Spectrum of the QCD Dirac operator and chiral random matrix theory Phys. Rev. Lett. 72 25312533 DOI:10.1103/PhysRevLett.72.2531
HalaszM AVerbaarschotJ J M1995 universal fluctuations in spectra of the lattice Dirac operator Phys. Rev. Lett. 74 39203923 DOI:10.1103/PhysRevLett.74.3920
BergB AMarkumHPullirschR1999 Quantum chaos in compact lattice QED Phys. Rev. D 59 097504 DOI:10.1103/PhysRevD.59.097504
AkemannGKanzieperE2000 Spectra of massive and massless QCD Dirac operators: a novel link Phys. Rev. Lett. 85 11741177 DOI:10.1103/PhysRevLett.85.1174
ToublanDVerbaarschotJ2001 Statistical properties of the spectrum of the QCD Dirac operator at low energy Nucl. Phys. B 603 343368 DOI:10.1016/S0550-3213(01)00093-1
ZhaoQMüllerC AGongJ2014 Quantum and classical superballistic transport in a relativistic kicked-rotor system Phys. Rev. E 90 022921 DOI:10.1103/PhysRevE.90.022921
YabanaKHoriuchiH1986 Adiabatic viewpoint for the WKB treatment of coupled channel system appearance of the Berry phase and another extra phase accompanying the adiabatic motion Prog. Theor. Phys. 75 592618 DOI:10.1143/PTP.75.592
BerryM VWilkinsonM1984 Diabolical points in the spectra of triangles Proc. R. Soc. A 392 1543 DOI:10.1098/rspa.1984.0022
WilczekFShapereA1989Geometric Phases in Physics vol 5SingaporeWorld Scientific
KuratsujiHIidaS1985 Effective action for adiabatic process dynamical meaning of Berry and Simon's phase Prog. Theor. Phys. 74 439445 DOI:10.1143/PTP.74.439
KuratsujiHIidaS1988 Deformation of symplectic structure and anomalous commutators in field theories Phys. Rev. D 37 441447 DOI:10.1103/PhysRevD.37.441
LittlejohnR GFlynnW G1991 Geometric phases in the asymptotic theory of coupled wave equations Phys. Rev. A 44 52395256 DOI:10.1103/PhysRevA.44.5239
LittlejohnR GFlynnW G1991 Geometric phases and the Bohr–Sommerfeld quantization of multicomponent wave fields Phys. Rev. Lett. 66 28392842 DOI:10.1103/PhysRevLett.66.2839
EmmrichCWeinsteinA1996 Geometry of the transport equation in multicomponent WKB approximations Commun. Math. Phys. 176 701711 DOI:10.1007/BF02099256
YajimaK1982 The quasiclassical approximation to Dirac equation. I J. Fac Sci. Univ. Tokyo Math. 29 161194Sect. 1 A
BagrovV GBelovV VTrifonovA YYevseyevichA A1994 Quantization of closed orbits in Dirac theory by Maslov's complex germ method J. Phys. A: Math. Gen. 27 10211043 DOI:10.1088/0305-4470/27/3/039
BagrovV GBelovV VTrifonovA YYevseyevichtA A1994 Quasi-classical spectral series of the Dirac operators corresponding to quantized two-dimensional Lagrangian tori J. Phys. A: Math. Gen. 27 52735306 DOI:10.1088/0305-4470/27/15/025
SpohnH2000 Semiclassical limit of the Dirac equation and spin precession Ann. Phys., NY 282 420431 DOI:10.1006/aphy.2000.6039
BolteJKeppelerS1999 A semiclassical approach to the Dirac equation Ann. Phys., NY 274 125162 DOI:10.1006/aphy.1999.5912
BolteJKeppelerS1999 Semiclassical form factor for chaotic systems with spin 1/2 J. Phys. A: Math. Gen. 32 8863 DOI:10.1088/0305-4470/32/50/307
BolteJKeppelerS1998 Semiclassical time evolution and trace formula for relativistic spin-1/2 particles Phys. Rev. Lett. 81 19871991 DOI:10.1103/PhysRevLett.81.1987
BolteJGlaserRKeppelerS2001 Quantum and classical ergodicity of spinning particles Ann. Phys., NY 293 114 DOI:10.1006/aphy.2001.6164
WurmJRichterKAdagideliI2011 Edge effects in graphene nanostructures: from multiple reflection expansion to density of states Phys. Rev. B 84 075468 DOI:10.1103/PhysRevB.84.075468 [Cited within: 1]
WurmJRichterKAdagideliI2011 Edge effects in graphene nanostructures: semiclassical theory of spectral fluctuations and quantum transport Phys. Rev. B 84 205421 DOI:10.1103/PhysRevB.84.205421 [Cited within: 1]
HuangLLaiY-CGrebogiC2011 Characteristics of level-spacing statistics in chaotic graphene billiards Chaos 21 013102 DOI:10.1063/1.3537814 [Cited within: 3]
DietzBKlausTMiski-OgluMRichterAWunderleMBouazzaC2016 Spectral properties of Dirac billiards at the van Hove singularities Phys. Rev. Lett. 116 023901 DOI:10.1103/PhysRevLett.116.023901 [Cited within: 3]
TanC-LTanZ-BMaLChenJYangFQuF-MLiuG-TYangH-FYangC-LLüL2009 Quantum chaos in graphene nanoribbon quantum dot Acta Phys. Sin. 58 5726 [Cited within: 1]
PoliniMGuineaFLewensteinMManoharanH CPellegriniV2013 Artificial honeycomb lattices for electrons, atoms and photons Nat. Nanotechnol. 8 625633 DOI:10.1038/nnano.2013.161 [Cited within: 1]
ZandbergenS Rde DoodM J A2010 Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene Phys. Rev. Lett. 104 043903 DOI:10.1103/PhysRevLett.104.043903
BittnerSDietzBMiski-OgluMOria IriartePRichterASchäferF2010 Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene Phys. Rev. B 82 014301 DOI:10.1103/PhysRevB.82.014301
KuhlUBarkhofenSTudorovskiyTStöckmannH-JHossainTde Forges de ParnyLMortessagneF2010 Dirac point and edge states in a microwave realization of tight-binding graphene-like structures Phys. Rev. B 82 094308 DOI:10.1103/PhysRevB.82.094308
BellecMKuhlUMontambauxGMortessagneF2013 Topological transition of Dirac points in a microwave experiment Phys. Rev. Lett. 110 033902 DOI:10.1103/PhysRevLett.110.033902
PooYWuR-XLinZYangYChanC T2011 Experimental realization of self-guiding unidirectional electromagnetic edge states Phys. Rev. Lett. 106 093903 DOI:10.1103/PhysRevLett.106.093903
BittnerSDietzBMiski-OgluMRichterA2012 Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard Phys. Rev. B 85 064301 DOI:10.1103/PhysRevB.85.064301
BellecMKuhlUMontambauxGMortessagneF2013 Tight-binding couplings in microwave artificial graphene Phys. Rev. B 88 115437 DOI:10.1103/PhysRevB.88.115437
WangXJiangH TYanCSunYLiY HShiY LChenH2013 Anomalous transmission of disordered photonic graphenes at the Dirac point Europhys. Lett. 103 17003 DOI:10.1209/0295-5075/103/17003
PlotnikYet al. 2013 Observation of unconventional edge states in photonic graphene Nat. Mater. 13 5762 DOI:10.1038/nmat3783
DietzBIachelloFMiski-OgluMPietrallaNRichterAvon SmekalLWambachJ2013 Lifshitz and excited-state quantum phase transitions in microwave Dirac billiards Phys. Rev. B 88 104101 DOI:10.1103/PhysRevB.88.104101
WangXJiangH TYanCDengF SSunYLiY HShiY LChenH2014 Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals Europhys. Lett. 108 14002 DOI:10.1209/0295-5075/108/14002
WangXJiangHLiYYanCDengFSunYLiYShiYChenH2015 Transport properties of disordered photonic crystals around a Dirac-like point Opt. Exp. 23 51265133 DOI:10.1364/OE.23.005126
DietzBKlausTMiski-OgluMRichterA2015 Spectral properties of superconducting microwave photonic crystals modeling Dirac billiards Phys. Rev. B 91 035411 DOI:10.1103/PhysRevB.91.035411 [Cited within: 1]
YuPLiZ-YXuH-YHuangLDietzBGrebogiCLaiY-C2016 Gaussian orthogonal ensemble statistics in graphene billiards with the shape of classically integrable billiards Phys. Rev. E 94 062214 DOI:10.1103/PhysRevE.94.062214 [Cited within: 3]
CabosartDFeltenAReckingerNIordanescuAToussaintSFanielSHackensB2017 Recurrent quantum scars in a mesoscopic graphene ring Nano Lett. 17 13441349 DOI:10.1021/acs.nanolett.6b03725 [Cited within: 1]
BarrosM S MJúniorA J NMacedo-JuniorA FRamosJ G G SBarbosaA L R2013 Open chaotic Dirac billiards: weak (anti)localization, conductance fluctuations, and decoherence Phys. Rev. B 88 245133 DOI:10.1103/PhysRevB.88.245133
SchomerusHMarcianiMBeenakkerC W J2015 Effect of chiral symmetry on chaotic scattering from Majorana zero modes Phys. Rev. Lett. 114 166803 DOI:10.1103/PhysRevLett.114.166803
RamosJ G G SHusseinM SBarbosaA L R2016 Fluctuation phenomena in chaotic Dirac quantum dots: Artificial atoms on graphene flakes Phys. Rev. B 93 125136 DOI:10.1103/PhysRevB.93.125136
VasconcelosT CRamosJ G G SBarbosaA L R2016 universal spin Hall conductance fluctuations in chaotic Dirac quantum dots Phys. Rev. B 93 115120 DOI:10.1103/PhysRevB.93.115120 [Cited within: 1]
DattaS1995Electronic Transport in Mesoscopic systemsCambridgeCambridge University Press [Cited within: 2]
LiT CLuS-P2008 Quantum conductance of graphene nanoribbons with edge defects Phys. Rev. B 77 085408 DOI:10.1103/PhysRevB.77.085408
HuangLLaiYFerryD KAkisRGoodnickS M2009 Transmission and scarring in graphene quantum dots J. Phys.: Condens. Matter 21 344203 DOI:10.1088/0953-8984/21/34/344203
YangRHuangLLaiY-CPecoraL M2012 Modulating quantum transport by transient chaos Appl. Phys. Lett. 100 093105 DOI:10.1063/1.3690046 [Cited within: 1]
YangRHuangLLaiY-CGrebogiCPecoraL M2013 Harnessing quantum transport by transient chaos Chaos 23 013125 DOI:10.1063/1.4790863 [Cited within: 1]
YingLHuangLLaiY-CGrebogiC2012 Conductance fluctuations in graphene systems: the relevance of classical dynamics Phys. Rev. B 85 245448 DOI:10.1103/PhysRevB.85.245448 [Cited within: 1]
LévyL PDolanGDunsmuirJBouchiatH1990 Magnetization of mesoscopic copper rings: evidence for persistent currents Phys. Rev. Lett. 64 20742077 DOI:10.1103/PhysRevLett.64.2074 [Cited within: 1]
ChandrasekharVWebbR ABradyM JKetchenM BGallagherW JKleinsasserA1991 Magnetic response of a single, isolated gold loop Phys. Rev. Lett. 67 35783581 DOI:10.1103/PhysRevLett.67.3578
Bleszynski-Jayich1A CShanksW EPeaudecerfBGinossarEvon OppenFGlazmanLHarrisJ G E2009 Persistent currents in normal metal rings Science 326 272275 DOI:10.1126/science.1178139
MaillyDchelierCBenoitA1993 Experimental observation of persistent currents in GaAs-AlGaAs single loop Phys. Rev. Lett. 70 20202023 DOI:10.1103/PhysRevLett.70.2020 [Cited within: 1]
ChakrabortyTPietiläinenP1995 Persistent currents in a quantum ring: effects of impurities and interactions Phys. Rev. B 52 19321935 DOI:10.1103/PhysRevB.52.1932 [Cited within: 1]
KawabataS1999 Persistent currents in quantum chaotic systems Phys. Rev. B 59 1225612259 DOI:10.1103/PhysRevB.59.12256
PershinY VPiermarocchiC2005 Persistent and radiation-induced currents in distorted quantum rings Phys. Rev. B 72 125348 DOI:10.1103/PhysRevB.72.125348
Bruno-AlfonsoALatgéA2008 Quantum rings of arbitrary shape and non-uniform width in a threading magnetic field Phys. Rev. B 77 205303 DOI:10.1103/PhysRevB.77.205303 [Cited within: 1]
LiJLouW-KZhangDLiX-JYangWChangK2014 Single- and few-electron states in topological-insulator quantum dots Phys. Rev. B 90 115303 DOI:10.1103/PhysRevB.90.115303
ZhaoYWyrickJNattererF DRodriguez-NievaJ FLewandowskiCWatanabeKTaniguchiTLevitovL SZhitenevN BStroscioJ A2015 Creating and probing electron whispering-gallery modes in graphene Science 348 672675 DOI:10.1126/science.aaa7469 [Cited within: 1]
XuH-YHuangLLaiY-CGrebogiC2015 Superpersistent currents and whispering gallery modes in relativistic quantum chaotic systems Sci. Rep. 5 8963 DOI:10.1038/srep08963 [Cited within: 2]
XuH-YHuangLLaiY-C2016 A robust relativistic quantum two-level system with edge-dependent currents and spin polarization Europhys. Lett. 115 20005 DOI:10.1209/0295-5075/115/20005 [Cited within: 1]
HanC-DXuH-YHuangLLaiY-C2019 Manifestations of chaos in relativistic quantum systems—a study based on out-of-time-order correlator Phys. Open 1 100001 DOI:10.1016/j.physo.2019.100001 [Cited within: 2]
ShenRShaoL BWangBXingD Y2010 Single Dirac cone with a flat band touching on line-centered-square optical lattices Phys. Rev. B 81 041410 DOI:10.1103/PhysRevB.81.041410 [Cited within: 2]
DóraBKailasvuoriJMoessnerR2011 Lattice generalization of the Dirac equation to general spin and the role of the flat band Phys. Rev. B 84 195422 DOI:10.1103/PhysRevB.84.195422 [Cited within: 1]
WangFRanY2011 Nearly flat band with Chern number c=2 on the dice lattice Phys. Rev. B 84 241103 DOI:10.1103/PhysRevB.84.241103
HuangXLaiYHangZ HZhengHChanC T2011 Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials Nat. Mater. 10 582586 DOI:10.1038/nmat3030 [Cited within: 1]
MeiJWuYChanC TZhangZ-Q2012 First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals Phys. Rev. B 86 035141 DOI:10.1103/PhysRevB.86.035141
MoitraPYangY-MAndersonZKravchenkoI IBriggsD PValentineJ2013 Realization of an all-dielectric zero-index optical metamaterial Nat. Photon. 7 791795 DOI:10.1038/nphoton.2013.214
RaouxAMorigiMFuchsJ-NPiéchonFMontambauxG2014 From dia- to paramagnetic orbital susceptibility of massless fermions Phys. Rev. Lett. 112 026402 DOI:10.1103/PhysRevLett.112.026402
Guzmán-SilvaDMejía-CortésCBandresM ARechtsmanM CWeimannSNolteSSegevMSzameitAVicencioR A2014 Experimental observation of bulk and edge transport in photonic Lieb lattices New J. Phys. 16 063061 DOI:10.1088/1367-2630/16/6/063061 [Cited within: 1]
RomhányiJPencKGaneshR2015 Hall effect of triplons in a dimerized quantum magnet Nat. Commun. 6 6805 DOI:10.1038/ncomms7805
GiovannettiGCaponeMvan den BrinkJOrtixC2015 Kekulé textures, pseudospin-one Dirac cones, and quadratic band crossings in a graphene-hexagonal indium chalcogenide bilayer Phys. Rev. B 91 121417 DOI:10.1103/PhysRevB.91.121417
MukherjeeSSpracklenAChoudhuryDGoldmanNÖhbergPAnderssonEThomsonR R2015 Observation of a localized flat-band state in a photonic Lieb lattice Phys. Rev. Lett. 114 245504 DOI:10.1103/PhysRevLett.114.245504 [Cited within: 1]
VicencioR ACantillanoCMorales-InostrozaLRealBMejía-CortésCWeimannSSzameitAMolinaM I2015 Observation of localized states in Lieb photonic lattices Phys. Rev. Lett. 114 245503 DOI:10.1103/PhysRevLett.114.245503 [Cited within: 1]
TaieSOzawaHIchinoseTNishioTNakajimaSTakahashiY2015 Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice Sci. Adv. 1 e1500854 DOI:10.1126/sciadv.1500854 [Cited within: 1]
FangAZhangZ QLouieS GChanC T2016 Klein tunneling and supercollimation of pseudospin-1 electromagnetic waves Phys. Rev. B 93 035422 DOI:10.1103/PhysRevB.93.035422 [Cited within: 1]
DiebelFLeykamDKroesenSDenzCDesyatnikovA S2016 Conical diffraction and composite Lieb Bosons in photonic lattices Phys. Rev. Lett. 116 183902 DOI:10.1103/PhysRevLett.116.183902 [Cited within: 1]
ZhuLWangS-SGuanSLiuYZhangTChenGYangS A2016 Blue phosphorene oxide: strain-tunable quantum phase transitions and novel 2D emergent fermions Nano Lett. 16 65486554 DOI:10.1021/acs.nanolett.6b03208
BradlynBCanoJWangZVergnioryM GFelserCCavaR JBernevigB A Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals Science 353 17 DOI:10.1126/science.aaf5037 [Cited within: 1]
FulgaI CSternA2017 Triple point fermions in a minimal symmorphic model Phys. Rev. B 95 241116 DOI:10.1103/PhysRevB.95.241116
EzawaM2017 Triplet fermions and Dirac fermions in borophene Phys. Rev. B 96 035425 DOI:10.1103/PhysRevB.96.035425
ZhongCChenYYuZ-MXieYWangHYangS AZhangS2017 Three-dimensional pentagon carbon with a genesis of emergent fermions Nat. Commun. 8 15641 DOI:10.1038/ncomms15641
ZhuY-QZhangD-WYanHXingD-YZhuS-L2017 Emergent pseudospin-1 Maxwell fermions with a threefold degeneracy in optical lattices Phys. Rev. A 96 033634 DOI:10.1103/PhysRevA.96.033634
DrostROjanenTHarjuALiljerothP2017 Topological states in engineered atomic lattices Nat. Phys. 13 668 DOI:10.1038/nphys4080 [Cited within: 1]
SlotM RGardenierT SJacobseP Hvan MiertG C PKempkesS NZevenhuizenS J MSmithC MVanmaekelberghDSwartI2017 Experimental realization and characterization of an electronic Lieb lattice Nat. Phys. 13 672 DOI:10.1038/nphys4105 [Cited within: 1]
XuH-YLaiY-C2016 Revival resonant scattering, perfect caustics, and isotropic transport of pseudospin-1 particles Phys. Rev. B 94 165405 DOI:10.1103/PhysRevB.94.165405 [Cited within: 2]
CheianovV VFal’koVAltshulerB L2007 The focusing of electron flow and a Veselago lens in graphene p–n junctions Science 315 12521255 DOI:10.1126/science.1138020 [Cited within: 1]
LeeG-HParkG-HLeeH-J2015 Observation of negative refraction of Dirac fermions in graphene Nat. Phys. 11 925929letter DOI:10.1038/nphys3460
AkisRBirdJ PFerryD K2002 The persistence of eigenstates in open quantum dots Appl. Phys. Lett. 81 129131 DOI:10.1063/1.1490404
HarayamaTDavisPIkedaK S2003 Stable oscillations of a spatially chaotic wave function in a microstadium laser Phys. Rev. Lett. 90 063901 DOI:10.1103/PhysRevLett.90.063901
LeeS-YKurdoglyanM SRimSKimC-M2004 Resonance patterns in a stadium-shaped microcavity Phys. Rev. A 70 023809 DOI:10.1103/PhysRevA.70.023809
FangWYamilovACaoH2005 Analysis of high-quality modes in open chaotic microcavities Phys. Rev. A 72 023815 DOI:10.1103/PhysRevA.72.023815
YusupovJOtajanovDEshniyazovVMatrasulovD2018 Classical and quantum dynamics of a kicked relativistic particle in a box Phys. Lett. A 382 633638 DOI:10.1016/j.physleta.2018.01.002 [Cited within: 1]
IhnatsenkaSKirczenowG2012 Effect of electron–electron interactions on the electronic structure and conductance of graphene nanoconstrictions Phys. Rev. B 86 075448 DOI:10.1103/PhysRevB.86.075448 [Cited within: 1]
,, Ying-Cheng Lai1School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, 
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
