Out-of-time-order correlators in the one-dimensional XY model
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Jia-Hui Bao,1, Cheng-Yong Zhang,2,31Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China 2School of Physics and Astronomy, Sun Yat-sen University, 2 Daxue Rd., Zhuhai 519082, China
First author contact:3Author to whom any correspondence should be addressed. Received:2020-01-15Revised:2020-04-2Accepted:2020-04-3Online:2020-07-20
Abstract We study the behavior of information spreading in the XY model, using out-of-time-order correlators (OTOCs). The effects of anisotropic parameter γ and external magnetic field λ on OTOCs are studied in detail within thermodynamical limits. The universal form which characterizes the wavefront of information spreading still holds in the XY model. The butterfly speed vB depends on (γ, λ). At a fixed location, the early-time evolution behavior of OTOCs agrees with the results of the Hausdorff–Baker–Campbell expansion. For long-time evolution, OTOCs with local operators decay as for power law t−1, but those with nonlocal operators show different and nontrivial power law behaviors. We also observe temperature dependence for OTOCs when (γ=0, λ=1). At low temperature, the OTOCs with nonlocal operators show divergence over time. Keywords：out-of-time-order correlators;information spreading;integrable model
PDF (2130KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Jia-Hui Bao, Cheng-Yong Zhang. Out-of-time-order correlators in the one-dimensional XY model. Communications in Theoretical Physics, 2020, 72(8): 085103- doi:10.1088/1572-9494/ab8a28
1. Introduction
Chaos is an interesting phenomenon in nature, and has attracted a lot of attention in the fields of condensed matter theory, quantum information theory, and high energy theory. A feature of chaos is the butterfly effect, which characterizes the chaos of a system as $\sim {{\rm{e}}}^{{\lambda }_{L}t}$ [1]. The Lyapunov exponent λL is unbounded for classical systems. For quantum systems, a conjecture was recently proposed which established a bound ${\lambda }_{L}\leqslant 2\pi {k}_{B}T/{\hslash }$ on strong quantum chaos [2]. This new conjecture encouraged further study of chaos as it relates to both high energy theory [3, 4] and condensed matter theory [5–9].
Quantum chaos theory in a quantum many-body system can be characterized by the out-of-time order correlator (OTOC) of the two related operators W and V. Let us consider the following ‘commutator function' first.$ \begin{eqnarray}C(l,t)\,=\,\displaystyle \frac{1}{2}\langle {\left[W\left(l,t\right),V\left(0\right)\right]}^{\dagger }[W(l,t),V(0)]\rangle ,\end{eqnarray}$here $\langle ...\rangle \equiv \langle {{\rm{e}}}^{-\beta H}...\rangle /\langle {{\rm{e}}}^{-\beta H}\rangle $ denotes the thermal average at temperature T=1/β and $W(l,t)\equiv {{\rm{e}}}^{{\rm{i}}{Ht}}W(l){{\rm{e}}}^{-{\rm{i}}{Ht}}$. l is the position of operator W, H is the Hamiltonian of the system. Assuming operators W and V are both unitary and Hermitian, we can rewrite C(l, t)=1−Re[F(l, t)]. Here, $F(l,t)=\langle W(l,t)V(0)W(l,t)V(0)\rangle $ denotes the so-called OTOC for special time ordering. It captures critical information different from the two-point correlation function. The OTOC can also be understood as the measurement of delocalization of spreading operators. It attracts a lot of attention not only because of its richness in terms of theoretical physics, but also because of its feasibility in experiments [10–15].
The behavior of OTOC has several interesting aspects. For example, the early-time behavior of the OTOC can usually be characterized by the Hausdorff–Baker–Campbell (HBC) formula. Around the wavefront of the spreading operators, there is a conjectured universal form describing the ballistic broadening of the OTOC [16, 17]$ \begin{eqnarray}C(l,t)\sim \exp \left(-c\displaystyle \frac{{\left(l-{v}_{B}t\right)}^{1+p}}{{t}^{p}}\right),\end{eqnarray}$here, c is constant and vB is the spreading velocity of the butterfly effect, which is decided by setting the velocity-dependent Lyapunov exponents λL(vB)=0. p is a coefficient related to models. For instance, p=1 for a random circuit model [18–20], p=1/2 for a non-interacting translation-invariant model [16, 21, 22], p=0 for a Sachdev–Ye–Kitaev (SYK) model [23], and chains of coupled SYK dots at large N [24]. The long-time aspect of OTOC also exhibits interesting behavior, which may reveal important information about how operators saturate bound chaos.
In order to understand how information spreads quantitatively, calculation of the OTOC in different systems, including integrable, or chaotic systems, is an appropriate place to begin. Recently, some work have been done to analyze the OTOC in conformal field theories [25–29], quantum phase transition [30, 31], Luttinger liquids [32], and also lattice integrable models such as the quantum Ising chain [33], the hard-core boson model [21], quadratic fermions [34], the random field XX spin chain [35], and the symmetric Kitaev chain [36]. Scrambling was observed at the critical point of the Ising spin chain. Weak chaos was also witnessed in some models [37].
It is well known that both the quantum Ising model and the XX model can be seen as special cases of the XY model [38]. This contains an anisotropic parameter γ that denotes the difference of components in the x and y direction for coupling of the two nearest neighbours. This feature accords the XY model several nontrivial quantum phase transitions and properties [39]. Therefore, it is interesting to study OTOCs in the XY model in order to learn how operators grow or how information scrambles.
In this paper we focus on the evolution of OTOCs in the XY model. We find that butterfly velocity is dependent on γ and λ, and that the universal form (2) holds for OTOCs in the XY model. We also study the early-time and long-time behavior of OTOCs. While the former is characterized by the HBC formula, the latter displays quite interesting and unusual power law behaviours. Moreover, we observe an interesting temperature dependence in OTOCs in particular cases.
This paper is organized as follows: we introduce the XY model in section 2, including its quantum phase transition, and procedures to diagonalize the Hamiltonian. In section 3, the calculation method of OTOC will be outlined. Next, we will show the exact evolutions of OTOCs with respect to time and location, and extract the information from them. Finally, in section 4 we will briefly discuss these results and conclude.
2. XY model
The XY model is one of the simplest nontrivial integrable models. It demonstrates rich phase diagram, and has the potential to enable the study of new effects. Its Hamiltonian is as follows:$ \begin{eqnarray}H=-\displaystyle \frac{J}{2}\sum _{j=0}^{N-1}\left[\displaystyle \frac{1+\gamma }{2}{\sigma }_{j}^{x}{\sigma }_{j+1}^{x}+\displaystyle \frac{1-\gamma }{2}{\sigma }_{j}^{y}{\sigma }_{j+1}^{y}+\lambda {\sigma }_{j}^{z}\right],\end{eqnarray}$where γ is the anisotropy coefficient describing the difference of interactive strength in the x and y components. λ describes the magnetic field along the z direction. These two parameters decide the phases and properties of this model. When γ=0, it becomes the isotropic XY model (also called XX model). When γ=1, it recovers the quantum Ising chain. The relationship between them is shown in figure 1. The shadow areas are the the corresponding critical regions for the different models.
Figure 1.
New window|Download| PPT slide Figure 1.Critical regions of XY model from [41]. Colored shaded areas show the corresponding critical region for each model. We only plot one quadrant due to the existence of symmetry $\gamma \to -\gamma $ and $\lambda \to -\lambda $.
The critical regions at λ=1 (blue line) or γ=0 (green line) are a conformal invariant, where the spectrum of the theory becomes gapless. They correspond to conformal theories with conformal charge c=1/2 CFT, and c=1 CFT, respectively. The XY chain inculdes two types of quantum phase transition, located at these two lines. The line located at λ=1 is a transition from a doubly degenerate state (λ<1) to a single ground state (λ>1). However, when (γ, λ)=(0, 1), it is not conformal, since the dynamical critical exponent is two [40].
In order to calculate the OTOC of an XY chain, we should diagonalize its Hamiltonian using the Jordan-Wigner transformation, and the Bogoliubov transformation. We set J=1 for convenience. Rewriting the Pauli matrices by spin operators ${\sigma }_{j}^{x}={a}_{j}^{\dagger }+{a}_{j}$, ${\sigma }_{j}^{y}=({a}_{j}^{\dagger }-{a}_{j})/{\rm{i}}$ and ${\sigma }_{j}^{z}\,=2{a}_{j}^{\dagger }{a}_{j}-1$, together with Jordan−Wigner transformation ${a}_{j}\,=\left[\exp \left({\rm{i}}\pi {\sum }_{l=1}^{j-1}{c}_{l}^{\dagger }{c}_{l}\right)\right]{c}_{j}$, the Hamiltonian becomes$ \begin{eqnarray}\begin{array}{rcl}H & = & -\displaystyle \frac{1}{2}\sum _{j=0}^{N-1}\left[({c}_{j}^{\dagger }{c}_{j+1}-{c}_{j}{c}_{j+1}^{\dagger })+\gamma ({c}_{j}^{\dagger }{c}_{j+1}^{\dagger }-{c}_{j}{c}_{j+1})\right.\\ & & \left.+\,\lambda (2{c}_{j}^{\dagger }{c}_{j}-1)\right]+\displaystyle \frac{\mu }{2}\left({c}_{N}^{\dagger }{c}_{0}+{c}_{0}^{\dagger }{c}_{N}+\gamma {c}_{N}^{\dagger }{c}_{0}^{\dagger }\right.\\ & & +\left.\,\gamma {c}_{0}{c}_{N}\right),\end{array}\end{eqnarray}$where $\mu ={\prod }_{j=1}^{N}{\sigma }_{j}^{z}$ is the parity operator. In order to deal with the boundary term, we separate the Hamiltonian as$ \begin{eqnarray}\displaystyle \begin{array}{rcl}H & = & \displaystyle \frac{1+\mu }{2}{H}^{+}+\displaystyle \frac{1-\mu }{2}{H}^{-},\\ {H}^{\pm } & = & -\displaystyle \frac{1}{2}\sum _{j=0}^{N-1}\left[({c}_{j}^{\dagger }{c}_{j+1}-{c}_{j}{c}_{j+1}^{\dagger })\right.\\ & & \left.+\gamma ({c}_{j}^{\dagger }{c}_{j+1}^{\dagger }-{c}_{j}{c}_{j+1})+\lambda (2{c}_{j}^{\dagger }{c}_{j}-1)\right],\end{array}\end{eqnarray}$since their even/odd parity is thereby conserved, respectively.
Now we can use the Fourier transform ${c}_{k}\,=\tfrac{{e}^{-{\rm{i}}\pi /4}}{\sqrt{N}}{\sum }_{j=0}^{N-1}{e}^{-{\rm{i}}{jk}}{c}_{j}$, and the Bogoliubov transformation ${\gamma }_{k}\,=\cos {\theta }_{k}{c}_{k}-\sin {\theta }_{k}{c}_{-k}^{\dagger }$ to complete the diagonalization, thus:$ \begin{eqnarray}\begin{array}{rcl}{H}^{\pm } & = & \sum _{{k}^{\pm }}{\epsilon }_{{k}^{\pm }}\left({\gamma }_{{k}^{\pm }}^{\dagger }{\gamma }_{{k}^{\pm }}-\displaystyle \frac{1}{2}\right),\\ {k}^{\pm } & = & \displaystyle \frac{2\pi \left[n+\tfrac{1\pm 1}{4}\right]}{N},n=0,1,\ldots N-1,\end{array}\end{eqnarray}$here ${\epsilon }_{k}={[{(\cos k-\lambda )}^{2}+{\gamma }^{2}{\sin }^{2}k]}^{1/2}$ is the dispersion of elementary excitations. The Bogoliubov angle θk satisfies $\tan (2{\theta }_{k})=\tfrac{\gamma \,\sin k}{\lambda -\cos k}.$
It has been observed in [33] that given a thermodynamical limit $N\to \infty $, we have $\langle O\rangle =\langle O{\rangle }_{+}=\langle O{\rangle }_{-}$ for OTOC with both local and nonlocal operators in the Ising model. Here, the subscript denotes the choice of k± corresponding to even/odd chain length N. We confirm that this conclusion still holds for the XY model. In this paper we will use k+ and even N for consistency.
3. Out-of-time-order correlator
Using the diagonalized Hamiltonian, we are now able to calculate the OTOC of the XY model. We consider the following term:$ \begin{eqnarray}{F}_{\mu \nu }(l,t)=\langle {\sigma }_{l}^{\mu }(t){\sigma }_{0}^{\nu }{\sigma }_{l}^{\mu }(t){\sigma }_{0}^{\nu }\rangle ,\end{eqnarray}$where $\mu ,\nu =x,y,z$. The Pauli matrices may be expressed by fermionic operators in the Majorana representation Aj=cj†+cj and ${B}_{j}={c}_{j}^{\dagger }-{c}_{j}$:$ \begin{eqnarray}\begin{array}{rcl}{\sigma }_{j}^{x} & = & \left(\prod _{j^{\prime} \lt j}{A}_{j^{\prime} }{B}_{j^{\prime} }\right){A}_{j},\ {\sigma }_{j}^{y}=-{\rm{i}}\left(\prod _{j^{\prime} \lt j}{A}_{j^{\prime} }{B}_{j^{\prime} }\right){B}_{j},\\ {\sigma }_{j}^{z} & = & -{A}_{j}{B}_{j}.\end{array}\end{eqnarray}$
All OTOCs may be expressed as the thermal average of Majorana fermion sequences. For instance, ${F}_{{zz}}(l,t)\,=\langle {A}_{l}(t){B}_{l}(t){A}_{0}{B}_{0}{A}_{l}(t){B}_{l}(t){A}_{0}{B}_{0}\rangle $. In terms of the thermodynamical limit, this can be computed using Wick's theorem, which turns the calculation of the long sequence into a combination of two-point correlation functions. We will use the Pfaffian method to complete the calculations numerically, with similar steps to [33].
The Pfaffian method [42, 43] can be expressed as$ \begin{eqnarray}F(l,t)=\pm \mathrm{Pf}({\rm{\Phi }})=\pm \sqrt{\mathrm{Det}({\rm{\Phi }})},\end{eqnarray}$where the matrix Φ is skew-symmetric, i.e. Φii=0 and Φij=−Φji. This form will be modified if we use ‘double trick' to deal with the calculation, in which case the sign of F(l, t) is not definitely positive. However, it can still be decided by requiring the ‘continuity' of the OTOC, which we will elaborate below. The matrix Φ is constructed in terms of Majorana correlation functions ${{\rm{\Phi }}}_{{ij}}=\langle {X}_{i}{X}_{j}\rangle $, where Xi is the ith element inside thermal average function $\langle {X}_{1}{X}_{2}...\rangle $.
The basic correlation functions are $\langle {A}_{m}(t){A}_{n}\rangle $, $\langle {A}_{m}(t){B}_{n}\rangle $, $\langle {B}_{m}(t){A}_{n}\rangle $, and $\langle {B}_{m}(t){B}_{n}\rangle $, which can be derived if we know the exact diagonalized form of the Hamiltonian. Their expressions are shown in appendix.
3.1. OTOC for local operators
OTOCs characterise information spreading and scrambling, in other words, the delocalization of operators. Since the behavior of many-body localized quantum systems can be revealed by the local operators, it will be interesting to study them.
For the XY model, the OTOC with local operators is Czz. As shown in equation (8), operator ${\sigma }_{j}^{z}=-{A}_{j}{B}_{j}$ is local, since it consists of fermions located at site j only. Operators ${\sigma }_{j}^{x}$ and ${\sigma }_{j}^{y}$ are nonlocal because they are related to all sites of fermions before site j. Combining equations (8) and (9), we obtain$ \begin{eqnarray}\begin{array}{rcl}{C}_{{zz}}(l,t) & = & 1-\mathrm{Re}\left[\langle {\left({A}_{l}\left(t\right){B}_{l}\left(t\right){A}_{0}{B}_{0}\right)}^{2}\rangle \right]\\ & = & 1-\mathrm{Re}\sqrt{\mathrm{Det}({{\rm{\Phi }}}_{{zz}})}.\end{array}\end{eqnarray}$With the help of the Pfaffian trick, we can compute this quantity numerically. The results in figure 2 illustrate how the parameters (γ, λ) affect the evolution of the OTOC and the spreading velocity of the butterfly effect. Here, we choose the system size N=500 and β=0. The lighter color in the figure denotes stronger C(t). Thus we can observe how the OTOC spreads.
Figure 2.
New window|Download| PPT slide Figure 2.General evolution of Czz with different choices of parameters in the XY model. We set the system size N=500, inverse temperature β=0. The space coordinate l ranges from −30 to 30. The time coordinate ranges from 0 to 30. Lighter color denotes a larger value of Czz, corresponding to stronger delocalization. These images clearly show how the spreading of the operator is bounded by a ‘cone structure'.
Several interesting properties can be observed from these results. Firstly, the cone structure indicates a binding of the butterfly effect, except in the model where γ=1, λ=0, which corresponds to the quantum Ising chain without external magnetic field. The C(t) of this model is always zero, except for sites l=0 and ±1. This indicates that the operators do not spread there. This may be further illustrated by considering the HBC formula as follows:$ \begin{eqnarray}\displaystyle \begin{array}{rcl}W(t) & = & \sum _{n=0}^{\infty }\displaystyle \frac{{\left({\rm{i}}t\right)}^{n}}{n!}{L}_{n}(W)\\ & = & W+{\rm{i}}t[H,W]+\displaystyle \frac{{\left({\rm{i}}t\right)}^{2}}{2!}[H,[H,W]]+....\end{array}\end{eqnarray}$When γ=1, λ=0, the commutator $[W(l,t),V(0)]\,=[{\sigma }_{l}^{z}(t),{\sigma }_{0}^{z}]$ vanishes for all sites except l=0,±1. Therefore $C(l,t)\equiv \tfrac{1}{2}\langle | [W(l,t),V(0)]{| }^{2}\rangle $ also vanishes.
Secondly, the butterfly effect of the spin chains where (γ=1, λ=1) and (γ=0, 0<λ<1) always satisfy vB=1. In the latter case, the OTOC is not influenced by the magnetic field λ if γ=0 and β=0. Nevertheless, the other positions display a narrower cone structure. This indicates that their speed of operator spreading is slower, and that the velocity vB depends on γ and λ. We can conclude that the existence of anisotropy will affect the speed of operator spreading. For models with a weak magnetic field, this will cause a reduction in speed.
Thirdly, by comparing figure 3 with figure 2, we note that temperature has a negligible effect on OTOCs with local operators, except for the model where (γ=0, λ=1). This model displays vanishing OTOC when the temperature falls to zero. The reason for this phenomenon is currently unclear.
Figure 3.
New window|Download| PPT slide Figure 3.The evolution of Czz at zero temperature T=1/β=0 with other configurations, as in figure 2. This shows that only the model where (γ=0, λ=1) displays vanishing Czz. All other models are not sensitive to temperature.
There is a conjecture that around the wavefront of information spreading, the wavefront of C(t) can be described by the universal form (2). To verify this conjecture in the XY model, we calculate the wavefront of Czz along fixed-velocity rays. The results are shown in figure 4.
Figure 4.
New window|Download| PPT slide Figure 4.Fitting of the universal form and numerical data of Czz. We select three models with different vB to confirm whether the conjecture holds in the XY model. The dots represent numerical data from the OTOC along velocity-fixed rays, where v=1.1, 1.2, and 1.5, respectively. The solid lines are fitting forms of −at+b. Coefficient a is the velocity-dependent Lyapunov exponents λL(v) that we need to extract. The insets show the three sets of extracted data fitted by $\sim {(v-{v}_{B})}^{1+p}$. We note that the numerical data fits quite well.
We posit that$ \begin{eqnarray}C(t)={const}.\,\times \,\exp (-{\lambda }_{L}(v)t).\end{eqnarray}$If the universal form indeed holds in the XY model, we should obtain the relation$ \begin{eqnarray}\begin{array}{rcl}\mathrm{Log}[C(t)] & = & -{\lambda }_{L}(v)t+{\rm{const.}},\\ {\lambda }_{L}(v) & \sim & {\left(v-{v}_{B}\right)}^{1+1/2},\end{array}\end{eqnarray}$as p=1/2 for XY model.
In figure 4, we use time t and Log[Czz] as coordinates, where the fixed velocity v=1.1, 1.2, and 1.5. Then we use form −λL(v)t + const. to fit the numerical results. In the insets, the values of λL(v) are extracted by fitting the numerical data. The power law relations are checked. For example, in the top picture, the coordinates of three points are (1.1, 0.38), (1.2, 0.54), and (1.5, 1.14), which fits function ${\lambda }_{L}{(v)=1.82(v-0.75)}^{3/2}$ quite well. In addition, we can see from figure 2 that the butterfly velocity of model (γ=0.5, λ=0.5) is indeed vB≈0.75. The other two figures also support this relation quite well. We conclude that for OTOC with local operators, the universal form is supported by XY model. On the other hand, this result also supports our former conclusion that vB depends on γ and λ.
Now we consider the time evolution of Czz at fixed sites. It will tell us exactly how the local operators will behave. Two meaningful areas must be considered: the early time and the long time. The results of models at four typical points are illustrated in figure 5 with sites l=1, 2, 3, 4. It is clear that for the early time, Czz is vanishing at (γ=1, λ=0), when l>1. We can also observe power law t4l−2 at (γ=1, λ=1) and at t2l for the remaining models. These behaviors can be understood by means of the HBC formula. Since t is small for early time, the description by HBC expansion remains quite accurate. Note that the lowest order of t, which makes C(t) nonzero, is decided by the lowest order of Ln(W) that satisfies $[{L}_{n}(W),V]\ne 0$. To be specific, the Hamiltonian of the XY model may be divided into six different types by selecting different values of γ and λ. They are (1). σxσx, (2). σyσy, (3). σxσx+σz, (4). σyσy+σz, (5). σxσx+σyσy, (6). σxσx+σyσy+σz. For zz OTOCs, the σxσx and σyσy exhibit the same behavior. Therefore, we can observe that there are four kinds of behaviors. We know that type (1) and (2) have vanishing C(t). For type (3), however, we see that$ \begin{eqnarray}\begin{array}{rcl}{L}_{1}({\sigma }_{0}^{z}) & = & [H,{\sigma }_{0}^{z}]\sim {\sigma }_{0}^{y}{\sigma }_{1}^{x},\\ {L}_{2}({\sigma }_{0}^{z}) & \sim & [H,{\sigma }_{0}^{y}{\sigma }_{1}^{x}]\sim {\sigma }_{0}^{y}{\sigma }_{1}^{y},\\ {L}_{3}({\sigma }_{0}^{z}) & \sim & [H,{\sigma }_{0}^{y}{\sigma }_{1}^{y}]\sim {\sigma }_{0}^{y}{\sigma }_{1}^{z}{\sigma }_{2}^{x}\,\ldots .\end{array}\end{eqnarray}$The lowest order $[{L}_{n}({\sigma }_{0}^{z}),{\sigma }_{l}^{z}]\ne 0$ and $[{\sigma }_{0}^{z}(t),{\sigma }_{l}]$ shows t2l−1 power law growth. Thus $C(t)=\tfrac{1}{2}\langle | [{\sigma }_{0}^{z}(t),{\sigma }_{l}]{| }^{2}\rangle $ evolves as t4l−2. The early-time behaviors of other types may also be checked in a similar fashion. Note that type (5) and type (6) show the same t2l growth behavior. This is because when x and y components exist at the same time, their behavior will dominate. Overall, we conclude that the early-time behaviors of Czz can be separated into three types: t4l−2 at ($\gamma =1,\lambda \ne 0$), t2l at $\gamma \ne 1$, and when l>1, vanishing will occur at (γ=1, λ=0).
Figure 5.
New window|Download| PPT slide Figure 5.Time evolution of Czz with four typical models. Yellow, blue, purple, and green lines correspond to fixed locations l=1, 2, 3, 4, respectively. Czz at (γ=1, λ=0) vanishes for l>1. The dashed lines are used to show power law fitting. We can see clearly from the figures that these models show t4l−2 and t2l power law growth at early time, and t−1 decay at late-time, independent of site l.
Next, we examine the late-time aspect of Czz. As exhibited in figure 5, all models decay as t−1, independent of γ, λ, site l, and temperature. This behavior may be understood by means of the stationary phase approximation of fermionic correlation functions, i.e. when $t\to \infty $, we obtain [33]$ \begin{eqnarray}{C}_{{zz}}(l,t)\sim (1-\langle {A}_{0}{B}_{0}{\rangle }^{2})\displaystyle \frac{2}{\pi | {\epsilon }_{\pi }^{{\prime\prime} }| t},\end{eqnarray}$here, ${\epsilon }_{\pi }^{{\prime\prime} }$ is the second derivative of εk with k=π.
3.2. OTOC with nonlocal operators
There are five types of OTOC with nonlocal operators in the XY model: i.e. Fxx, Fyy, Fxy, Fxz, and Fyz. However, since the operators ${\sigma }_{j}^{x}$ and ${\sigma }_{j}^{y}$ change the fermion parity, their Heisenberg evolution cannot simply obtained from Aj(t) and Bj(t). Following the method used in [33, 44], we will use the ‘double trick' to calculate the OTOC. Define quantity$ \begin{eqnarray}{{\rm{\Gamma }}}_{\mu \nu }(j,t)\equiv \langle {\left({\sigma }_{\tfrac{N}{2}}^{\mu }(t){\sigma }_{N-j}^{\mu }(t){\sigma }_{0}^{\nu }{\sigma }_{\tfrac{N}{2}-j}^{\nu }\right)}^{2}\rangle ,\end{eqnarray}$for large enough N. Invoking Lieb-Robinson bound and cluster property [44], we obtain$ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{\mu \nu }(j,t) & \approx & \langle {\left({\sigma }_{\tfrac{N}{2}}^{\mu }(t){\sigma }_{\tfrac{N}{2}-j}^{\nu }\right)}^{2}\rangle \langle {\left({\sigma }_{N-j}^{\mu }(t){\sigma }_{0}^{\nu }\right)}^{2}\rangle \\ & = & {F}_{\mu \nu }(j,t){F}_{\mu \nu }(-j,t)={F}_{\mu \nu }^{2}(j,t),\end{array}\end{eqnarray}$here, ${F}_{\mu \nu }(j,t)={F}_{\mu \nu }(-j,t)$ due to mirror symmetry.
Taking xx OTOC as an example, quantity Γxx(j, t) can be expressed as$ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{{xx}}(j,t) & = & \left\langle \Space{0ex}{1.0ex}{0ex}\left[\left(\prod _{j^{\prime} =\tfrac{N}{2}}^{N-j-1}{B}_{j^{\prime} }(t){A}_{j^{\prime} +1}(t\right)\right.\right.\\ & & \left.{\left.\times \left(\prod _{j^{\prime} =0}^{\tfrac{N}{2}-j-1}{B}_{j^{\prime} }{A}_{j^{\prime} +1}\right)\right]}^{2}\right\rangle .\end{array}\end{eqnarray}$We can use the Pfaffian method to calculate this. We construct a matrix Φxx of dimension 4(N−2j)×4(N−2j). Then Fxx(j, t) can be calculated as$ \begin{eqnarray}{F}_{{xx}}(j,t)=\pm \sqrt{| \mathrm{Pf}({{\rm{\Phi }}}_{{xx}})| }=\pm {\left[\mathrm{Det}\left({{\rm{\Phi }}}_{{xx}}\right)\right]}^{\tfrac{1}{4}}.\end{eqnarray}$Since the quantity is doubled, we do not know the sign of it directly. However, this can be recovered by requiring the ‘continuity' of ${F}_{{xx}}(j,t)$. More specifically, there is a critical rule for all the points on site j and time t: on the premise of turning least directions, choose closest distance. With this rule we can check how the curve is finally organized with all the points from calculation. It should be noted that when j>vt, ${F}_{{xx}}\to 1$, due to the existence of cone structure. OTOCs with other operators may be calculated in the same way. Combining equations (8) and (17) gives$ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{{xy}}(j,t) & = & \left\langle \Space{0ex}{1.0ex}{0ex}\left[\left(\prod _{j^{\prime} =\tfrac{N}{2}}^{N-j-1}{B}_{j^{\prime} }(t){A}_{j^{\prime} +1}(t\right)\right.\right.\\ & & \left.{\left.\times \left(\prod _{j^{\prime} =0}^{\tfrac{N}{2}-j-1}{A}_{j^{\prime} }{B}_{j^{\prime} +1}\right)\right]}^{2}\right\rangle ,\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{{xz}}(j,t) & = & \left\langle \left[\left(\prod _{j^{\prime} =\tfrac{N}{2}}^{N-j-1}{B}_{j^{\prime} }(t){A}_{j^{\prime} +1}(t\right)\right.\right.\\ & & \left.{\left.\times {A}_{0}{A}_{N/2-l}{B}_{0}{B}_{N/2-l}\right]}^{2}\right\rangle .\end{array}\end{eqnarray}$Γyy and Γyz can also be constructed in the same way.
The general behavior of Cxx with system size N=100 and β=0 is illustrated in figure 6. Other C(t)s with different operators show similar behaviors, so we will not show them here. We can see that while C(t) in figure 3 is vanishing at (γ=0, λ=1), it does not vanish with nonlocal operators. From these figures, we see that the butterfly velocity is the same as for local operators. This indicates that the butterfly velocity only depends on the model, not the operators, in OTOC function. Moreover, scrambling is observed for all sites inside the ‘light cone'. When γ=0, β=0, λ has tiny effect on the OTOC. The observation of scrambling in OTOCs with nonlocal operators illustrates their main differences compared with local operators. This can easily be understood, since nonlocal operators possess nonlocal information about operators, which leads to delocalization once they spread inside the light cone.
Figure 6.
New window|Download| PPT slide Figure 6.General evolution of Cxx with different choices of parameters in the XY model. We set system size N=100, inverse temperature β=0. Coordinates are given as space ranges from −30 to 30, time ranges from 0 to 30, respectively. Lighter color indicates a larger value of Cxx, corresponding to stronger delocalization. These images show similar light cone binding for the different models, but some differences may be observed, as compared with local operators inside the light cone: relatively speaking, they appear more scrambled.
Knowing that the butterfly velocity of the different operators remains unchanged within the same model, we further examine whether the universal form regarding wavefront behavior holds in the case of nonlocal operators. Here we only show the results Cxx due to space contstraints, but we have confirmed that all nonlocal C(t) support the conjecture quite well. The results are illustrated in figure 7. We can clearly see the fitting results of numerical data in the insets.
Figure 7.
New window|Download| PPT slide Figure 7.Fitting results for the universal form and numerical data of Cxx. We select three models with different vB ; note that all other models of OTOC with nonlocal operators have also been verified. The outside dots represent numerical data obtained by calculating the OTOC along velocity-fixed rays where v=1.1, 1.2, and 1.5 respectively. The solid lines are fitting forms of −at+b, and a is the velocity-dependent Lyapunov exponents λL(v) that we need to extract. The inset shows how these three sets of extracted data fitted with $\sim {(v-{v}_{B})}^{1+p}$ as a function of v. We observe that the numerical data fits quite well.
The evolution of OTOCs with nonlocal operators can also be analyzed, including their early-time and long-time power law behaviors. With regard to early-time observations, detailed plots are included in figures 9–13, and summarize the results in table.1. Note that since l=1 is no longer a special case for nonlocal operators, we only plot results of l=2, 3 , and 4, for clarity. Here, the sign of the OTOC is no longer a problem; since C(t) are relatively close to 0 at early time, the sign of $| F(t)| $ must be positive. These results are actually not beyond our expectation because all of them agree with the HBC formula. Thus, this aspect requires no further consideration.
Table 1. Table 1.Summary of early-time power law growth of OTOCs with both local and nonlocal operators in the XY model.
It should be noted, however, that long-time behavior is in some ways more subtle. In [33], the author found that for a quantum Ising chain at the critical point, $| {F}_{{xx}}(t)| $ exhibits nontrivial t−1/4 decay at long-time. Here, we confirm this result, and examine further long-time behaviors in other regions. The results are shown in figures 9–13 and table 2. Surprisingly, they show quite different power law behaviors for different selections of γ and λ. We know that at late-time, $| F(t)| $ with nonlocal operators approach 0, which indicates that C(t) is appoaching saturation value 1. Thus, the power law behaviors indicate that different operators will exhibit different rates of saturation. In particular $| {F}_{{xz}}(t)| $ and $| {F}_{{yz}}(t)| $, which describe how nonlocal operators and local operators interact with each other, show no decay at the critical point. This observation indicates that for these two kinds of OTOC, C(t) will be constant at very long time, and will not grow at all.
Table 2. Table 2.Summary of long-time power law growth of OTOC for both local and nonlocal operators in the XY model.
Moreover, our results appear to be independent of location l when β=0. However, if we set β to a bigger value, the pattern of the OTOC becomes quite complex. Some results of $| {F}_{{xx}}| $ are shown in figure 8. Divergence can be observed at late-time. The reason for this is currently unclear, as we used numerical method rather than analytical form to complete the calculations. As a result, we have been unable to obtain a universal description of the long-time behaviors of OTOCs with nonlocal operators, but we do hope that our calculations will contribute towards achieving the final form.
Figure 8.
New window|Download| PPT slide Figure 8.Late-time behavior of $| {F}_{{xx}}| $ at low temperature β=500. Blue, purple, and green lines correspond to fixed locations l=2, 3, and 4 respectively. The OTOC is observed to be divergent at late-time.
4. Discussion and conclusion
The study of OTOCs in integrable systems is a relatively new area of research, and may reveal a great deal of interesting information about how operators evolve in such systems, and how scrambling happens. In this work we mainly focus on the behaviors of OTOCs in the XY model, including early time, long time, and wavefront aspects, together with an examination of the conjectured universal form (2). Careful calculation and analysis reveals some interesting points relating to OTOCs in this system. We found that the butterfly velocity in the XY model depends on its anisotropy parameter γ and magnetic field λ, but is independent of the locality of the operators in the OTOC. Based on this observation, we proved that for all kinds of OTOC, with all choices of parameters in the XY model, the conjectured form (2) regarding wavefront behavior holds true. Therefore, it is indeed a viable description of OTOCs around wavefront (v>vB), at least insofar as it relates to the XY model.
Furthermore, we conducted a comprehensive study of the time and space evolution of OTOCs with both local and nonlocal operators in the XY model. We find some interesting points about their general behavior: (1) When γ=β=0, OTOCs with local operators are independent of the external magnetic field λ; (2) For the noncritical point γ=0, λ=1, OTOCs with local operators vanish when the temperature tends to zero. This phenomenon does not occur with other typical sets of parameters. In addition to these results, the early-time, and long-time evolution of OTOCs at fixed locations have also been studied. We found that while early-time behavior agrees with the power law results of the HBC formula, the long-time behaviors exhibit nontrivial saturation rates for different operators and models. These are independent of location l when β=0. However, when temperature decreases, their behaviors become very complex. This is not easy to analyze by means of numerical calculation. In addition, OTOCs with both local and nonlocal operators, i.e. $| {F}_{{xz}}| $ and $| {F}_{{yz}}| $ show t0 at long time. Their long-time evolutions are constant, rather than approaching 0 as other systems do.
Overall, we have studied many aspects of OTOCs in the XY model, and provide some evidence to support the conjecture regarding information spreading around wavefronts. Further research is required in order to better understand the underlying values of these observations and conclusions, which may be explored in future experiments. Equally, further analysis of OTOCs in other systems is also required in order to arrive at a more profound understanding of their behaviour in both chaotic and integrable systems.
Acknowledgments
We thank Peng-Cheng Li for helpful discussions. This work is funded by the China Postdoctoral Science Foundation. Cheng-Yong Zhang is supported by NSFC Grant No. 11 947 067.
,1, Cheng-Yong Zhang
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