Slow relaxation of out-of-time-ordered correlators in
interacting integrable and nonintegrable spin-$\tfrac{1}{2}$ XYZ chains

Out-of-time-ordered correlators (OTOCs) are a diagnostic of quantum information scrambling [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], can be used to detect quantum phases [15, 16, 17], and can be measured experimentally [18, 19, 20, 21, 22, 23, 24, 25, 26]. In quantum systems with a classical limit, OTOCs grow exponentially fast as a result of chaos [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] or of instabilities in integrable systems [40, 41, 42, 43, 44, 45]. Due to the relationship with chaos in classical systems, connections between the behavior of OTOCs and the onset of thermalization have been discussed [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. In nonintegrable models with local conserved quantities, OTOCs have been shown to exhibit a slower (algebraic) growth when the operators involved have an overlap with a conserved quantity [46, 47, 48, 49, 50, 51, 52]. This behavior has been explained in terms of the Lieb-Robinson bounds [53, 54] and of the eigenstate thermalization hypothesis (ETH) [51, 52]. In this work, we study the unitary dynamics of OTOCs in interacting integrable and nonintegrable spin-$\tfrac{1}{2}$ chains with time-independent Hamiltonians in regimes that do not have a classical counterpart. Our goal is to understand the effect that the overlap of OTOC operators with the Hamiltonian has on the relaxation dynamics of OTOCs.

Interacting integrable systems have been extensively studied theoretically in recent years because of their relevance to experiments [55, 56], and this has resulted in remarkable progress in understanding their dynamics [57]. In those systems, after reaching equilibrium following unitary dynamics, observables are not described by traditional Gibbs ensembles [58, 59, 60], but by generalized Gibbs ensembles that incorporate all the conserved quantities that make the models integrable [61, 62, 63]. Furthermore, as a result of the presence of an extensive number of conservation laws, the large-distance dynamics of integrable models is different from regular hydrodynamics [64, 65, 66], and it is described by a generalized hydrodynamics [67, 68].

We compare the time-evolution of OTOCs in interacting integrable and nonintegrable spin-$\tfrac{1}{2}$ XYZ chains. Our analytical insights and numerical results elucidate the role that the overlap between the OTOC operators and the Hamiltonian has on the dynamics of OTOCs. We make those overlaps nonvanishing using symmetries, specifically, $U(1)$ symmetry and supersymmetry. We first highlight that, when the OTOC operators overlap with the Hamiltonian, the OTOC dynamics can mirror the behavior of two-time correlators for long times. We then show that, within the system sizes and timescales that we consider, the dynamics of OTOCs for both interacting integrable and nonintegrable many-body quantum systems can be qualitatively and quantitatively analogous. The OTOCs for interacting integrable and nonintegrable models can exhibit a slow (algebraic-like) relaxation if the OTOC operators overlap with the Hamiltonian, and a faster (exponential-like) relaxation if they do not. Analytical insights are obtained using the relation between the dynamics of the OTOCs and the behavior of the matrix elements of the operators involved in the OTOCs in the energy eigenbasis; specifically, the behavior of the diagonal matrix elements in finite systems with short-range interaction, and the behavior of the off-diagonal elements in the thermodynamic limit.

The presentation is organized as follows. In Sec. II, we use the properties of the matrix elements of the operators in the OTOCs, written in the eigenenergy basis, to provide a general understanding of the relaxation dynamics of the OTOCs. The spin-$\tfrac{1}{2}$ XYZ chains in which the OTOCs are studied and the symmetries of their Hamiltonians are discussed in Sec. III. The numerical results are presented in Sec. IV. A summary of our results is then provided in Sec. V.

Because of its nontrivial dynamics and universality, we study the infinite-temperature OTOC. For operators $\hat{A}$ and $\hat{B}$, it is defined as

where $\hat{A}(t)=\hat{U}^{\dagger}(t)\hat{A}\hat{U}(t)$ is the time-evolved operator $\hat{A}$ under the unitary evolution operator $\hat{U}(t)=\mathcal{T}e^{-i\int_{0}^{t}\hat{H}(\tau)d\tau}$, with $\mathcal{T}$ indicating the time-ordered integration, and $\hat{H}(t)$ being a general time-dependent Hamiltonian. By infinite-temperature we mean that $\langle\ldots\rangle={\rm tr}(\ldots)/\mathcal{V}$, where $\mathcal{V}$ is the relevant Hilbert space dimension. In the context of such an average, one says that two operators $\hat{A}$ and $\hat{B}$ have a nonzero overlap whenever $\langle\hat{A}\hat{B}\rangle\neq 0$. Equation (1) can be rewritten as

where

equals 1 for unitary operators, which are the focus of this work, and

In what follows, we study the time evolution of $F^{AB}(t)$.

For a time-independent Hamiltonian (our focus here), with eigenenergies $E_{\alpha}$ and eigenkets $|\alpha\rangle$, Eq. (4) can be written as

where $A_{\alpha\beta}=\langle\alpha|\hat{A}|\beta\rangle$, $B_{\beta\gamma}=\langle\beta|\hat{B}|\gamma\rangle$. We work in units in which $\hbar=1$.

Next, we discuss two ways in which one can gain an analytic understanding of the OTOCs decay. The first one involves infinite-time averages in finite systems, and the second one involves the structure of the off-diagonal matrix elements of the operators of interest in the energy eigenbasis in the thermodynamic limit. Our focus is on traceless operators $\hat{A}$ and $\hat{B}$ that, in addition to being unitary, are local and Hermitian (i.e., that are physical observables).

We consider the spin-$\tfrac{1}{2}$ XYZ model described by the following Hamiltonian,

where $J^{a}$ are the coupling strengths along the $a=x$, $y$, and $z$ directions, $\hat{\sigma}^{a}_{l}$ is represented by the $a$ Pauli matrix for site $l$, and $L$ is the number of lattice sites. $\hat{H}_{I}$ is an interacting integrable Hamiltonian that can be solved analytically using the eight vertex model [91, 92]. The addition of the staggered interactions, $\hat{H}_{NI}$, of magnitude $\Lambda$ along the $z$-direction, breaks integrability.

The symmetries of the total Hamiltonian $\hat{H}$ are:

(i) The parity operators along each axis, $\hat{P}^{a}=\prod_{l}\hat{\sigma}^{a}_{l}$, commute with the total Hamiltonian. (ia) When $L$ is even, the parity operators also commute with each other. As a result, the diagonal matrix elements of the local operators $\hat{\sigma}^{a}_{l}$ in the energy eigenstates are exactly zero (numerically, we find them to be of the order of $10^{-14}$), unless there are other symmetries such as those discussed next in (ii) and (iii). We can understand this as follows: For $L$ even, the energy eigenkets $|\alpha\rangle$ (where $\hat{H}|\alpha\rangle=E_{\alpha}|\alpha\rangle$) are simultaneous eigenkets of the three operators $\hat{P}^{a}$. Hence, we can write

where we used that $\hat{P}^{x}\hat{\sigma}^{z}_{l}\hat{P}^{x}=-\hat{\sigma}^{z}_{l}$ and that $\hat{P}^{x}|\alpha\rangle=\pm|\alpha\rangle$. (ib) When the system size $L$ is odd, the parity operators anticommute with each other. As a result, one finds that the diagonal matrix elements of the local operators $\hat{\sigma}^{a}_{l}$ in the energy eigenstates need not vanish, which is what we observe numerically. We can understand this as follows: For $L$ odd, the energy eigenkets $|\alpha\rangle$ are not simultaneous eigenkets of the three operators $\hat{P}^{a}$. Let us choose the energy eigenkets to be simultaneous eigenkets of $\hat{P}^{z}$. If $|\alpha_{+}\rangle$ is an energy eigenket with eigenenergy $E_{\alpha}$ satisfying $\hat{P}^{z}|\alpha_{+}\rangle=|\alpha_{+}\rangle$, then $\hat{P}^{x}|\alpha_{+}\rangle=|\alpha_{-}\rangle$ is an energy eigenket with eigenenergy $E_{\alpha}$ (because $\hat{P}^{x}$ and $\hat{H}$ commute) satisfying $\hat{P}^{z}|\alpha_{-}\rangle=-|\alpha_{-}\rangle$ (because $\hat{P}^{x}$ and $\hat{P}^{z}$ anticommute), i.e., the energy spectrum is doubly degenerate [93]. For the diagonal matrix elements of the local operators, instead of Eq. (19) we have

i.e., they need not vanish. With increasing system size, one expects that the difference between the results obtained for chains with $L$ and $L+1$ sites decreases, and vanish in the thermodynamic limit. Our numerical results are consistent with that expectation.

(ii) In general, the integrable Hamiltonian $\hat{H}_{I}$ does not conserve total magnetization in any direction, except when two of the coupling parameters are equal, in which case the model exhibits a $U(1)$ symmetry and reduces to the spin-$\tfrac{1}{2}$ XXZ chain (upon relabeling, if needed, $x$, $y$, and $z$ so that $J^{z}\neq J^{x}=J^{y}$). For $\Lambda\neq 0$, the $U(1)$ symmetry is only present for $\hat{H}$ if $J^{x}=J^{y}$.

One can show that if $\hat{H}$ has $U(1)$ symmetry, i.e., if $\hat{{\cal S}}^{z}=\sum_{l}\hat{\sigma}^{z}_{l}$ is conserved, then the diagonal matrix elements of the local operators $\hat{\sigma}^{z}_{l}$ in the energy eigenstates need not vanish. Considering $L$ even, we have already mentioned that $\hat{P}^{x}$, $\hat{P}^{y}$, and $\hat{P}^{z}$ commute with each other and with the Hamiltonian. However, only $\hat{P}^{z}$ commutes with $\hat{{\cal S}}^{z}=\sum_{l}\hat{\sigma}^{z}_{l}$. If we choose the energy eigenstates to be simultaneous eigenstates of $\hat{{\cal S}}^{z}$, then we have

Since $\hat{P}^{x}$ and $\hat{{\cal S}}^{z}$ do not commute, $\hat{P}^{x}|\alpha\rangle$ does not need to be $e^{i\phi}|\alpha\rangle$, so $\langle\alpha|\hat{\sigma}^{z}_{l}|\alpha\rangle$ does not need to vanish. However, one can show that the matrix elements $\langle\alpha|\hat{\sigma}^{x}_{l}|\alpha\rangle$ must vanish

(iii) The integrable Hamiltonian $\hat{H}_{I}$ exhibits supersymmetry when [94, 95]

Defining the operator $\hat{P}^{a}_{e}=\prod_{l=1}^{L/2}\hat{\sigma}^{a}_{2l}$, which acts only on even sites, we get $\hat{P}^{z}_{e}\hat{H}_{I}(J^{x},J^{y},J^{z})\hat{P}^{z}_{e}=\hat{H}_{I}(-J^{x},-J^{y},J^{z})$, and similarly for $a=x,y$. Hence, the supersymmetry also holds for

The previous symmetry analysis shows that the $\hat{\sigma}^{a}_{l}$ (where $a=x$, $y$, or $z$) operators have vanishing diagonal matrix elements, i.e., a vanishing overlap with the Hamiltonian (18), unless the Hamiltonian has special symmetries such as $U(1)$ or supersymmetry. This means that, by tuning the Hamiltonian parameters to be at those special symmetry points (or away from them), we can test the effect that the overlap (or lack thereof) of operators with the Hamiltonian has on the relaxation dynamics of the OTOCs at and away from integrability.

Throughout this paper, we fix $J^{y}=1.0$ to be our energy scale, $J^{z}=1.5$, and scan across $J^{x}$. For the integrable chain $\Lambda=0$, and for the nonintegrable one we choose $\Lambda=0.2$.

The fact that the diagonal matrix elements of $\hat{\sigma}^{a}_{l}$ are vanishing away from the points at which the Hamiltonian of the spin-$\tfrac{1}{2}$ XYZ chain exhibits $U(1)$ symmetry or supersymmetry, and can be nonvanishing as one approaches those points, is illustrated in Figs. 1(a) and 1(b). There we plot the average (over the entire energy spectrum) of the absolute values of the diagonal matrix elements of $\hat{\sigma}^{z}_{L/2}$ and $\hat{\sigma}^{x}_{L/2}$, respectively, as functions of $J^{x}$ for $L=14$. The averages are labeled as

and

When $J^{x}=1$, since $J^{x}=J^{y}$, the Hamiltonian has $U(1)$ symmetry and conserves total magnetization in the $z$-direction. For values of $J^{x}$ close to 1 the total magnetization is a nearly-conserved quantity, which is known to result in prethermal behavior both in the integrable and nonintegrable cases [96]. This explains the “broad” peak of $S^{z}$ at $J^{x}=1$ in Fig. 1(a) for both the integrable and the nonintegrable chains. In the discussions that follow, we refer to the $J^{x}=1.0$, $\Lambda=0$ point as the “$U(1)$ integrable point” and to the $J^{x}=1.0$, $\Lambda=0.2$ point as the “$U(1)$ nonintegrable point.” The sharper peaks at $J^{x}=0.6$ for both $S^{z}$ in Fig. 1(a) and $S^{x}$ in Fig. 1(b) occur only in the integrable chain and are caused by the supersymmetric point satisfying Eq. (24). In the discussions that follow, we refer to $J^{x}=0.6$, $\Lambda=0$ point as the “SUSY point.”

The results in the main panels in Figs. 1(a) and 1(b) are for a chain in which $L$ is even. For an odd number of sites, as mentioned before, the diagonal matrix elements of $\hat{\sigma}^{a}_{l}$ need not vanish even if they vanish for an even number of sites. The inset in Fig. 1(a) [Fig. 1(b)] shows $S^{z}$ [$S^{x}$] as a function of $L$ for $J^{x}=1,\;1.6$ and $\Lambda=0,\;0.2$ [$J^{x}=0.6,\;1.3$ and $\Lambda=0,\;0.2$]. At the $U(1)$ integrable and nonintegrable points in the inset of Fig. 1(a), and at the SUSY point in the inset of Fig. 1(b), both the $L$ even and $L$ odd systems have nonzero diagonal values. For the other parameters we only show results for $L$ odd. In the latter cases we see that $S^{z}$ and $S^{x}$ decrease rapidly as $L$ (odd) increases, i.e., the results for chains with $L$ odd approach the vanishing results for $L$ even. Hence, we expect that Fig. 1(a) and Fig. 1(b) are representative of what happens in large chains no matter whether $L$ is even or odd. (See Appendix A for a comparison between the dynamics of OTOCs in systems with even and odd numbers of lattice sites.)

Our discussion thus far allows us to conclude that the peaks in Figs. 1(a) and 1(b) establish the Hamiltonian parameters for which the corresponding operators exhibit an overlap with the Hamiltonian. Next, we study how such an overlap (or lack thereof) affects the time evolution of the OTOCs involving those operators.

The scrambling of quantum information as characterized by the relaxation dynamics of the OTOCs has been mainly investigated in nonintegrable models. Here, we report a study of OTOCs in interacting integrable and nonintegrable spin-$\tfrac{1}{2}$ XYZ chains, in regimes without a classical counterpart.

We show that the main factor determining the behavior of the OTOCs in those chains (for the timescales and chain sizes explored) is the overlap between its constituent operators and conserved quantities of the model, rather than integrability or lack thereof. If the overlap is nonzero, then the OTOCs decay slowly (algebraically-like) for both the integrable and nonintegrable chains, while if there is no overlap, then the OTOCs decay rapidly (exponentially-like) for both the integrable and the nonintegrable chains. For the model Hamiltonians and operators considered, nonzero overlaps are obtained only in the presence of $U(1)$ symmetry and supersymmetry. Our numerical results for the dynamics of the OTOCs can be understood, in a complementary manner, in terms of the behavior of the off-diagonal matrix elements of the OTOCs operators at low frequencies and of the scaling with the system size of the diagonal matrix elements.

Furthermore, our analytical results (confirmed numerically) also indicate that when the operators of interest overlap with the Hamiltonian (a conserved quantity under the dynamics studied), the infinite-time average of the two-time correlators in finite (but sufficiently large) chains determines the infinite-time average of the OTOCs. Hence, in the thermodynamic limit (via the Lieb-Robinson bound argument presented in Sec. II), the slow relaxation of the OTOC can be understood to be a consequence of the slow relaxation of two-time correlators, all resulting from the presence of a conservation law and independent of the integrable or nonintegrable nature of the system.