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arXiv:2607.25496

Asymmetric information scrambling and eigenstate thermalization in inhomogeneous XXZ spin chains

Shivam Mishra, Ravi Prakash

quant-phSubmitted Jul 28, 2026• Announced Jul 29, 2026• 0 citations
v1 · Jul 28, 2026v2 · Jul 31, 2026v3 · Sep 23, 2026v4 · Sep 28, 2026
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Abstract

Deterministic spatial inhomogeneity has become increasingly relevant in experimentally engineered quantum many-body systems, where interaction gradients can strongly influence nonequilibrium dynamics. Motivated by this, we investigate out-of-time-ordered correlators (OTOCs) and their connection to the eigenstate thermalization hypothesis (ETH) in inhomogeneous XXZ spin chains. Using a deterministic spatially varying interaction profile, we show that finite interaction gradients ($δ>0$) induce a pronounced left--right asymmetry in information scrambling, as quantified by OTOCs. This asymmetry persists even when the system exhibits spectral signatures of quantum chaos, with operators on the strongly interacting side exhibiting suppressed scrambling. To elucidate the origin of the asymmetric finite-size long-time saturation value of OTOCs, we employ two complementary approaches. First, we analyze the diagonal matrix elements of the OTOC observables in the energy eigenbasis within the ETH framework. Second, we derive an analytical expression for the finite-size saturation value based on the overlap between the Hamiltonian and the OTOC observables, which explicitly incorporates the spatial interaction profile. The analytical prediction is fully consistent with the numerical results and provides a microscopic explanation for how deterministic interaction gradients generate the observed asymmetry in the long-time saturation of OTOCs.

Comments
12 pages,9 figures

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Deterministic spatial inhomogeneity has become increasingly relevant in experimentally engineered quantum many-body systems, where interaction gradients can strongly influence nonequilibrium dynamics. Motivated by this, we investigate out-of-time-ordered correlators (OTOCs) and their connection to the eigenstate thermalization hypothesis (ETH) in inhomogeneous XXZ spin chains. Using a deterministic spatially varying interaction profile, we show that finite interaction gradients (δ>0δ>0δ>0) induce a pronounced left--right asymmetry in information scrambling, as quantified by OTOCs. This asymmetry persists even when the system exhibits spectral signatures of quantum chaos, with operators on the strongly interacting side exhibiting suppressed scrambling. To elucidate the origin of the asymmetric finite-size long-time saturation value of OTOCs, we employ two complementary approaches. First, we analyze the diagonal matrix elements of the OTOC observables in the energy eigenbasis within the ETH framework. Second, we derive an analytical expression for the finite-size saturation value based on the overlap between the Hamiltonian and the OTOC observables, which explicitly incorporates the spatial interaction profile. The analytical prediction is fully consistent with the numerical results and provides a microscopic explanation for how deterministic interaction gradients generate the observed asymmetry in the long-time saturation of OTOCs.
Out-of-time-ordered correlators (OTOCs) help characterize the scrambling of quantum information and are usually studied in the context of nonintegrable systems. In this work, we compare the relaxation dynamics of OTOCs in interacting integrable and nonintegrable spin-12\frac{1}{2}21​ XYZ chains in regimes without a classical counterpart. In both kinds of chains, using the presence of symmetries such as U(1)U(1)U(1) and supersymmetry, we consider regimes in which the OTOC operators overlap or not with the Hamiltonian. We show that the relaxation of the OTOCs is slow (fast) when there is (there is not) an overlap, independently of whether the chain is integrable or nonintegrable. When slow, we show that the OTOC dynamics follows closely that of the two-point correlators. We study the dynamics of OTOCs using numerical calculations, and gain analytical insights from the properties of the diagonal and of the off-diagonal matrix elements of the corresponding local operators in the energy eigenbasis.
Despite the fact that power-law interactions occur in a plethora of physical systems, their many-body dynamics is far less understood than that of nearest-neighbor interacting systems. Here, we study information scrambling in strongly disordered spin systems with power-law interactions via out-of-time-order correlators (OTOCs). Numerically, we find pronounced differences in the dynamical spreading of OTOCs between nearest-neighbor and power-law interacting systems. This deviation persists even for short-range interactions, opposing the common view that these interactions produce dynamics equivalent to the nearest-neighbor case. In a detailed experimental proposal, tailored but not limited to Rydberg tweezer setups, we present a protocol to extract OTOCs in XXZ Heisenberg spin systems with tunable anisotropy and programmable disorder based on currently available techniques.
Out-of-time-order correlators (OTOCs) serve as a proxy for quantum information scrambling, which refers to the process where information stored locally disperses across the many-body degrees of freedom in a quantum system, rendering it inaccessible to local probes. Most experimental implementations of OTOCs to probe information scrambling rely on indirect measurements based on global observables, using techniques such as Loschmidt echoes andmultiple quantum coherences, via time-reversal evolutions. In this article, we establish a direct connection between OTOCs with global and local observables in the context of NMR experiments, where the observable is the total magnetization of the system. We conduct a numerical analysis to quantify the differences in the evolution of both magnitudes, evaluating the excitation dynamics in spin ring systems with 8 to 16 spins, using a many-body Hamiltonian and long-range interactions. Our analysis decomposes the global echo into a sum of local echoes and cross-contributions, leading to local and global OTOCs. The results indicate that, after an initial transient period, local OTOCs determine the global ones. We observe that the difference between the average of local OTOCs and the global one, as well as their fluctuations, becomes negligible as the system size increases. Thus, for large homogeneous systems, global and local OTOCs become equivalent. This behavior aligns with that observed in highly interacting or chaotic systems in several experiments.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-1/21/21/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law (1/t)(1/t)(1/t) relaxation for integrable systems and a higher-degree power-law decay (1/tα;α≥1)(1/t^α; α\ge 1)(1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
Abstract The action of any local operator on a quantum system propagates through the system carrying the information of the operator. This is usually studied via the out-of-time-order correlator (OTOC). We numerically study the information propagation from one end of a periodically driven spin-1/2 XY chain with open boundary conditions using the Floquet infinite-temperature OTOC. We calculate the OTOC for two different spin operators, σ x and σ z . For sinusoidal driving, the model can be shown to host different types of edge states, namely, topological (Majorana) edge states and non-topological edge states. We observe a localization of information at the edge for both σ z and σ x OTOCs whenever edge states are present. In addition, in the case of non-topological edge states, we see oscillations of the OTOC in time near the edge, the oscillation period being inversely proportional to the gap between the Floquet eigenvalues of the edge states. We provide an analytical understanding of these effects due to the edge states. It was known earlier that the OTOC for the spin operator which is local in terms of Jordan–Wigner fermions (σ z ) shows no signature of information scrambling inside the light cone of propagation, while the OTOC for the spin operator which is non-local in terms of Jordan–Wigner fermions (σ x ) shows signatures of scrambling. We report a remarkable ‘unscrambling effect’ in the σ x OTOC after reflections from the ends of the system. Finally, we demonstrate that the information propagates into the system mainly via the bulk states with the maximum value of the group velocity, and we show how this velocity is controlled by the driving frequency and amplitude.