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arXiv:1909.09376 · Physical review. B./Physical review. B

Out-of-time-ordered commutators in Dirac–Weyl systems

Zoltán Okvátovity, Balázs Dóra

Published Jun 5, 2020• 1 citation
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Abstract

Quantum information stored in local operators spreads over other degrees of freedom of the system during time evolution, known as scrambling. This process is conveniently characterized by the out-of-time-order commutator (OTOC), the time dependence of which reveals salient aspects of the system's dynamics. Here we study the spatially local spin-correlation function, i.e., the expectation value of the spin commutator and the corresponding OTOC of Dirac-Weyl systems in one, two, and three spatial dimensions. The OTOC can be written as the square of the expectation value of the commutator and the variance of the commutator. In principle, the problem features two energy scales, the chemical potential and the high-energy cutoff. We find that only the latter is dominant, therefore the time evolution is separated into only two different regions. The spin-correlation function grows linearly with time initially and decays as ${t}^{\ensuremath{-}2}$ for late times. The OTOC reveals a universal ${t}^{2}$ initial growth from both the commutator and the variance while its late time decay ${t}^{\ensuremath{-}2}$ originates from the variance of the commutator. This late time decay is identified as a characteristic signature of Dirac-Weyl fermions. These features remain present also at finite temperatures. Our results indicate that Dirac-Weyl systems are slow information scramblers and are essential when additional channels for scrambling, i.e., interaction and disorder, are analyzed.

DOI
10.1103/physrevb.101.245125

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Quantum information stored in local operators spreads over other degrees of freedom of the system during time evolution, known as scrambling. This process is conveniently characterized by the out-of-time-order commutator (OTOC), the time dependence of which reveals salient aspects of the system's dynamics. Here we study the spatially local spin-correlation function, i.e., the expectation value of the spin commutator and the corresponding OTOC of Dirac-Weyl systems in one, two, and three spatial dimensions. The OTOC can be written as the square of the expectation value of the commutator and the variance of the commutator. In principle, the problem features two energy scales, the chemical potential and the high-energy cutoff. We find that only the latter is dominant, therefore the time evolution is separated into only two different regions. The spin-correlation function grows linearly with time initially and decays as t\ensuremath−2{t}^{\ensuremath{-}2}t\ensuremath−2 for late times. The OTOC reveals a universal t2{t}^{2}t2 initial growth from both the commutator and the variance while its late time decay t\ensuremath−2{t}^{\ensuremath{-}2}t\ensuremath−2 originates from the variance of the commutator. This late time decay is identified as a characteristic signature of Dirac-Weyl fermions. These features remain present also at finite temperatures. Our results indicate that Dirac-Weyl systems are slow information scramblers and are essential when additional channels for scrambling, i.e., interaction and disorder, are analyzed.
We study the finite-temperature scrambling behavior of a quantum system described by a Hamiltonian chosen from a random matrix ensemble. This effectively (0+1)-dimensional model admits an exact calculation of various ensemble-averaged out-of-time-ordered correlation functions in the large-NNN limit, where NNN is the Hilbert space dimension. For a Hamiltonian drawn from the Gaussian unitary ensemble, we calculate the ensemble averaged OTOC at all temperatures. In addition to an early time quadratic growth of the averaged out-of-time-ordered commutator (OTOC), we determine that the OTOC saturates to its asymptotic value as a power-law in time, with an exponent that depends both on temperature, and on one of four classes of operators appearing in the correlation function, that naturally emerge from this calculation. Out-of-time-ordered correlation functions of operators that are distributed around the thermal circle take a time ts∼βt_{s}\sim βts​∼β to decay at low temperatures. We generalize these exact results, by demonstrating that out-of-time-ordered correlation functions averaged over any ensemble of Hamiltonians that are invariant under unitary conjugation H→UHU†H \rightarrow {U} H {U}^{\dagger}H→UHU†, exhibit power-law decay to an asymptotic value. We argue that this late-time behavior is a generic feature of unitary dynamics with energy conservation. Finally, by comparing the OTOC with a commutator-anticommutator correlation function, we examine whether there is a time window over which a typical Hamiltonian behaves as a "coherent scrambler" in the language of Ref. \cite{Kitaev_IAS, Kitaev_Suh}.
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.
Out-of-time-ordered correlators (OTOC) have been proposed to characterize quantum chaos in generic systems. However, they can also show interesting behavior in integrable models, resembling the OTOC in chaotic systems in some aspects. Here we study the OTOC for different operators in the exactly-solvable one-dimensional quantum Ising spin chain. The OTOC for spin operators that are local in terms of the Jordan-Wigner fermions has a ``shell-like'' structure: After the wavefront passes, the OTOC approaches its original value in the long-time limit, showing no signature of scrambling; the approach is described by a ${t}^{\ensuremath{-}1}$ power law at long time ttt. On the other hand, the OTOC for spin operators that are nonlocal in the Jordan-Wigner fermions has a ``ball-like'' structure, with its value reaching zero in the long-time limit, looking like a signature of scrambling; the approach to zero, however, is described by a slow power law ${t}^{\ensuremath{-}1/4}$ for the Ising model at the critical coupling. These long-time power-law behaviors in the lattice model are not captured by conformal field theory calculations. The mixed OTOC with both local and nonlocal operators in the Jordan-Wigner fermions also has a ``ball-like'' structure, but the limiting values and the decay behavior appear to be nonuniversal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.
Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator Cr(t)C_r(t)Cr​(t), its spatial sum Cˉ(t)=∑rCr(t)\bar C(t) = \sum_r C_r(t)Cˉ(t)=∑r​Cr​(t), and the symmetrically regularised connected dOTOC D(t)D(t)D(t) of two extensive observables. We show that Lieb--Robinson locality gives Cˉ(t)≲td\bar C(t) \lesssim t^dCˉ(t)≲td. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, D(t)≲tdD(t) \lesssim t^dD(t)≲td, improving our earlier suboptimal t3dt^{3d}t3d bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates D(t)D(t)D(t) to Cˉ(t)\bar C(t)Cˉ(t) and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.
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.
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.