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arXiv:1801.01636 · Physical review. B./Physical review. B

Out-of-time-ordered correlators in a quantum Ising chain

Cheng-Ju Lin, Olexei I. Motrunich

Published Apr 18, 2018• 192 citations
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Abstract

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 $t$. 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.

DOI
10.1103/physrevb.97.144304

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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 $t$. 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{t}^{\ensuremath{-}1/4}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.
The non-integrability of quantum systems, often associated with chaotic behavior, is a concept typically applied to cases with a high-dimensional Hilbert space. Among different indicators signaling this behavior, the study of the long-time oscillations of the Out-of-Time Ordered Correlator (OTOC) appears as a versatile tool, that can be adapted to the case of systems with a small number of degrees of freedom. Using such an approach, we consider the oscillations observed after the scrambling time in the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator ( Li J. et al., Phys. Rev. X , 7 (2017) 031011). We show that the systematic of the OTOC oscillations describes qualitatively well, in a chain with only 4 spins, the integrability-to-chaos transition inherited from the infinite chain.
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.
The non-integrability of quantum systems, often associated with chaotic behavior, is a concept typically applied to cases with a high-dimensional Hilbert space. Among different indicators signaling this behavior, the study of the long-time oscillations of the Out-of-Time Ordered Correlator (OTOC) appears as a versatile tool, that can be adapted to the case of systems with a small number of degrees of freedom. Using such an approach, we consider the oscillations observed after the scrambling time in the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator (Li J. et al., Phys. Rev. X, 7 (2017) 031011). We show that the systematic of the OTOC oscillations describes qualitatively well, in a chain with only 4 spins, the integrability-to-chaos transition inherited from the infinite chain.
We study out-of-time-ordered correlators (OTOCs) in hard-core boson models with short-range and long-range hopping and compare the results to the OTOCs in the Luttinger-liquid model. For density-density correlations, a related expectation value of the squared commutator starts at zero and decays back to zero after the passage of the wavefront in all three models, while the wavefront broadens as t1/3{t}^{1/3}t1/3 in the short-range model and shows no broadening in the long-range model and the Luttinger-liquid model. For the boson creation operator, the corresponding commutator function shows saturation inside the light cone in all three models, with similar wavefront behavior as in the density-density commutator function, despite the presence of a nonlocal string in terms of Jordan-Wigner fermions. For the long-range model and the Luttinger-liquid model, the commutator function decays as a power law outside the light cone in the long-time regime when following different fixed-velocity rays. In all cases, the OTOCs approach their long-time values in a power-law fashion, with different exponents for different observables and short-range versus long-range cases. Our long-range model appears to capture exponents in the Luttinger-liquid model (which are found to be independent of the Luttinger parameter in the model). This conclusion also comes to bear on the OTOC calculations in conformal field theories, which we propose correspond to long-ranged models.
Out-of-time-order correlators (OTOC) in the Ising Floquet system, which can be both integrable and nonintegrable, are studied. Instead of localized spin observables, we study contiguous symmetric blocks of spins or random operators localized on these blocks as observables. We find only power-law growth of OTOC in both integrable and nonintegrable regimes. In the nonintegrable regime, beyond the scrambling time, there is an exponential saturation of the OTOC to values consistent with random matrix theory. This motivates the use of ``prescrambled'' random block operators as observables. A pure exponential saturation of OTOC in both integrable and nonintegrable system is observed, without a scrambling phase. Averaging over random observables from the Gaussian unitary ensemble, the OTOC is found to be exactly same as the operator entanglement entropy, whose exponential saturation has been observed in previous studies of such spin chains.
Out-of-time-ordered correlators (OTOCs) are useful tools for investigating foundational questions such as thermalization in closed quantum systems because they can potentially distinguish between integrable and nonintegrable dynamics. Here, we discuss the properties of wavefronts of OTOCs by focusing on the region around the main wavefront at x=vBtx={v}_{B}tx=vB​t, where vB{v}_{B}vB​ is the butterfly velocity. Using a Heisenberg spin model as an example, we find that the leading edge of a propagating Gaussian with the argument $\ensuremath{-}m(x){(x\ensuremath{-}{v}_{B}t)}^{2}+b(x)t$ gives an excellent fit to the region around x=vBtx={v}_{B}tx=vB​t for both the free and chaotic cases. However, the scaling in these two regimes is very different: in the free case the coefficients m(x)m(x)m(x) and b(x)b(x)b(x) have an inverse power law dependence on xxx, whereas in the chaotic case they decay exponentially. We conjecture that this result is universal by using catastrophe theory to show that, on the one hand, the wavefront in the free case has to take the form of an Airy function and its local expansion shows that the power law scaling seen in the numerics holds rigorously and, on the other hand, an exponential scaling of the OTOC wavefront must be a signature of nonintegrable dynamics. We find that the crossover between the two regimes is smooth and characterized by an S\mathsf{S}S-shaped curve giving the lifting of Airy nodes as a function of a chaos parameter. This shows that the Airy form is qualitatively stable against weak chaos and consistent with the concept of a quantum Kolmogorov-Arnold-Moser theory.