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arXiv:1807.08826 · Physical review. B./Physical review. B

Out-of-time-ordered correlators in short-range and long-range hard-core boson models and in the Luttinger-liquid model

Cheng-Ju Lin, Olexei I. Motrunich

Published Oct 15, 2018• 32 citations
arXivHTMLPDF

Abstract

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 ${t}^{1/3}$ 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.

DOI
10.1103/physrevb.98.134305

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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-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.
Information scrambling and the butterfly effect in chaotic quantum systems can be diagnosed by out-of-time-ordered (OTO) commutators through an exponential growth and large late time value. We show that the latter feature shows up in a strongly correlated many-body system, a Luttinger liquid, whose density fluctuations we study at long and short wavelengths, both in equilibrium and after a quantum quench. We find rich behavior combining robustly universal and nonuniversal features. The OTO commutators display temperature- and initial-state-independent behavior and grow as t^{2} for short times. For the short-wavelength density operator, they reach a sizable value after the light cone only in an interacting Luttinger liquid, where the bare excitations break up into collective modes. This challenges the common interpretation of the OTO commutator in chaotic systems. We benchmark our findings numerically on an interacting spinless fermion model in 1D and find persistence of central features even in the nonintegrable case. As a nonuniversal feature, the short-time growth exhibits a distance-dependent power.
In a free Luttinger liquid, we calculate the exact out-of-time-ordered correlator (OTOC) for vertex operators, which is valid for arbitrary operator charges without small-charge approximations. The OTOC grows purely as a power law [Formula: see text] inside the light cone without exponential spreading, characteristic of integrable systems. We develop a systematic perturbative expansion in the interaction strength [Formula: see text] and construct the full structure of first- and second-order corrections via Wick’s theorem. At finite temperature [Formula: see text], thermal effects modify the late-time scaling from power-law to exponential growth [Formula: see text]; however, this reflects thermal statistics rather than quantum chaos. These results lay a solid foundation for studies of information scrambling and quantum chaos in one-dimensional integrable systems.
The out-of-time-order correlator (OTOC) is studied for a bosonic quantum lattice model. We gain its classical analog through the replacement of both commutators appearing in the quantum correlator by a corresponding Poisson bracket. The evaluation of the Poisson bracket is then performed in a complex-valued description of the Hamiltonian dynamics and, for the initial choice of quantum operators to be site-specific annihilators, turns out to be given by the expectation value of the absolute square of a specific element of the complex-valued monodromy matrix. The growth rate of this expectation value is compared to a typical chaos indicator, the mean finite-time Lyapunov exponent (FTLE). In both cases the numerical phase-space average is weighted by a Wigner function corresponding to a multimode coherent state. For a three-well Bose-Hubbard model in the Mott insulator regime, it is found that although, on the level of single trajectories, FTLE and classical OTOC show similar long-time behavior, after averaging, they exhibit a marked difference [which for purely chaotic initial conditions is close to ln(sqrt[2])], rooting in the different order the logarithm and the average are taken. This observation is an example of the relevance of the fluctuations of the FTLE to correctly explain the quantitative difference between OTOC growth rate and FTLE in a prototypical many-body system.
Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems.We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators C(r, t ) = [A i (t ), B i+r ] of local operators.Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law 1/r α with distance r.Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for α < 1 and to linear light cones for α > 1.The asymptotic form of the tails of C(r, t ) ∝ t/r α is described by short-time perturbation theory, which is valid at short times and long distances.
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 v B 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.