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arXiv:1712.06054 · Journal of Statistical Mechanics Theory and Experiment

Measuring the distance between quantum many-body wave functions

Xiao Chen, Tianci Zhou, Cenke Xu

Published Jul 1, 2018• 13 citations
arXivHTMLPDF

Abstract

Abstract We study the distance of two wave functions under chaotic time evolution. The two initial states differ only by a local perturbation. To be entitled ‘chaos’ the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d 2 ( t ) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d 2 ( t ) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d 2 ( t ) to the square of commutator (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A . Among various one-dimensional spin- models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d 2 ( t ) when the local perturbation and subsystem A are well separated. This so-called Lyapunov regime is absent in the spin- chain with local interaction only. After sufficiently long time evolution, d 2 ( t ) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d 2 ( t ) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.

DOI
10.1088/1742-5468/aace1f

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Abstract We study the distance of two wave functions under chaotic time evolution. The two initial states differ only by a local perturbation. To be entitled ‘chaos’ the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d 2 ( t ) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d 2 ( t ) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d 2 ( t ) to the square of commutator (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A . Among various one-dimensional spin- models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d 2 ( t ) when the local perturbation and subsystem A are well separated. This so-called Lyapunov regime is absent in the spin- chain with local interaction only. After sufficiently long time evolution, d 2 ( t ) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d 2 ( t ) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.
We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n = 2 , this ansatz reproduces ETH. For n = 4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024
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.
Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-1/21/21/2 chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite different, a simple Pauli spin operator can eventually approach a "highly entangled" operator with operator entanglement entropy taking a volume law value (close to the Page value). Meanwhile, the spectrum of the operator reduced density matrix develops a universal spectral correlation which can be characterized by the Wishart random matrix ensemble. In the second example, we further connect the 2-local model into a one dimensional chain and briefly discuss the operator scrambling there. In contrast, in the quantum linear map, although the square of commutator can increase exponentially with time, a simple operator does not scramble but performs chaotic motion in the operator basis space determined by the classical linear map. We show that once we modify the quantum linear map such that operator can mix in the operator basis, the operator entanglement entropy can grow and eventually saturate to its Page value, thus making it a truly quantum chaotic model.
We view the operator spreading in chaotic evolution as a stochastic process of height growth. The height of an operator represents the size of its support and chaotic evolution increases the height. We consider N-spin models with all two-body interactions and embody the height picture in a random model. The exact solution shows that the mean height, being proportional to the squared commutator, grows exponentially within logN scrambling time and saturates in a manner of logistic function. We propose that the temperature dependence of the chaos bound could be due to initial height biased toward high operators, which has a smaller Lyapunov exponent.
We propose a characterization of quantum many-body chaos: given a collection of simple operators, the set of all possible pair correlations between these operators can be organized into a matrix with a random-matrix-like spectrum. This approach is particularly useful for locally interacting systems, which do not generically show exponential Lyapunov growth of out-of-time-ordered correlators. We demonstrate the validity of this characterization by numerically studying the Sachdev-Ye-Kitaev model and a one-dimensional spin chain with random magnetic field (XXZ model).