Chaos and high temperature pure state thermalization

Abstract

Classical arguments for thermalization of isolated systems do not apply in a straightforward way to the quantum case. Recently, there has been interest in diagnostics of quantum chaos in manybody systems. In the classical case, chaos is a popular explanation for the legitimacy of the methods of statistical physics. In this work, we relate a previously proposed criteria of quantum chaos in the unitary time evolution operator to the entanglement entropy growth for a far-from-equilibrium initial pure state. By mapping the unitary time evolution operator to a doubled state, chaos can be characterized by suppression of mutual information between subsystems of the past and that of the future. We show that when this mutual information is small, a typical unentangled initial state will evolve to a highly entangled final state. Our result provides a more concrete connection between quantum chaos and thermalization in many-body systems. * ydl@stanford.edu 1 arXiv:1805.03675v2 [cond-mat.stat-mech]

New characteristic of quantum many-body chaotic systems

“…We provide analytic and numerical evidence that γ 0 is at most polynomially dependent on L −1 . It has been shown recently that γ (x), which has the dimension of energy, is related to the characteristic timescale of thermalization t 0 [21], thus connecting chaos with thermalization dynamics [9,14,[22][23][24][25][26][27]. Finally, we note that E(x) provides an efficient way to distinguish chaotic systems from the nonchaotic ones.…”

Fast scrambling on sparse graphs

“…To show this, we relate the decay of the functions α i (t) to the decay of twopoint correlation functions, and argue on general grounds that such correlators cannot decay arbitrarily quickly. We note that other recent work [7,8,9,10] has also discussed contributions to scrambling arising from the decay of two-point functions.…”

“…We conjecture that at all temperatures and for all initially unentangled states, the scrambling time is limited by the decay of thermal two point functions. A proof likely requires more sophisticated methods than what we have developed here: see [10] for some preliminary directions. More interestingly, our explicit construction of random circuits with λ L = ∞ is surprising given that, under mild assumptions, Lyapunov exponents generally are bounded from above by λ L 2πk B T / at finite temperature T [42], where = h/2π and k B and h are Boltzmann's and Planck's constants, respectively.…”

“…The exponential decay of two-point functions is necessary but not sufficient for fast scrambling: for example, a system made of two decoupled fast scramblers is not itself a fast scrambler (information cannot spread between decoupled systems). To answer this question, therefore, we now turn to another probe of fast scrambling: the growth of operators [10,11,12]. In particular, we bound the growth of operators, and t s , by how the Hamiltonian connects the N degrees of freedom.…”

Scrambling in Yang-Mills

“…The bound limits the growth of commutators [O i (0), O j (t)] where O i ∈ F i so that we are bounding the growth of operators [52]. The growth of operators is a reliable probe of scrambling [27,[53][54][55]. To obtain our bounds, we use arguments of [56], used to derive Lieb-Robinson bounds for general harmonic systems on general lattices.…”