Locality, Quantum Fluctuations, and Scrambling

Abstract

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 wavefront and other random circuit models showing a diffusively broadened wavefront. 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 wavefront. 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 cross-over to a diffusively broadened wavefront. 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 time-independent Hamiltonian model are also consistent with a diffusively-broadened wavefront. CONTENTS

Transport and chaos in lattice Sachdev-Ye-Kitaev models

“…Finally, it is useful to compare our results with recent studies of operator spreading using random unitary circuits [38][39][40][41][42] , which report a broadening of the chaos wavefront. In our model, the phonons remain essentially free oscillators, and the N 2 oscillators on each island can have consequences similar to a random unitary perturbation.…”

Scrambling and complexity in phase space

“…< l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > 1/ p n th < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > (a) This motivates us to turn our attention to OTOCs. In thermalizing many-body systems, the decay of OTOCs detects the spread of local operators in real space [6,8,9,[24][25][26][27][28][29][43][44][45][46][47]. In few-body CV systems, OTOCs have also found use due to their correspondence with diagnostics of classical chaos [37].…”

“…One of the defining features of such DV scrambling is the notion of operator growth, where the time evolution of an initially simple, local operator V, yields a more complex, late-time operator, V(t) = U † (t) VU (t) [4], whose decomposition is dominated by non-local operator strings. A particularly powerful quantitative diagnostic of operator growth is provided by the so-called out-of-time-order correlation (OTOC) function V † (t)W † (0)V(t)W(0) , which measures the spreading of V(t) via another local probe operator W [23][24][25][26][27][28][29]. In addition to its use on the theory front, OTOCs have also attracted a significant amount of experimental interest and attention [30][31][32][33][34].…”

“…they simply map one displacement operator to another), useful insights into genuine scrambling can be obtained by studying the stochastic evolution of quasi scramblers. Indeed, recent progress [9,[43][44][45][46][47] on the interplay between holography and quantum information theory has revealed that quantum circuits composed of short-range random Clifford unitaries can provide useful intuition for understanding entanglement and operator propagation in DV many-body systems [24][25][26][27][28][29].…”

Exact dynamics in dual-unitary quantum circuits

“…Recently, increasing attention has been devoted to the class of random unitary quantum circuits, which provide an alternative theoretical laboratory for the study of the many-body dynamics [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] . The main appeal of these systems is that they represent minimally structured dynamical models where analytic results can be obtained beyond the realm of integrability.…”