Quantum butterfly effect in polarized Floquet systems Xiao Chen,1, 2 Rahul M. Nandkishore,1 and Andrew Lucas1 1Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA 2Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA (Dated: February 26, 2020) We explore quantum dynamics in Floquet many-body systems with local conservation laws in one spatial dimension, focusing on sectors of the Hilbert space which are highly polarized. We numerically compare the predicted charge diffusion constants and quantum butterfly velocity of operator growth between models of chaotic Floquet dynamics (with discrete spacetime translation invariance) and random unitary circuits which vary both in space and time. We find that for small but nonzero density of charge (in the thermodynamic limit), the random unitary circuit correctly predicts the scaling of the butterfly velocity but incorrectly predicts the scaling of the diffusion constant. We argue that this is a consequence of quantum coherence on short time scales. Our work clarifies the settings in which random unitary circuits provide correct physical predictions for non-random chaotic systems, and sheds light into the origin of the slow down of the butterfly effect in highly polarized systems or at low temperature. 1. INTRODUCTION a Heisenberg operator O1(0; t) = U yO1(0)U and a time independent operator O2(r; 0) initially separated by a dis- Understanding the approach to equilibrium in strongly tance r, where U is the time evolution operator. In RUC interacting many-body systems has become a problem of models, the growth of the Heisenberg operator O(t) re- significant interest in the past few years. It is widely ex- duces to the solution of a classical stochastic problem: pected that certain features of dynamics will be universal the biased random walkp [4,5]. This directly implies that among many different quantum systems { for example, C(r; t) ∼ C((r − pvBt)= t), where vB is the butterfly ve- the emergence of diffusion and hydrodynamics in systems locity and the 1= t factor indicates the diffusive broad- with conserved quantities [1], or the quantum butterfly ening of the front. Inspired by this work, a series of RUCs effect in which the domain of support of operators ex- with different symmetries have been proposed [6,7, 11], pands ballistically in systems with sufficiently local in- which introduce extra conservation laws in the quantum teractions [2]. Much recent work has accordingly focused dynamics and give rise to diffusive transport on top of on cartoon models of quantum dynamics consisting of ballistic information propagation in conventional models. random unitary gates applied in discrete time steps: the With the right conservation laws [12{14], localization is random unitary circuit (RUC) [3{7]. These RUCs often also possible. provide an analytical solution for many-body quantum Here, we consider both random and non-random dis- dynamics and, it is hoped, shed light on the dynamics crete time quantum circuits with U(1) symmetry and ex- for more general quantum systems. Yet in all these RUC plore operator growth and OTOCs in distinct symmetry models, a key feature is randomness in both time and sectors associated to the conserved charge. To be pre- spatial directions. Averaging over this randomness leads cise, we consider a one dimensional chain of length L, 1 to substantial decoherence in the quantum unitary evolu- with a spin- 2 degree of freedom on every site, and a con- z 2 L tion and, in many simple cases, maps quantum dynamics served S . The Hilbert space H = (C )⊗ can be written to a classical stochastic process. However, in systems (in the obvious product state basis) as the direct sum of without randomness, it is less clear whether quantum subspaces with a fixed number of up spins: coherence is as negligible as RUCs suggest, and it is im- L " portant to learn whether and when non-random chaotic H = HN ; (1.2) systems have different dynamical behavior than RUCs. N "=1 Indeed, there are simple models where such random cir- M cuits fail to properly describe chaotic Hamiltonian quan- with L tum evolution [8]. N " arXiv:1912.02190v2 [cond-mat.stat-mech] 25 Feb 2020 In this paper, we focus on a specific question: do ran- H = span js1s2 ··· sLi : si = 2N " − L : (1.3) ( i=1 ) dom circuits correctly describe the growth of operators in X one dimensional Floquet systems? The operator growth Here si = 1 or −1 represents whether a spin is up (") or " can be (partially) diagnosed by the out-of-time-ordered down (#) respectively. Define projectors PN onto sub- " correlator (OTOC)[9, 10], spaces HN , we are interested in discrete time quantum 2 C(r; t) = −Tr [O1(0; t);O2(r; 0)] =Tr(I): (1.1) evolution arising from a many-body unitary matrix U(t) obeying where Tr( ) is the dimension of the total Hilbert space. I " This quantity measures the non-commutativity between [PN ;U(t)] = 0: (1.4) 2 vB (RUC) D (RUC) vB (CFC) D (CFC) (a) (b) short time: O(1) ∼ α O(1) ∼ 1/α long time: ∼ α t=3 t=2 TABLE I. The comparison of the quantum dynamics between t=2 RUC and CFC. t=1 t=1 In particular, we will focus on the regime with small but finite density α = N "=L of conserved charge in the time space large L limit. 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