Semiclassical limit for the many-body localization transition

Abstract

We introduce a semi-classical limit for many-body localization in the absence of global symmetries. Microscopically, this limit is realized by disordered Floquet circuits composed of Clifford gates. In d = 1, the resulting dynamics are always many-body localized with a complete set of strictly local integrals of motion. In d ≥ 2, the system realizes both localized and delocalized phases separated by a continuous transition in which ergodic puddles percolate. We argue that the phases are stable to deformations away from the semi-classical limit and estimate the resulting phase boundary. The Clifford circuit model is a distinct tractable limit from that of free fermions and suggests bounds on the critical exponents for the generic transition.PACS numbers: 05.30. Rt,72.15.Rn The central assumption of statistical mechanics is that interactions between particles establish local equilibrium. It is now believed that this assumption of ergodicity breaks down in isolated quantum systems in the presence of sufficient quenched disorder [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The fundamental feature of such many-body localized (MBL) systems is the slowness of the spread of entanglement [19,20] due to the presence of local integrals of motion (IOM) [8,12,[21][22][23][24]. These local observables remember initial conditions forever and thereby block both equilibration and transport. At lower disorder strengths, the system delocalizes and ergodicity is restored. There is now much theoretical and numerical evidence for this picture in one dimension [4][5][6][7][8][9][10][11][12][13][14][15][16][19][20][21][22][23][24][25][26] and the experimental search in quantum optical and atomic systems is underway [25,[27][28][29].The nature of the two phases and the dynamical transition between them is poorly understood in dimensions greater than one. In this article, we present a semiclassical theory of the delocalization transition in the absence of global symmetries in arbitrary spatial dimension. In the semi-classical limit, the transition maps precisely onto classical site percolation. Each lattice site either blocks the flow of quantum information or permits it to spread. Clusters of unblocked sites form 'ergodic puddles', as such regions locally thermalize. Blockage sites, on the other hand, host local integrals of motion. Thus, at large density of blockage sites p b , the system is localized. As p b decreases, the ergodic puddles grow and eventually percolate (p c b ) leading to delocalization. The sharp distinction between blocked and unblocked sites is a consequence of the lack of quantum tunneling in our semi-classical limit. More precisely, the localization length ξ q of the integrals of motion on blocked sites is identically zero. The delocalization transition is instead driven by the divergence of the percolation length ξ e characterizing the size of ergodic puddles. Perturbing away from the semi-classical limit, both length scales be- * Electronic address: achandran@perimeterinstitute.ca come imp...

Low-frequency conductivity in many-body localized systems

“…It is not presently clear whether or not the stretched-exponential effective interactions that occur at putative critical points within the MBL phase [9,14,21,22] substantially modify the above story. Also, our analysis of near-transition behavior assumes that the delocalized phase is thermal, and thus may not apply to hypothesized transitions between an MBL phase and a nonthermal delocalized phase [47][48][49].…”

Quantum Entanglement Growth under Random Unitary Dynamics

“…(31)] and the single-site Hadamard and phase gates R H and R P [Eqs. (29) and (30)]. For circuits built from these gates, time evolving the state on L spins up to a time t takes a computational time of order Lt, and measuring the entanglement across a given bond in the final state takes a time of order L 3 at most.…”

“…These operators, denoted In the following, we focus on evolution of the initial state with unitary gates in the Clifford group [80]. Such gates have recently been used in toy models for many-body localization [29]. Entanglement generation in non-random Clifford circuits has also been studied [43].…”

“…The single-site gates include the eightfold rotations mentioned above, together with the Hadamard gate R H [. (29)]. R H is applied with probability 1=2 and the rotations with probability 1=16 each.…”

“…This irreversible growth of entanglementquantified by the growth of the von Neumman entropyis important for several reasons. It is an essential part of thermalization, and as a result has been addressed in diverse contexts ranging from conformal field theory [1][2][3][4] and holography [5][6][7][8][9][10][11][12] to integrable [13][14][15][16][17][18][19], nonintegrable [20][21][22][23], and strongly disordered spin chains [24][25][26][27][28][29][30]. Entanglement growth is also of practical importance as the crucial obstacle to simulating quantum dynamics numerically, for example, using matrix product states or the density matrix renormalization group [31].…”