Scaling Theory of Entanglement at the Many-Body Localization Transition

Abstract

We study the universal properties of eigenstate entanglement entropy across the transition between many-body localized (MBL) and thermal phases. We develop an improved real space renormalization group approach that enables numerical simulation of large system sizes and systematic extrapolation to the infinite system size limit. For systems smaller than the correlation length, the average entanglement follows a sub-thermal volume law, whose coefficient is a universal scaling function. The full distribution of entanglement follows a universal scaling form, and exhibits a bimodal structure that produces universal subleading power-law corrections to the leading volume-law. For systems larger than the correlation length, the short interval entanglement exhibits a discontinuous jump at the transition from fully thermal volume-law on the thermal side, to pure area-law on the MBL side.Recent experimental advances in synthesizing isolated quantum many-body systems, such as cold-atoms [1][2][3][4], trapped ions [5,6], or impurity spins in solids [7,8], have raised fundamental questions about the nature of statistical mechanics. Even when decoupled from external sources of dissipation, large interacting quantum systems tend to act as their own heat-baths and reach thermal equilibrium. This behavior is formalized in the eigenstate thermalization hypothesis (ETH) [9,10]. Generic excited eigenstates of such thermal systems are highly entangled, with the entanglement of a subregion scaling as the volume of that region ("volume law"). This results in incoherent, classical dynamics at long times. In contrast, strong disorder can dramatically alter this picture by pinning excitations that would otherwise propagate heat and entanglement [11][12][13][14][15][16][17]. In such many-body localized (MBL) systems [18][19][20], generic eigenstates have properties akin to those of ground states. They exhibit short-range entanglement that scales like the perimeter of the subregion [17] ("area law"), and have quantum coherent dynamics up to arbitrarily long time scales [21][22][23][24][25][26][27], even at high energy densities [17,26,[28][29][30][31].A transition between MBL and thermal regimes requires a singular rearrangement of eigenstates from area-law to volume-law entanglement. This many-body (de)localization transition (MBLT) represents an entirely new class of critical phenomena, outside the conventional framework of equilibrium thermal or quantum phase transitions. Developing a systematic theory of this transition promises not only to expand our understanding of possible critical phenomena, but also to yield universal insights into the nature of the proximate MBL and thermal phases.The eigenstate entanglement entropy can be viewed as a non-equilibrium analog of the thermodynamic free energy for a conventional thermal phase transition, and plays a central role in our conceptual understanding of the MBL and ETH phases. Describing the entanglement across the MBLT requires addressing the challenging combination of disorder, interactio...

Renormalization-group study of the many-body localization transition in one dimension

“…Many approximate RG studies [2,36,37,39] and one recent ED study [31] of the MBL transition seem to agree on the prediction that physical lengths of thermal inclusions in critical systems are distributed according to a power law ∝ (L T ) −α , with the exponent taking an apparently universal value near α = 2. In addition, and consistent with a KT-type scenario, is the possibility of an intermediate critical MBL phase [39] where the lengths of thermal inclusions are power-law distributed with a continuously varying exponent [2,31,39].…”

“…As mentioned earlier, before a KT-like scenario for the MBL transition was proposed [2,39], studies using approximate RG treatments of the MBL transition assumed a one-parameter scaling ansatz [1,[34][35][36][37][38]. These works reported apparent correlation-length critical exponents in the range ν ∼ = 2.5 to ν ∼ = 3.5.…”

Toy model for anomalous transport and Griffiths effects near the many-body localization transition

“…It is thus reasnoable to assume that keeping only a small portion of the total Hilbert space that grows linearly with the number of blocks might provide a plausible starting point for a zero-th order simplified description. In this sense, our model is very similar in spirit to the effective coarsegrained models introduced and studied in the context of the strong disorder RG approach to MBL [32,33,[36][37][38] (see also Ref. [47]).…”

Characterizing the many-body localization transition using the entanglement spectrum

“…However, the quantum phase transition between a 'thermal' phase where statistical mechanics is obeyed and a 'many body localized' (MBL) phase where it is not continues to be an open problem. This is a dynamical transition which lies outside the usual thermodynamic frameworks, and while it has been attacked with a variety of techniques, from mean field theory [24,25] to the strong disorder renormalization group [26][27][28][29][30], and using both analytic arguments [31][32][33][34] and numerics [35][36][37], (see [38] for a review), a complete understanding of the transition remains elusive.…”