Percolation in Fock space as a proxy for many-body localization

Abstract

We study classical percolation models in Fock space as proxies for the quantum many-body localisation (MBL) transition. Percolation rules are defined for two models of disordered quantum spin-chains using their microscopic quantum Hamiltonians and the topologies of the associated Fock-space graphs. The percolation transition is revealed by the statistics of Fock-space cluster sizes, obtained by exact enumeration for finite-sized systems. As a function of disorder strength, the typical cluster size shows a transition from a volume law in Fock space to sub-volume law, directly analogous to the behaviour of eigenstate participation entropies across the MBL transition. Finitesize scaling analyses for several diagnostics of cluster size statistics yield mutually consistent critical properties. We show further that local observables averaged over Fock-space clusters also carry signatures of the transition, with their behaviour across it in direct analogy to that of corresponding eigenstate expectation values across the MBL transition. The Fock-space clusters can be explored under a mapping to kinetically constrained models. Dynamics within this framework likewise show the ergodicity-breaking transition via Monte Carlo averaged local observables, and yield critical properties consistent with those obtained from both exact cluster enumeration and analytic results derived in our recent work [arXiv:1812.05115]. This mapping allows access to system sizes two orders of magnitude larger than those accessible in exact enumerations. Simple physical pictures based on freezing of local real-space segments of spins are also presented, and shown to give values for the critical disorder strength and correlation length exponent ν consistent with numerical studies.

Superfluid weight and Berezinskii-Kosterlitz-Thouless transition temperature of twisted bilayer graphene

“…Our results for the pairing symmetry are in agreement with Ref. 54 and they differ from the topological d + id symmetry predicted in many TBG studies [31,34,35,39,40,44,51,57,58,82] and also from other proposed symmetries which include s-wave [48-50, 53, 55, 57], extended s-wave [38,41,46,82], p-wave [53,56], p + ip-wave [37], d-wave [50,53,56], and f -wave [36,39,40,53]. Apart from Ref.…”

“…Importantly, we show that including only the few flat bands is not sufficient but one needs also a number of dispersive bands to correctly predict the geometric contribution. Therefore, approximate models of TBG such as those with only flat bands, as used for deriving upper [29] and lower [30] bounds of the superfluid weight and in many other works [31][32][33][34][35][36][37][38][39][40][41][42][43], may not be suited for quantitative predictions of TBG superconductivity. Moreover, we predict that, in the flat-band regime, local (s-wave) and non-local interactions yield distinct behavior, namely an anisotropic superfluid weight in the latter case.…”

“…An outstanding problem in describing the TBG physics theoretically [31][32][33][34][35][36][37][38][39][40][41][42][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] is the fact that the unit cell of the moiré superlattice with twist angles close to θ * contains a large amount of carbon atoms [ Fig. 1(a)], and therefore TBG theory should take a stand on how to describe the interlayer couplings within this unit cell.…”

Fock-space correlations and the origins of many-body localization

“…In recent work focussing on short-ranged models, a classical percolation model on the Fock space was constructed, acting as a proxy for quantum many-body localisation [40,41]. The percolation picture was based on a competition between Fock-space site-energy differences, and the hopping energy scale on the Fock-space.…”

Harmonic fingerprint of unconventional superconductivity in twisted bilayer graphene

“…What makes TBG so challenging for a SC description is, in particular, that it has flat bands, which yield very high density of states (van-Hove type of singularities) at a very low carrier density. Accordingly, a considerable amount of work has so far been devoted to understanding insulating [6][7][8][9][10][11][12][13][14][15][16] and superconducting 6,9,10,[14][15][16][17][18][19][20][21][22][23][24][25][26][27] phases from an electronic correlation point of view. Typically here, again in some analogy to the cuprates, a standard (multi-orbital) Hubbard model, only including the on-site interactions, has been employed.…”