Modelling equilibration of local many-body quantum systems by random graph ensembles

Abstract

We introduce structured random matrix ensembles, constructed to model many-body quantum systems with local interactions. These ensembles are employed to study equilibration of isolated many-body quantum systems, showing that rather complex matrix structures, well beyond Wigner's full or banded random matrices, are required to faithfully model equilibration times. Viewing the random matrices as connectivities of graphs, we analyse the resulting network of classical oscillators in Hilbert space with tools from network theory. One of these tools, called the maximum flow value, is found to be an excellent proxy for equilibration times. Since maximum flow values are less expensive to compute, they give access to approximate equilibration times for system sizes beyond those accessible by exact diagonalisation.

Typical relaxation of perturbed quantum many-body systems

“…As discussed above (46), there is a third pole which happens to be exactly at −1 for all β. More generally, it thus seems reasonable to expect that the order-M approximation of (30) will exhibit M + 1 real-and positive-valued poles close to 0, −1, −2, ..., −M for small β, which turn step by step into complex conjugated pairs upon increasing β.…”

“…On the other hand, we found in Sec. IV B that the second-order approximation (46) reproduces the exact behavior remarkably well for perturbation profiles f (x) of the form (19) with N = 1. Rewriting this approximation (46) in terms of the two parameters α and ∆ v according to (79)-( 81) thus yields as a very general prediction for the function g(t) the approximation (16) as stated in Sec.…”

Modification of quantum many-body relaxation by perturbations exhibiting a banded matrix structure

“…To conclude this section, we remark that the true perturbation will usually exhibit correlations (i.e., functional interdependencies) between the matrix elements V µν , which may arise, for example, due to the locality and few-body character of interactions [34,35]. Since such correlations are not accounted for in the considered perturbation ensembles, it is implicitly assumed that their effect on the dynamics is negligible.…”

“…FIG.1: Overlap distribution u(E) from(7) for three different perturbation profiles σ 2 (E) as depicted in the insets of the left-most panel of each row, namely a step profile(33) in (a), an exponential profile(34) in (b), and a double-Breit-Wigner profile(35) with b1 = 0.45, b2 = 0.9, d = 3.5 in (c). In all three cases, we employed the same parameter values ε = 1/512, σ2 = 0.2, and ∆v = 750ε = 1.46.…”

Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories

“…The study of quantum nonequilibrium dynamics has only recently become experimentally feasible [8][9][10][11][12][13], raising questions surrounding the process and conditions in which isolated many-body quantum systems equilibrate to a thermal state [14][15][16][17][18][19][20][21][22], a process known as quantum thermalization [23][24][25][26][27]. Important related questions remain surrounding relaxation timescales and the route to equilibrium of complex quantum systems [28][29][30][31][32][33][34][35][36], as well as the emergence of thermodynamical laws [37][38][39][40]. A useful approach to the description of generic nonintegrable quantum systems can be developed from quantum chaos [41,42] and the eigenstate thermalization hypothesis (ETH), which in turn can be derived from an underlying random matrix theory (RMT) [32,33,[43][44][45][46].…”