Ergodicity breaking transition in finite disordered spin chains

Abstract

We study disorder-induced ergodicity breaking transition in high-energy eigenstates of interacting spin-1/2 chains. Using exact diagonalization we introduce a cost function approach to quantitatively compare different scenarios for the eigenstate transition. We study ergodicity indicators such as the eigenstate entanglement entropy and the spectral level spacing ratio, and we consistently find that an (infinite-order) Kosterlitz-Thouless transition yields a lower cost function when compared to a finite-order transition. Interestingly, we observe that the transition point in finite systems exhibits nearly thermal properties, i.e., ergodicity indicators at the transition are close to the random matrix theory predictions.

Quantum chaos challenges many-body localization

“…Intriguingly, the crossing points W * (L) in the inset of Fig. 2(b) agree both qualitatively and quantitatively with those obtained using other ergodicity indicators such as the eigenstate entanglement entropy S and the average level spacing ratio r [33], where a much larger interval of disorders has been taken into account. This suggests that the ergodicity breaking transition described here corresponds to the onset of departure of S from the maximal value (calculated in [79,80]), and of r from the corresponding random matrix theory prediction r GOE [67], see Fig.…”

“…2(a)-2(b) and 2(d)-2(e). Using the cost function minimization approach [33] we quantitatively verified that the scaling solution for g as a function of L/ξ BKT is more favorable than as a function of L/ξ 0 (see Appendix D for details).…”

“…Despite the considerable ongoing efforts, universal properties of the MBL transition are still intensively debated. From the perspective of exact numerical solutions, there is still lack of consensus about the position and the nature of the transition point in paradigmatic models (in particular, disordered spin-1/2 chains, our focus here) [13,16,[21][22][23][24][25][26][27][28][29][30][31][32][33]. Moreover, phenomenological theories such as the real-space renormalization group are also experiencing rapid development [34][35][36][37][38][39][40][41], with the current understanding that the transition exhibits some properties of the Berezinskii-Kosterlitz-Thouless transition [38][39][40][41].…”

“…Signatures of the transition are usually explored by studying the structure of Hamiltonian eigenstates [13,42,45] and matrix elements of local observables [13,[46][47][48][49]. Another convenient approach to characterize the MBL transition, which is inspired by successful analyses of the Anderson localization transition [50,51], is based on studies of spectral properties [16,33,[52][53][54]. The key reference point for the latter is provided by the random matrix theory and the so-called quantum chaos conjecture [55][56][57].…”

“…Exact numerical calculations, however, seemed to indicate [23,26,45,78] that the optimal scaling collapse of ergodicity indicators is obtained by using the power-law correlation length ξ 0 = |W − W * | −ν . Nevertheless, it was recently argued that the BKT correlation length ξ BKT yields a more favorable scaling solution than ξ 0 [33].…”