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arXiv:2502.16219 · Physical review. B./Physical review. B

Scaling of many-body localization transitions: Quantum dynamics in Fock space and real space

Thibault Scoquart, I. V. Gornyi, A. D. Mirlin

Published Jul 14, 2025• 3 citations
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Abstract

Many-body-localization (MBL) transitions are studied in a family of single-spin-flip spin- 1 2 models, including the one-dimensional (1D) chain with nearest-neighbor interactions, the quantum dot (QD) model with all-to-all pair interactions, and the quantum random energy model (QREM). We investigate the generalized imbalance that characterizes propagation in Fock space out of an initial basis state and, at the same time, can be efficiently probed by real-space measurements. For all models considered, the average imbalance and its quantum and mesoscopic fluctuations provide excellent indicators for the position of the MBL transition W c ( n ) , where n is the number of spins. Combining these findings with earlier results on level statistics, we determine phase diagrams of the MBL transitions in the n − W plane. Our results provide evidence for a direct transition between the ergodic and MBL phases for each of the models, without any intermediate phase. For QREM and QD model, W c ( n ) grows as a power law of n (with logarithmic corrections), in agreement with analytical predictions W c QREM ( n ) ∼ n 1 / 2 ln n and W c QD ( n ) ≳ n 3 / 4 ln 1 / 2 n . This growth is in stark contrast to the 1D model, where W c ( n ) is essentially independent of n , consistent with the analytic expectation W c 1 D ( n → ∞ ) = const . We also determine the scaling of the transition width Δ W ( n ) / W c ( n ) and estimate the system size n needed to study the asymptotic scaling behavior. Although the corresponding values of n are larger than those accessible to exact simulations on a classical computer, they are within the reach of quantum simulators. Our results indicate feasibility of experimental studies of n − W phase diagrams and scaling properties of MBL transitions in models of 1D and QD type and in their extensions to other spatial geometry or distance-dependent interactions.

DOI
10.1103/qcvd-n2yk

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We explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin-12\frac{1}{2}21​ chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.
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