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arXiv:2404.07975 · Physical review. B./Physical review. B

Spectral multifractality and emergent energy scales across the many-body localization transition

Sthitadhi Roy

Published May 8, 2025• 2 citations
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Abstract

We present a scaling theory of the many-body localization transition in terms of emergent, characteristic energy scales. The analysis is based on the decomposition of the eigenstates in the basis of trivially localized states, resolved in the energies of the latter, which we refer to as the spectral decomposition of the eigenstates. The characteristic energy scales emerge when the multifractal properties, or lack thereof, of the spectral decomposition are studied at different scales. These characteristic scales correspond to the ones, above which the spectral decompositions exhibit their global behavior, namely full ergodicity in the ergodic phase and multifractality in the many-body localized phase. On the other hand, at scales below the characteristic ones, the decomposition in the ergodic phase shows finer (multi)fractal structures whereas in the localized phase, the decomposition picks out well-separated, localized resonant peaks. The scaling of these characteristic energy scales across the many-body localization transition admits a scaling theory consistent with a Kosterlitz-Thouless-type scenario and bears striking resemblances to that of inverse participation ratios of eigenstates.

DOI
10.1103/physrevb.111.184201

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We present a scaling theory of the many-body localization transition in terms of emergent, characteristic energy scales. The analysis is based on the decomposition of the eigenstates in the basis of trivially localized states, resolved in the energies of the latter, which we refer to as the spectral decomposition of the eigenstates. The characteristic energy scales emerge when the multifractal properties, or lack thereof, of the spectral decomposition are studied at different scales. These characteristic scales correspond to the ones, above which the spectral decompositions exhibit their global behavior, namely full ergodicity in the ergodic phase and multifractality in the many-body localized phase. On the other hand, at scales below the characteristic ones, the decomposition in the ergodic phase shows finer (multi)fractal structures whereas in the localized phase, the decomposition picks out well-separated, localized resonant peaks. The scaling of these characteristic energy scales across the many-body localization transition admits a scaling theory consistent with a Kosterlitz-Thouless-type scenario and bears striking resemblances to that of inverse participation ratios of eigenstates.
We study spectral and wave-function statistics for many-body localization transition in systems with long-range interactions decaying as $1/{r}^{\ensuremath{\alpha}}$ with an exponent $\ensuremath{\alpha}$ satisfying $d\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}2d$, where ddd is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder ${W}^{*}=W/{L}^{2d\ensuremath{-}\ensuremath{\alpha}}lnL$, where WWW is the disorder strength and LLL the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder WWW with the system size LLL and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×10^{6}). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.
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.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum f(0≤α≤2)=α2f(0 \leq \alpha \leq 2)=\frac{\alpha}{2}f(0≤α≤2)=2α​, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.
Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
The authors construct an efficient algorithm for extracting the entanglement structure of wave functions in the form of a cluster decomposition. They apply this to the many-body localization problem by analyzing entanglement clusters in exact eigenstates. Strongly entangled ergodic clusters are connected. Their size is distributed according to a power law at the transition, while at higher disorder the distribution is a stretched exponential. This constrains possible phenomenological renormalization group descriptions of the transition, and is consistent with a Kosterlitz-Thouless type scaling.