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arXiv:2410.12430 · Physical review. B./Physical review. B

Renormalization group analysis of the many-body localization transition in the random-field XXZ chain

Jacopo Niedda, Giacomo Bracci-Testasecca, Giuseppe Magnifico, Federico Balducci, Carlo Vanoni, Antonello Scardicchio

Published Sep 17, 2025• 6 citations
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Abstract

The spectral properties of the Heisenberg spin-1/2 chain with random fields are analyzed in light of recent works on the renormalization-group flow of the Anderson model in infinite dimension. We reconstruct the β function of the order parameter from the numerical data, and observe that it may not admit a one-parameter scaling form and a simple Wilson-Fisher fixed point. Rather, it appears to be more compatible with a two-parameter, Berezinskii–Kosterlitz-Thouless-like flow with a line of fixed points (the many-body localized phase) terminating at the localization transition critical point. We argue that this renormalization group framework provides a more coherent and intuitive explanation of numerical data, up to the system sizes available with the present technology.

DOI
10.1103/gcwf-jdlr

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The spectral properties of the Heisenberg spin-1/2 chain with random fields are analyzed in light of recent works on the renormalization-group flow of the Anderson model in infinite dimension. We reconstruct the β function of the order parameter from the numerical data, and observe that it may not admit a one-parameter scaling form and a simple Wilson-Fisher fixed point. Rather, it appears to be more compatible with a two-parameter, Berezinskii–Kosterlitz-Thouless-like flow with a line of fixed points (the many-body localized phase) terminating at the localization transition critical point. We argue that this renormalization group framework provides a more coherent and intuitive explanation of numerical data, up to the system sizes available with the present technology.
We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.
The Anderson transition in random graphs has raised great interest, partly out of the hope that its analogy with the many-body localization (MBL) transition might lead to a better understanding of this hotly debated phenomenon. Unlike the latter, many results for random graphs are now well established, in particular, the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. In this paper, we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ than the one in the transverse direction ${\ensuremath{\xi}}_{\ensuremath{\perp}}$. Importantly, these two lengths have different critical behaviors: ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ diverges with a critical exponent ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=1$, while ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ reaches a finite universal value ${\ensuremath{\xi}}_{\ensuremath{\perp}}^{c}$ at the transition point Wc{W}_{c}Wc​. Indeed, ${\ensuremath{\xi}}_{\ensuremath{\perp}}^{\ensuremath{-}1}\ensuremath{\approx}{{\ensuremath{\xi}}_{\ensuremath{\perp}}^{c}}^{\ensuremath{-}1}+{\ensuremath{\xi}}^{\ensuremath{-}1}$, with $\ensuremath{\xi}\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\perp}}}$ associated with a new critical exponent ${\ensuremath{\nu}}_{\ensuremath{\perp}}=\frac{1}{2}$, where $exp(\ensuremath{\xi})$ controls finite-size effects. The delocalized phase inherits the strongly nonergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $\ensuremath{\nu}=\frac{1}{2}$. This shows a very strong analogy with the MBL transition: the behavior of ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a small-world complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.
We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions DqD_qDq​. The RG flows recover the qualitative structure previously reported for the information dimension D1D_1D1​, while their extension to other DqD_qDq​ with smaller moment orders q<1/2q<1/2q<1/2 confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension D1D_1D1​ need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large q>1/2q>1/2q>1/2 and small q<1/2q<1/2q<1/2 eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.
Despite tremendous theoretical efforts to understand subtleties of the many-body localization (MBL) transition, many questions remain open, in particular concerning its critical properties.Here we make the key observation that MBL in one dimension is accompanied by a spin freezing mechanism which causes chain breaks in the thermodynamic limit.Using analytical and numerical approaches, we show that such chain breaks directly probe the typical localization length and that their scaling properties at the MBL transition agree with the Kosterlitz-Thouless scenario predicted by phenomenological renormalization group approaches.
We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the many-body localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior.
Using a new approximate strong-randomness renormalization group (RG), we study the many-body localized (MBL) phase and phase transition in one-dimensional quantum systems with short-range interactions and quenched disorder. Our RG is built on those of Zhang et al. [Phys. Rev. B 93, 224201 (2016)] and Goremykina et al. [Phys. Rev. Lett. 122, 040601 (2019)], which are based on thermal and insulating blocks. Our main addition is to characterize each insulating block with two lengths: a physical length and an internal decay length $\ensuremath{\zeta}$ for its effective interactions. In this approach, the MBL phase is governed by a RG fixed line that is parametrized by a global decay length $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$, and the rare large thermal inclusions within the MBL phase have a fractal geometry. As the phase transition is approached from within the MBL phase, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$ approaches the finite critical value corresponding to the avalanche instability, and the fractal dimension of large thermal inclusions approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG flow, with no intermediate critical MBL phase.