Renormalization-Group Analysis of the Many-Body Localization Transition in the Random-Field XXZ Chain

TL;DR

This work investigates the MBL transition in the random-field XXZ chain by reconstructing the renormalization-group beta function from exact-diagonalization data. It demonstrates that a simple one-parameter scaling with an isolated fixed point cannot capture the observed flow, instead revealing a two-parameter, BKT-like renormalization-group structure with a line of localized fixed points terminating at a critical point (or, if no transition exists, a confining flow toward ergodicity). By mapping the two-parameter RG to Newtonian dynamics in a logarithmic coordinate, the authors fit a non-confining potential with $n\approx 1$ and argue that the critical disorder lies between $W_c\approx 4$ and $W_c\approx 5$, with critical scaling $\phi(L)\sim L^{-2}$ at criticality. They also analyze large-disorder statistics and show that resolving a true transition requires massive sampling, highlighting practical challenges in numerically resolving MBL transitions. Overall, the paper provides a two-parameter RG framework for interacting disordered quantum systems and clarifies the scaling structure underpinning MBL phenomena.

Abstract

We analyze the spectral properties of the Heisenberg spin-1/2 chain with random fields in light of recent works of the renormalization-group flow of the Anderson model in infinite dimension. We reconstruct the beta function of the order parameter from the numerical data, and show that it does not admit a one-parameter scaling form and a simple Wilson-Fisher fixed point. Rather, it is compatible with a two-parameter, Berezinskii-Kosterlitz-Thouless-like flow with a line of fixed points (the many-body localized phase), which terminates into the localization transition critical point. Therefore, we argue that previous studies, which assumed the existence of an isolated Wilson- Fisher fixed point and performed one-parameter finite-size scaling analysis, could not explain the numerical data in a coherent way.

Figures (12)

• Figure 1: System-size dependence of the rescaled $r$-parameter $\phi$, Eq. \ref{['eq:def_phi']}, in the XXZ spin chain. Different colors correspond to different values of the disorder strength $W$. Dots represent numerical data, while solid lines are polynomial interpolating curves, later used for the $\beta$-function computations. Dashed lines represent the asymptotic values: $\phi=0$ for the localized and $\phi=1$ for the ergodic phase.
• Figure 2: Sketch of a one-parameter-scaling RG flow with irrelevant corrections, similar to what can be observed in the Anderson model in $d$-dimensional space altshuler2024renormalization. The solid black curve represents the one-parameter-scaling beta function $\beta_0(A)$, which vanishes at the critical point $A_c$. The colored solid lines represent the beta function for different choices of the initial parameters, as described by Eq. \ref{['eq:1PS_irrelevant_corrections']}. The starting point of these lines occurs at $N=N_0$ and $N$ grows in the direction indicated by the arrows. Around the critical point $A_c$, the beta function can be linearized as described by Eq. \ref{['eq:beta_crit_1PS']}.
• Figure 3: Naive finite-size scaling for the average rescaled gap ratio $\phi$. In the main plot, the collapse is shown for the larger sizes, $L>12$, while the inset contains data before the collapse down to size $L=8$. The collapse is obtained by fixing $\phi_c=0$, and it yields a critical disorder $W_c \simeq 8.0$ and exponents $\nu \simeq 5.5$, $\omega \simeq 2.0$. We will present a different explanation for the critical exponent $\omega=2$ in the next section. Notice also the similarity with the data for the Anderson model on the RRG in sierant23a. For a collapse in which we allow $\phi_c$ to vary, we refer to App. \ref{['app:sec:datacollapse']}.
• Figure 4: (Left) Beta function for a two-parameter scaling transition scenario. In the localized phase, the RG flow reaches $A=0$ at a finite and negative $\beta = -1/\xi_\mathrm{loc}$, and its value depends on $W$, thus generating a line of fixed points (solid red). In the ergodic phase, instead, the curves flow to $A=1$ as prescribed by random matrix theory. The RG flow around the ergodic fixed point is described by a one-parameter scaling, meaning that all the RG curves for $W<W_c$ ultimately approach a universal curve (dashed black). The red dotted line is the critical line for $W=W_c$, which separates the localized and ergodic phases. (Right) Beta function in the absence of a localized phase. For any finite value of $W$, the RG flow never reaches $A=0$ and flows to the ergodic fixed point at large enough sizes.
• Figure 5: (Left) Example of potential function for the one-dimensional dynamical model in presence of a localization transition, similarly to what happens for the RRG. For small disorder, corresponding to energies $E<E_c$ ($E_c=0$ in the sketch), the presence of the potential confines the trajectories, excluding the point $\alpha=-\infty$ (i.e. $A=0$): the original system is ergodic. For large enough disorder, corresponding to energies $E>E_c$, the potential does not confine the classical trajectories, and $\alpha \to -\infty$ when $t\to \infty$: the localized phase is approached in the thermodynamic limit. Notice also that, at $t=0$, particles move from right to left, since $\beta<0$, being the beta function negative at small enough sizes. (Right) When the potential is such that $V(\alpha \to -\infty)\to + \infty$, the classical trajectories are always reflected and $\alpha \to 0$ (equivalently $A \rightarrow 1$) when $t \to \infty$. In this case there is no localized phase.