Extreme statistics as a probe of the superfluid to Bose-glass Berezinskii-Kosterlitz-Thouless transition

TL;DR

The authors address how to detect a delocalization-localization transition in a disordered 1D quantum system by exploiting extreme statistics of local observables. They study the random-field XXZ chain in the ground state and at high energy, using DMRG and exact diagonalization to show that the typical minimal deviation δ_min^typ of local magnetization decays as a power law in the Bose-glass phase and exhibits healing in the superfluid phase, enabling a BKT-like analysis in the strong-disorder regime. They identify a critical disorder W_c ≈ 0.38 with a local-exponent γ_c ≈ 0.294 and a localization-length exponent ν_loc ≈ 0.67, consistent with a BKT scenario; a finite-size scaling collapse supports this picture. The weak-disorder (Giamarchi–Schulz) regime remains challenging for extreme-statistics probes due to phase-slip–driven transitions and finite-size effects, but the work lays a solid foundation for using extreme local observables as practical, experimentally accessible probes of delocalization-localization transitions in disordered quantum chains.

Abstract

Recent studies of delocalization-localization transitions in disordered quantum chains have highlighted the role of rare, chain-breaking events that favor localization, in particular for high-energy eigenstates related to many-body localization. In this context, we revisit the random-field XXZ spin-1/2 chain at zero temperature with ferromagnetic interactions, equivalent to interacting fermions or hard-core bosons in a random potential with attractive interactions. We argue that localization in this model can be characterized by chain-breaking events, which are probed by the extreme values of simple local observables, such as the on-site density or the local magnetization, that are readily accessible in both experiments and numerical simulations. Adopting a bosonic language, we study the disorder-induced Berezinskii-Kosterlitz-Thouless (BKT) quantum phase transition from superfluid (SF) to Bose glass (BG), and focus on the strong disorder regime where localization is driven by weak links. Based on high-precision density matrix renormalization group simulations, we numerically show that extreme local densities accurately capture the BKT transition, even for relatively short chains ranging from a few dozen to a hundred sites. We also discuss the SF-BG transition in the weak disorder regime, where finite-size effects pose greater challenges. Overall, our work seeks to establish a solid foundation for using extreme statistics of local observables, such as density, to probe delocalization-localization transitions in disordered quantum chains, both in the ground state and at high energy.

Figures (18)

• Figure 1: Sketch of the phase diagram of the spin-1/2 XXZ chain in a random field, Eq. \ref{['eq:ham']}, from Doggen et al.doggen_weak_2017. The green line indicates the Luttinger liquid regime ($-1 < \Delta < 1$) at $W=0$, which is a superfluid (SF) in the bosonic language. For $-1 < \Delta < -0.5$ the SF phase (in gray) remains stable against weak enough random field, while for $\Delta > -0.5$, any finite disorder immediately gives rise to a localized, gapless Bose glass (BG). The half-bandwidth $W^{\star} = 1+\Delta$ is expected to play a qualitative role in controlling the nature of the BKT transition from the SF to the BG phase doggen_weak_2017. At strong disorder $W>W^\star$ the SF is destroyed by weak-links, while at weak disorder, it is destroyed by quantum phase slips (dubbed Giamarchi-Schulz giamarchi_localization_1987). In this work, we use extreme magnetization statistics to revisit the phase diagram along the vertical dotted lines.
• Figure 2: Extreme magnetization in the ground-state of the XXZ chain in random field at $\Delta = -0.75$. (a) Distribution of the expectation value of the local magnetization for $L = 100$ sites (5k independent samples). (b) Example of the expectation value of the local magnetization and (c) corresponding deviation for one sample, at $W=0.7$. (d) Decay of the typical minimal deviation in the SF phase ($W=0.1,\,0.15$), averaged over 10k samples for $L\leq 64$ and 5k samples otherwise. The dashed lines are fits to pure power laws, while the solid lines are fits to Eq. \ref{['eq:deltafit']} allowing a non-zero $\delta_{\infty}$. (e) The decay in the BG ($W=0.7$) is well captured by a pure power-law decay. Note that both panels (d) and (e) are in log-log scale.
• Figure 3: Decay of the typical minimal deviation at $\Delta = -0.75$ (weak-link/strong disorder transition) and effective exponent. The critical disorder strength is $W = 0.375 \pm 0.015$doggen_weak_2017. (a) and (b) Decay of the typical minimal deviation, in log-log scale. Lines are fits for $\ln \delta_{\min}^{\rm{typ}} = \ln A - \gamma_{\rm eff} \ln L$ for $L \in [14,28]$ (dashed) and $L \in [48,100]$ (full). Below the critical disorder strength (a), the decay is slower than a pure power-law decay (in agreement with $\gamma_{\rm eff} \rightarrow 0$, see panel (c)), whereas at and above the critical disorder strength (b), $\delta_{\min}^{\rm{typ}}(L)$ decays as (or slightly faster than) a pure power-law. (c) Effective power-law exponents $\gamma_{\rm eff}$ from sliding fits, log-linear scale. The sizes for sliding fits are chosen such that $\ln L_{\max} - \ln L_{\min}$ is roughly constant, and the fits for the smallest sizes and largest sizes correspond to those shown in the first panels. Inset: same data, shown for various disorders as a function of the mid-point of the fitting interval and highlighting the trend inversion between increasing and decreasing $\gamma_{\rm eff}$, and the saturation for $\gamma_{\rm eff}$ for large system sizes above the transition.
• Figure 4: $\Delta = -0.75$: Comparing pure power-law fits with exponent $\gamma$ to fits with an additional constant and exponent $\tilde{\gamma}$. (a) Exponents comparison. $\tilde{\gamma}$ is obtained from a 3-parameter fits, hence the larger errors, but all sizes from $L_{\min}\in {14, 20}$ to $L_{\mathrm{max}} = 100$ are included in the fit. $\gamma$ is obtained as in Fig. \ref{['fig:Delta-075']}. Inset: optimal value for $\tilde{\gamma}$ at $W = 0.1$. (b) Constant $\delta_{\infty}$ from the fit Eq. \ref{['eq:deltafit']} (crosses) for a fixed exponent $\tilde{\gamma} = 0.29$, compared to the clean value for $\delta = 1/2-m$ (red dashed line). Inset: comparison of fit qualities between this and a pure power-law fit.
• Figure 5: Finite-size scaling analysis of the typical minimal deviation in the strong disorder regime of the transition, on the BG side. The scaling analysis is performed for various candidate values for the critical disorder $W_c$. (a) Scaling collapse for fixed $W_c = 0.38$. Different symbol shapes correspond to different disorder strengths, and different colors to different system sizes. For this value of $W_c$, corresponding to the critical disorder discussed in the previous section, the best collapse is obtained with $\gamma(W_c) = \gamma_c = 0.294 \pm 0.002$ and $\nu_{\rm loc} = 0.67 \pm 0.03$ (errors are estimated from a parametric bootstrap). The red line is the scaling function Eq. \ref{['eq:scalingfit']} with fitted parameters. (b) Reduced chi-square for the best fit as a function of the chosen critical disorder strength $W_c$. A very large $\chi^2_{\rm red}$ indicates a poor fit, while one of order 1 indicates a reasonable fit: for $W_c \in [0.35, 0.45]$ and in particular for $W_c = 0.38$, the fit is good. (c) Exponent $\nu_{\rm loc}(W_c)$ and exponent $\gamma_c(W_c)$ as a function of the same critical disorder strength.