From Bose glass to many-body localization in a one-dimensional disordered Bose gas

TL;DR

The paper investigates the finite‑temperature phase diagram of a one‑dimensional disordered Bose gas using bosonization, replica methods, and a nonperturbative FRG with two truncations of the effective action. One truncation yields a Bose glass destabilized at any finite temperature in favor of a normal fluid, with quantum and classical glassy crossovers governing intermediate scales. The other truncation reveals a finite‑temperature fluid–insulator transition at Tc, with a low‑temperature localized (MBL‑like) phase described by a droplet picture and exhibiting nonergodic behavior, slow dynamics, and a nontrivial many‑body spectrum; this scenario aligns with MAAS. The work connects disorder‑driven Bose gas physics to MBL phenomenology, highlighting intermediate‑scale glassiness and a robust finite‑T transition, and discusses limitations and avenues for further refinement beyond the current derivative expansion.

Abstract

We determine the finite-temperature phase diagram of a one-dimensional disordered Bose gas using bosonization and the nonperturbative functional renormalization group (RG). We discuss two different scenarios, based on distinct truncations of the effective action. In the first scenario, the Bose glass is destabilized at any finite temperature, giving rise to a normal fluid. Nevertheless, one can distinguish a low-temperature glassy regime, where disorder plays an important role on intermediate length and time scales, from a high-temperature regime, where disorder becomes irrelevant. In the second scenario, below a temperature $T_c$, the RG flow exhibits a singularity at a finite value of the RG momentum scale. We propose that this singularity signals a lack of thermalization and the existence of a localized phase for $T<T_c$. We provide a description of this low-temperature localized phase within a droplet picture and highlight a number of possible similarities with characteristics of many-body localized phases, including non-thermal behavior, avalanche instabilities and many-body resonances, the structure of the many-body spectrum, and slow dynamics in the ergodic phase. The normal fluid above $T_c$, and below a crossover temperature $T_g$, exhibits glassy properties on intermediate scales.

Figures (11)

• Figure 1: Phase diagram of a one-dimensional disordered Bose gas, obtained from a truncation of the effective action that neglects the (quantum) time-derivative term in the two-replica component. The Luttinger parameter is varied at fixed dimensionless disorder (see Sec. \ref{['sec_finiteT']}). The ground state is a Bose glass for $K<3/2$ (in the limit of infinitesimal disorder), while the three finite-temperature regimes correspond to the quantum glassy normal fluid (QGNF), the classical glassy normal fluid (CGNF), and the nonglassy normal fluid (NGNF). In the glassy normal-fluid regimes, the RG flow is controlled by the zero-temperature Bose-glass fixed point on intermediate length and time scales. The correlation length is set by the zero-temperature localization length $\xi_{\rm loc}$ in the quantum glassy regime, and by the thermal length in the classical glassy regime.
• Figure 2: Phase diagram obtained from a truncation of the effective action that includes all second-order derivative terms in the one- and two-replica components. The existence of a finite-temperature transition between a low-temperature localized (MBL) phase and a high-temperature normal fluid is consistent with the conclusions of Michal, Aleiner, Altshuler, and Shlyapnikov (MAAS) Michal16. Possible similarities between the low-temperature localized phase and the characteristic properties of MBL phases are discussed in Sec. \ref{['sec_mbl']}.
• Figure 3: Flow of the Luttinger parameter $K_k$ and the exponent $\theta_k=z_k-1=\partial_t \ln K_k$ (inset) as functions of $\ln(\Lambda/k)$ for $\tilde{W}_{3,k}=0$ and $T=0$ ($\tilde{\Delta}_{1,\Lambda}=0.002$) .
• Figure 4: $\tilde{\Delta}_k(u)$ and $\delta\tilde{W}_{1,k}(u)=-\delta\tilde{W}_{2,k}(u)$ for $\ln(\Lambda/k)=0/1.0/1.54/2.34/2.75/3.82$, for $\tilde{W}_{3,k}=0$ and $T=0$ ($K=0.4$, $\tilde{\Delta}_{1,\Lambda}=0.002$). The dash-dotted (green) curves show the initial conditions $\tilde{\Delta}_\Lambda(u)=2\tilde{{\cal D}}_\Lambda \cos(2u)$, $\delta\tilde{W}_{1,\Lambda}(u)=0$, and the dashed (black) curves show the fixed-point solutions (\ref{['fppot']}).
• Figure 5: Dimensionless disorder correlator $\tilde{\Delta}_k(u)$, and the derivative terms $\delta\tilde{W}_{1,k}(u)$ and $\delta\tilde{W}_{3,k}(u)$ for $\ln(\Lambda/k)\simeq 4.504/4.655/4.693/4.694$ at zero temperature and near the singularity at $k_c$ ($\tilde{\Delta}_{1,\Lambda}=0.02$, $K=1.4$). As the singularity is approached, cusps form in $\tilde{\Delta}_k(u)$ and $\delta\tilde{W}_{1,k}(u)$ around $u=0$ and $u=\pi$, and $\delta\tilde{W}_{3,k}(u)$ grows large. The dashed (black) line shows the fixed-point solution $\tilde{\Delta}^*(u)$ obtained in the absence of $\tilde{W}_{3,k}(u)$ [Eq. (\ref{['fppot']})].