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

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.

💡 Research Summary

In this work the authors investigate the finite‑temperature phase diagram of a one‑dimensional disordered Bose gas by combining bosonization, the replica formalism, and a non‑perturbative functional renormalization group (FRG) approach. Two distinct truncations of the effective action are considered, leading to two competing physical scenarios.

The first truncation neglects the quantum time‑derivative term in the two‑replica sector and retains only second‑order spatial derivatives. Within this scheme the FRG flow reproduces the well‑known perturbative RG result: disorder is irrelevant at any non‑zero temperature, so the Bose‑glass phase is destabilized and the system behaves as a normal fluid. Nevertheless, three temperature regimes emerge. At the lowest temperatures (T < Tₓ) the flow is temporarily controlled by the zero‑temperature Bose‑glass fixed point, giving rise to a “quantum glassy normal fluid” (QGNF) in which the correlation length is set by the zero‑temperature localization length ξ_loc. In an intermediate window (Tₓ < T < T_g) thermal fluctuations dominate the correlation length while the flow still feels the zero‑temperature fixed point, producing a “classical glassy normal fluid” (CGNF). At high temperatures (T > T_g) disorder becomes completely irrelevant and a conventional nonglassy normal fluid (NGNF) is obtained.

The second truncation includes the time‑derivative term in the two‑replica sector. Here a qualitatively different behavior appears when the Luttinger parameter K < 3/2: the RG flow develops a finite‑scale singularity at a momentum scale k_c. The renormalized disorder correlator acquires a cusp at this scale, signalling the breakdown of the Matsubara formalism and the loss of thermalization. The authors interpret this singularity as the hallmark of a many‑body localized (MBL‑like) phase. The singularity persists up to a critical temperature T_c ≈ v/ξ_loc (v is the sound velocity of the clean system), in agreement with the finite‑temperature fluid‑insulator transition proposed by Michal, Aleiner, Altshuler and Shlyapnikov (MAAS). Below T_c the system is argued to reside in a localized phase that can be described by a droplet picture: quantum‑active droplets have a maximal size ∼ 1/k_c and are responsible for the non‑thermal behavior. Above T_c the singularity disappears, the flow can be integrated to k = 0, and disorder becomes irrelevant, yielding again a normal fluid that nevertheless exhibits the same three sub‑regimes (QGNF, CGNF, NGNF) as in the first scenario.

The paper then draws several parallels between the low‑temperature localized phase obtained with the second ansatz and the phenomenology of MBL. First, the lack of thermalization implies that the many‑body spectrum does not follow a Gibbs distribution; instead level statistics become Poissonian, reflecting the presence of localized integrals of motion. Second, the cusp in the disorder correlator can be interpreted as an “avalanche” instability: a rare thermal region can destabilize surrounding localized regions, a mechanism that has been identified as crucial for the stability of MBL in higher dimensions. Third, the droplet picture naturally incorporates many‑body resonances, i.e., rare configurations where two distant droplets become resonant and hybridize, leading to a proliferation of low‑energy excitations. Fourth, the structure of the many‑body spectrum near the transition shows a dense set of low‑lying states separated by exponentially small gaps, reminiscent of the “Griffiths” regime in disordered spin chains. Fifth, even in the ergodic (normal‑fluid) phase close to T_c, transport is sub‑diffusive and relaxation times grow anomalously, mirroring the slow dynamics observed experimentally in cold‑atom realizations of MBL.

Methodologically, the work demonstrates that the FRG, when equipped with a sufficiently rich ansatz for the effective action, can capture non‑perturbative features such as finite‑scale singularities and cusp formation, which are invisible to conventional perturbative RG. The comparison of the two truncations highlights how the inclusion of time‑derivative terms in the replica sector dramatically changes the physical picture, suggesting that higher‑order replica couplings (beyond two replicas) may be essential for a complete description.

In the conclusion the authors summarize the two possible phase diagrams (Fig. 1 and Fig. 2 in the paper), emphasize the consistency of the second scenario with the MAAS prediction of a finite‑temperature fluid‑insulator transition, and outline future directions: extending the truncation to include higher‑order replica interactions, performing numerical FRG studies to test the robustness of the singularity, and comparing with experimental platforms such as ultracold atoms in disordered optical lattices or quasi‑periodic potentials. The study thus provides a unified framework that bridges the traditional Bose‑glass physics of disordered bosons with the modern theory of many‑body localization, opening avenues for both theoretical refinement and experimental verification.