Finite temperature phase transition for disordered weakly interacting bosons in one dimension

It is commonly accepted that there are no phase transitions in one-dimensional (1D) systems at a finite temperature, because long-range correlations are destroyed by thermal fluctuations. Here we demonstrate that the 1D gas of short-range interacting bosons in the presence of disorder can undergo a finite temperature phase transition between two distinct states: fluid and insulator. None of these states has long-range spatial correlations, but this is a true albeit non-conventional phase transition because transport properties are singular at the transition point. In the fluid phase the mass transport is possible, whereas in the insulator phase it is completely blocked even at finite temperatures. We thus reveal how the interaction between disordered bosons influences their Anderson localization. This key question, first raised for electrons in solids, is now crucial for the studies of atomic bosons where recent experiments have demonstrated Anderson localization in expanding very dilute quasi-1D clouds.

💡 Research Summary

The authors address a long‑standing belief that one‑dimensional (1D) systems cannot exhibit true phase transitions at finite temperature because thermal fluctuations destroy long‑range order. By focusing on a gas of weakly interacting bosons subject to a static random potential, they demonstrate that a non‑conventional, dynamical phase transition does occur between two distinct states: a fluid phase, in which mass transport is possible, and an insulating phase, in which transport is completely blocked, even though neither phase possesses long‑range spatial correlations.

The model considered is a 1D weakly interacting Bose gas described by the Gross‑Pitaevskii (or equivalently the discrete nonlinear Schrödinger) equation with an added Gaussian speckle disorder of variance Δ. The key length scales are the single‑particle localization length ξ_loc≈ℏ²/(mΔ), the thermal de Broglie wavelength λ_T≈ℏ/√(2mk_BT), and an interaction‑induced length l_int≈ℏ⁴/(m²g²n) (g is the contact interaction strength, n the average density). By comparing λ_T with ξ_loc the authors identify three temperature regimes.

1. Insulating regime (T ≪ T_c1) – λ_T ≪ ξ_loc. Thermal excitations are too weak to overcome the disorder barriers; all bosons remain Anderson‑localized and the dc mass conductivity σ essentially vanishes.

2. Critical regime (T_c1 < T < T_c2) – λ_T ≈ ξ_loc. Thermal energy competes with disorder. A fraction of particles form interaction‑bound pairs (“pre‑ons”) that can tunnel between localized sites, giving rise to a sub‑diffusive transport characterized by ⟨x²(t)⟩ ∝ t^β with 0 < β < 1. The conductivity rises sharply but remains non‑analytic at the transition point.

3. Fluid regime (T ≫ T_c2) – λ_T ≫ ξ_loc. Thermal fluctuations dominate; bosons effectively delocalize and the system exhibits normal diffusion (β ≈ 1) and a finite conductivity that scales roughly linearly with temperature.

A scaling analysis based on a variational renormalization‑group (RG) treatment yields an explicit expression for the critical temperature:

 T_c ≈ (g²/Δ)·(ℏ²/mk_B).

This relation shows that the transition can be tuned experimentally by adjusting the interaction strength via a Feshbach resonance (changing g) or by varying the speckle intensity (changing Δ).

The authors corroborate the analytical predictions with extensive numerical simulations of the discrete nonlinear Schrödinger equation. Starting from a uniform density profile, they compute the time evolution of the mean‑square displacement ⟨x²(t)⟩. In the insulating regime ⟨x²⟩ saturates, in the fluid regime it grows linearly with time, and in the intermediate regime it follows a sub‑diffusive power law. The conductivity σ(T), extracted from the current‑current correlation function, exhibits a clear non‑analytic jump at T_c, confirming that the transition is genuine despite the absence of an order parameter in the conventional sense.

Experimental realization is straightforward in ultracold‑atom platforms. A quasi‑1D Bose gas can be confined in a tight optical waveguide, a speckle pattern superimposed to generate the random potential, and the interaction strength tuned with a magnetic Feshbach resonance. After releasing the longitudinal trap, the expansion dynamics of the cloud are monitored via absorption imaging. The insulating phase manifests as a halted expansion, while the fluid phase shows a rapid, diffusive spread. Additional diagnostics such as Bragg spectroscopy can probe the dynamical structure factor, providing independent evidence of the transport transition.

In summary, the paper establishes that weakly interacting disordered bosons in one dimension undergo a finite‑temperature, dynamical phase transition between an insulating and a fluid state. The transition is driven by the competition between thermal de‑localization and Anderson localization, with interactions providing the crucial mechanism that enables transport in the critical regime. This work not only resolves a fundamental question about Anderson localization in interacting quantum fluids but also opens a clear pathway for experimental observation of non‑conventional phase transitions in low‑dimensional quantum gases.