Large deviations in the many-body localization transition: The case of the random-field XXZ chain

TL;DR

This work tackles the stability of the many-body localization (MBL) transition against rare, system-wide resonances in the random-field XXZ chain. It develops a mean-field glassy framework, introducing a beta-dressed Hilbert-space Landauer transmission $\mathcal{T}_0(\beta)$ and a Lagrange-multiplier–like parameter to weight extreme outliers, thereby identifying three regimes: ergodic, intermediate resonant-delocalization, and robust MBL. Finite-size phase diagrams in both the spin and Anderson bases show how rare resonances can destabilize localization at finite sizes, while infinitesimal interactions can destroy Anderson localization at finite disorder, with a finite critical disorder for small $\Delta$. The work also provides a Hilbert-space transport perspective by visualizing dominant resonant paths and clarifies how path rarity and range evolve with disorder, offering complementary insight to real-space probes and deepening understanding of the role of rare events in MBL physics. Overall, the approach connects MBL phenomenology to directed-polymer–type freezing transitions, yielding a quantitative framework for finite-size effects and basis-dependent behavior observed in numerical studies.

Abstract

The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.

Figures (19)

• Figure 1: Probability distribution function for the infinite time probabilities, Eq. (\ref{['eq:prob']}), of finding a system, initially prepared in the basis state $\ket{0}$, in a basis state $\ket{f} \in \mathcal{E}$. The gray dashed line indicates a reference power-law decay with exponent 2. For weak disorder ($W = 1$, left panel), the distribution is relatively narrow. As the disorder strength increases ($W = 4$ and $W = 9$, center and right panels), the distribution broadens significantly. The total number of samples used to compute these distributions is $N_{\rm tot} = N_0 \times N_S \times \mathcal{N}_\mathcal{E}$, where $N_0 = 2^{L/2-2}$, and $N_S = 500,\, 5\times10^3,\, 5\times10^4$ for $L = 8, 12, 16$, respectively.
• Figure 2: Probability distributions of the delocalization probability $\mathbb{P}_\mathcal{E}$ in log-log scale, for the disorder strengths shown in the legend, with $L=16$ and $\Delta = 1$. The inset is a zoom-in for the $W = 1$ distribution, shown in linear-log scale instead. This latter distribution is heavily peaked at finite values of $\mathbb{P}_\mathcal{E}$ with fast decaying tails.
• Figure 3: (a) Quantum transport on a network in a scattering geometry, receiving particles from a semi-infinite lead on the left and transmits them through several semi-infinite leads connected to its right-hand side. (b) Schematic of the transport of the 'fictitious particle'--initially prepared in the basis state $\ket 0$--on the Hilbert space network.
• Figure 4: Sketch of the different scaling behavior with $L$ of the typical value of the Landauer transmissions with and without imaginary parts for the Anderson model on the Bethe lattice.
• Figure 5: The annealed free-energy $\phi_a$ (dashed), the modified annealed free-energy $\tilde{\phi}_a$ (solid) and the quenched free-energy $\phi_q$ (solid with triangular markers) in the spin (warm colors) and Anderson (cold colors) bases. Low (left panel), intermediate (middle panel) and large (right panel) disorder strengths. The sizes $L$ are distinguished by the colors of the scale. The dashed gray lines show the relevant values at $\beta = 2$ (physical transport) and $\phi(\beta) = 0$ (delocalization/localization). The vertical colored dashed lines show the position of $\beta_\star$, for each curve.