The eigenstate thermalization hypothesis (ETH) is a successful theory that provides sufficient criteria for ergodicity in quantum many-body systems. Most studies were carried out for Hamiltonians relevant for ultracold quantum gases and single-component systems of spins, fermions, or bosons. The paradigmatic example for thermalization in solid-state physics are phonons serving as a bath for electrons. This situation is often viewed from an open-quantum system perspective. Here, we ask whether a minimal microscopic model for electron-phonon coupling is quantum chaotic and whether it obeys ETH, if viewed as a closed quantum system. Using exact diagonalization, we address this question in the framework of the Holstein polaron model. Even though the model describes only a single itinerant electron, whose coupling to dispersionless phonons is the only integrability-breaking term, we find that the spectral statistics and the structure of Hamiltonian eigenstates exhibit essential properties of the corresponding random-matrix ensemble. Moreover, we verify the ETH ansatz both for diagonal and offdiagonal matrix elements of typical phonon and electron observables, and show that the ratio of their variances equals the value predicted from random-matrix theory.

We experimentally and numerically investigate the sudden expansion of fermions in a homogeneous one-dimensional optical lattice. For initial states with an appreciable amount of doublons, we observe a dynamical phase separation between rapidly expanding singlons and slow doublons remaining in the trap center, realizing the key aspect of fermionic quantum distillation in the strongly-interacting limit. For initial states without doublons, we find a reduced interaction dependence of the asymptotic expansion speed compared to bosons, which is explained by the interaction energy produced in the quench.Many-body physics in one dimension (1D) differs in numerous essential aspects from its higher-dimensional counterparts. Several familiar concepts, such as Fermiliquid theory [1,2], are not applicable in 1D. Moreover, many 1D models are integrable, meaning that there exist exact solutions. Examples include the Lieb-Liniger model [3], the Heisenberg chain [4] or the 1D Fermi-Hubbard model (FHM) [5]. These models exhibit extensive sets of conserved quantities that prevent thermalization [6][7][8][9][10][11] and can, in lattice systems, lead to anomalous transport properties [12][13][14][15]. Coldatom experiments offer the possibility to study transport properties of strongly-correlated quantum gases in a clean environment. Their excellent controllability enabled far-from-equilibrium experiments [16][17][18][19][20] as well as close-to-equilibrium measurements in the linear-response regime [21][22][23][24] both in extended lattices and mesoscopic systems [25][26][27].Here, we investigate mass transport in the 1D FHM in far-from-equilibrium expansion experiments [18][19][20], where an initially trapped gas is suddenly released into a homogeneous potential landscape as illustrated in Fig. 1. There are two distinct regimes of interest in suddenexpansion studies: the asymptotic one, where the expanding gas has become dilute and effectively noninteracting [28][29][30][31][32][33][34][35][36] and the transient regime, where the dynamical quasi-condensation of hardcore bosons [37-41] and quantum distillation [20, 42-44] have been found. Quantum distillation occurs for large interactions. It relies on the dynamical demixing of fast singlons (one atom per site) and slow doublons (two atoms per site) during the expansion: while isolated doublons only move with a small effective second-order tunneling matrix element J eff = 2J 2 /U J for U J [46, 47], neighboring singlons and doublons can exchange their positions via fast, resonant first-order tunneling processes. Thus, after opening the trap, singlons escape from regions of the cloud initially occupied by singlons and doublons, leading (a) Initial state with doublons (b) Initial state without doublons x d U J x J J 10 20 30 40 50 Site index i 2 4 6 8 Time t (¿) 10 20 30 40 50 Site index i 0.0 0.4 0.8 1.2 1.6 n i ® FIG. 1. Schematics of the expansion experiment. Top: Initial state of the harmonically trapped two-component Fermi gas with (a) singlons (red) and doublons (blue) and (b) o...

We study the Holstein model of spinless fermions, which at half filling exhibits a quantum phase transition from a metallic Tomonaga-Luttinger liquid phase to an insulating charge-density-wave (CDW) phase at a critical electron-phonon coupling strength. In our work, we focus on the realtime evolution starting from two different types of initial states that are CDW ordered: (i) ideal CDW states with and without additional phonons in the system and (ii) correlated ground states in the CDW phase. We identify the mechanism for CDW melting in the ensuing real-time dynamics and show that it strongly depends on the type of initial state. We focus on the far-from-equilibrium regime and emphasize the role of electron-phonon coupling rather than dominant electronic correlations, thus complementing a previous study of photo induced CDW melting [H. Hashimoto and S. Ishihara, Phys. Rev. B 96, 035154 (2017)]. The numerical simulations are performed by means of matrix-product-state based methods with a local basis optimization (LBO). Within these techniques, one rotates the local (bosonic) Hilbert spaces adaptively into an optimized basis that can then be truncated while still maintaining a high precision. In this work, we extend the time-evolving block decimation (TEBD) algorithm with LBO, previously applied to single-polaron dynamics, to a half-filled system. We demonstrate that in some parameter regimes, a conventional TEBD method without LBO would fail. Furthermore, we introduce and use a ground-state densitymatrix renormalization group method for electron-phonon systems using local basis optimization.In our examples, we account for up to M ph = 40 bare phonons per site by working with O(10) optimal phonon modes.

Quantum lattice models with large local Hilbert spaces emerge across various fields in quantum many-body physics. Problems such as the interplay between fermions and phonons, the BCS-BEC crossover of interacting bosons, or decoherence in quantum simulators have been extensively studied both theoretically and experimentally. In recent years, tensor network methods have become one of the most successful tools to treat such lattice systems numerically. Nevertheless, systems with large local Hilbert spaces remain challenging. Here, we introduce a mapping that allows to construct artificial U(1)U(1) symmetries for any type of lattice model. Exploiting the generated symmetries, numerical expenses that are related to the local degrees of freedom decrease significantly. This allows for an efficient treatment of systems with large local dimensions. Further exploring this mapping, we reveal an intimate connection between the Schmidt values of the corresponding matrixproductstate representation and the singlesite reduced density matrix. Our findings motivate an intuitive physical picture of the truncations occurring in typical algorithms and we give bounds on the numerical complexity in comparison to standard methods that do not exploit such artificial symmetries. We demonstrate this new mapping, provide an implementation recipe for an existing code, and perform example calculations for the Holstein model at half filling. We studied systems with a very large number of lattice sites up to L=501L=501 while accounting for N_{\rm ph}=63 phonons per site with high precision in the CDW phase.