Using the transfer matrix method, we numerically investigate the structure of spatial coherences and their fluctuations in the three-dimensional (3D) Anderson model in the metallic phase when driven out of equilibrium by external leads at zero temperature and in linear response. We find that the stationary state entails nonlocal non-Gaussian correlations in the longitudinal direction, which are characteristic of diffusive nonequilibrium steady states. These correlations are quantitatively matched, at least up to third order, by those analytically derived in the quantum symmetric simple exclusion process (QSSEP) which describes diffusive fermions in 1D subject to dynamical disorder. Furthermore, the large deviation scaling and $U(1)$ invariance of these correlations imply a link between the Anderson model and free probability theory. Our findings suggest the existence of a universal structure of correlations in noninteracting diffusive quantum systems that might be captured by QSSEP.
Using the transfer matrix method, we numerically investigate the structure of spatial coherences and their fluctuations in the three-dimensional (3D) Anderson model in the metallic phase when driven out of equilibrium by external leads at zero temperature and in linear response. We find that the stationary state entails nonlocal non-Gaussian correlations in the longitudinal direction, which are characteristic of diffusive nonequilibrium steady states. These correlations are quantitatively matched, at least up to third order, by those analytically derived in the quantum symmetric simple exclusion process (QSSEP) which describes diffusive fermions in 1D subject to dynamical disorder. Furthermore, the large deviation scaling and U(1) invariance of these correlations imply a link between the Anderson model and free probability theory. Our findings suggest the existence of a universal structure of correlations in noninteracting diffusive quantum systems that might be captured by QSSEP.
Using the transfer matrix method, we numerically investigate the structure of spatial coherences and their fluctuations in the 3d Anderson model in the metallic phase when driven out-of-equilibrium by external leads at zero temperature and in linear response. We find that the stationary state entails non-local non-Gaussian correlations in the longitudinal direction, which are characteristic of diffusive non equilibrium steady states. These correlations are quantitatively matched, at least up to third order, by those analytically derived in the Quantum Symmetric Simple Exclusion Process (QSSEP) which describes diffusive fermions in 1d subject to dynamical disorder. Furthermore, the large deviation scaling and U(1) invariance of these correlations imply a link between the Anderson model and free probability theory. Our findings suggest the existence of a universal structure of correlations in non-interacting diffusive quantum systems that might be captured by QSSEP.
The motivation for this thesis is to find a fluctuating hydrodynamic description of quantum coherent effects in mesoscopic quantum systems with diffusive transport properties. Coherent effects are inscribed into the coherences (two-point Green's function) on which we focus as the building blocks of such a theory. Our approach is rather mathematical and not related to concrete experiments. We study the quantum symmetric simple exclusion process (QSSEP), a potentially iconic model describing transport of noisy free fermions on a 1D lattice, which has the minimal structure to capture what we are interested in: Long-ranged coherences and diffusive transport. In mean, QSSEP reduces to the symmetric simple exclusion process (SSEP), a classical toy model that was important in the development of the macroscopic fluctuation theory, a fluctuating hydrodynamic description of diffusive transport in classical systems. Studying QSSEP we hope to make progress towards a quantum coherent extension of the macroscopic fluctuation theory. The thesis summarizes many results we have obtained about QSSEP in the last three years, such as the dynamical equation and an exact stationary solution for correlation functions of coherences at hydrodynamic scales, the distribution of entanglement in QSSEP and an argument why QSSEP might be an effective noisy description for more generic mesoscopic quantum systems. Many of these results are due to a relation between the statistical properties of coherences in QSSEP and free probability theory. We devote a whole chapter to this relation and present, as a by-product, a method to characterize the spectrum of subblocks of a large class of structured random matrices.
We analyze the effects of various forms of noise on one-dimensional systems of noninteracting fermions. In the strong noise limit, we demonstrate, under mild assumptions, that the statistics of the fermionic correlation matrix in the thermodynamic limit follow a universal form described by the recently introduced quantum simple symmetric exclusion process (QSSEP). For charge transport, we show that QSSEP, along with all models in its universality class, shares the same large deviation function for the transferred charge as the classical SSEP model. The method we introduce to derive this result relies on a gaugelike invariance associated with the choice of the bond where the current is measured. This approach enables the explicit calculation of the cumulant-generating function for both QSSEP and SSEP and establishes an exact correspondence between them. These analytical findings are validated by extensive numerical simulations. Our results establish that a wide range of noisy free-fermionic models share the same QSSEP universality class and show that their transport properties are essentially classical.
We study the fluctuation statistics of integrated currents in noisy quantum diffusive systems, focusing on the Quantum Symmetric Simple Exclusion and Inclusion Processes (QSSEP/QSSIP). These one-dimensional fermionic (QSSEP) and bosonic (QSSIP) models feature stochastic nearest-neighbor hopping driven by Brownian noise, together with boundary injection and removal processes. They provide solvable microscopic settings in which quantum coherence coexists with diffusion. Upon noise averaging, their dynamics reduce to those of the classical SSEP/SSIP. We show that the cumulant-generating function of the integrated current, at large scales, obeys a large deviation principle. To leading order in system size and for each noise realization, it converges to that of the corresponding classical process, establishing a classical typicality of current fluctuations in these noisy quantum systems. We further demonstrate a direct connection with Macroscopic Fluctuation Theory (MFT), showing that the large-scale equations satisfied by biased quantum densities coincide with the steady-state Hamilton equations of MFT, thereby providing a microscopic quantum justification of the MFT framework in these models. Finally, we identify the leading finite-size corrections to the current statistics. We show the existence of subleading contributions of purely quantum origin, which are absent in the corresponding classical setting, and provide their explicit expressions for the second and third current cumulants. These quantum corrections are amenable to direct experimental or numerical verification, provided sufficient control over the noise realizations can be achieved. Their presence points toward the necessity of a quantum extension of MFT.
The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying nonequilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its nonequilibrium steady-state density matrix. We show that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.
We consider the boundary driven Quantum Symmetric Simple Inclusion Process (QSSIP) which describes a one-dimensional system of bosonic particles with stochastic nearest-neighbor hopping, modeled as a Brownian motion, with gain/loss processes at the endpoints of the chain driving the system out-of-equilibrium. Although the averaged QSSIP dynamics differs from that of the Quantum Symmetric Simple Exclusion Process (QSSEP)-the analogous system where bosons are replaced by fermions-we show that, paradoxically, the dynamics of their matrices of two-point functions, along with all their fluctuations, coincide. In contrary, the underlying classical models differs significantly, as the bosonic statistics allow the inclusion of multiple particles at the same site, in contrast to (symmetric) simple exclusion processes (SSEP). We provide an exact derivation of the large deviation function of density fluctuations in QSSIP and, as a consequence, in the classical inclusion process (SSIP) by exploiting its quantum formulation. Remarkably, our study highlights that, both in QSSEP and QSSIP, fluctuations of the local densities are typically classical, i.e., the cumulant generating functions of the local densities are asymptotically self-averaging and converge toward those of the classical SSEP and SSIP, realization-wise. This provides a test of the conjectured almost sure classical behavior of transport fluctuations, at leading order in the system size, in noisy diffusive quantum many-body systems.