arXiv:1811.09427 · SciPost Physics

Equilibrium fluctuations in maximally noisy extended quantum systems

Michel Bauer, Denis Bernard, Tony Jin

Published Apr 16, 2019• 57 citations

Abstract

We introduce and study a class of models of free fermions hopping between neighbouring sites with random Brownian amplitudes. These simple models describe stochastic, diffusive, quantum, unitary dynamics. We focus on periodic boundary conditions and derive the complete stationary distribution of the system. It is proven that the generating function of the latter is provided by an integral with respect to the unitary Haar measure, known as the Harish-Chandra-Itzykson-Zuber integral in random matrix theory, which allows us to access all fluctuations of the system state. The steady state is characterized by non trivial correlations which have a topological nature. Diagrammatic tools appropriate for the study of these correlations are presented. In the thermodynamic large system size limit, the system approaches a non random, self averaging, equilibrium state plus occupancy and coherence fluctuations of magnitude scaling proportionally with the inverse of the square root of the volume. The large deviation function for those fluctuations is determined. Although decoherence is effective on the mean steady state, we observe that sub-leading fluctuating coherences are dynamically produced from the inhomogeneities of the initial occupancy profile.

DOI: 10.21468/scipostphys.6.4.045

Related papers

Open Quantum Symmetric Simple Exclusion Process

Denis Bernard, Tony Jin

We present the solution to a model of fermions hopping between neighboring sites on a line with random Brownian amplitudes and open boundary conditions driving the system out of equilibrium. The average dynamics reduces to that of the symmetric simple exclusion process. However, the full distribution encodes for a richer behavior, entailing fluctuating quantum coherences which survive in the steady limit. We determine exactly the steady statistical distribution of the system state. We show that the out-of-equilibrium quantum coherence fluctuations satisfy a large-deviation principle, and we present a method to recursively compute exactly the large-deviation function. As a by-product, our approach gives a solution of the classical symmetric simple exclusion process based on fermion technology. Our results open the route towards the extension of the macroscopic fluctuation theory to many-body quantum systems.

arXiv:1904.01406Physical Review LettersAug 21, 201980 citations

Universal classical and quantum fluctuations in the large deviations of current of noisy quantum systems: Quantum symmetric simple exclusion and inclusion processes

Mathias Albert, Denis Bernard, Tony Jin +2 more

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.

cond-mat.stat-mechCross-listedarXiv:2601.16883Jan 26, 2026

Dynamics of fluctuations in quantum simple exclusion processes

Denis Bernard, Fabian H. L. Eßler, Ludwig Hruza +1 more

We consider the dynamics of fluctuations in the quantum asymmetric simple exclusion process (Q-ASEP) with periodic boundary conditions. The Q-ASEP describes a chain of spinless fermions with random hoppings that are induced by a Markovian environment. We show that fluctuations of the fermionic degrees of freedom obey evolution equations of Lindblad type, and derive the corresponding Lindbladians. We identify the underlying algebraic structure by mapping them to non-Hermitian spin chains and demonstrate that the operator space fragments into exponentially many (in system size) sectors that are invariant under time evolution. At the level of quadratic fluctuations we consider the Lindbladian on the sectors that determine the late time dynamics for the particular case of the quantum symmetric simple exclusion process (Q-SSEP). We show that the corresponding blocks in some cases correspond to known Yang-Baxter integrable models and investigate the level-spacing statistics in others. We carry out a detailed analysis of the steady states and slow modes that govern the late time behaviour and show that the dynamics of fluctuations of observables is described in terms of closed sets of coupled linear differential-difference equations. The behaviour of the solutions to these equations is essentially diffusive but with relevant deviations, that at sufficiently late times and large distances can be described in terms of a continuum scaling limit which we construct. We numerically check the validity of this scaling limit over a significant range of time and space scales. These results are then applied to the study of operator spreading at large scales, focusing on out-of-time ordered correlators and operator entanglement.

arXiv:2107.02662SciPost PhysicsJan 28, 202234 citations

Typicality of entanglement growth and transport fluctuations in quantum symmetric exclusion processes

Ewan McCulloch

We use macroscopic fluctuation theory (MFT) to establish the typicality of charge-transfer statistics and entanglement dynamics in the quantum symmetric exclusion process, a model of stochastically hopping free fermions with diffusive transport and long-range quantum coherence. We show that the leading long-time charge-transfer full counting statistics of individual noise realizations coincide almost surely with those of the classical symmetric exclusion process (SSEP), in equilibrium and following quenches from arbitrary occupation-diagonal product states. Our results apply to transient fluctuations in the thermodynamic limit, extending previous work that has primarily focused on stationary regimes. Our approach compares the noise averages of the charge-transfer generating function and its inverse. The former reduces to SSEP, while the latter maps to an auxiliary Brownian energy process whose MFT saddle-point problem is related to that of SSEP, yielding matching large-deviation exponents. Through free-fermion identities relating entanglement to subsystem-number fluctuations, this establishes the self-averaging of the covariance-matrix eigenvalue distribution and of Rényi and von Neumann entropies. We obtain asymptotically exact predictions for entanglement growth from domain-wall and Néel initial states, in excellent agreement with numerical simulations.

cond-mat.stat-mechCross-listedarXiv:2610.36048Oct 28, 20260 citations

Extension of quantum symmetric simple exclusion process model for entanglement spreading in stochastic diffusive quantum systems

Vincenzo Alba

We investigate out-of-equilibrium entanglement dynamics in a generalization of the so-called Q S S E P model, which is a free-fermion chain with stochastic in space and time hopping amplitudes. In our setup, the noisy amplitudes are spatially modulated satisfying a ν -site translation invariance but retaining their randomness in time. For each noise realization, the dynamics preserves Gaussianity, which allows to obtain noise-averaged entanglement-related quantities. The statistics of the steady-state correlators satisfy nontrivial relationships that are of topological nature. They reflect the Haar invariance under multiplication with structured momentum-dependent random S U ( ν ) matrices. We discuss in detail the case with ν = 1 and ν = 2 . For ν = 1 , i.e., spatially homogeneous noise, we show that the entanglement dynamics is describable by a stochastic generalization of the quasiparticle picture. Precisely, entanglement is propagated by pairs of quasiparticles. The entanglement content of the pairs is the same as for the deterministic chain. However, the trajectories of the quasiparticles are random walks, giving rise to diffusive entanglement growth.

arXiv:2507.11674Physical review. B./Physical review. BJan 22, 20264 citations

Fluctuation of coherences in noisy mesoscopic quantum systems with diffusive transport

Ludwig Hruza

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.

arXiv:2408.03917Aug 7, 20241 citation