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Denis Bernard

Publications and source records attributed to Denis Bernard.

At least 19 recordsLinked to original sources
arXiv2026

Nonlinear Fluctuating Hydrodynamics from Interacting Noisy Quantum Matter

A universal characterization of non-equilibrium steady states in interacting quantum many-body systems remains one of the central challenges of statistical physics. Here, we address this problem for a paradigmatic model of diffusive interacting quantum matter---the boundary-driven XXZ spin chain with bulk dephasing---and derive, directly from its microscopic Lindblad dynamics, an emergent classical Macroscopic Fluctuation Theory (MFT) governing its large-scale fluctuations. Crucially, the resulting hydrodynamics carries a density-dependent diffusivity and mobility as the fingerprint of interactions. This effective description enables the exact computation of the stationary density profile, long-range correlations, and the full counting statistics of the current, in excellent agreement with tensor-network simulations. Our work demonstrates that noisy quantum many-body systems can realize the universality class of genuinely interacting diffusive matter, beyond the constant-diffusivity class of the symmetric simple exclusion process, and establishes MFT as a powerful universal framework for interacting diffusive quantum systems.

cond-mat.stat-mech↗
arXiv2026

The Renormalization Group as a Stochastic Exploration Process

The Renormalization Group (RG) is a powerful and versatile framework for analyzing complex physical systems. Here, we reinterpret it as a stochastic process that explores physical phase spaces, scale by scale, progressively revealing finer details of small-scale structures. This perspective establishes a natural connection to other random exploration processes, such as the Schramm-Loewner evolution, and is more suited for a probabilist audience. It links RG concepts such as RG transformations, effective actions, etc, to usual probabilistic tools such as conditional expectation values, martingales, etc, but also makes contact with stochastic quantization.

math-ph↗
arXiv2026

Interacting Quantum Symmetric Exclusion Process

We introduce and solve the Interacting Quantum Symmetric Exclusion Process (IQSEP), a family of models describing the stochastic quantum hopping of charged particles along the edges of a lattice, with hopping amplitudes that depend on the occupations of neighbouring sites. In the absence of interactions, they reduce to the standard quantum simple symmetric exclusion process, exhibiting coherent diffusive transport. For interactions of order one, they capture incoherent diffusive transport and its fluctuations, characterized by density-dependent diffusivity and mobility, making contact with the macroscopic fluctuation theory. By rescaling the interaction strength appropriately with the lattice mesh, we define a mesoscopic scaling regime that retains a finite coherence length in the continuous thermodynamic limit. This regime interpolates between coherent behavior at small length scales and incoherent behavior at large scales. The resulting scaling theory accounts for fluctuations of quantum coherences in interacting diffusive systems, going beyond the scope of standard fluctuating hydrodynamics.

cond-mat.stat-mech↗
arXiv2026

Domain-wall melting in all-to-all QSSEP from random-matrix theory

We study the melting of a domain wall in the quantum simple exclusion process with all-to-all hoppings (a.k.a. the charged SYK$_2$ model). We show that the real-time dynamics of physical quantities of interest can be obtained exploiting spectral results in random matrix theory. We first show that the eigenvalues of the correlation matrix corresponding to the initially charged subsystem evolve according to a Jacobi process, which is defined in terms of a closed system of stochastic differential equations. In turn, this observation allows us to obtain the real-time dynamics of all the eigenvalue moments. We present two physical applications. First, we study the dynamics of the averaged von Neumann entanglement entropy, arriving at a fully explicit expression in the thermodynamic limit. Second, we compute analytically the full-counting statistics of the charge. Our formula allows us to perform a thorough comparison with the full-counting statistics of the classical simple exclusion process. Notably, we show that, in the thermodynamic limit, the quantum and classical full-counting statistics coincide, with no finite-time corrections.

cond-mat.stat-mech↗
arXiv2026

The Quantum Symmetric Simple Exclusion Process in the Continuum and Free Processes

The quantum symmetric simple exclusion process (QSSEP) is a recent extension of the symmetric simple exclusion process, designed to model quantum coherent fluctuating effects in noisy diffusive systems. It models stochastic nearest-neighbor fermionic hopping on a lattice, possibly driven out-of-equilibrium by boundary processes. We present a direct formulation in the continuum, and establish how this formulation captures the scaling limit of the discrete version. In the continuum, QSSEP emerges as a non-commutative process, driven by free increments, conditioned on the algebra of functions on the ambiant space to encode spatial correlations. We actually develop a more general framework dealing with conditioned orbits with free increments which may find applications beyond the present context. We view this construction as a preliminary step toward formulating a quantum extension of the macroscopic fluctuation theory.

math-ph↗
arXiv2026

Universal classical and quantum fluctuations in the large deviations of current of noisy quantum systems: The case of QSSEP and QSSIP

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 Macroscopic Fluctuation Theory.

cond-mat.stat-mech↗
arXiv2025

Introduction to quantum exclusion processes

The QSSEP, short for quantum symmetric simple exclusion process, is a paradigm model for stochastic quantum dynamics. Averaging over the noise, the quantum dynamics reduce to the well-studied SSEP (symmetric simple exclusion process). These notes provide an introduction to quantum exclusion processes, focusing on the example of QSSEP and its connection to free probability, with an emphasis on mathematical aspects.

math.PR↗
arXiv2025

Addition to "Structured random matrices and cyclic cumulants: A free probability approach"

We give a refined definition of the class of random matrix ensembles introduced in our paper "Structured random matrices and cyclic cumulants: A free probability approach" (arXiv:2309.14315) by extending the so-called fourth axiom to deal with cumulants of disjoint cycles. We argue that the theorems concerning the stability of such ensembles under non-linear transformations still hold with these refined axioms.

math.PR↗
arXiv2025

Large deviations of density fluctuations in the boundary driven Quantum Symmetric Simple Inclusion Process

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.

cond-mat.stat-mech↗
arXiv2024

Monitored fermions with conserved $\mathrm{U}(1)$ charge

We study measurement-induced phases of free fermion systems with U(1) symmetry. Following a recent approach developed for Majorana chains, we derive a field theory description for the purity and bipartite entanglement at large space and time scales. We focus on a multi-flavor one-dimensional chain with random complex hoppings and continuous monitoring of the local fermion density. By means of the replica trick, and using the number of flavors as a large parameter controlling our approximations, we derive an effective field theory made up of a SU(N) non-linear sigma model (NL$\sigma$M) coupled to fluctuating hydrodynamics. Contrary to the case of non-interacting Majorana fermions, displaying no U(1) symmetry, we find that the bipartite entanglement entropy satisfies an area law for all monitoring rates, but with a nontrivial scaling of entanglement when the correlation length is large. We provide numerical evidence supporting our claims. We briefly show how imposing a reality condition on the hoppings can change the NL$\sigma$M and also discuss higher dimensional generalizations.

cond-mat.stat-mech↗
arXiv2024

Symmetry classes of classical stochastic processes

We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions.

cond-mat.stat-mech↗
arXiv2023

Structured random matrices and cyclic cumulants: A free probability approach

We introduce a new class of large structured random matrices characterized by four fundamental properties which we discuss. We prove that this class is stable under matrix-valued and pointwise non-linear operations. We then formulate an efficient method, based on an extremization problem, for computing the spectrum of subblocks of such large structured random matrices. We present different proofs -- combinatorial or algebraic -- of the validity of this method, which all have some connection with free probability. We illustrate this method with well known examples of unstructured matrices, including Haar randomly rotated matrices, as well as with the example of structured random matrices arising in the quantum symmetric simple exclusion process. tured random matrices arising in the quantum symmetric simple exclusion process.

math.PR↗
arXiv2023

On the Angular Resolution of Pair-Conversion $\gamma$-Ray Telescopes

I present a study of the several contributions to the single-photon angular resolution of pair telescopes in the MeV energy range. I examine some test cases, the presently active {\sl Fermi} LAT, the ``pure-silicon'' projects ASTROGAM and AMEGO-X, and the emulsion-based project GRAINE.

astro-ph.IM↗
arXiv2023

Exact Entanglement in the Driven Quantum Symmetric Simple Exclusion Process

Entanglement properties of driven quantum systems can potentially differ from the equilibrium situation due to long range coherences. We confirm this observation by studying a suitable toy model for mesoscopic transport~: the open quantum symmetric simple exclusion process (QSSEP). We derive exact formulae for its mutual information between different subsystems in the steady state and show that it satisfies a volume law. Surprisingly, the QSSEP entanglement properties only depend on data related to its transport properties and we suspect that such a relation might hold for more general mesoscopic systems. Exploiting the free probability structure of QSSEP, we obtain these results by developing a new method to determine the eigenvalue spectrum of sub-blocks of random matrices from their so-called local free cumulants -- a mathematical result on its own with potential applications in the theory of random matrices. As an illustration of this method, we show how to compute expectation values of observables in systems satisfying the Eigenstate Thermalization Hypothesis (ETH) from the local free cumulants.

cond-mat.stat-mech↗
arXiv2023

Nonlinear sigma models for monitored dynamics of free fermions

We derive field theory descriptions for measurement-induced phase transitions in free fermion systems. We focus on a multi-flavor Majorana chain, undergoing Hamiltonian evolution with continuous monitoring of local fermion parity operators. Using the replica trick, we map the dynamics to the imaginary time evolution of an effective spin chain, and use the number of flavors as a large parameter for a controlled derivation of the effective field theory. This is a nonlinear sigma model for an orthogonal $N\times N$ matrix, in the replica limit $N\to 1$. (On a boundary of the phase diagram, another sigma model with higher symmetry applies.) Together with known results for the renormalization-group beta function, this derivation establishes the existence of stable phases -- nontrivially entangled and disentangled respectively -- in the physically-relevant replica limit $N\to 1$. In the nontrivial phase, an asymptotically exact calculation shows that the bipartite entanglement entropy for a system of size $L$ scales as $(\log L)^2$, in contrast to findings in previously-studied models. Varying the relative strength of Hamiltonian evolution and monitoring, as well as a dimerization parameter, the model's phase diagram contains transitions out of the nontrivial phase, which we map to vortex-unbinding transitions in the sigma model, and also contains separate critical points on the measurement-only axis. We highlight the close analogies as well as the differences with the replica approach to Anderson transitions in disordered systems.

cond-mat.stat-mech↗
arXiv2023

Spacetime picture for entanglement generation in noisy fermion chains

Studies of random unitary circuits have shown that the calculation of Renyi entropies of entanglement can be mapped to classical statistical mechanics problems in spacetime. In this paper, we develop an analogous spacetime picture of entanglement generation for random free or weakly interacting fermion systems without conservation laws. We first study a free-fermion model, namely a 1D chain of Majorana modes with nearest neighbour hoppings, random in both space and time. We analyze the Nth Renyi entropy of entanglement using a replica formalism, and we show that the effective model is equivalent to an SO(2N) Heisenberg spin chain evolving in imaginary time. By applying a saddle-point approximation to the coherent states path integral for the N = 2 case, we arrive at a semiclassical picture for the dynamics of the entanglement purity, in terms of two classical fields in spacetime. The classical solutions involve a smooth domain wall that interpolates between two values, with this domain wall relaxing diffusively in the time direction. We then study how adding weak interactions to the free-fermion model modifies this spacetime picture, reflecting a crossover from diffusive to ballistic spreading of information.

cond-mat.stat-mech↗
arXiv2022

Bernoulli variables, classical exclusion processes and free probability

We present a new description of the known large deviation function of the classical symmetric simple exclusion process by exploiting its connection with the quantum symmetric simple exclusion processes and using tools from free probability. This may seem paradoxal as free probability usually deals with non commutative probability while the simple exclusion process belongs to the realm of classical probability. On the way, we give a new formula for the free energy -- alias the logarithm of the Laplace transform of the probability distribution -- of correlated Bernoulli variables in terms of the set of their cumulants with non-coinciding indices. This latter result is obtained either by developing a combinatorial approach for cumulants of products of random variables or by borrowing techniques from Feynman graphs.

math-ph↗