arXiv subjects

Sylvain Prolhac

Publications and source records attributed to Sylvain Prolhac.

At least 19 recordsLinked to original sources

arXiv2026

Logarithmic singularity in a dynamical quantum phase transition for free fermions

We study the Loschmidt echo in a system of N non-interacting spinless lattice fermions released from a double-domain-wall initial state. In the large-N limit, the return probability is characterized by a large-deviation rate function known as the dynamical free energy. The Loschmidt echo is dominated by a complex instanton configuration, and the dynamical free energy develops a logarithmic singularity at the dynamical quantum phase transition. We obtain analytical results in the short-time and long-time regimes and support them with numerical computations. We show that the transition can be understood as a topological change of the dominant instanton configuration, analogous to the emergence of a cut in random matrix theory. We further show that post-selection can drive the system through two successive DQPTs, associated with successive topological changes of the dominant instanton configuration in complex time.

cond-mat.stat-mech↗

arXiv2026

Cheaper access to universal fluctuations in integrable spin chains from boundary effects

Observing super-diffusive fluctuations from Kardar-Parisi-Zhang (KPZ) universality in isotropic integrable spin chains is usually challenging as it requires a fairly large number of spins in interaction. We demonstrate in this paper, in the context of classical spins, that accounting for boundary effects lowers the bar, down to a few dozen spins in some cases. Additionally, boundaries control the relaxation to stationarity, which leads to many new universal scaling functions to explore, both in periodic spin chains and for open chains with magnetization imposed by reservoirs at the ends.

cond-mat.stat-mech↗

arXiv2025

Current fluctuations for the second class particle : joint statistics

We consider TASEP with a single second class particle and periodic boundary conditions. Using Bethe ansatz, we compute stationary large deviations for the joint statistics of the current of first and second class particles. At large scales, the generating function of the joint cumulants shows an unexpected connection to current fluctuations of TASEP with open boundaries.

cond-mat.stat-mech↗

arXiv2024

Comment on the paper "Exact decay of the persistence probability in the Airy$_1$ process" by Ferrari and Liu

We point out that the non-trivial function obtained by Ferrari and Liu for the persistence probability of the Airy$_1$ process has a strikingly similar form as a large deviation function found earlier by the author for current fluctuations of the totally asymmetric exclusion process with periodic boundaries conditioned on flat initial and final states. A proposed explanation for this observation, which relates similar yet clearly distinct quantities, is that both results pertain to conditioning on the same kind of rare events where a current larger than typical is maintained throughout the system.

cond-mat.stat-mech↗

arXiv2024

Approach to stationarity for the KPZ fixed point with boundaries

Current fluctuations for the one-dimensional totally asymmetric exclusion process (TASEP) connected to reservoirs of particles, and their large scale limit to the KPZ fixed point in finite volume, are studied using exact methods. Focusing on the maximal current phase for TASEP, corresponding to infinite boundary slopes for the KPZ height field, we obtain for general initial condition an exact expression for the late time correction to stationarity, involving extreme value statistics of Brownian paths. In the special cases of stationary and narrow wedge initial conditions, a combination of Bethe ansatz and numerical conjectures alternatively provide fully explicit exact expressions.

cond-mat.stat-mech↗

arXiv2024

KPZ fluctuations in finite volume

These lecture notes, adapted from the habilitation thesis of the author, survey in a first part various exact results obtained in the past few decades about KPZ fluctuations in one dimension, with a special focus on finite volume effects describing the relaxation to its stationary state of a finite system starting from a given initial condition. The second part is more specifically devoted to an approach allowing to express in a simple way the statistics of the current in the totally asymmetric simple exclusion process in terms of a contour integral on a compact Riemann surface, whose infinite genus limit leads to KPZ fluctuations in finite volume.

math.PR↗

arXiv2024

Probability of a single current

The Riemann surface associated with counting the current between two states of an underlying Markov process is hyperelliptic. We explore the consequences of this property for the time-dependent probability of that current for Markov processes with generic transition rates. When the system is prepared in its stationary state, the relevant meromorphic differential is in particular fully characterized by the precise identification of all its poles and zeroes.

math-ph↗

arXiv2022

Riemann surfaces for integer counting processes

Integer counting processes increment of an integer value at transitions between states of an underlying Markov process. The generator of a counting process, which depends on a parameter conjugate to the increments, defines a complex algebraic curve through its characteristic equation, and thus a compact Riemann surface. We show that the probability of a counting process can then be written as a contour integral on that Riemann surface. Several examples are discussed in details.

cond-mat.stat-mech↗

arXiv2021

From the Riemann surface of TASEP to ASEP

We consider the asymmetric simple exclusion process (ASEP) with forward hopping rate 1, backward hopping rate q and periodic boundary conditions. We show that the Bethe equations of ASEP can be decoupled, at all order in perturbation in the variable q, by introducing a formal Laurent series mapping the Bethe roots of the totally asymmetric case q=0 (TASEP) to the Bethe roots of ASEP. The probability of the height for ASEP is then written as a single contour integral on the Riemann surface on which symmetric functions of TASEP Bethe roots live.

cond-mat.stat-mech↗

arXiv2021

Riemann surface crossover for the spectral gaps of open TASEP

We consider the totally asymmetric simple exclusion process with open boundaries, at the edge of the maximal current phase. Using analytic continuations from the known stationary eigenvalue, we obtain exact expressions for the spectral gaps in the limit of large system size. The underlying Riemann surface,generated by modified Lambert functions, interpolates between the one for periodic TASEP and the one for open TASEP in the maximal current phase.

cond-mat.stat-mech↗

arXiv2020

Riemann surface for TASEP with periodic boundaries

The Bethe ansatz solution of periodic TASEP is formulated in terms of a ramified covering from a Riemann surface to the sphere. The joint probability distribution of height fluctuations at $n$ distinct times has in particular a relatively simple expression as a function of $n$ variables on the Riemann surface built from exponentials of Abelian integrals, traced over the ramified covering and integrated on $n$ nested contours in the complex plane.

cond-mat.stat-mech↗

arXiv2020

Spectral gaps of open TASEP in the maximal current phase

We study spectral gaps of the one-dimensional totally asymmetric simple exclusion process (TASEP) with open boundaries in the maximal current phase. Earlier results for the model with periodic boundaries suggest that the gaps contributing to the universal KPZ regime may be understood as points on an infinite genus Riemann surface built from a parametric representation of the cumulant generating function of the current. We perform explicit analytic continuations from the known large deviations of the current for open TASEP, and confirm the results for the gaps by an exact Bethe ansatz calculation, with additional checks using high precision extrapolation numerics.

cond-mat.stat-mech↗

arXiv2019

Riemann surfaces for KPZ with periodic boundaries

The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to one-dimensional KPZ universality in finite volume. Known exact results for fluctuations of the KPZ height with periodic boundaries are expressed in terms of meromorphic functions on this Riemann surface, summed over all the sheets of a covering map to an infinite cylinder. Connections to stationary large deviations, particle-hole excitations and KdV solitons are discussed.

cond-mat.stat-mech↗

arXiv2018

Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume

Height fluctuations are studied in the one-dimensional totally asymmetric simple exclusion process with periodic boundaries, with a focus on how late time relaxation towards the non-equilibrium steady state depends on the initial condition. Using a reformulation of the matrix product representation for the dominant eigenstate, the statistics of the height at large scales is expressed, for arbitrary initial conditions, in terms of extremal values of independent standard Brownian bridges. Comparison with earlier exact Bethe ansatz asymptotics leads to explicit conjectures for some conditional probabilities of non-intersecting Brownian bridges with exponentially distributed distances between the endpoints.

cond-mat.stat-mech↗

arXiv2018

Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions

We present a systematic short time expansion for the generating function of the one point height probability distribution for the KPZ equation with droplet initial condition, which goes much beyond previous studies. The expansion is checked against a numerical evaluation of the known exact Fredholm determinant expression. We also obtain the next order term for the Brownian initial condition. Although initially devised for short time, a resummation of the series allows to obtain also the \textit{long time large deviation function}, found to agree with previous works using completely different techniques. Unexpected similarities with stationary large deviations of TASEP with periodic and open boundaries are discussed. Two additional applications are given. (i) Our method is generalized to study the linear statistics of the {Airy point process}, i.e. of the GUE edge eigenvalues. We obtain the generating function of the cumulants of the empirical measure to a high order. The second cumulant is found to match the result in the bulk obtained from the Gaussian free field by Borodin and Ferrari, but we obtain systematic corrections to the Gaussian free field (higher cumulants, expansion towards the edge). This also extends a result of Basor and Widom to a much higher order. We obtain {large deviation functions} for the {Airy point process} for a variety of linear statistics test functions. (ii) We obtain results for the \textit{counting statistics of trapped fermions} at the edge of the Fermi gas in both the high and the low temperature limits.

cond-mat.stat-mech↗

arXiv2017

Perturbative solution for the spectral gap of the weakly asymmetric exclusion process

We consider the weakly asymmetric exclusion process with $N=L/2$ particles on a periodic lattice of $L$ sites, and hopping rates $1$ and $q=1-μ/\sqrt{L}$ respectively in the forward and in the backward direction. Using Bethe ansatz, we obtain a systematic perturbative expansion of the spectral gap near $μ=0$ by solving order by order a simple functional equation. A key point is that when $μ\to0$, Bethe roots at a distance $1/\sqrt{L}$ from the edge of the Fermi sea should not be considered as a continuum, but converge instead at large $L$ to the complex zeroes of $1+\mathrm{erf}(x)$ after a rescaling by $\sqrt{L}$.

math-ph↗

arXiv2017

Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $λH<0$

We study atypically large fluctuations of height $H$ in the 1+1-dimensional Kardar-Parisi-Zhang (KPZ) equation at long times $t$, when starting from a "droplet" initial condition. We derive exact large deviation function of height for $λH<0$, where $λ$ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy-Widom distribution tail at small $|H|/t$, which scales as $|H|^3/t$, to a different tail at large $|H|/t$, which scales as $|H|^{5/2}/t^{1/2}$. The latter tail exists at all times $t>0$. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at $|H|\sim t$ as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.

cond-mat.stat-mech↗

arXiv2017

Ground state energy of the $δ$-Bose and Fermi gas at weak coupling from double extrapolation

We consider the ground state energy of the Lieb-Liniger gas with $δ$ interaction in the weak coupling regime $γ\to0$. For bosons with repulsive interaction, previous studies gave the expansion $e_{\text{B}}(γ)\simeqγ-4γ^{3/2}/3π+(1/6-1/π^{2})γ^{2}$. Using a numerical solution of the Lieb-Liniger integral equation discretized with $M$ points and finite strength $γ$ of the interaction, we obtain very accurate numerics for the next orders after extrapolation on $M$ and $γ$. The coefficient of $γ^{5/2}$ in the expansion is found approximately equal to $-0.00158769986550594498929$, accurate within all digits shown. This value is supported by a numerical solution of the Bethe equations with $N$ particles followed by extrapolation on $N$ and $γ$. It was identified as $(3ζ(3)/8-1/2)/π^{3}$ by G. Lang. The next two coefficients are also guessed from numerics. For balanced spin $1/2$ fermions with attractive interaction, the best result so far for the ground state energy was $e_{\text{F}}(γ)\simeqπ^{2}/12-γ/2+γ^{2}/6$. An analogue double extrapolation scheme leads to the value $-ζ(3)/π^{4}$ for the coefficient of $γ^{3}$.

cond-mat.stat-mech↗