Showing papers by "Pierre Le Doussal published in 2021"

Counting statistics for noninteracting fermions in a d-dimensional potential.

Abstract: We develop a first-principles approach to compute the counting statistics in the ground state of $N$ noninteracting spinless fermions in a general potential in arbitrary dimensions $d$ (central for $dg1$). In a confining potential, the Fermi gas is supported over a bounded domain. In $d=1$, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions ${\mathcal{N}}_{\mathcal{D}}$ in a domain $\mathcal{D}$ of macroscopic size in the bulk of the support. We show that the variance of ${\mathcal{N}}_{\mathcal{D}}$ grows as ${N}^{(d\ensuremath{-}1)/d}({A}_{d}logN+{B}_{d})$ for large $N$, and obtain the explicit dependence of ${A}_{d},{B}_{d}$ on the potential and on the size of $\mathcal{D}$ (for a spherical domain in $dg1$). This generalizes the free-fermion results for microscopic domains, given in $d=1$ by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem $\mathcal{D}$, in any dimension, supported by exact results for $d=1$.

Inverse Scattering of the Zakharov-Shabat System Solves the Weak Noise Theory of the Kardar-Parisi-Zhang Equation.

Abstract: We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic, and integrable techniques We expand the program of the weak noise theory, which maps the large deviations onto a nonlinear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions

Condensation transition in the late-time position of a run-and-tumble particle.

Abstract: We study the position distribution $P(\stackrel{P\vec}{R},N)$ of a run-and-tumble particle (RTP) in arbitrary dimension $d$, after $N$ runs. We assume that the constant speed $vg0$ of the particle during each running phase is independently drawn from a probability distribution $W(v)$ and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, $P(\stackrel{P\vec}{R},N)\ensuremath{\rightarrow}P(R,N)$ where $R=|\stackrel{P\vec}{R}|$. We show that, under certain conditions on $d$ and $W(v)$ and for large $N$, a condensation transition occurs at some critical value of $R={R}_{c}\ensuremath{\sim}O(N)$ located in the large-deviation regime of $P(R,N)$. For $Rl{R}_{c}$ (subcritical fluid phase), all runs are roughly of the same size in a typical trajectory. In contrast, an RTP trajectory with $Rg{R}_{c}$ is typically dominated by a ``condensate,'' i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions $W(v)=\ensuremath{\alpha}{(1\ensuremath{-}v/{v}_{0})}^{\ensuremath{\alpha}\ensuremath{-}1}/{v}_{0}$, parametrized by $\ensuremath{\alpha}g0$, we show that, for large $N, P(R,N)\ensuremath{\sim}exp[\ensuremath{-}N{\ensuremath{\psi}}_{d,\ensuremath{\alpha}}(R/N)]$, and we compute exactly the rate function ${\ensuremath{\psi}}_{d,\ensuremath{\alpha}}(z)$ for any $d$ and $\ensuremath{\alpha}$. We show that the transition manifests itself as a singularity of this rate function at $R={R}_{c}$ and that its order depends continuously on $d$ and $\ensuremath{\alpha}$. We also compute the distribution of the condensate size for $Rg{R}_{c}$. Finally, we study the model when the total duration $T$ of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision $\ensuremath{\sim}{10}^{\ensuremath{-}100}$.

Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics, and spiked random matrices: Pinning and localization.

Abstract: We revisit the problem of an elastic line (such as a vortex line in a superconductor) subject to both columnar disorder and point disorder in dimension $d=1+1$. Upon applying a transverse field, a delocalization transition is expected, beyond which the line is tilted macroscopically. We investigate this transition in the fixed tilt angle ensemble and within a ``one-way'' model where backward jumps are neglected. From recent results about directed polymers in the mathematics literature, and their connections to random matrix theory, we find that for a single line and a single strong defect this transition in the presence of point disorder coincides with the Baik--Ben Arous--P\'ech\'e (BBP) transition for the appearance of outliers in the spectrum of a perturbed random matrix in the Gaussian unitary ensemble. This transition is conveniently described in the polymer picture by a variational calculation. In the delocalized phase, the ground state energy exhibits Tracy-Widom fluctuations. In the localized phase we show, using the variational calculation, that the fluctuations of the occupation length along the columnar defect are described by ${f}_{\mathrm{KPZ}}$, a distribution which appears ubiquitously in the Kardar-Parisi-Zhang universality class. We then consider a smooth density of columnar defect energies. Depending on how this density vanishes at its lower edge we find either (i) a delocalized phase only or (ii) a localized phase with a delocalization transition. We analyze this transition which is an infinite-rank extension of the BBP transition. The fluctuations of the ground state energy of a single elastic line in the localized phase (for fixed columnar defect energies) are described by a Fredholm determinant based on a new kernel, closely related to the kernel describing the largest real eigenvalues of the real Ginibre ensemble. The case of many columns and many nonintersecting lines, relevant for the study of the Bose glass phase, is also analyzed. The ground state energy is obtained using free probability and the Burgers equation. Connections with recent results on the generalized Rosenzweig-Porter model suggest that the localization of many polymers occurs gradually upon increasing their lengths.

Spatial Clustering of Depinning Avalanches in Presence of Long-Range Interactions.

Abstract: Disordered elastic interfaces display avalanche dynamics at the depinning transition. For short-range interactions, avalanches correspond to compact reorganizations of the interface well described by the depinning theory. For long-range elasticity, an avalanche is a collection of spatially disconnected clusters. In this Letter we determine the scaling properties of the clusters and relate them to the roughness exponent of the interface. The key observation of our analysis is the identification of a Bienayme-Galton-Watson process describing the statistics of the number of clusters. Our work has concrete importance for experimental applications where the cluster statistics is a key probe of avalanche dynamics.

Full counting statistics for interacting trapped fermions

Abstract: We study $N$ spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index $\beta$. In the fermion model $\beta$ controls the strength of the interaction, $\beta=2$ corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the Fermi gas. We predict that for general $\beta$ the variance of ${\cal N}_{\cal D}$ grows as $A_{\beta} \log N + B_{\beta}$ for $N \gg 1$ and we obtain a formula for $A_\beta$ and $B_\beta$. This is based on an explicit calculation for $\beta\in\left\{ 1,2,4\right\} $ and on a conjecture that we formulate for general $\beta$. This conjecture further allows us to obtain a formula for the higher cumulants of ${\cal N}_{\cal D}$. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter $K = 2/\beta$, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter $\beta$. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

Spherical Spin Glass Model with External Field

Abstract: We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.

Impurities in systems of noninteracting trapped fermions

Abstract: We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Peche (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.

Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions

Abstract: We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The non-linear hydrodynamic equations of the WNT are solved analytically through a connexion to the Zakharov-Shabat (ZS) system using its classical integrability. This approach is based on a recently developed Fredholm determinant framework previously applied to the droplet IC. The flat IC provides the case for a non-vanishing boundary condition of the ZS system and yields a richer solitonic structure comprising the appearance of multiple branches of the Lambert function. As a byproduct, we obtain the explicit solution of the WNT for the Brownian IC, which undergoes a dynamical phase transition. We elucidate its mechanism by showing that the related spontaneous breaking of the spatial symmetry arises from the interplay between two solitons with different rapidities.

Thermal Buckling Transition of Crystalline Membranes in a Field

Abstract: Two-dimensional crystalline membranes in isotropic embedding space exhibit a flat phase with anomalous elasticity, relevant, e.g., for graphene. Here we study their thermal fluctuations in the absence of exact rotational invariance in the embedding space. An example is provided by a membrane in an orientational field, tuned to a critical buckling point by application of in-plane stresses. Through a detailed analysis, we show that the transition is in a new universality class. The self-consistent screening method predicts a second-order transition, with modified anomalous elasticity exponents at criticality, while the RG suggests a weakly first-order transition.

Kernels for non interacting fermions via a Green’s function approach with applications to step potentials

Abstract: The quantum correlations of $N$ noninteracting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general trapping potential in terms of the Green's function for the corresponding single particle Schr\"odinger equation. For smooth potentials the method allows a simple alternative derivation of the local density approximation for the density and of the sine kernel in the bulk part of the trap in the large $N$ limit. It also recovers the density and the kernel of the so-called {\em Airy gas} at the edge. This method allows to analyse the quantum correlations in the ground state when the potential has a singular part with a fast variation in space. For the square step barrier of height $V_0$, we derive explicit expressions for the density and for the kernel. For large Fermi energy $\mu>V_0$ it describes the interpolation between two regions of different densities in a Fermi gas, each described by a different sine kernel. Of particular interest is the {\em critical point} of the square well potential when $\mu=V_0$. In this critical case, while there is a macroscopic number of fermions in the lower part of the step potential, there is only a finite $O(1)$ number of fermions on the shoulder, and moreover this number is independent of $\mu$. In particular, the density exhibits an algebraic decay $\sim 1/x^2$, where $x$ is the distance from the jump. Furthermore, we show that the critical behaviour around $\mu = V_0$ exhibits universality with respect with the shape of the barrier. This is established (i) by an exact solution for a smooth barrier (the Woods-Saxon potential) and (ii) by establishing a general relation between the large distance behavior of the kernel and the scattering amplitudes of the single-particle wave-function.

Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall.

Abstract: We present an exact solution for the height distribution of the KPZ equation at any time $t$ in a half space with flat initial condition. This is equivalent to obtaining the free-energy distribution of a polymer of length $t$ pinned at a wall at a single point. In the large $t$ limit a binding transition takes place upon increasing the attractiveness of the wall. Around the critical point we find the same statistics as in the Baik-Ben--Arous-P\'ech\'e transition for outlier eigenvalues in random matrix theory. In the bound phase, we obtain the exact measure for the endpoint and the midpoint of the polymer at large time. We also unveil curious identities in distribution between partition functions in half-space and certain partition functions in full space for Brownian-type initial condition.

Wigner function for noninteracting fermions in hard-wall potentials

Abstract: The Wigner function ${W}_{N}(\mathbf{x},\mathbf{p})$ is a useful quantity to characterize the quantum fluctuations of an $N$-body system in its phase space. Here we study ${W}_{N}(\mathbf{x},\mathbf{p})$ for $N$ noninteracting spinless fermions in a $d$-dimensional spherical hard box of radius $R$ at temperature $T=0$. In the large-$N$ limit, the local-density approximation predicts that ${W}_{N}(\mathbf{x},\mathbf{p})\ensuremath{\approx}1/{(2\ensuremath{\pi}\ensuremath{\hbar})}^{d}$ inside a finite region of the $(\mathbf{x},\mathbf{p})$ plane, namely, for $|\mathbf{x}|lR$ and $|\mathbf{p}|l{k}_{F}$, where ${k}_{F}$ is the Fermi momentum, while ${W}_{N}(\mathbf{x},\mathbf{p})$ vanishes outside this region, or droplet, on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)$ and find a striking algebraic tail for very large momenta ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)\ensuremath{\propto}1/{p}^{4}$, well beyond ${k}_{F}$, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher $d$ and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension $d$.

Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential

Abstract: The random variable $1+z_1+z_1z_2+\dots$ appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to $N \times N$ matrices either real symmetric $\beta=1$ or complex Hermitian $\beta=2$. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to $N$ fermions in a Morse potential, which are non-interacting for $\beta=2$. At finite $N$ the distribution of eigenvalues exhibits heavy tails, generalizing Kesten's results in the scalar case. The density of fermions in this potential is studied for large $N$, and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large $N$ limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valko, Grabsch and Texier, as well as Ossipov, is discussed.

Steady state of the KPZ equation on an interval and Liouville quantum mechanics

Abstract: We obtain a simple formula for the stationary measure of the height field evolving according to the Kardar-Parisi-Zhang equation on the interval $[0,L]$ with general Neumann type boundary conditions and any interval size. This is achieved using the recent results of Corwin and Knizel (arXiv:2103.12253) together with Liouville quantum mechanics. Our formula allows to easily determine the stationary measure in various limits: KPZ fixed point on an interval, half-line KPZ equation, KPZ fixed point on a half-line, as well as the Edwards-Wilkinson equation on an interval.

Matrix Kesten recursion, inverse-Wishart ensemble and fermions in a Morse potential

Abstract: The random variable 1 + z1 + z1z2 +. .. appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to N × N matrices either real symmetric β = 1 or complex Hermitian β = 2. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to N fermions in a Morse potential, which are non-interacting for β = 2. At finite N the distribution of eigenvalues exhibits heavy tails, generalizing Kesten's results in the scalar case. The density of fermions in this potential is studied for large N , and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large N limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valko, Grabsch and Texier, as well as Ossipov, is discussed.

Hole probability for noninteracting fermions in a $d$-dimensional trap

Abstract: The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability $P(R)$ for a spherical region of radius $R$ in the case of $N$ noninteracting fermions in their ground state in a $d$-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large $N$ and in the bulk of the Fermi gas, $P(R)$ is described by a universal scaling function of $k_F R$, for which we obtain an exact formula ($k_F$ being the local Fermi wave-vector). It exhibits a super exponential tail $P(R)\propto e^{- \kappa_d (k_F R)^{d+1}}$ where $\kappa_d$ is a universal amplitude, in good agreement with existing numerical simulations. When $R$ is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.

Ranked diffusion, delta Bose gas and Burgers equation

Abstract: We study the diffusion of $N$ particles in one dimension interacting via a drift proportional to their rank. In the attractive case (self-gravitating gas) a mapping to the Lieb Liniger quantum model allows to obtain stationary time correlations, return probabilities and the decay rate to the stationary state. The rank field obeys a Burgers equation, which we analyze. It allows to obtain the stationary density at large $N$ in an external potential $V(x)$ (in the repulsive case). In the attractive case the decay rate to the steady state is found to depend on the initial condition if its spatial decay is slow enough. Coulomb gas methods allow to study the final equilibrium at large $N$.

Thermal buckling transition of crystalline membranes

Abstract: Two dimensional crystalline membranes in isotropic embedding space exhibit a flat phase with anomalous elasticity, relevant e.g., for graphene. Here we study their thermal fluctuations in the absence of exact rotational invariance in the embedding space. An example is provided by a membrane in an orientational field, tuned to a critical buckling point by application of in-plane stresses. Through a detailed analysis, we show that the transition is in a new universality class. The self-consistent screening method predicts a second order transition, with modified anomalous elasticity exponents at criticality, while the RG suggests a weakly first order transition.