Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time

Abstract

We consider the system of $N$ one-dimensional free fermions confined by a harmonic well $V(x)=m{\omega }^2{x}^2/2$ at finite inverse temperature $\beta =1/T$. The average density of fermions ${\rho }_N(x,T)$ at position $x$ is derived. For $N\gg 1$ and $\beta \sim \mathcal{O}(1/N)$, ${\rho }_N(x,T)$ is given by a scaling function interpolating between a Gaussian at high temperature, for $\beta \ll 1/N$, and the Wigner semi-circle law at low temperature, for $\beta \gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $\beta \hslash \omega =b{N}^{-1/3}$, where the fluctuations close to the edge of the support, at $x\sim \pm \sqrt{2\hslash N/(m\omega )}$, are described by a limiting kernel $K_b^{\mathrm{ff}}(s,s^{\prime })$ that depends continuously on $b$ and is a generalization of the Airy kernel, found in the Gaussian Unitary Ensemble of random matrices. Remarkably, exactly the same kernel $K_b^{\mathrm{ff}}(s,s^{\prime })$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence $t=b^3$.

5 pages + 6 pages of supplementary material, 3 figures, published version

References in corpus (9)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The entanglement entropy of one-dimensional gases
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• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
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Cited by in corpus (50)

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• Inhomogeneous quenches in the transverse field Ising chain: scaling and front dynamics
• Noninteracting fermions in a trap and random matrix theory
• Ballistic front dynamics after joining two semi-infinite quantum Ising chains
• Exact extremal statistics in the classical $1d$ Coulomb gas
• Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions
• Coulomb-gas electrostatics controls large fluctuations of the KPZ equation
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• Wigner function of noninteracting trapped fermions
• Lower tail of the KPZ equation
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• Universal ground state properties of free fermions in a $d$-dimensional trap
• Extremes of $2d$ Coulomb gas: universal intermediate deviation regime
• Extreme statistics and index distribution in the classical $1d$ Coulomb gas
• The periodic Schur process and free fermions at finite temperature
• Simple derivation of the $(-\lambda H{)}^{5/2}$ tail for the 1D KPZ equation
• Determinantal structures in the O'Connell-Yor directed random polymer model
• Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
• Periodic Airy process and equilibrium dynamics of edge fermions in a trap
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• Statistics of fermions in a $d$-dimensional box near a hard wall
• Free fermions and the classical compact groups
• Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap
• Non-interacting fermions in hard-edge potentials
• Kinetic energy of a trapped Fermi gas at finite temperature
• Fluctuations of observables for free fermions in a harmonic trap at finite temperature
• Wigner function for noninteracting fermions in hard wall potentials
• Free fermions and $\alpha$-determinantal processes
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
• Impurities in systems of noninteracting trapped fermions
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Width and extremal height distributions of fluctuating interfaces with window boundary conditions
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• KPZ fluctuations in finite volume
• Incomplete determinantal processes: from random matrix to Poisson statistics
• Connected correlations in partitioning protocols: a case study and beyond
• Kernels for noninteracting fermions via a Green's function approach with applications to step potentials
• Large deviation bounds for the Airy point process
• General truncated linear statistics for the top eigenvalues of random matrices
• Nonequilibrium dynamics of noninteracting fermions in a trap
• Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Narain transform for spectral deformations of random matrix models
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"