Non-interacting fermions in hard-edge potentials

Abstract

We consider the spatial quantum and thermal fluctuations of non-interacting Fermi gases of $N$ particles confined in $d$-dimensional non-smooth potentials. We first present a thorough study of the spherically symmetric pure hard-box potential, with vanishing potential inside the box, both at $T=0$ and $T>0$. We find that the correlations near the wall are described by a "hard edge" kernel, which depend both on $d$ and $T$, and which is different from the "soft edge" Airy kernel, and its higher $d$ generalizations, found for smooth potentials. We extend these results to the case where the potential is non-uniform inside the box, and find that there exists a family of kernels which interpolate between the above "hard edge" kernel and the "soft edge" kernels. Finally, we consider one-dimensional singular potentials of the form $V(x)\sim |x{|}^{-\gamma }$ with $\gamma >0$. We show that the correlations close to the singularity at $x=0$ are described by this "hard edge" kernel for $1\le \gamma <2$ while they are described by a broader family of "hard edge" kernels known as the Bessel kernel for $\gamma =2$ and, finally by the Airy kernel for $\gamma >2$. These one-dimensional kernels also appear in random matrix theory, and we provide here the mapping between the $1d$ fermion models and the corresponding random matrix ensembles. Part of these results were announced in a recent Letter, EPL 120, 10006 (2017).

arXiv comment56 pages, 12 figures

References in corpus (44)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Bose-Einstein condensation of atoms in a uniform potential
• Determinantal Processes and Independence
• Matrix Models for Beta Ensembles
• Single-atom imaging of fermions in a quantum-gas microscope
• A Quantum Gas Microscope for Fermionic Atoms
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• Fredholm Determinants, Differential Equations and Matrix Models
• Site-resolved Imaging of Fermionic Lithium-6 in an Optical Lattice
• Level-Spacing Distributions and the Bessel Kernel
• Growing interfaces uncover universal fluctuations behind scale invariance
• Homogeneous Atomic Fermi Gases
• Edge Electron Gas
• The entanglement entropy of one-dimensional gases
• Two-Dimensional Homogeneous Fermi Gases
• A generalized ensemble of random matrices
• Universality in the full counting statistics of trapped fermions
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time
• Random matrices and entanglement entropy of trapped Fermi gases
• Exact solution of the Schrödinger equation for the inverse square root potential $V_0/\sqrt{x}$
• Non-intersecting Brownian Interfaces and Wishart Random Matrices
• Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
• Quantum Joule-Thomson Effect in a Saturated Homogeneous Bose Gas
• Wigner function of noninteracting trapped fermions
• Universal ground state properties of free fermions in a $d$-dimensional trap
• Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable Bosons
• Fermi distribution of semicalssical non-eqilibrium Fermi states
• Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity
• Random matrix ensembles with an effective extensive external charge
• Chiral Random Matrix Model for Critical Statistics
• Universal Parametric correlations at the soft edge of the spectrum of random matrix ensembles
• Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles
• On certain functionals of the maximum of Brownian motion and their applications
• 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 fermions in a $d$-dimensional box near a hard wall
• Free fermions and the classical compact groups
• Kinetic energy of a trapped Fermi gas at finite temperature
• Fluctuations of observables for free fermions in a harmonic trap at finite temperature
• On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators
• On the spatial persistence for Airy processes

Cited by in corpus (11)

• Extreme value statistics of correlated random variables: a pedagogical review
• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• The Elliptic Ginibre Ensemble: A Unifying Approach to Local and Global Statistics for Higher Dimensions
• Wigner function for noninteracting fermions in hard wall potentials
• Stampedes I: Fishnet OPE and Octagon Bootstrap with Nonzero Bridges
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
• Quench dynamics of noninteracting fermions with a delta impurity
• High-momentum oscillating tails of strongly interacting 1D gases in a box
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution