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 (LDA) predicts that $W_N(\mathbf{x},\mathbf{p})\approx 1/(2\pi \hslash {)}^d$ inside a finite region of the $(\mathbf{x},\mathbf{p})$ plane, namely for $|\mathbf{x}|<R$ and $|\mathbf{p}|<k_F$ 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 ${\hat{\rho }}_N(p)$ and find a striking algebraic tail for very large momenta ${\hat{\rho }}_N(p)\propto 1/p^4$, 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~$d$.

arXiv comment31 pages, 14 figures

References in corpus (37)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Negativity of the Wigner function as an indicator of nonclassicality
• Energetics of a strongly correlated Fermi gas
• A Quantum Gas Microscope for Fermionic Atoms
• Site-resolved Imaging of Fermionic Lithium-6 in an Optical Lattice
• Homogeneous Atomic Fermi Gases
• Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases
• Edge Electron Gas
• The entanglement entropy of one-dimensional gases
• Tan relations in one dimension
• Universality in the full counting statistics of trapped fermions
• High-momentum tail in the Tonks gas under harmonic confinement
• 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
• Wigner function approach to single electron coherence in quantum Hall edge channels
• Weak-to-strong transition of quantum measurement in a trapped-ion system
• Observation of Pauli Crystals
• Breakdown of Tan's relation in lossy one-dimensional Bose gases
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Wigner function of noninteracting trapped fermions
• Momentum distribution of a trapped Fermi gas with large scattering length
• Universal ground state properties of free fermions in a $d$-dimensional trap
• Synchrotron radiation representation in phase space
• Fermi distribution of semicalssical non-eqilibrium Fermi states
• Free fermions at the edge of interacting systems
• Wigner quasi-probability distribution for the infinite square well: energy eigenstates and time-dependent wave packets
• Quantum quench and thermalization of one-dimensional Fermi gas via phase space hydrodynamics
• Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes
• Statistics of fermions in a $d$-dimensional box near a hard wall
• Non-interacting fermions in hard-edge potentials
• Universal edge scaling in random partitions
• Phase space tomography of cold-atoms dynamics in a weakly corrugated potential
• Classical phase-space approach for coherent matter waves
• Kernels for noninteracting fermions via a Green's function approach with applications to step potentials

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