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arXiv:1810.12583 · Journal of Physics A Mathematical and Theoretical

Noninteracting fermions in a trap and random matrix theory

David S. Dean, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr

Published Feb 22, 2019• 84 citations
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Abstract

Abstract We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions , the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the ‘Airy kernel’) and hard edge (described by the ‘Bessel kernel’) universality classes. We further discuss extensions of these results to higher dimensions and to finite temperature. Finally, we discuss correlations in the phase space, i.e. in the space of positions and momenta, characterized by the so called Wigner function.

DOI
10.1088/1751-8121/ab098d

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Abstract We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions , the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the ‘Airy kernel’) and hard edge (described by the ‘Bessel kernel’) universality classes. We further discuss extensions of these results to higher dimensions and to finite temperature. Finally, we discuss correlations in the phase space, i.e. in the space of positions and momenta, characterized by the so called Wigner function.
We study a system of NNN noninteracting spinless fermions trapped in a confining potential, in arbitrary dimensions ddd and arbitrary temperature TTT. The presence of the confining trap breaks the translational invariance and introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so-called ``bulk regime''), where the fermions do not feel the curvature of the trap, physical properties of the fermions have traditionally been understood using the local density (or Thomas-Fermi) approximation. However, these approximations drastically fail near the edge where the density vanishes and thermal and quantum fluctuations are thus enhanced. The main goal of this paper is to show that, even near the edge, novel universal properties emerge, independently of the details of the shape of the confining potential. We present a unified framework to investigate both the bulk and the edge properties of the fermions. We show that for large NNN, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any nnn-point correlation function, including the average density profile, can be expressed as an $n\ifmmode\times\else\texttimes\fi{}n$ determinant whose entry is called the kernel, a central object for such processes. Near the edge, we derive the large-NNN scaling form of the kernels, parametrized by ddd and TTT. In d=1d=1d=1 and T=0T=0T=0, this reduces to the so-called Airy kernel, that appears in the Gaussian unitary ensemble (GUE) of random matrix theory. In d=1d=1d=1 and T>0T>0T>0 we show a remarkable connection between our kernel and the one appearing in the (1+11+11+1)-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently, our result provides a finite-TTT generalization of the Tracy-Widom distribution, that describes the fluctuations of the position of the rightmost fermion at T=0T=0T=0, or those of the largest single-fermion momentum. In d>1d>1d>1 and $T\ensuremath{\ge}0$, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the 1d1d1d Airy kernel at T=0T=0T=0 obtained in random matrix theory. Some of our finite-temperature results should be testable in present-day cold-atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.
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 with . We show that the correlations close to the singularity at x = 0 are described by this ‘hard edge’ kernel for while they are described by a broader family of ‘hard edge’ kernels known as the Bessel kernel for and, finally by the Airy kernel for . These one-dimensional kernels also appear in random matrix theory, and we provide here the mapping between the 1 d fermion models and the corresponding random matrix ensembles. Part of these results were announced in a recent letter, Lacroix-A-Chez-Toine et al (2017 Europhys. Lett . 120 10006).
We consider the real time dynamics of NNN noninteracting fermions in d=1d=1d=1. They evolve in a trapping potential V(x)V(x)V(x), starting from the equilibrium state in a potential V0(x)V_0(x)V0​(x). We study the time evolution of the Wigner function W(x,p,t)W(x,p,t)W(x,p,t) in the phase space (x,p)(x,p)(x,p), and the associated kernel which encodes all correlation functions. At t=0t=0t=0 the Wigner function for large NNN is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale eNe_NeN​ being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several V0(x)V_0(x)V0​(x). In the large NNN limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time dependent rescaling. For non-harmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a non-trivial time and space dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the "classical" regime ℏ∼1/N\hbar \sim 1/Nℏ∼1/N. Using the diagonal ensemble we compute the Wigner function in the quantum case (large NNN, fixed ℏ\hbarℏ) and show that it agrees with the GGE. We also obtain the higher order (non-local) correlations in the diagonal ensemble.
We analyze a class of one-dimensional quantum systems characterized by a position-dependent kinetic term arising as the continuum limit of an inhomogeneous tight-binding model with spatially varying hopping amplitudes. In this limit, the Schrodinger equation takes the so-called BenDaniel-Duke form with an effective mass, scaling as meff(x)=meff∣x∣αm_{eff}(x) = m_{eff}|x|^αmeff​(x)=meff​∣x∣α with α>0α> 0α>0, leading to a framework we term sluggish quantum mechanics, where particle motion is progressively suppressed at larger distances. Both without any external potential and with Vext(x)=12meffω2∣x∣α+2V_{ext}(x)=\frac{1}{2}m_{eff}ω^2 |x|^{α+2}Vext​(x)=21​meff​ω2∣x∣α+2, we obtain the eigenfunctions and the quantum propagators exactly. We then investigate the problem of NNN noninteracting spinless fermions in the trap, determining the many-body ground-state wavefunction and the joint probability density function of the positions of the NNN fermions. We show that the many-body quantum probability density in the ground state forms a determinantal point process whose correlation kernel can be computed for any NNN, giving access to the average density as well as higher order correlation functions for any finite NNN. Moreover, we analyze the scaling form of this kernel in the large NNN limit in the bulk, near the edge, and close to the origin. Our results show that the scaled average density profile for large NNN has a finite support symmetric with respect to the origin, but has a non-monotonic shape with a vanishing minimum at the origin for any α>0α>0α>0. One of the key findings of our work is that the scaled kernel near the origin x=0x=0x=0 for α>0α>0α>0 is neither the Bessel nor the Airy kernel (that are standard for trapped fermions), but is new, and is given by a sum of two Bessel kernels with different indices. Our results thus provide a framework relevant to engineered optical lattices with position-dependent tunneling.
We consider the real-time dynamics of N noninteracting fermions in d = 1. They evolve in a trapping potential V ( x ), starting from the equilibrium state in a potential V 0 ( x ). We study the time evolution of the Wigner function W ( x , p , t ) in the phase space ( x , p ), and the associated kernel which encodes all correlation functions. At t = 0 the Wigner function for large N is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale e N being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several V 0 ( x ). In the large- N limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time-dependent rescaling. For nonharmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a nontrivial time- and space-dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large-time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the “classical” regime . Using the diagonal ensemble we compute the Wigner function in the quantum case (large N , fixed ℏ ) and show that it agrees with the GGE. We also obtain the higher order (nonlocal) correlations in the diagonal ensemble.
We study the distribution of particle number in extended subsystems of a one-dimensional noninteracting Fermi gas confined in a potential well at zero temperature. Universal features are identified in the scaled bulk and edge regions of the trapped gas where the full counting statistics are given by the corresponding limits of the eigenvalue statistics in Gaussian unitary random matrix ensembles. The universal limiting behavior is confirmed by the bulk and edge scaling of the particle number fluctuations and the entanglement entropy.