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arXiv:1902.02594

Nonequilibrium dynamics of noninteracting fermions in a trap

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

Published Feb 7, 2019• 4 citations
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Abstract

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 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 $\hbar \sim 1/N$. Using the diagonal ensemble we compute the Wigner function in the quantum case (large $N$, fixed $\hbar$) and show that it agrees with the GGE. We also obtain the higher order (non-local) correlations in the diagonal ensemble.

DOI
10.48550/arxiv.1902.02594

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The Wigner function WN(x,p){W}_{N}(\mathbf{x},\mathbf{p})WN​(x,p) is a useful quantity to characterize the quantum fluctuations of an NNN-body system in its phase space. Here we study WN(x,p){W}_{N}(\mathbf{x},\mathbf{p})WN​(x,p) for NNN noninteracting spinless fermions in a ddd-dimensional spherical hard box of radius RRR at temperature T=0T=0T=0. In the large-NNN 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 (x,p)(\mathbf{x},\mathbf{p})(x,p) plane, namely, for ∣x∣<R|\mathbf{x}|<R∣x∣<R and ∣p∣<kF|\mathbf{p}|<{k}_{F}∣p∣<kF​, where kF{k}_{F}kF​ is the Fermi momentum, while WN(x,p){W}_{N}(\mathbf{x},\mathbf{p})WN​(x,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 kF{k}_{F}kF​, 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 ddd and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension ddd.
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 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 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.
In the first part of the article, we study one-dimensional noninteracting fermions in the continuum and in the presence of the repulsive inverse power law potential, with an emphasis on the Wigner function in the semiclassical limit. In this limit, the Wigner function exhibits an edge called the Fermi surf that depends only on the classical one-particle Hamiltonian. Around the Fermi surf, under a well-defined semiclassical limit, the Wigner function can be expressed in terms of Airy functions which yield a smooth matching between the two regions delimited by the Fermi surf. In the second part of the article, the system is prepared in the ground state of the inverse power law potential where only the left half line is filled with fermions. Then the potential is switched off, resulting in the emergence of a propagating quantum front. We show that the power law decay of the pre-quench potential that separates the left and right half systems leads to the emergence of the Airy kernel (well known in Random Matrix Theory) at the quantum front in the long-time limit. This also comes with anomalous diffusive spreading around the front.