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arXiv:2009.07251 · Physical Review A

Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap

Manas Kulkarni, Satya N. Majumdar, Grégory Schehr

Published Mar 22, 2021• 17 citations
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Abstract

We compute exactly the average spatial density for $N$ spinless noninteracting fermions in a $2d$ harmonic trap rotating with a constant frequency $\mathrm{\ensuremath{\Omega}}$ in the presence of an additional repulsive central potential $\ensuremath{\gamma}/{r}^{2}$. We find that in the large-$N$ limit, the bulk density has a rich and nontrivial profile---with a hole at the center of the trap and surrounded by a multilayered ``wedding cake'' structure. The number of layers depends on $N$ and on the two parameters $\mathrm{\ensuremath{\Omega}}$ and $\ensuremath{\gamma}$ leading to a rich phase diagram. Zooming in on the edge of the $k\mathrm{th}$ layer, we find that the edge density profile exhibits $k$ kinks located at the zeros of the $k\mathrm{th}$ Hermite polynomial. Interestingly, in the large-$k$ limit, we show that the edge density profile approaches a limiting form, which resembles the shape of a propagating front, found in the unitary evolution of certain quantum spin chains. We also study how a newly formed droplet grows in size on top of the last layer as one changes the parameters.

DOI
10.1103/physreva.103.033321

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We compute exactly the average spatial density for NNN spinless noninteracting fermions in a 2d2d2d harmonic trap rotating with a constant frequency \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ in the presence of an additional repulsive central potential \ensuremathγ/r2\ensuremath{\gamma}/{r}^{2}\ensuremathγ/r2. We find that in the large-NNN limit, the bulk density has a rich and nontrivial profile---with a hole at the center of the trap and surrounded by a multilayered ``wedding cake'' structure. The number of layers depends on NNN and on the two parameters \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ and \ensuremathγ\ensuremath{\gamma}\ensuremathγ leading to a rich phase diagram. Zooming in on the edge of the kthk\mathrm{th}kth layer, we find that the edge density profile exhibits kkk kinks located at the zeros of the kthk\mathrm{th}kth Hermite polynomial. Interestingly, in the large-kkk limit, we show that the edge density profile approaches a limiting form, which resembles the shape of a propagating front, found in the unitary evolution of certain quantum spin chains. We also study how a newly formed droplet grows in size on top of the last layer as one changes the parameters.
We study the average density of NNN spinless noninteracting fermions in a two-dimensional harmonic trap rotating with a constant frequency $\mathrm{\ensuremath{\Omega}}$ and in the presence of an additional repulsive central potential $\ensuremath{\gamma}/{r}^{2}$. The average density at zero temperature was recently studied in Phys. Rev. A 103, 033321 (2021), and an interesting multilayered ``wedding-cake'' structure with a ``hole'' at the center was found for the density in the large-NNN limit. In this paper, we study the average density at finite temperature. We demonstrate how this ``wedding-cake'' structure is modified at finite temperature. We show that, while the bulk density profile gets smeared significantly at a temperature $T\ensuremath{\sim}O(1)$, the edge profile already acquires significant smearing at a much lower temperature $T\ensuremath{\sim}O(1/\sqrt{N})$. These large-NNN results warrant going much beyond the standard local density approximation. We also generalize our results to a wide variety of trapping potentials, and we demonstrate the universality of the associated scaling functions both in the bulk and at the edges of the ``wedding cake.''
Density profile of noninteracting fermions in a rotating 2d2d2d trap at finite temperature
We study the average density of NNN spinless noninteracting fermions in a 2d2d2d harmonic trap rotating with a constant frequency ΩΩΩ and in the presence of an additional repulsive central potential γ/r2γ/r^2γ/r2. The average density at zero temperature was recently studied in Phys. Rev. A 103\textbf{103}103, 033321 (2021) and an interesting multi-layered "wedding cake" structure with a "hole" at the center was found for the density in the large NNN limit. In this paper, we study the average density at finite temperature. We demonstrate how this "wedding-cake" structure is modified at finite temperature. These large NNN results warrant going much beyond the standard Local Density Approximation. We also generalize our results to a wide variety of trapping potentials and demonstrate the universality of the associated scaling functions both in the bulk and at the edges of the "wedding-cake".
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 the ground state of $N\ensuremath{\gg}1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $\mathrm{\ensuremath{\Omega}}>0$. The support of the density of the Fermi gas is a disk of radius Re{R}_{e}Re​. We calculate the variance of the number of fermions, NR{\mathcal{N}}_{R}NR​, inside a disk of radius RRR centered at the origin for RRR in the bulk of the Fermi gas. We find rich and interesting behaviors in two different scaling regimes, (i) $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}<1$ and (ii) $1\ensuremath{-}\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1/N)$, where $\ensuremath{\omega}$ is the angular frequency of the oscillator. In the first regime we find that $\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}\ensuremath{\simeq}(AlnN+B)\sqrt{N}$ and we calculate AAA and BBB as functions of R/ReR/{R}_{e}R/Re​, $\mathrm{\ensuremath{\Omega}}$, and $\ensuremath{\omega}$. We also predict the higher cumulants of NR{\mathcal{N}}_{R}NR​ and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime, the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling kkk successive Landau levels, as found in previous studies. Here, we show that VarNR\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}VarNR​ is a discontinuous piecewise linear function of $\ensuremath{\sim}(R/{R}_{e})\sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any kkk. We argue that a similar piecewise linear behavior extends to all the cumulants of NR{\mathcal{N}}_{R}NR​ and to the entanglement entropy. We show that these results match smoothly at large kkk with the above results for $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of VarNR\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}VarNR​ near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the zzz direction.
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.
We establish an exact mapping between the positions of NNN noninteracting fermions in a two-dimensional rotating harmonic trap in its ground state and the eigenvalues of the $N\ifmmode\times\else\texttimes\fi{}N$ complex Ginibre ensemble of random matrix theory (RMT). Using RMT techniques, we make precise predictions for the statistics of the positions of the fermions, both in the bulk as well as at the edge of the trapped Fermi gas. In addition, we compute exactly, for any finite NNN, the R\'enyi entanglement entropy and the number variance of a disk of radius rrr in the ground state. We show that while these two quantities are proportional to each other in the (extended) bulk, this is no longer the case very close to the trap center nor at the edge. Near the edge, and for large NNN, we provide exact expressions for the scaling functions associated with these two observables.