Density profile of noninteracting fermions in a rotating two-dimensional trap at finite temperature

We study the average density of $N$ spinless noninteracting fermions in a two-dimensional harmonic trap rotating with a constant frequency $\mathrm{\Omega }$ and in the presence of an additional repulsive central potential $\gamma /r^2$. 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-$N$ 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 $T\sim O\left(1\right)$, the edge profile already acquires significant smearing at a much lower temperature $T\sim O(1/\sqrt{N})$. These large-$N$ 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.”

Figure 1

Top: The external potential $V\left(r\right)=\frac{1}{2}{\omega }^2{r}^2+\frac{\gamma }{2r^2}$ from Eq. (2) is plotted in the 2D plane. We choose $c=10$ and $N=400$, which means $\gamma =cN=4\times {10}^3$. We see the highly repulsive central potential that creates a hole (a depleted region devoid of particles) at the center in the density profile shown in the bottom figure. Bottom: A 3D representation of the exact density in Eq. (13). A hole around the origin is surrounded by a multilayered “wedding-cake” structure. We choose $c=0.5$, $M=30$, and $N=8000$, which leads to three layers in this case. The normalization condition Eq. (27) gives $\mu =9.48$.

Figure 2

Energy levels $E_{k,l}$ in Eq. (7) vs $l$ for $k=0,1,2,3,4$, and with the choice of parameters $\gamma =10$ and $\nu =0.9$. The black (dashed) horizontal line marks the Fermi level, $\mu =7.5$ (at zero temperature, i.e., when $\beta \to \infty$). Only the states with energy below $\mu$ contribute to the ground-state density profile. Hence, in this case, $k^{*}=2$, where $k^{*}$ denotes the highest occupied Landau level.

Figure 3

Finite-temperature density profile along a radial cut as a function of the scaled radial distance $z=r/\sqrt{N}$ obtained by direct enumeration of Eqs. (28) and (29). The values of the parameters used are $c=0.5,\nu =0.998,\beta =15.0,N=8000$. This is a three-layered smeared “wedding cake.” From the normalization condition [Eq. (27)], the corresponding chemical potential turns out to be ${\mu }_{\beta }\sim 9.48$.

Figure 4

Top: Large-$N$ bulk density profile using Eqs. (30) and (31) for three different values of temperature $T=0$ (red, dashed), $T=1/15$ (black, dotted), and $T=1/4$ (green, solid). The values of the parameters used are $c=1.0$ and $M=30$. This is a three-layered smeared “wedding cake.” Bottom: A 3D representation of the above large-$N$ bulk density profile for $T=1/15$. The smearing of the “wedding cake” can be clearly seen. In the bottom figure, the density profile along a cut in a radial direction is given by the top panel of the figure.

Figure 5

Plot of ${\mu }_{\beta }$ vs $\beta$ obtained by solving numerically Eqs. (31) and (32) for $c=1$ and $M=10$.

Figure 6

Finite-temperature large-$N$ edge density profile (for the inner edge) using Eq. (42) for $k=0,1,2$ (red, black and blue from bottom to top). The values of the parameters used are $c=1.0,M=30,\beta =50,N=4000,\tilde{\beta }=\beta /\sqrt{N}=0.56$. From the normalization condition [Eq. (32)], the corresponding chemical potential is ${\mu }_{\beta }\approx 10.73$. Note that the plots are vertically translated by $1/\pi$ for a better visualization.

Figure 7

Finite-temperature large-$N$ edge density profile (for the inner edge) for $k=2$. The values of the parameters used are $c=1.0,M=30,N=4000$. Blue, red, and black lines (bottom to top) represent $\beta =25,50,500$, respectively. As temperature decreases, the edge density profile becomes steeper and steeper. Note that the plots are vertically translated by $1/\pi$ for a better visualization.

Figure 8

A schematic plot of $g_k\left(z\right)$ vs $z$ given by Eq. (D24) for a general function $v\left(z\right)$. The support of the $k\mathrm{th}$ layer's density is the set of points with $g_k\left(z\right)<\mu$ (shaded green region). The filled circles denote the locations $z=z_e$ of the edges of this support, where $g_k\left(z_e\right)=\mu$ (purple dashed line). It is easy to see that for every interval (which is defined as a segment between a green and a red circle), the support of the outer boundary (red) has $b>0$ [Eq. (D33)], and the support of the inner boundary (green) has $b<0$.