Number statistics for $\beta$-ensembles of random matrices: Applications to trapped fermions at zero temperature

Let ${\mathcal{P}}_{\beta }^{\left(V\right)}\left(N_{\mathcal{I}}\right)$ be the probability that a $N\times N\beta$-ensemble of random matrices with confining potential $V\left(x\right)$ has $N_{\mathcal{I}}$ eigenvalues inside an interval $\mathcal{I}=[a,b]$ on the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically ${\mathcal{P}}_{\beta }^{\left(V\right)}\left(N_{\mathcal{I}}\right)$ for large $N$. We show that this probability scales for large $N$ as ${\mathcal{P}}_{\beta }^{\left(V\right)}\left(N_{\mathcal{I}}\right)\approx exp\left[-\beta N^2{\psi }^{\left(V\right)}(N_{\mathcal{I}}/N)\right]$, where $\beta$ is the Dyson index of the ensemble. The rate function ${\psi }^{\left(V\right)}\left(k_{\mathcal{I}}\right)$, independent of $\beta$, is computed in terms of single integrals that can be easily evaluated numerically. The general formalism is then applied to the classical $\beta$-Gaussian ($\mathcal{I}=[-L,L]$), $\beta$-Wishart ($\mathcal{I}=[1,L]$), and $\beta$-Cauchy ($\mathcal{I}=[-L,L]$) ensembles. Expanding the rate function around its minimum, we find that generically the number variance $var\left(N_{\mathcal{I}}\right)$ exhibits a nonmonotonic behavior as a function of the size of the interval, with a maximum that can be precisely characterized. These analytical results, corroborated by numerical simulations, provide the full counting statistics of many systems where random matrix models apply. In particular, we present results for the full counting statistics of zero-temperature one-dimensional spinless fermions in a harmonic trap.

Figure 1

Average density of scaled Gaussian eigenvalues with a representation of bulk and edge regimes. The bulk is the scale of typical fluctuations of eigenvalues, whose width is of order $1/N$. The edge corresponds to the scale of typical fluctuations ($\sim O\left(N^{-2/3}\right)$) of the largest (or symmetrically the smallest) eigenvalue.

Figure 2

Results for the number variance of GUE eigenvalues when $\mathcal{I}=[-L,L]$. The theoretical result is in Eq. (3) of letter [8] for $\beta =2$.

Figure 3

Regimes of behavior of the number variance for the Gaussian ensemble. The solid blue line represents the interval $\mathcal{I}=[-L,L]$. (a) Bulk regime; (b) extended bulk regime; (c) edge regime; (d) tail regime.

Figure 4

(a) Average density of eigenvalues for the Wishart ensemble (and $\beta =2$), which is the Marčenko-Pastur distribution, for the $c=1$ case, and (b) results for the Wishart number variance when $\mathcal{I}=[1,1+l]$. The theoretical result is in Eq. (19). We write ${\tilde{V}}_2\left(s\right)=\frac{1}{2{\pi }^2}lnN+\frac{1}{2}{V}_2\left(s\right)$.

Figure 5

(a) Average density ${\rho }_C\left(x\right)=(1/\pi )1/(1+x^2)$ in the Cauchy ensemble (and $\beta =2$) and (b) results for the Cauchy number variance when $\mathcal{I}=[-L,L]$. The theoretical result is Eq. (21).

Figure 6

Sketch of the expected behavior of the density ${\rho }^{☆}\left(x\right)$ for the Gaussian ensemble when $\mathcal{I}=[a,b]$ and $-\sqrt{2}<a<b<\sqrt{2}$. The dashed orange line is the Wigner semicircle law, when $k_{\mathcal{I}}=\overline{k_{\mathcal{I}}}$. (a) Constraining $k_{\mathcal{I}}<\overline{k_{\mathcal{I}}}$; (b) constraining $k_{\mathcal{I}}>\overline{k_{\mathcal{I}}}$.

Figure 7

Behavior of the rate function ${\psi }^{\left(G\right)}\left(k_{\mathcal{I}}\right)$ as a function of $k_{\mathcal{I}}\in [0,1]$ for two different values of $L$: $L=0.6<\sqrt{2}$ and $L=1.6>\sqrt{2}$. $k_{\mathcal{I}}$ is the fraction of eigenvalues inside the interval $\mathcal{I}=[-L,L]$. The solid gray line in the plane $(L,k_{\mathcal{I}})$ is the critical line $\overline{k_{\mathcal{I}}}$ given in Eq. (63), where ${\psi }^{\left(G\right)}\left(k_{\mathcal{I}}\right)$ has a minimum (zero).

Figure 8

Phase transition of the average density ${\rho }^{☆}\left(x\right)$, considering the interval $\mathcal{I}=[-L,L]$. The critical line $k_{\mathcal{I}}=\overline{k_{\mathcal{I}}}$ [where $\overline{k_{\mathcal{I}}}$ is given in Eq. (63)] separates two phases. For $k_{\mathcal{I}}<\overline{k_{\mathcal{I}}}$ (left side), the saddle-point density ${\rho }^{*}\left(x\right)$ has nonzero support over $[-b_2,-L]\cup [-a_2,a_2]\cup [L,b_2]$. For $k_{\mathcal{I}}>\overline{k_{\mathcal{I}}}$ (right side), in contrast, ${\rho }^{*}\left(x\right)$ is nonzero over $[-b_2,-a_2]\cup [-L,L]\cup [a_2,b_2]$. Exactly at $k_{\mathcal{I}}=\overline{k_{\mathcal{I}}}$, ${\rho }^{*}\left(x\right)=(1/\pi )\sqrt{2-x^2}$ is the Wigner semicircular law, which is shown by the dashed (orange) line.

Figure 9

Extended bulk regime for the Gaussian ensemble.

Figure 10

Edge regime for the Gaussian ensemble.

Figure 11

Tail regime for the Gaussian ensemble.

Figure 12

Energy required to pull one eigenvalue from the semicircle sea and place it at $L>\sqrt{2}$.

Figure 13

Sketch of the expected behavior of the density ${\rho }^{☆}\left(x\right)$ for the Wishart ensemble when $\mathcal{I}=[1,L]$ and $1<L<4$. The dashed orange line is the Marčenko-Pastur distribution for $N=M$ [Eq. (18)], when $k_{\mathcal{I}}=\overline{k_{\mathcal{I}}}$. (a) Constraining $k_{\mathcal{I}}<\overline{k_{\mathcal{I}}}$; (b) constraining $k_{\mathcal{I}}>\overline{k_{\mathcal{I}}}$.

Figure 14

Regimes of behavior of the number variance for the Wishart ensemble. The solid blue line represents the interval $\mathcal{I}=[1,L]=[1,1+l]$. (a) Bulk regime; (b) extended bulk regime; (c) edge regime; (d) tail regime.

Figure 15

Results for the variance of $N_{\mathcal{I}}$ for the Wishart ensemble (and $\beta =2$) when $\mathcal{I}=[1,1+l]$ and $l<3$. Theory is Eq. (117). (Inset) Results for the edge regime, where $s=(l-3){(N/4)}^{2/3}$ and theory is Eq. (93). We write ${\tilde{V}}_2\left(s\right)=\frac{1}{2{\pi }^2}lnN+\frac{1}{2}{V}_2\left(s\right)$.

Figure 16

Sketch of the expected behavior of ${\rho }^{☆}\left(x\right)$ for the Cauchy ensemble when $\mathcal{I}=[-L,L]$. The dashed orange line is Cauchy distribution [Eq. (132)], when $k_{\mathcal{I}}=\overline{k_{\mathcal{I}}}$. (a) Constraining $k_{\mathcal{I}}<\overline{k_{\mathcal{I}}}$; (b) constraining $k_{\mathcal{I}}>\overline{k_{\mathcal{I}}}$.

Figure 17

Regimes of behavior of the number variance for the Cauchy ensemble. The solid blue line represents the interval $\mathcal{I}=[-L,L]$. (a) Bulk regime; (b) extended bulk regime; (c) edge regime; (d) tail regime.

Figure 18

Results for the variance of $N_{\mathcal{I}}$ for the Cauchy ensemble (and $\beta =2$) for different values of $N$ when $\mathcal{I}=[-L,L]$. The theoretical result is in Eq. (148).

Figure 19

Numerical simulations of the number variance for GUE as a function of the scaled variable ${\mathrm{\Delta }}_G=NL{(2-L^2)}^{3/2}$, where $L$ is varied over a range $L\in [{10}^{-4},\sqrt{2}-{10}^{-3}]$, and for different values of $N=50,500,5000$. The solid line represents $(1/{\pi }^2)ln\left({\mathrm{\Delta }}_G\right)+C_2$, where $C_2=(1+\gamma +ln2)/{\pi }^2$ [see Eq. (9)] is the Dyson-Mehta constant.