Finite-range Coulomb gas models. II. Applications to quantum kicked rotors and banded random matrices

In part I of this two-stage exposition [Pandey, Kumar, and Puri, preceding paper, Phys. Rev. E 101, 022217 (2020)], we introduced finite-range Coulomb gas (FRCG) models, and developed an integral-equation framework for their study. We obtained exact analytical results for $d=0,1,2$, where $d$ denotes the range of eigenvalue interaction. We found that the integral-equation framework was not analytically tractable for higher values of $d$. In this paper, we develop a Monte Carlo (MC) technique to study FRCG models. Our MC simulations provide a solution of FRCG models for arbitrary $d$. We show that, as $d$ increases, there is a transition from Poisson to Wigner-Dyson classical random matrix statistics. Thus FRCG models provide a route for transition from Poisson to Wigner-Dyson statistics. The analytical formulation obtained in part I, and MC techniques developed in this paper, are used to study banded random matrices (BRMs) and quantum kicked rotors (QKRs). We demonstrate that, for a BRM of bandwidth $b$ and a QKR of chaos parameter $\alpha$, the appropriate FRCG model has range $d=b^2/N={\alpha }^2/N$, for $N\to \infty$. Here, $N$ is the dimensionality of the matrix in the BRM, and the evolution operator matrix in the QKR.

Figure 1

Schematic of the MC technique to generate eigenvalue spectra for FRCG models. We consider the case of linear ensembles. The particles on the line denote the positions of the eigenvalues $\left\{x_j\right\}$. The eigenvalues interact up to a range $d=2$ in the case shown. This range is measured in terms of the particle indices. In an attempted MC move, the particle $x_k$ is displaced with uniform probability to $x_k^{\prime }$ in the range $(x_{k-1},x_{k+1})$. The move is accepted with a probability $exp(-\beta ▵W)$, where $\mathrm{\Delta }W$ is the change in the potential in Eq. (2). An MCS corresponds to $N$ attempted moves. The independent spectra are realized by sampling the MC evolution at suitable intervals.

Figure 2

Level density for the $d=2$ FRCG model with $\beta =2$. We used the quartic potential with $\kappa =1$, and (a) $\alpha =-2$, (b) $\alpha =0$, and (c) $\alpha =2$. The filled circles correspond to MC results, and the solid lines are analytical results from Eq. (29) in Ref. [13].

Figure 3

Level density for the $d=750$ FRCG model with $\beta =2$. We used the quartic potential with $\kappa =1$, and (a) $\alpha =-2$, (b) $\alpha =0$, and (c) $\alpha ={\alpha }_c=1.414$. The filled circles correspond to MC results, and the solid lines are analytical results from Eqs. (46) and (47) in Ref. [13].

Figure 4

Comparison of nearest-neighbor spacing distribution $[p_0\left(s\right)$ vs $s]$ for FRCG models with $d=0,1,2,5,10$. We show results for (a) $\beta =1$, (b) $\beta =2$, and (c) $\beta =4$. In each frame, the dashed line denotes the classical Gaussian ensemble result.

Figure 5

Comparison of the two-point correlation function $[R_2\left(s\right)$ vs $s]$ for FRCG models with $d=0,1,2,3,5$. We show results for (a) $\beta =1$, (b) $\beta =2$, and (c) $\beta =4$. In each frame, the dashed line denotes the classical Gaussian ensemble result.

Figure 6

Analogous to Fig. 5, but for the two-point cluster function: $Y_2\left(s\right)=1-R_2\left(s\right)$.

Figure 7

Number variance $[{\mathrm{\Sigma }}^2\left(r\right)$ vs $r]$ for FRCG models with $d=0,1,2,5,10,25$. We show results for (a) $\beta =1$, (b) $\beta =2$, and (c) $\beta =4$. The dashed line denotes the classical Gaussian ensemble result.

Figure 8

Spacing density $[p_{n-1}\left(s\right)$ vs $s]$ for FRCG models. We consider various values of $d,\beta ,\text{and}n$ as indicated. The filled circles correspond to MC results, and the solid lines denote the MF result from Eq. (91) of Ref. [13].

Figure 9

(a) Plot of $\text{var}\left(L\right)/\text{var}\left(1\right)$ vs $L$ for the QKR momentum operator in the $U$-diagonal basis. We have $\gamma =0$, and the values of $\alpha$ are such that $d={\alpha }^2/N=1,5,10,25,50$. (b) Data in panel (a), plotted against the scaled variable $L/d$. (c) Analogous to panel (a), but for a large value of $\alpha$ so that $d=O\left(N\right)$.

Figure 10

(a) Plot of $\text{var}\left(L\right)/\text{var}\left(1\right)$ vs $L$ for the BRM. The bandwidth $b=32$, so that $d=b^2/N=1$. (b) Plot of $\mathrm{\Gamma }\left(L\right)$ vs $L$ for the data in panel (a). (c) Superposition of $\mathrm{\Gamma }\left(L\right)$ vs $L/d$ for $b=32,72,100$ so that $d=1,5,10$.

Figure 11

Typical eigenvector of a BRM with bandwidth $b=5$. (a) Plot of $|{\psi }_n{|}^2$ vs $n$ on a direct scale. Here, $n$ is the component of the eigenvector. The inset shows an expanded region near the point of localization. (b) Data in panel (a), plotted on a linear-log scale. The eigenfunction decays exponentially from the point of localization.

Figure 12

Typical eigenvector of a BRM with bandwidth $b=250$ so that $d=63$. (a) Plot of $|{\psi }_n{|}^2$ vs $n$. This corresponds to an extended state. The inset shows the eigenvector when $d=N-1$. (b) Data in panel (a), plotted on a linear-log scale.

Figure 13

Spacing density $p_0\left(s\right)$ vs $s$ for the QKR and BRM. The solid line denotes the FRCG result. We show data for various values of $d$ and $\beta$, as indicated.

Figure 14

Spacing density $p_{n-1=k}\left(s\right)$ vs $s$ for the QKR. The solid line denotes the FRCG result. (a) Data for $d=3,\beta =1$ with $k=0,1,2,3,4,5$. (b) Data for $d=3,\beta =2$ with $k=0,1,2,3,4,5$. (c) Data for $d=3,\beta =1$ with $k=7$. We also superpose the MF result.

Figure 15

Spacing variance ${\sigma }^2(n-1)$ vs $n$ for the QKR and BRM. We show data for $d=0,1,2,5,10,25,N-1$ (from top to bottom), and (a) $\beta =1$ and (b) $\beta =2$. The solid lines denote the corresponding FRCG results. The dashed lines denote the classical ensemble results.

Figure 16

Two-point correlation function $R_2\left(s\right)$ vs $s$ for the QKR. The solid lines denote the corresponding FRCG results. The dashed lines denote the GOE result. We show data for $\beta =1$ and (a) $d=1$, (b) $d=2$, and (c) $d=3$.

Figure 17

Analogous to Fig. 16, but for the $\beta =2$ case.

Figure 18

Two-point correlation function $R_2\left(s\right)$ vs $s$ for the BRM. The solid lines denote the corresponding FRCG results. The dashed lines denote the GOE result. We show data for $\beta =1$ and (a) $d=1$, (b) $d=2$, and (c) $d=3$.

Figure 19

Analogous to Fig. 18, but for the $\beta =2$ case.

Figure 20

Plot of $p_0\left(s\right)$ vs $s$ for the QKR with fractional values of $d$. The solid lines denote the appropriate FRCG result. We show results for various values of $\beta$ and $d$, as indicated.