Nearest-neighbor distributions and tunneling splittings in interacting many-body two-level boson systems

We study the nearest-neighbor distributions of the $k$-body embedded ensembles of random matrices for $n$ bosons distributed over two-degenerate single-particle states. This ensemble, as a function of $k$, displays a transition from harmonic-oscillator behavior $\left(k=1\right)$ to random-matrix-type behavior $\left(k=n\right)$. We show that a large and robust quasidegeneracy is present for a wide interval of values of $k$ when the ensemble is time-reversal invariant. These quasidegenerate levels are Shnirelman doublets which appear due to the integrability and time-reversal invariance of the underlying classical systems. We present results related to the frequency in the spectrum of these degenerate levels in terms of $k$ and discuss the statistical properties of the splittings of these doublets.

Figure 1

Nearest-neighbor distribution $P_k\left(s\right)$ for the $k$-body interacting two-level boson ensemble for $\beta =1$, $n=2000$, and (a) $k=1$, (b) $k=2$, (c) $k=10$, (d) $k=200$, (e) $k=1000$, (f) $k=1150$, (g) $k=1850$, and (h) $k=2000$. Notice the large peak observed at $s=0$, which is linked with the occurrence of quasidegenerate levels. The dashed curve corresponds to the Wigner surmise for $\beta =1$, while the dotted curve is the Poisson distribution.

Figure 2

Same as Fig. 1 for $\beta =2$, $n=1000$, and (a) $k=1$, (b) $k=2$, (c) $k=10$, (d) $k=200$, (e) $k=500$, (f) $k=700$, (g) $k=800$, and (h) $k=940$. Notice that the strong peak observed in Fig. 1 at $s=0$ for $\beta =1$ is absent in this case, indicating that its origin is due to time-reversal symmetry. Yet, certain degree of level clustering is still observed on a wide interval of $k$.

Figure 3

(Color online) Relative measure ${\mu }_k$ of the number of levels contained within the first four bins of the nearest-neighbor spacing distributions as a function of $k/n$. The blue curve (dotted curve with squares) corresponds to the time-reversal invariant case $\left(\beta =1\right)$ and the red curve (dashed curve with triangles) corresponds to the broken time-reversal case $\left(\beta =2\right)$. The continuous black curve (full circles) represents the average number of Shnirelman doublets $\left(\beta =1\right)$ obtained using the symmetry properties of the eigenfunctions. Inset shows details for small values of $k$.

Figure 4

(Color online) Phase space representation (level curves) of the reduced Hamiltonian ${\mathcal{H}}_k^{\left(\beta \right)}\left(J,K,\psi \right)$ for $\beta =1$ and $n=100$. (a) $k=1$, (b) $k=2$, (c) $k=4$, (d) $k=13$, (e) $k=29$, (f) $k=56$, (g) $k=81$, and (h) $k=100$. The continuous lines represent three groups of ten levels, taken from either edge and from the center of the spectrum. The color (gray) palette displays the overall energy scale of ${\mathcal{H}}_k^{\left(\beta =1\right)}\left(J,K,\psi \right)$, while the other one displays the full spectrum in the same scale. Note the clear appearance of ladders of levels associated with certain organizing centers in phase space. As $k$ is increased, the level curves spread over all the available phase space, while the width of the spectrum decreases.

Figure 5

Log-log plot showing the growth of the number of stable fixed points in terms of $k$ for $n=100$. Straight lines included have slope equal to 1 or 2. Around $k\gtrsim 20$, the initial linear growth rate becomes quadratic.

Figure 6

(Color online) (a) Classical phase-space representation for $k=3$ showing two pairs of non-self-retracing tori. The labels indicate the symmetry-related eigenfunctions. The color (gray) code is related to the full classical energy-scale of this case. (b) Reduced representation of the square modulus of the eigenfunction $A$. (c) Linear combinations (10, 11) with respect to levels $A$ and $A^{\prime }$; the figure shows that the linear combinations lie on opposite sides of the $\psi$ axis and therefore the eigenfunctions are a Shnirelman doublet. (d) Result of the linear combinations when considering two non-symmetry-related nearby levels $A$ and $B$.

Figure 7

Distribution $P_k\left(\xi \right)$ of the normalized spacings of the Shnirelman doublets for (a) $k=2$, (b) $k=3$, (c) $k=250$, (d) $k=750$, (e) $k=1200$, and (f) $k=1800$.