Semiclassical Identification of Periodic Orbits in a Quantum Many-Body System

While a wealth of results has been obtained for chaos in single-particle quantum systems, much less is known about chaos in quantum many-body systems. We contribute to recent efforts to make a semiclassical analysis of such systems feasible, which is nontrivial due to the exponential proliferation of orbits with increasing particle number. Employing a recently discovered duality relation, we focus on the collective, coherent motion that together with the also present incoherent one typically leads to a mixture of regular and chaotic dynamics. We investigate a kicked spin chain as an example of a presently experimentally and theoretically much studied class of systems.

Figure 1

For $T=1$, $|\rho (S)|$ as black curve. (a) $N=7$ spins with $J=0.75$ and $b^x=b^z=0.9$, cutoff $j_{\mathrm{cut}}=800$. (b) $N=19$ spins with $J=0.7$, $j_{\mathrm{cut}}=4650$, and $\mathbf{b}$ as in (a). The positions of the classical POs as red lines for $N_{\gamma }^{(P)}=1$, as blue lines for $N_{\gamma }^{(P)}=N$.

Figure 2

Scaling exponent $\alpha$ of $|\rho (S_{\gamma })|\sim (j_{\mathrm{cut}}{)}^{\alpha }$ at the position $S_{\gamma }$ of the largest peak, versus the number of spins. For $T=1$, 2, $J=0.7$, $b^x=b^z=0.9$, and $N=4k$.

Figure 3

For $T=2$, $|\rho (S)|$ as black curve with parameters $J=0.7$, $b^x=b^z=0.9$, and $j_{\mathrm{cut}}=114$. The peaks for $N_{\gamma }^{(P)}=N$ are blue, for $N_{\gamma }^{(P)}=1$ red, and yellow otherwise. For $N=4$, 8, 20, 100, very large peaks, marked green, due to collective motion occur at actions $S_{\mathrm{man}}$.

Figure 4

Difference $\mathrm{\Delta }(j)$ divided by $N$, for the manifolds with $N=4$ as orange circles and with $N=80$ as green squares, and for $N=3$ as blue diamonds.