Spins coupled to a spin bath: From integrability to chaos

Motivated by the hyperfine interaction of electron spins with surrounding nuclei, we investigate systems of central spins coupled to a bath of noninteracting spins in the framework of random matrix theory. With increasing number of central spins a transition from Poissonian statistics to the Gaussian orthogonal ensemble occurs which can be described by a generalized Brody distribution. These observations are unaltered upon applying an external magnetic field. In the transition region, the classical counterparts of the models studied have mixed phase space.

Figure 1

(Color online) Nearest-neighbor level spacing for a system of a single central spin $S_1=1/2$ and 13 bath spins of length $I=1/2$. The red curve is the exponential $p\left(s\right)=e^{-s}$.

Figure 2

(Color online) Nearest-neighbor level spacing for a system of a two central spins $S_1=S_2=1/2$ and 12 bath spins of length $I=1/2$. The red curve is the GOE Wigner surmise.

Figure 3

(Color online) The same data as in the bottom right panel of Fig. 2 but for $N=10$ and $N=11$ bath spins.

Figure 4

(Color online) Joint level spacing distribution for $J=0,\dots ,4$ and increasing number of central spins. The red dashed lines are the exponential function $\left(N_c=1\right)$ and the Wigner surmise $\left(N_c>1\right)$. The green solid lines are a fit to generalized Brody distribution (2).

Figure 5

A cut through the plane $I_1^z=I_2^z=0$ of two different phase-space orbits demonstrating the close vicinity of regular and chaotic dynamics in a system of two central spins (see text).