Global and local relaxation of a spin chain under exact Schrödinger and master-equation dynamics

We solve the Schrödinger equation for an interacting spin chain locally coupled to a quantum environment with a specific degeneracy structure. The reduced dynamics of the whole spin chain as well as of single spins is analyzed. We show that the total spin chain relaxes to a thermal equilibrium state independently of the internal interaction strength. In contrast, the asymptotic states of each individual spin are thermal for weak but nonthermal for stronger spin-spin coupling. The transition between both scenarios is found for couplings of the order of $0.1\times \Delta E$, with $\Delta E$ denoting the Zeeman splitting. We compare these results with a master-equation treatment; when time averaged, both approaches lead to the same asymptotic state and finally with analytical results.

Figure 1

Coupling topology of the spin chain to the quantum environment.

Figure 2

Relaxation of the spin chain $s$ into equilibrium under Schrödinger dynamics for $\lambda =0.4$ and $\kappa =0.001$ with a random interaction. The horizontal lines are the expected probabilities from (11) to find the system at the corresponding energies. These are in good accordance with the time-averaged values of ${\rho }_{\mathrm{s}}\left(t\right)$. $\lambda$ and $\kappa$ are energies taken in units of the Zeeman splitting. [Note that $W^d\left(E_4^{\mathrm{s}}\right)=W^d\left(E_5^{\mathrm{s}}\right)$, Eq. (11).]

Figure 3

Relaxation into equilibrium under Schrödinger dynamics as of Fig. 2 but with antiferromag netic Heisenberg interaction.

Figure 4

Global and local temperatures as a function of $\lambda$ for a random interaction.

Figure 5

Global and local temperatures as a function of $\lambda$ for antiferromagnetic Heisenberg interaction.

Figure 6

Correlation $C$ [Eq. (26)] as a function of $\lambda$ for a random interaction.

Figure 7

Correlation $C$ [Eq. (26)] as a function of $\lambda$ for a Heisenberg interaction.

Figure 8

Time evolution of ${\rho }^{\mathrm{s}}$ under a master equation with random interaction. The equilibrium reached is the same as that under Schrödinger dynamics (Fig. 2) and the one predicted by Eq. (11) (horizontal lines).

Figure 9

Time evolution of ${\rho }^{\mathrm{s}}$ under a master equation with antiferromagnetic Heisenberg interaction. (compare Fig. 3).

Figure 10

Spectral temperatures under a master equation with random interaction as function of $\lambda$. The solid line is the spectral temperature $T^{\mathrm{s}}$ of the total spin chain. The dashed line is the local temperature $T_1^{\mathrm{loc}}$ of spin 1, the narrower dashed line $T_2^{\mathrm{loc}}$ the one of spin 2, and the dotted line $T_3^{\mathrm{loc}}$ the one of spin 3.

Figure 11

Spectral temperatures under a master equation with antiferromagnetic Heisenberg interaction in dependence on $\lambda$ (compare Fig. 10).

Figure 12

Analytical results for local spectral temperatures with the global inverse temperature $\beta =\ln 2$ as function of $\lambda$ for ferromagnetic $(\lambda <0)$ and antiferromagnetic $(\lambda >0)$ Heisenberg coupling.

Figure 13

Correlations $C$ as function of $\lambda (\beta =\ln 2)$ for ferromagnetic $(\lambda <0)$ and antiferromagnetic $(\lambda >0)$ Heisenberg coupling.

Figure 14

Correlations $C$ as a function of $\beta$ and $\lambda$ for ferromagnetic $(\lambda <0)$ and antiferromagnetic $(\lambda >0)$ Heisenberg coupling.