Nonequilibrium current-carrying steady states in the anisotropic $XY$ spin chain

Out-of-equilibrium behavior is explored in the one-dimensional anisotropic $XY$ model. Initially preparing the system in the isotropic $XX$ model with a linearly varying magnetic field to create a domain-wall magnetization profile, dynamics is generated by rapidly changing the exchange interaction anisotropy and external magnetic field. Relaxation to a nonequilibrium steady state is studied analytically at the critical transverse Ising point, where correlation functions may be computed in closed form. For arbitrary values of anisotropy and external field, an effective generalized Gibbs' ensemble is shown to accurately describe observables in the long-time limit. Additionally, we find spatial oscillations in the exponentially decaying, transverse spin-spin correlation functions with wavelength set by the magnetization jump across the initial domain wall. This wavelength depends only weakly on anisotropy and magnetic field in contrast to the current, which is highly dependent on these parameters.

Figure 1

Domain-wall broadening in the isotropic $XY$ model. As a domain-wall magnetization profile spreads, the central region relaxes to a nonequilibrium steady state. The presence of the polarized edges drives a current through the central region. This situation may be viewed as expansion of a quantum gas of interacting ($XXZ$ chain) or noninteracting ($XY$ chain) fermions in one dimension.

Figure 2

Long-time limit of transverse correlations ${\mathcal{C}}_{j,j+n}^{xx}\left(t\right)$ for $Jt\gg \left|j\right|,\left|n\right|$ after a quench to the transverse-Ising limit with an initial magnetization domain wall of height $m_0=\frac{1}{2}$. Oscillations are superimposed on an overall exponential decay (inset). The system retains memory of its initial state through both the short wavelength of oscillations and the equivalence between ${\mathcal{C}}_{j,j+n}^{xx}\left(t\right)$ and ${\mathcal{C}}_{j,j+n}^{yy}\left(t\right)$ (not shown).

Figure 3

Long-time limit of transverse correlations ${\mathcal{C}}_{j,j+n}^{xx}\left(t\right)$ for $Jt\gg \left|j\right|,\left|n\right|$ for the ground state of the $XX$ model after a rapid anisotropy and magnetic field quench to the critical transverse-Ising limit $\gamma =\frac{h}{J}=1$. Note that ${\mathcal{C}}_n^{xx}=0$ for $\text{mod}(n,4)=2,3$, which is not captured by Eq. (93). Such oscillations are not present in ${\mathcal{C}}^{yy}$, which possesses the same decay envelope in Eq. (93).

Figure 4

Residual current as a fraction of that which appears in the $XX$ model at long times with the full domain-wall initial state, $m_0=\frac{1}{2}$. Here ${\mathcal{J}}_0=\frac{J}{\pi }$ is the corresponding current for $\gamma =\frac{h}{J}=0$.

Figure 5

Dependence of ratio of residual current to ${\mathcal{J}}_0$ on energy gap $\mathrm{\Delta }=2\gamma \sqrt{1-\frac{h^2}{J^2}}$ in the final Hamiltonian.

Figure 6

Current approaching nonequilibrium steady-state value for $m_0=0.25, h=J, \gamma =1$ (main). Magnetization also relaxes on either side of the domain wall and within the center region (inset). For $m_0<0.5$, the central region relaxes to nonzero magnetization. Plots depict the average of instantaneous magnetization or current on six neighboring lattice sites obtained by applying Eqs. (43) and (44) to the initial state studied in Ref. [34], which corresponds to a sharp domain wall of height $m_0<\frac{1}{2}$.

Figure 7

Magnitude of ${\mathcal{C}}^{xx}\left(n\right)$ in central nonequilibrium steady state for $m_0=0.05, h=0$, and various values of $\gamma$. Power-law decay is known to form the envelope of oscillations for $\gamma =0$ whereas exponential decay sets in for increasing $\gamma$. While the current drops rapidly as $\gamma$ increases, the spatial oscillation wavelength is only weakly affected by the anisotropy. We note that for $h=0$ the magnetization in the central region approaches zero.