Nonequilibrium variational cluster perturbation theory: Quench dynamics of the quantum Ising model

We introduce a variational implementation of cluster perturbation theory (CPT) to address the dynamics of spin systems driven out of equilibrium. We benchmark the method with the quantum Ising model subject to a sudden quench of the transverse magnetic field across the transition or within a phase. We treat both the one-dimensional case, for which an exact solution is available, as well the two-dimensional case, for which we have to resort to numerical results. Comparison with exact results shows that the approach provides a quite accurate description of the real-time dynamics up to a characteristic timescale $\tau$ that increases with the size of the cluster used for CPT. In addition, and not surprisingly, $\tau$ is small for quenches across the equilibrium phase transition point, but can be quite larger for quenches within the ordered or disordered phases.

Figure 1

The $L$-shaped Kadanoff–Baym contour $\mathcal{C}$. The arrows indicate the contour ordering. For example, $t^{\prime }$ lies ahead of $t$ in the ordering ($t>t^{\prime }$), i.e., operators at $t^{\prime }$ are sorted to the right by the contour ordering.

Figure 2

Partitioning of a $D=1$ lattice into clusters with size $L_c=4$. The intercluster hopping is denoted by $V$.

Figure 3

Magnetization along the $z$ direction versus magnetic field for different sites on a lattice of size $L=8$ with open boundary conditions. Cluster size in CPT is $L_c=4$. Exact results are also being reported for comparison.

Figure 4

(left panel) Ground-state energy from VCPT compared with the exact value for lattice of size $L=8$ with open boundary conditions. (right panel) $f=J\langle S_{1B}^x\rangle =J\langle S_{4A}^x\rangle$.

Figure 5

Magnetization of different sites versus magnetic field for a lattice of size $L=8$ with open boundary conditions. The cluster size in VCPT is $L_c=4$. Exact results are also reported for comparison.

Figure 6

Time dependence of the magnetization in the $z$ direction for different sites on a lattice of size $L=8$. The cluster size in NE-VCPT is $L_c=4$. The magnetic field has been suddenly changed from $h_0=0.2$ to $h=1.2$. The exact dynamics is shown for comparison.

Figure 7

Time dependence of the magnetization for an infinite Ising chain evaluated within NE-VCPT with different cluster sizes $L_c$ compared with exact results for a chain of length $L=400$.

Figure 8

Dynamics for the magnetization compared with Lanczos results for different types of quenches in two dimension. NE-VCPT calculation is for an infinite lattice with the cluster size of $L_c=2\times 2$.