Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm

We present an algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems. The algorithm can be used, e.g., to construct thermal states or to simulate real time evolution given by a generic master equation. Its two main ingredients are (i) a superoperator renormalization scheme to efficiently describe the state of the system and (ii) the time evolving block decimation technique to efficiently update the state during a time evolution. The computational cost of a simulation increases significantly with the amount of correlations between subsystems, but it otherwise depends only linearly on the system size. We present simulations involving quantum spins and fermions in one spatial dimension.

Figure 1

Quantum Ising chain with transverse magnetic field, Eq. (17), at finite temperature. Local dimension $d=2$, $n=100$ sites, and effective ${\chi }_{♯}=80$. At zero temperature, $\beta \to \infty$, this model corresponds to a quantum critical point. The spectrum $\{\lambda _{♯\alpha }{}^2\}$ of the reduced superoperator $\mathcal{Q}_{♯}{}^{[L]}$ for the left half chain is plotted as a function of $\beta \in [0,10/J]$ (only the $52$ largest eigenvalues are shown). For any inverse temperature $\beta$, a fast decay of $\lambda _{♯\alpha }{}^2$ in $\alpha$ ensures that the state can be accurately approximated by a MPD with small effective ${\chi }_{♯}$.

Figure 2

Fermionic lattice of Eq. (18) at finite temperature $\beta =1/J$, dephasing $\gamma =0.4J$, bias ${\mu }_0=0.1J$, and with $d=2$, $n=100$. The particle current [see Eq. (20)] is due to a time-dependent applied bias $\mu (t)$ with turn-on time $t_0=2/J$ and rise time $t_s=0.1/J$. The exact solution is obtained by numerically integrating the (Gaussian) time evolution for two-point correlators. Instead, the numerical simulations are achieved after mapping the fermion lattice into a spin lattice with nearest neighbor couplings by means of a Jordan-Wigner transformation. The time evolution is then broken into small gates for pairs of nearest neighbor spins and implemented using the TEBD algorithm. The simulations with an effective ${\chi }_{♯}=30,40,50$ show rapid convergence to the exact solution. Figure 3 justifies such convergence. This will also be addressed in more detail in 13.

Figure 3

Same system as in Fig. 2. The spectrum $\{\lambda _{♯\alpha }{}^2\}$ of the reduced superoperator $\mathcal{Q}_{♯}{}^{[L]}$ for the left $n/2$ sites is plotted as a function of time. The number of relevant eigenvalues $\lambda _{♯\alpha }{}^2$, say above ${10}^{-6}$, increases as the applied bias is turned on, but remains small throughout the evolution, and it even decreases for long times.

Figure 4

Fermionic lattice of Eq. (18) at finite temperature $\beta =1/J$, dephasing $\gamma =0.4J$, and with $d=2$, $n=100$, and no applied bias, $\mu (t)=0$. Unequal-time, two-point correlator (21) for $l=2,4,6,8,10$ and $t\in [0,10/J]$. The results corresponding to an effective ${\chi }_{♯}=40$ and $50$ practically overlap at all times, as the inset shows.