Time evolution of an infinite projected entangled pair state: An efficient algorithm

An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension $D$. Its real, Lindbladian, or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension $D^{\prime }>D$, which is approximated by an iPEPS with the original $D$. In P. Czarnik and J. Dziarmaga, Phys. Rev. B 98, 045110 (2018), an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension $D^{\prime }$. In this paper, we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required $D$. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field $h_x$ in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: $0.1\%$ for $h_x=2.5$ and $0.5\%$ for the more challenging case of $h_x=2.9$ close to the quantum critical point at $h_x=3.04438\left(2\right)$.

Figure 1

In (a), an elementary rank-6 tensor $A$ of a purification is shown. The top (orange) index numbers ancilla states $a=0,1$, the bottom (red) index numbers spins states $s=0,1$, the four (black) bond indices have a bond dimension $D$. In (b), an iPEPS representation of the purification. Here pairs of elementary tensors at nearest-neighbor sites are contracted through their connecting bond indices. The dotted lines connect with the rest of the infinite lattice. We obtain the iPEPS representation of a pure state by reducing the dimension of ancilla indices to 1 or, equivalently, erasing the ancilla lines altogether.

Figure 2

In (a), an infinite square lattice is divided into two sublattices with tensors $A$ (brighter green) and $B$ (green). In (b), SVD decomposition of the nearest-neighbor gate in Eq. (11) is applied to spin indices of every considered nearest-neighbor pair of tensors $A$ and $B$. At the same time, a disentangler $g$ is applied to their ancilla indices. The result is an exact purification $|{\psi }^{\prime }\rangle$. In (c), the tensors $A$ and $B$ can be conveniently fused with their respective pair of $z$ tensors becoming new tensors $A^{\prime }$ and $B^{\prime }$ with a doubled bond dimension $2D$. In (d), the algorithm optimizes $g$ together with new tensors $A^{″}$ and $B^{″}$—that have the original bond dimension $D$—so that an approximate new iPEPS $|{\psi }^{″}\rangle$ made of $A^{″}$ and $B^{″}$ has a maximal fidelity to the exact $|{\psi }^{\prime }\rangle$.

Figure 3

In (a), tensor $A^{\prime }$ is contracted with its complex conjugate into a transfer tensor $t_A$. In (b), tensor $B^{\prime }$ is contracted with its complex conjugate into a transfer tensor $t_B$. In (c), the tensor environment for the bond $A^{″}-B^{″}$. The environment is a rank-6 tensor, each index has dimension $D^2$.

Figure 4

In all panels, each of the six indices of the environment in Fig. 3 was split into a ket (upper) and a bra (lower) index, each of dimension $D$. In (a), metric tensor $G_A$. In (b), gradient $V_A$. In (c), tensor environment for the disentangler $E\left(g\right)$.

Figure 5

Transverse magnetization $\langle X\rangle$ (left column) and relative energy error per site (right column) after a sudden quench from a ground state in a limit of $h_x\to \infty$, with all spins pointing along $x$, down to a finite $h_x=2h_0$ (top row), $h_x=h_0$ (middle row), and $h_x=h_0/10$ (bottom row). With increasing bond dimension $D=2,...,8$ the magnetization appears converged for increasingly long times. The relative energy error is defined as $\mathrm{\Delta }E/E_0$, where $E_0=-h_x$ is the initial energy at $t=0^{+}$ right after the quench and $\mathrm{\Delta }E=E\left(t\right)-E_0$ is the energy error. The energy conservation shows improvement with increasing $D$.

Figure 6

Time evolution of the longitudinal magnetization $\langle Z\rangle$ with the Markovian master Eq. (19) initialized with all spins pointing down: $\langle Z\rangle =-1$. Different colors correspond to different bond dimensions. With increasing $D$ the curves appear converged over an increasing range of evolution time.

Figure 7

Thermal states obtained by imaginary time eeFU evolution of a purification. In (a) and (b), the longitudinal magnetization $m=\langle Z\rangle$ as a function of the inverse temperature $\beta$ for longitudinal symmetry breaking bias field $h_z=0.01$ and two different values of the transverse field $h_x=2.5$ (a) and $h_x=2.9$ (b). With increasing $D$, the magnetization plots appear to converge. The $D=5$ magnetization plot appears to be already converged. The convergence is slower for the larger transverse field $h_x=2.9$ for which quantum fluctuations are stronger. In (c), we show $m$ for smaller $h_z=3\times {10}^{-4}$ and $h_x=2.9$. Smaller $h_z$ makes simulations more demanding, but again the $D=5$ magnetization plot appears to be converged.

Figure 8

A comparison of the eeFU with variational tensor network renormalization (VTNR) [38, 46, 47] for $h_x=2.9$ and $h_z=0.01$. Both methods converge to each other with increasing $D$. The convergence is faster for the eeFU algorithm.

Figure 9

In (a), $m^{\prime }\left(T\right)$ for $h_x=2.5$ and $D=5$. The blue dashed line shows the QMC $T_c$. In (b), $T_c$ is estimated by fitting the scaling Eq. (23) of $m^{\prime }$ peaks $T^{*}\left(h_z\right)$. Here we use $0.0005\le h_z\le 0.01$ and $D=5$. We obtain $T_c=1.2745\left(7\right)$ and $1/\tilde{\beta }\delta =0.549\left(4\right)$. The obtained $T_c$ agrees with the QMC $T_c=1.2737\left(6\right)$ [80], while obtained $1/\tilde{\beta }\delta$ is close to the exact value $1/\tilde{\beta }\delta =8/15\approx 0.533$ (it differs from the exact value by $3\%$).

Figure 10

In (a), $m^{\prime }\left(T\right)$ for $h_x=2.9$ and $D=5$. The blue dashed line shows the QMC $T_c$. In (b) $T_c$ and $1/\tilde{\beta }\delta$ are estimated by fitting the scaling Eq. (23) of $m^{\prime }$ peaks $T^{*}\left(h_z\right)$. Here we use $0.0005\le h_z\le 0.01$ and $D=5$ obtaining $T_c=0.6055\left(10\right)$ and $1/\tilde{\beta }\delta =0.563\left(4\right)$.

Figure 11

In (a) and (b), we show the longitudinal magnetization $m=\langle Z\rangle$ in function of the inverse temperature $\beta$ obtained with (eeFUd) and without (eeFU) the disentanglers for the transverse field $h_x=2.9$. In (a) results for the longitudinal symmetry breaking field $h_z=0.01$. In (b) results for more demanding $h_z=5\times {10}^{-4}$. In both cases the $D=4$ results with the disentanglers are much closer to the converged $D=5-6$ results. We see also that the $D=5$ results with the disentanglers are equivalent to the $D=5-6$ results obtained without them. In (c) $T_c$ and and $1/\tilde{\beta }\delta$ estimation by the scaling Eq. (23) for $0.0005\le h_z\le 0.01$ and $D=4,5$. The results corroborate the conclusions drawn from (a) and (b). Numerical values of the fitted $T_c$ and $1/\tilde{\beta }\delta$ are listed in Table 1.

Figure 12

A comparison of thermal states obtained by the FU and the eeFU algorithms. Here $h_x=2.9$ and $h_z=3\times {10}^{-4}$. Both algorithms give similar results.

Figure 13

A comparison of thermal states obtained by the full update (FU) algorithm and the simple update (SU) evolution of a thermal state purification for $h_x=2.9$ and $h_z=5\times {10}^{-4}$. With increasing $D$, the SU magnetization moves slowly toward the converged FU magnetization ($D=5,6$) but even for the largest $D=14$ it is still far from it.

Figure 14

Tensors $A$ and $B$ are decomposed into isometries $Q_A$ and $Q_B$ and smaller reduced tensors $A_{\mathrm{red}}$ and $B_{\mathrm{red}}$. A similar decomposition, with the same $Q_A$ and $Q_B$, applies to the pairs $A^{\prime },B^{\prime }$ and $A^{″},B^{″}$. We show two versions of reduced tensors. The larger ones are intended for simulations with disentanglers while the smaller ones, which leave ancilla indices with the isometries, for simulations without the disentanglers.