Classical Simulation of Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions

We present an algorithm to simulate two-dimensional quantum lattice systems in the thermodynamic limit. Our approach builds on the projected entangled-pair state algorithm for finite lattice systems [F. Verstraete and J. I. Cirac, arxiv:cond-mat/0407066] and the infinite time-evolving block decimation algorithm for infinite one-dimensional lattice systems [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)]. The present algorithm allows for the computation of the ground state and the simulation of time evolution in infinite two-dimensional systems that are invariant under translations. We demonstrate its performance by obtaining the ground state of the quantum Ising model and analyzing its second order quantum phase transition.

Figure 1

Diagrammatic representations of (a) a PEPS tensor $A_{sudlr}$ with one physical index $s$ and four inner indices $u$, $d$, $l$ and $r$; (b) local detail of the tensor network $\mathcal{P}$ for an iPEPS. Copies of tensors $A$ and $B$ are connected through four types of links; (c) reduced tensor $a$ of Eq. (2); and (d) local detail of the tensor network $\mathcal{E}$.

Figure 2

The environment ${\mathcal{E}}^{[{\overrightarrow{r}}_1,{\overrightarrow{r}}_2]}$ for a link of type $r$ is first approximated by an infinite strip ${\mathcal{F}}^{[{\overrightarrow{r}}_1,{\overrightarrow{r}}_2]}$ and then by a six-tensor network ${\mathcal{G}}^{[{\overrightarrow{r}}_1,{\overrightarrow{r}}_2]}$. These reductions can be performed according to either a vertical or horizontal scheme (b) or a diagonal scheme (c). Tensors $A$, $A^{\star }$, $B$, and $B^{\star }$ are not part of the environment.

Figure 3

Transfer matrices $R$ (a) and $S$ (b) for the vertical or horizontal contraction scheme. Multiplication of an iMPS by $R$ using the iTEBD algorithm comes at a computational time that scales as $O({\chi }^3{D}^6+{\chi }^2{D}^{8}d)$ (similar costs apply to multiplying by $S$). This cost is lower in diagonal contraction scheme (c) and (d), namely $O({\chi }^3{D}^4+{\chi }^2{D}^{6}d)$, but a slightly larger $\chi$ is required in order to retain the same accuracy.

Figure 4

(a) Transverse magnetization $m_x$ and energy per site $e$ as a function of the transverse magnetic field $h$. The continuous line shows series expansion results (to 26th and 16th order in perturbation theory) for $h$ smaller and larger than $h_c\approx 3.044$ [13]. Increasing $D$ leads to a lower energy per site $e$. For instance, at $h=3.1$, $e(D=2)\approx -1.6417$ and $e(D=3)\approx -1.6423$. (b) Two-point correlator $S_{zz}(l)$ near the critical point, $\lambda =3.05$. For nearest neighbors, the correlator quickly converges as a function of $D$, whereas for long distances we expect to see convergence for larger values of $D$.

Figure 5

Magnetization $m_z(\lambda )$ as a function of the transverse magnetic field $\lambda$. Dashed lines are a guide to the eye. We have used the diagonal scheme for $(D,\chi )=(2,20)$, (3,25) and (4,35) [15] (the vertical or horizontal scheme leads to comparable results with slightly smaller $\chi$.) The inset shows a log plot of $m_z$ versus $|\lambda -{\lambda }_c|$, including our estimate of ${\lambda }_c$ and $\beta$. The continuous line shows the linear fit.