Gradient methods for variational optimization of projected entangled-pair states

We present a conjugate-gradient method for the ground-state optimization of projected entangled-pair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimization is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.

Figure 1

The magnetization curve for the transverse Ising model with bond dimensions $D=2$ (blue) and $D=3$ (orange). We nicely capture the phase transition, although the critical point has been slightly shifted. The critical point can be estimated as the point where the slope of the curve is maximal; we arrive at ${\lambda }_c\approx 3.09$ ($D=2$) and ${\lambda }_c\approx 3.054$ ($D=3$).

Figure 2

Our variational results compared to the results that are obtained with imaginary-time evolution using the full-update algorithm; the comparison is done for the transverse Ising model at $\lambda =3.04$ for bond dimensions $D=2$ and $D=3$. On the left we have plotted the convergence for the energy (magnetization) as a function of the Trotter step size of the full-update scheme (blue points), and our results (red line). For both plots, the upper (lower) lines are for $D=2$ ($D=3$).

Figure 3

Details on the convergence of the optimization algorithm for the Ising model at $\lambda =3$. Upper panel: The convergence of the norm of the gradient $\parallel g\parallel =\sqrt{g^{†}g}$ (blue), the error in the energy (red), and the error in the magnetization (yellow), as a function of the iteration. The errors are computed as the relative error with respect to the last iteration. In this $D=2$ simulation the convergence criterion was $\parallel g\parallel \le {10}^{-5}$, a value for which the two plotted observables have clearly converged. Lower panel: The convergence of the norm of the gradient as a function of the bond dimension $\chi$ of the corner environment, at a particular iteration of the conjugate-gradient scheme for $D=3$ (close to convergence). This plot shows that large values of $\chi$ are needed to obtain a required tolerance on the norm of the gradient (in this case $\chi \approx 100$).

Figure 4

Results for the $\mathit{XY}$ model ($J_z=0$), compared to the Monte Carlo results in Ref. [60]. Left: The relative error $\mathrm{\Delta }e=\left|(e_{\text{var}}-e_{\text{MC}})/e_{\text{MC}}\right|$ as a function of the bond dimension. Right: The staggered magnetization as a function of the bond dimension; the red line is the Monte Carlo result with error bars.

Figure 5

Results for the Heisenberg antiferromagnet ($J_z=1$), compared to the Monte Carlo results in Refs. [61] and [62]. Left: The relative error $\mathrm{\Delta }e=\left|(e_{\text{var}}-e_{\text{MC}})/e_{\text{MC}}\right|$ as a function of the bond dimension. Right: The staggered magnetization as a function of the bond dimension; the red line is the Monte Carlo result for which the error bars are too small to plot.

Figure 6

The energy expectation value as a function of the variance per site for the isotropic ($J_z=1$) Heisenberg antiferromagnet, for four values of the bond dimension ($D=2\to 5$). The red line represents the Monte Carlo result for the ground-state energy [61]. A linear extrapolation with respect to the two best points ($D=4,5$) gives an energy with a relative error of $\mathrm{\Delta }e\approx 4.7\times {10}^{-5}$. The striped line is drawn between the exact MC result and the $D=5$ point and serves only as a guide to the eye.

Figure 7

Structure factor $s\left(\vec{q}\right)$ of the isotropic ($J_z=1$) Heisenberg antiferromagnet along a path through the Brillouin zone for optimized PEPS states with bond dimensions $D=2$ (blue), $D=3$ (red), $D=4$ (orange), and $D=5$ (purple), in agreement with the results in Refs. [63] and [64]. The divergence around the $X$ point and the zero around the $\mathrm{\Gamma }$ point are better reproduced as $D$ increases, although the improvement as a function of $D$ seems not to be smooth.