Accurate computation of the energy variance and $\langle\langle \mathcal{L}^\dagger \mathcal{L} \rangle\rangle$ using iPEPS

TL;DR

The paper tackles the challenge of accurately estimating the energy variance for iPEPS in two-dimensional quantum systems to enable variational extrapolations to the exact ground state. It introduces a large-cell CTMRG-based contraction (LC-CTMRG) that computes correlators between Hamiltonian terms inside a big unit cell, reducing the required contraction dimension $\chi$ and improving accuracy. Benchmark results on the Heisenberg model, a gapped free-fermion system, and the Shastry-Sutherland model show that variance extrapolation yields energies in agreement with exact or high-precision results, with substantially faster convergence than previous methods. The authors also extend the approach to open quantum systems by evaluating $\epsilon = \langle\langle \mathcal{L}^agger \mathcal{L} \rangle\rangle$, providing a practical diagnostic for steady-state quality and a means to locate first-order dissipative phase transitions in models like the dissipative quantum Ising system. Overall, the LC-CTMRG method integrates with existing iPEPS workflows to offer a robust tool for precise ground-state energies and steady-state analyses in 2D quantum many-body problems.

Abstract

Infinite projected entangled-pair states (iPEPS) provide a powerful tensor network ansatz for two-dimensional quantum many-body systems in the thermodynamic limit. In this paper we introduce an approach to accurately compute the energy variance of an iPEPS, enabling systematic extrapolations of the ground-state energy to the exact zero-variance limit. It is based on the contraction of a large cell of tensors using the corner transfer matrix renormalization group (CTRMG) method, to evaluate the correlator between pairs of local Hamiltonian terms. We show that the accuracy of this approach is substantially higher than that of previous methods, and we demonstrate the usefulness of variance extrapolation for the Heisenberg model, for a free fermionic model, and for the Shastry-Sutherland model. Finally, we apply the approach to compute $\langle \langle \mathcal{L}^\dagger \mathcal{L} \rangle \rangle$ for an open quantum system described by the Liouvillian $\mathcal{L}$, in order to assess the quality of the steady-state solution and to locate first-order phase transitions, using the dissipative quantum Ising model as an example.

Figures (7)

• Figure 1: (a) iPEPS with a $3 \times 2$ unit cell. (b) Double-layer tensor $a$ at coordinate $[x,y]$, defined as the contraction of the iPEPS tensor $A$ with its conjugate $A^\dagger$ along the physical leg. (c) Norm of the state, represented by an infinite square-lattice tensor network of the $a$ tensors. (d) In CTMRG the tensor network surrounding a central site is effectively encoded in four corner ($C$) and edge ($T$) environment tensors. The coordinates $[x,y]$ denote the relative position in the unit cell. (e) Example of the relevant tensor-network diagram for the expectation value of a nearest-neighbor Hamiltonian term in vertical direction (orange rectangle), which is sandwiched between the physical legs of the $A$ and $A^\dagger$ tensors.
• Figure 2: (a) Translationally invariant iPEPS with the unit cell enlarged to size $L\times L$. (b) iPEPS wavefunction from (a) with a Hamiltonian term $H_0$ applied in the center of the cell, resulting in $| \Psi' \rangle$. (c) The overlap $\langle \Psi | \Psi' \rangle$, which we contract using CTMRG, starting from the central columns.
• Figure 3: (a) Convergence of the variance as a function of inverse $\chi$ for $D=4$ and large $L=30$, comparing the previous $H$-environment approach from Ref. corboz16b with the LC-CTMRG method introduced in this work. (b) Convergence of the variance with cell size $L$ for different values of $D$. (c) Energy as a function of the variance. Using a second-order polynomial extrapolation yields a value compatible with the QMC result. The inset shows a zoomed-in view. (d) Energy as a function of inverse bond dimension, showing a non-monotonic behavior, which makes it difficult to extrapolate to the infinite $D$ limit.
• Figure 4: Energy per site as a function of the variance (left panels) and inverse bond dimension $1/D$ (right panels), for two different values of $\Delta$. The extrapolated results based on the variance are in agreement with the exact results, indicated by the green lines. The dotted lines in the right panels are a guide to the eye.
• Figure 5: Convergence of the variance as a function of cell size $L$, for different values of $\chi$. A small value of $\chi$ can lead to an artificial drift of the variance value with increasing cell size, as can be seen in the $\chi=40$ data.