Computing excited states with isometric tensor networks in two-dimensions

TL;DR

This work develops block-isoPEPS, a block-structured isometric tensor-network ansatz for two-dimensional quantum lattices, to compute several low-lying eigenpairs via an inexact subspace iteration driven by imaginary-time evolution. Central to the approach are the block Moses Move and isometric tensor ring decompositions, which enable controlled truncations and efficient evaluation of observables while maintaining an orthogonal block basis. The authors analyze error sources and provide cost estimates that scale favorably (roughly as a seventh power of bond dimensions) compared with standard PEPS tangent-space methods, and they validate the method on the 2D transverse-field Ising and Heisenberg models, demonstrating accurate excitations beyond the ground state. The results suggest block-isoPEPS as a scalable pathway to study excitations in 2D quantum many-body systems, with potential for symmetry extensions, 3D generalizations, and accelerated implementations.

Abstract

We present a new subspace iteration method for computing low-lying eigenpairs (excited states) of high-dimensional quantum many-body Hamiltonians with nearest neighbor interactions on two-dimensional lattices. The method is based on a new block isometric projected entangled pair state (block-isoPEPS) ansatz that generalizes the block matrix product state (MPS) framework, widely used for Hamiltonians defined on one-dimensional chains, to two-dimensions. The proposed block-isoPEPS ansatz offers several attractive features for PEPS-based algorithms, including exact block orthogonalization, controlled local truncation via singular value decompositions, and efficient evaluation of observables. We demonstrate the proposed inexact subspace iteration for block-isoPEPS by computing excitations of the two-dimensional transverse-field Ising and Heisenberg models and compare our results with existing PEPS methods. Our results demonstrate that block isometric tensor networks provide a scalable framework for studying excitations in quantum many-body systems beyond one dimension.

Figures (4)

• Figure 1: Tensor network diagrams for representing three tensors. The block MPS network, shown here with 3 sites, provides a compact representation of states with limited entanglement in 1D geometries. In this paper we propose the block (isometric) PEPS ansatz, shown here with 9 sites, as a compact representation of several states with limited entanglement in 2D geometries.
• Figure 2: Relative eigenvalue error for the 2D transverse field Ising (TFI) model with $g=3.5$ on a $6 \times 6$ lattice. Eigenvalues were computed using the block-isoPEPS subspace iteration algorithm with bond dimensions $\chi = 8$ and $\eta = 16$. Results are shown for block sizes $p = 1$, $2$, and $3$.
• Figure 3: Relative eigenvalue error for the transverse field Ising (TFI) model \ref{['eq:TFI']} with $g=3.5$ and $g=3.0$ as a function of lattice side length $L_x$. Approximate eigenvalues were computed using $50$ iterations of block iso-PEPS subspace iteration with bond dimensions $\chi = 12$ and $\eta = 20$ and $\tau = 0.1$. Results are shown for block sizes $p = 1$ and $p = 2$.
• Figure 4: Relative eigenvalue error for the Heisenberg model \ref{['eq:heis']} versus lattice side length $L_x$. Approximate eigenvalues were computed using $50$ iterations of block iso-PEPS subspace iteration with bond dimensions $\chi = 12$ and $\eta = 36$ and $\tau = 0.1$. Results are shown for block sizes $p = 1$ (left) and $p = 2$ (right).