Accurate determination of low-energy eigenspectra with multi-target matrix product states

TL;DR

This paper introduces two innovative algorithms using multi-target matrix product states to accurately compute low-energy eigenspectra of quantum many-body systems, demonstrating high precision and efficiency on the transverse-field Ising model.

Contribution

The work presents novel canonicalization and variational algorithms for multi-target MPS, improving convergence speed and accuracy in low-energy spectrum calculations.

Findings

Algorithms achieve high accuracy matching exact solutions.

Eigenenergies converge uniformly in gapped phases.

Methods are versatile and applicable to quantum many-body systems.

Abstract

Determining the low-energy eigenspectra of quantum many-body systems is a long-standing challenge in physics. In this work, we solve this problem by introducing two novel algorithms to determine low-energy eigenstates based on a compact matrix product state (MPS) representation of the multiple targeted eigenstates. The first algorithm utilizes a canonicalization approach that takes advantage of the imaginary-time evolution of multi-target MPS, offering faster convergence and ease of implementation. The second algorithm employs a variational approach that optimizes local tensors on the Grassmann manifold, capable of achieving higher accuracy. These algorithms can be used independently or combined to enhance convergence speed and accuracy. We apply them to the transverse-field Ising model and demonstrate that the calculated low-energy eigenspectra agree remarkably well with the exact solution. Moreover, the eigenenergies exhibit uniform convergence in gapped phases, suggesting that the low-energy excited eigenstates have nearly the same level of accuracy as the ground state. Our results highlight the accuracy and versatility of multi-target MPS-based algorithms for determining low-energy eigenspectra and their potential applications in quantum many-body physics.