Excited states from local effective Hamiltonians of matrix product states and their entanglement spectrum transition

TL;DR

This paper explores the theoretical foundation of using local effective Hamiltonians derived from matrix product states to access excited states in one-dimensional critical systems, revealing an entanglement spectrum transition explained by conformal field theory.

Contribution

It provides a conformal field theory perspective on the method of obtaining excited states from local effective Hamiltonians of MPS, elucidating the underlying mechanism and entanglement spectrum transition.

Findings

The method effectively uses a truncated basis of Schmidt vectors to represent excited states.

The entanglement spectrum reorganizes into conformal towers as the subsystem size ratio varies.

Numerical results support the CFT-based prediction of an entanglement-spectrum transition.

Abstract

Solving excited states is a challenging task for interacting systems. For one-dimensional critical systems, however, excited states can be directly accessed from the eigenvectors of the local effective Hamiltonian that is constructed from the ground state obtained by variational matrix product state (MPS) optimization. Despite its numerical success, the theoretical mechanism underlying this method has remained largely unexplored. In this work, we provide a conformal field theory (CFT) perspective that helps elucidate this connection. The key insight is that this construction effectively uses a truncated basis of ground-state Schmidt vectors to represent excited states, where the contribution of each Schmidt vector can be expressed as a CFT correlation function and shown to decay with increasing Schmidt index. The CFT analysis further predicts an entanglement-spectrum transition of excited states as the ratio of the subsystem size to the total system size is varied. Our numerical results support this picture and demonstrate a reorganization of the entanglement spectrum into distinct conformal towers as this ratio changes.