Tensor product representation of topological ordered phase: necessary symmetry conditions

TL;DR

This paper investigates the tensor product state representation of topological ordered phases, revealing that only tensor variations respecting certain symmetries correspond to physically realizable ground state changes, guiding variational studies.

Contribution

It demonstrates that tensor variations in TPS for topological phases must respect specific symmetries to be physically meaningful, providing a key insight for variational methods.

Findings

Not all tensor variations correspond to local Hamiltonian perturbations.

A necessary condition for physical tensor variations is symmetry preservation.

Explicit calculation shows how topological entanglement entropy changes with tensor variations.

Abstract

The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long range entanglement. However, an important issue arises when we use tensor product states (TPS) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with $\mathbb{Z}_2$ topological order, we show that, surprisingly, not all variations of the tensors correspond to the variation of the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations of the tensors to be physical is that they respect certain $\mathbb{Z}_2$ symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order.