Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes

TL;DR

This paper presents an exact classification of quantum phases in a specific subclass of frustration-free Hamiltonians, using logical operators from quantum coding theory to distinguish phases and identify quantum phase transitions.

Contribution

It introduces a model covering many stabilizer Hamiltonians with locality, translation, and scale symmetries, and establishes a method to classify quantum phases via logical operators.

Findings

Quantum phases in 2D models can be classified exactly.

Logical operators' shapes determine quantum phase distinctions.

Models with topologically different logical operators are separated by phase transitions.

Abstract

Searches for possible new quantum phases and classifications of quantum phases have been central problems in physics. Yet, they are indeed challenging problems due to the computational difficulties in analyzing quantum many-body systems and the lack of a general framework for classifications. While frustration-free Hamiltonians, which appear as fixed point Hamiltonians of renormalization group transformations, may serve as representatives of quantum phases, it is still difficult to analyze and classify quantum phases of arbitrary frustration-free Hamiltonians exhaustively. Here, we address these problems by sharpening our considerations to a certain subclass of frustration-free Hamiltonians, called stabilizer Hamiltonians, which have been actively studied in quantum information science. We propose a model of frustration-free Hamiltonians which covers a large class of physically realistic stabilizer Hamiltonians, constrained to only three physical conditions; the locality of interaction terms, translation symmetries and scale symmetries, meaning that the number of ground states does not grow with the system size. We show that quantum phases arising in two-dimensional models can be classified exactly through certain quantum coding theoretical operators, called logical operators, by proving that two models with topologically distinct shapes of logical operators are always separated by quantum phase transitions.