[Paper Review] Towards a mathematical formalism for classifying phases of matter

Andreas Bauer, Jens Eisert+1 more|arXiv (Cornell University)|Mar 13, 2019

Quantum many-body systems81 references49 citations

TL;DR

The paper proposes a unified mathematical framework based on tensor lattices and local moves to classify phases of matter, encompassing symmetry-breaking, topological phases, and related structures, with extensions to boundaries, defects, and anyons in 2+1D.

ABSTRACT

We propose a unified mathematical framework for classifying phases of matter. The framework is based on different types of combinatorial structures with a notion of locality called lattices. A tensor lattice is a local prescription that associates tensor networks to those lattices. Different lattices are related by local operations called moves. Those local operations define consistency conditions for the tensors of the tensor network, the solutions to which yield exactly solvable models for all kinds of phases. We implement the framework to obtain models for symmetry-breaking and topological phases in up to three space-time dimensions, their boundaries, defects, domain walls and symmetries, as well as their anyons for 2+1-dimensional systems. We also deliver ideas of how other kinds of phases, like SPT/SET, fermionic, free-fermionic, chiral, and critical phases, can be described within our framework. We also define another structure called contracted tensor lattices which generalize tensor lattices: The former associate tensors instead of tensor networks to lattices, and the consistency conditions for those tensors are defined by another kind of local operation called gluings. Using this generalization, our framework also covers mathematical structures like axiomatic (non-fully extended or defective) TQFTs, that do not directly describe phases on a microscopic physical level, but formalize certain aspects of potential phases, like the anyon statistics of 2+1-dimensional phases. We also introduce the very powerful concept of (contracted) tensor lattice mapping, unifying a lots of different operations, such as stacking, anyon fusion, anyon condensation, equivalence of different fixed point models, taking the Drinfel'd centre, trivial defects or interpreting a bosonic model as a fermionic model.

Motivation & Objective

• Develop a unified mathematical framework for classifying phases of matter.
• Introduce tensor lattices as local prescriptions for tensor networks on lattices.
• Define local moves that relate different lattices and impose consistency conditions.
• Demonstrate the framework on symmetry-breaking and topological phases in up to 3 space-time dimensions.
• Explore extensions to boundaries, defects, domain walls, symmetries, and anyons; outline descriptions for SPT/SET, fermionic, chiral, and critical phases.

Proposed method

• Define tensor lattices as local prescriptions associating tensor networks to lattices.
• Impose consistency via local moves that relate lattices and constrain tensor networks.
• Introduce contracted tensor lattices with gluings and a corresponding notion of consistency.
• Show how the framework yields exactly solvable models for various phases in 2+1 and 3+1 dimensions.
• Develop the concept of (contracted) tensor lattice mapping to unify operations like stacking, anyon fusion/condensation, and Drinfeld center interpretations.
• Relate to axiomatic TQFTs to formalize certain phase aspects beyond direct microscopic models.

Experimental results

Research questions

• RQ1 Can a single mathematical formalism classify both symmetry-breaking and topological phases across dimensions?
• RQ2 How do local moves and consistency conditions yield exactly solvable models for different phases?
• RQ3 What is the role of contracted tensor lattices and gluings in encompassing broader phase structures (e.g., TQFTs, anyon statistics)?

Key findings

• A unified tensor-lattice framework can describe symmetry-breaking and topological phases up to 3 space-time dimensions.
• Local moves define equivalence relations between lattice representations and enforce tensor-network consistency.
• Contracted tensor lattices generalize tensor lattices and connect to gluings, enabling descriptions of boundaries, defects, and anyons.
• The approach provides pathways to describe SPT/SET, fermionic, free-fermionic, chiral, and critical phases within the same formalism.
• Connections to axiomatic TQFTs are established, formalizing aspects like anyon statistics in 2+1D.
• A tensor-lattice mapping unifies operations such as stacking, anyon fusion, condensation, and fixed-point model equivalences.