Topological Field Theory and Matrix Product States

Abstract

It is believed that most (perhaps all) gapped phases of matter can be described at long distances by Topological Quantum Field Theory (TQFT). On the other hand, it has been rigorously established that in 1+1d ground states of gapped Hamiltonians can be approximated by Matrix Product States (MPS). We show that the state-sum construction of 2d TQFT naturally leads to MPS in their standard form. In the case of systems with a global symmetry $G$, this leads to a classification of gapped phases in 1+1d in terms of Morita-equivalence classes of $G$-equivariant algebras. Non-uniqueness of the MPS representation is traced to the freedom of choosing an algebra in a particular Morita class. In the case of Short-Range Entangled phases, we recover the group cohomology classification of SPT phases.

14 pages, references added and minor clarifications made

References in corpus (3)

• Entropy and Entanglement in Quantum Ground States
• D-branes and K-theory in 2D topological field theory
• Spin Topological Field Theory and Fermionic Matrix Product States

Cited by in corpus (17)

• Symmetry Protected Topological Order in Open Quantum Systems
• On lattice models of gapped phases with fusion category symmetries
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Topological field theories and symmetry protected topological phases with fusion category symmetries
• Spin Topological Field Theory and Fermionic Matrix Product States
• Fermionization of fusion category symmetries in 1+1 dimensions
• Fermionic Matrix Product States and One-Dimensional Short-Range Entangled Phases with Anti-Unitary Symmetries
• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• The structure of fixed-point tensor network states characterizes patterns of long-range entanglement
• Diagrammatic State Sums for 2D Pin-Minus TQFTs
• Non-local Order Parameters and Quantum Entanglement for Fermionic Topological Field Theories
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• Preparing matrix product states via fusion: constraints and extensions
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• A Classifying Space for Phases of Matrix Product States
• Non-local order parameters for fermion chains via the partial transpose
• Detection of 2D SPT Order with Partial Symmetries