Topological Field Theory and Matrix Product States

TL;DR

The paper builds a precise bridge between 1D gapped phases described by Matrix Product States and 2D Topological Quantum Field Theories. It shows that RG-fixed MPS correspond to modules over finite-dimensional semisimple algebras, with the phase determined solely by the center $Z(A)$, and that unitary 2D TQFTs are classified by this center via the state-sum construction. Extending to finite symmetry $G$, the authors identify $G$-equivariant algebras and Morita equivalence as the organizing principle, reproducing the known bosonic 1+1D phase classification and the group-cohomology classification of SPT phases through $H^2(G,U(1))$. Twisted sectors, boundary data, and stacking rules are treated within a unified TQFT/state-sum framework, providing a systematic map from algebraic data to physical gapped phases. The results yield a cohesive, algebraic route to understanding how 1D gapped matter phases are captured by 2D TQFTs and clarify the role of symmetry in organizing these phases.

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.

Figures (11)

• Figure 1: An MPS represented as a tensor network
• Figure 2: The 2-2 and the 3-1 Pachner moves
• Figure 3: An elementary shelling representing $T^\mu_{\rho i} T^\rho_{\nu j} =C^k_{ij} T^\mu_{\nu k}$\ref{['b2-2']}. The thick line is a physical boundary.
• Figure 4: An elementary shelling representing $T^\mu_{\rho j}T^\rho_{\nu k} C^{jk}_i = T^\mu_{\nu i}$\ref{['b3-1']}.
• Figure 5: The Poincare dual of a triangle