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. Nonuniqueness 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.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. Nonuniqueness 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.

arXiv:1607.06766Physical review. B./Physical review. BAug 14, 201718 citations

A new class of (2 + 1)-dimensional topological superconductors with $\mathbb {Z}_8$ topological classificationA new class of (2 + 1)-dimensional topological superconductors with $\mathbb {Z}_8$ topological classification

Xiao-Liang Qi

The classification of topological states of matter depends on spatial dimension and symmetry class. For non-interacting topological insulators and superconductors, the topological classification is obtained systematically and non-trivial topological insulators are classified by either integer or Z 2 . The classification of interacting topological states of matter is much more complicated and only special cases are understood. In this paper we study a new class of topological superconductors in (2 + 1) dimensions which has time-reversal symmetry and a spin conservation symmetry. We demonstrate that the superconductors in this class are classified by when electron interaction is considered, while the classification is without interaction.The classification of topological states of matter depends on spatial dimension and symmetry class. For non-interacting topological insulators and superconductors, the topological classification is obtained systematically and non-trivial topological insulators are classified by either integer or Z 2 . The classification of interacting topological states of matter is much more complicated and only special cases are understood. In this paper we study a new class of topological superconductors in (2 + 1) dimensions which has time-reversal symmetry and a spin conservation symmetry. We demonstrate that the superconductors in this class are classified by when electron interaction is considered, while the classification is without interaction.

arXiv:1202.3983New Journal of PhysicsJun 3, 2013137 citations

Topological States of Quantum Condensed Matter

F. D. M. Haldane

Bulletin of the American Physical SocietyMar 15, 20170 citations

Classification of (2+1) D invertible fermionic topological phases with symmetry

Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin +1 more

Integer quantum Hall states and their generalization to topological insulators and superconductors are paradigmatic examples of invertible fermionic topological states of matter. Here, the authors develop a comprehensive characterization and classification of invertible fermionic topological phases of matter in two spatial dimensions for general symmetry groups. The results are nonperturbative and apply to strongly interacting systems. In particular, they extend previous classification results to account for the possible chiral nature of invertible phases.Integer quantum Hall states and their generalization to topological insulators and superconductors are paradigmatic examples of invertible fermionic topological states of matter. Here, the authors develop a comprehensive characterization and classification of invertible fermionic topological phases of matter in two spatial dimensions for general symmetry groups. The results are nonperturbative and apply to strongly interacting systems. In particular, they extend previous classification results to account for the possible chiral nature of invertible phases.

arXiv:2109.11039Physical review. B./Physical review. BJun 29, 202264 citations

Dynamics of the Topological States of Matter

Minchul Lee, Mahn‐Soo Choi

Physics and High TechnologyMay 31, 20160 citations

G

, this leads to a classification of gapped phases in 1+1d in terms of Morita-equivalence classes of

G

-equivariant algebras. Nonuniqueness 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.

A new class of (2 + 1)-dimensional topological superconductors with

\mathbb {Z}_8

topological classification