TY - JOUR

T1 - Many-body topological invariants in fermionic symmetry-protected topological phases

T2 - Cases of point group symmetries

AU - Shiozaki, Ken

AU - Shapourian, Hassan

AU - Ryu, Shinsei

N1 - Funding Information:
This work was supported in part by the National Science Foundation Grant No. DMR-1455296, by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing (SciDAC) program under Award No. FG02-12ER46875, and by Alfred P. Sloan foundation. K.S. is supported by JSPS Postdoctoral Fellowship for Research Abroad.
Publisher Copyright:
©2017 American Physical Society.

PY - 2017/5/25

Y1 - 2017/5/25

N2 - We propose the definitions of many-body topological invariants to detect symmetry-protected topological phases protected by point group symmetry, using partial point group transformations on a given short-range entangled quantum ground state. Here, partial point group transformations gD are defined by point group transformations restricted to a spatial subregion D, which is closed under the point group transformations and sufficiently larger than the bulk correlation length ξ. By analytical and numerical calculations, we find that the ground state (GS) expectation value of the partial point group transformations behaves generically as (GS|gD|GS)∼exp[iθ+γ-αArea(∂D)ξd-1]. Here, Area(∂D) is the area of the boundary of the subregion D, and α is a dimensionless constant. The complex phase of the expectation value θ is quantized and serves as the topological invariant, and γ is a scale-independent topological contribution to the amplitude. The examples we consider include the Z8 and Z16 invariants of topological superconductors protected by inversion symmetry in (1+1) and (3+1) dimensions, respectively, and the lens space topological invariants in (2+1)-dimensional fermionic topological phases. Connections to topological quantum field theories and cobordism classification of symmetry-protected topological phases are discussed.

AB - We propose the definitions of many-body topological invariants to detect symmetry-protected topological phases protected by point group symmetry, using partial point group transformations on a given short-range entangled quantum ground state. Here, partial point group transformations gD are defined by point group transformations restricted to a spatial subregion D, which is closed under the point group transformations and sufficiently larger than the bulk correlation length ξ. By analytical and numerical calculations, we find that the ground state (GS) expectation value of the partial point group transformations behaves generically as (GS|gD|GS)∼exp[iθ+γ-αArea(∂D)ξd-1]. Here, Area(∂D) is the area of the boundary of the subregion D, and α is a dimensionless constant. The complex phase of the expectation value θ is quantized and serves as the topological invariant, and γ is a scale-independent topological contribution to the amplitude. The examples we consider include the Z8 and Z16 invariants of topological superconductors protected by inversion symmetry in (1+1) and (3+1) dimensions, respectively, and the lens space topological invariants in (2+1)-dimensional fermionic topological phases. Connections to topological quantum field theories and cobordism classification of symmetry-protected topological phases are discussed.

UR - http://www.scopus.com/inward/record.url?scp=85023620136&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85023620136&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.95.205139

DO - 10.1103/PhysRevB.95.205139

M3 - Article

AN - SCOPUS:85023620136

VL - 95

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 20

M1 - 205139

ER -