Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)-dimensional topological order
- Univ. of Maryland, College Park, MD (United States). Joint Quantum Institute; Institute for Advanced Study, Princeton, NJ (United States)
- University of California, Berkeley, CA (United States); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States); Univ. of California, Santa Barbara, CA (United States). Kavli Institute for Theoretical Physics
- Harvard Univ., Cambridge, MA (United States)
- Univ. of Pittsburgh, PA (United States)
- Princeton Univ., NJ (United States)
A (2+1)D topologically ordered phase with U(1) symmetry may or may not have a symmetric gapped edge state, even if both thermal and electric Hall conductivity are vanishing. It has recently been discovered that there are “higher” versions of Hall conductivity valid for fermionic fractional quantum Hall (FQH) states that obstruct symmetry-preserving gapped edge states beyond thermal and electric Hall conductivity. In this paper, we show that one can extract higher Hall conductivity from a single wave function of an FQH state, by evaluating the expectation value of the “partial rotation” unitary, which is a combination of partial spatial rotation and a U(1) phase rotation. This result is verified numerically with the fermionic Laughlin state with 𝜈=1/3 and 1/5, as well as the non-Abelian Moore-Read state. Together with topological entanglement entropy, we prove that the expectation values of the partial rotation completely determine if a bosonic/fermionic Abelian topological order with U(1) symmetry has a symmetry-preserving gappable edge state or not. We also show that thermal and electric Hall conductivity of Abelian topological order can be extracted by partial rotations. Even in non-Abelian FQH states, partial rotation provides the Lieb-Schultz-Mattis type theorem constraining the low-energy spectrum of the bulk-boundary system. The generalization of higher Hall conductivity to the case with Lie group symmetry is also presented.
- Research Organization:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation; Gordon and Betty Moore Foundation; Simons Foundation
- Grant/Contract Number:
- AC02-05CH11231
- OSTI ID:
- 2571939
- Journal Information:
- Physical Review. B, Journal Name: Physical Review. B Journal Issue: 24 Vol. 111; ISSN 2469-9950; ISSN 2469-9969
- Publisher:
- American Physical Society (APS)Copyright Statement
- Country of Publication:
- United States
- Language:
- English
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