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Title: Invariance of Topological Indices Under Hilbert Space Truncation

Abstract

Here, we show that the topological index of a wave function, computed in the space of twisted boundary phases, is preserved under Hilbert space truncation, provided the truncated state remains normalizable. If truncation affects the boundary condition of the resulting state, the invariant index may acquire a different physical interpretation. If the index is symmetry protected, the truncation should preserve the protecting symmetry. We discuss implications of this invariance using paradigmatic integer and fractional Chern insulators, Z 2 topological insulators, and spin-1 Affleck-Kennedy-Lieb-Tasaki and Heisenberg chains, as well as its relation with the notion of bulk entanglement. As a possible application, we propose a partial quantum tomography scheme from which the topological index of a generic multicomponent wave function can be extracted by measuring only a small subset of wave function components, equivalent to the measurement of a bulk entanglement topological index.

Authors:
ORCiD logo [1]; ORCiD logo [1];  [2]; ORCiD logo [1]; ORCiD logo [3]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Univ. of California San Diego, La Jolla, CA (United States)
  3. Los Alamos National Lab. (LANL), Los Alamos, NM (United States); NORDITA, Stockholm (Sweden)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1418765
Alternate Identifier(s):
OSTI ID: 1415875
Report Number(s):
LA-UR-17-24060
Journal ID: ISSN 0031-9007; PRLTAO; TRN: US1801298
Grant/Contract Number:  
AC52-06NA25396; BES E3B7
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Physical Review Letters
Additional Journal Information:
Journal Volume: 120; Journal Issue: 1; Journal ID: ISSN 0031-9007
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Material Science

Citation Formats

Huang, Zhoushen, Zhu, Wei, Arovas, Daniel P., Zhu, Jian -Xin, and Balatsky, Alexander V. Invariance of Topological Indices Under Hilbert Space Truncation. United States: N. p., 2018. Web. doi:10.1103/PhysRevLett.120.016403.
Huang, Zhoushen, Zhu, Wei, Arovas, Daniel P., Zhu, Jian -Xin, & Balatsky, Alexander V. Invariance of Topological Indices Under Hilbert Space Truncation. United States. doi:10.1103/PhysRevLett.120.016403.
Huang, Zhoushen, Zhu, Wei, Arovas, Daniel P., Zhu, Jian -Xin, and Balatsky, Alexander V. Fri . "Invariance of Topological Indices Under Hilbert Space Truncation". United States. doi:10.1103/PhysRevLett.120.016403. https://www.osti.gov/servlets/purl/1418765.
@article{osti_1418765,
title = {Invariance of Topological Indices Under Hilbert Space Truncation},
author = {Huang, Zhoushen and Zhu, Wei and Arovas, Daniel P. and Zhu, Jian -Xin and Balatsky, Alexander V.},
abstractNote = {Here, we show that the topological index of a wave function, computed in the space of twisted boundary phases, is preserved under Hilbert space truncation, provided the truncated state remains normalizable. If truncation affects the boundary condition of the resulting state, the invariant index may acquire a different physical interpretation. If the index is symmetry protected, the truncation should preserve the protecting symmetry. We discuss implications of this invariance using paradigmatic integer and fractional Chern insulators, Z2 topological insulators, and spin-1 Affleck-Kennedy-Lieb-Tasaki and Heisenberg chains, as well as its relation with the notion of bulk entanglement. As a possible application, we propose a partial quantum tomography scheme from which the topological index of a generic multicomponent wave function can be extracted by measuring only a small subset of wave function components, equivalent to the measurement of a bulk entanglement topological index.},
doi = {10.1103/PhysRevLett.120.016403},
journal = {Physical Review Letters},
issn = {0031-9007},
number = 1,
volume = 120,
place = {United States},
year = {2018},
month = {1}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Figures / Tables:

FIG. 1 FIG. 1: Chern number density in the space of parton boundary phases for (a) the parent (untruncated) parton mean-field state, and (b) the bosonic fractional Chern insulator state obtained via Gutzwiller projection. In both cases, the Chern number density integrates to the same CMF = 2 over the parton BZ,more » as required by truncation invariance. The physical Hall conductance of the FCI state is given by C = $\frac{C_{MF}}{m^2}$ m=2 = $\frac{1}{2}$ with m = 2 parton species, see text for detail. Calculation is done with a 4 x 4 lattice and a 40 x 40 grid of (kx; ky).« less

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    Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.