Abstract
In this paper we propose a Hamiltonian approach to gapped topological phases on open surfaces. Our setting is an extension of the Levin-Wen model to a 2d graph on an open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected. It is shown that the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system on a disk as well as on a cylinder. For boundary quasiparticle excitations, we are able to construct their creation, annihilation, measuring and hopping operators etc. Given a bulk string-net theory, our approach provides a classification scheme of possible types of gapped boundary conditions by Frobenius algebras (modulo Morita equivalence) of the bulk fusion category; the boundary quasiparticles are characterized by bimodules of the pertinent Frobenius algebras. Our approach also offers a set of concrete tools for computations. We illustrate our approach by a few examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
X.G. Wen, Topological Order in Rigid States, Int. J. Mod. Phys. B 4 (1990) 239 [INSPIRE].
R. Tao and Y.-S. Wu, Gauge invariance and fractional quantum Hall effect, Phys. Rev. B 30 (1984) 1097 [INSPIRE].
Q. Niu, D.J. Thouless and Y.-S. Wu, Quantized hall conductance as a topological invariant, Phys. Rev. B 31 (1985) 3372 [INSPIRE].
X.G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B 41 (1990) 9377 [INSPIRE].
Y.-S. Wu, General Theory for Quantum Statistics in Two-Dimensions, Phys. Rev. Lett. 52 (1984) 2103 [INSPIRE].
E. Dennis, A. Kitaev, A. Landahl and J. Preskill, Topological quantum memory, J. Math. Phys. 43 (2002) 4452 [quant-ph/0110143] [INSPIRE].
A.Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
M. Freedman, A.Y. Kitaev, J. Preskill and Z. Wang, Topological Quantum Computation, Bull. Amer. Math. Soc. 40 (2003) 31 [quant-ph/0101025].
A. Stern and B.I. Halperin, Proposed Experiments to Probe the Non-Abelian ν = 5/2 Quantum Hall State, Phys. Rev. Lett. 96 (2006) 016802.
C. Nayak, S.H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083 [INSPIRE].
Z. Luo et al., Experimental Preparation of Topologically Ordered States via Adiabatic Evolution, arXiv:1608.06963.
Z. Luo et al., Experimentally Probing Topological Order and Its Breakdown via Modular Matrices, Nature Phys. (2017) [arXiv:1608.06978].
I. Lesanovsky and H. Katsura, Interacting Fibonacci anyons in a Rydberg gas, Phys. Rev. A 86 (2012) 041601.
K. Li et al., Experimental Identification of Non-Abelian Topological Orders on a Quantum Simulator, Phys. Rev. Lett. 118 (2017) 080502.
X.G. Wen and A. Zee, Quantum Statistics and Superconductivity in Two Spatial Dimensions, Nucl. Phys. Proc. Suppl. 15 (1990) 135 [INSPIRE].
X.-G. Wen, Topological orders and Chern-Simons theory in strongly correlated quantum liquid, Int. J. Mod. Phys. B 5 (1991) 1641 [INSPIRE].
A. Kitaev and L. Kong, Models for gapped boundaries and domain walls, Commun. Math. Phys. 313 (2012) 351 [arXiv:1104.5047].
M. Levin, Protected edge modes without symmetry, Phys. Rev. X 3 (2013) 021009 [arXiv:1301.7355] [INSPIRE].
L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436 [arXiv:1307.8244] [INSPIRE].
M. Barkeshli, C.-M. Jian and X.-L. Qi, Classification of Topological Defects in Abelian Topological States, Phys. Rev. B 88 (2013) 241103 [arXiv:1304.7579] [INSPIRE].
M. Barkeshli, C.-M. Jian and X.-L. Qi, Theory of defects in Abelian topological states, Phys. Rev. B 88 (2013) 235103 [arXiv:1305.7203] [INSPIRE].
T. Lan and X.-G. Wen, Topological quasiparticles and the holographic bulk-edge relation in (2 + 1)-dimensional string-net models, Phys. Rev. B 90 (2014) 115119 [arXiv:1311.1784] [INSPIRE].
J. Wang and X.-G. Wen, Boundary Degeneracy of Topological Order, Phys. Rev. B 91 (2015) 125124 [arXiv:1212.4863] [INSPIRE].
L.-Y. Hung and Y. Wan, K matrix Construction of Symmetry-Enriched Phases of Matter, Phys. Rev. B 87 (2013) 195103 [arXiv:1302.2951] [INSPIRE].
T. Iadecola, T. Neupert, C. Chamon and C. Mudry, Accessing topological order in fractionalized liquids with gapped edges, Phys. Rev. B 90 (2014) 205115 [arXiv:1407.4129] [INSPIRE].
M. Barkeshli, Y. Oreg and X.-L. Qi, Experimental Proposal to Detect Topological Ground State Degeneracy, arXiv:1401.3750 [INSPIRE].
M. Barkeshli, E. Berg and S. Kivelson, Coherent Transmutation of Electrons into Fractionalized Anyons, Science 346 (2014) 6210 [arXiv:1402.6321] [INSPIRE].
L.-Y. Hung and Y. Wan, Ground State Degeneracy of Topological Phases on Open Surfaces, Phys. Rev. Lett. 114 (2015) 076401 [arXiv:1408.0014] [INSPIRE].
T. Lan, J.C. Wang and X.-G. Wen, Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy, Phys. Rev. Lett. 114 (2015) 076402 [arXiv:1408.6514] [INSPIRE].
L.-Y. Hung and Y. Wan, Generalized ADE classification of topological boundaries and anyon condensation, JHEP 07 (2015) 120 [arXiv:1502.02026] [INSPIRE].
R.B. Laughlin, Anomalous quantum Hall effect: An Incompressible quantum fluid with fractionallycharged excitations, Phys. Rev. Lett. 50 (1983) 1395 [INSPIRE].
M.A. Levin and X.-G. Wen, String net condensation: A physical mechanism for topological phases, Phys. Rev. B 71 (2005) 045110 [cond-mat/0404617] [INSPIRE].
Y. Hu, Y. Wan and Y.-S. Wu, Twisted quantum double model of topological phases in two dimensions, Phys. Rev. B 87 (2013) 125114 [arXiv:1211.3695] [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
J. Kock, Frobenius Algebras and 2-D Topological Quantum Field Theories, Cambridge University Press (2004), 1st edition, ISBN: 978-0521540315.
S.B. Bravyi and A.Yu. Kitaev, Quantum codes on a lattice with boundary, quant-ph/9811052 [INSPIRE].
S. Beigi, P.W. Shor and D. Whalen, The Quantum Double Model with Boundary: Condensations and Symmetries, Commun. Math. Phys. 306 (2011) 663.
I. Cong, M. Cheng and Z. Wang, Topological Quantum Computation with Gapped Boundaries, arXiv:1609.02037.
I. Cong, M. Cheng and Z. Wang, Defects between gapped boundaries in two-dimensional topological phases of matter, Phys. Rev. B 96 (2017) 195129 [arXiv:1703.03564] [INSPIRE].
A. Bullivant, Y. Hu and Y. Wan, Twisted quantum double model of topological order with boundaries, Phys. Rev. B 96 (2017) 165138 [arXiv:1706.03611] [INSPIRE].
Y. Hu, Y. Wan and Y.-s. Wu, Boundary Hamiltonian Theory for Gapped Topological Orders, Chin. Phys. Lett. 34 (2017) 077103.
Y. Hu, S.D. Stirling and Y.-S. Wu, Ground State Degeneracy in the Levin-Wen Model for Topological Phases, Phys. Rev. B 85 (2012) 075107 [arXiv:1105.5771] [INSPIRE].
V. Ostrik, Module categories, weak Hopf algebras and modular invariants, math/0111139.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding authors
Additional information
ArXiv ePrint: 1706.03329
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hu, Y., Luo, ZX., Pankovich, R. et al. Boundary Hamiltonian theory for gapped topological phases on an open surface. J. High Energ. Phys. 2018, 134 (2018). https://doi.org/10.1007/JHEP01(2018)134
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2018)134