Abstract
We extend the twisted gauge theory model of topological orders in three spatial dimensions to the case where the three spaces have two dimensional boundaries. We achieve this by systematically constructing the boundary Hamiltonians that are compatible with the bulk Hamiltonian. Given the bulk Hamiltonian defined by a gauge group G and a four-cocycle ω in the fourth cohomology group of G over U(1), we construct a gapped boundary Hamiltonian using {K, α}, with a subgroup K ⊆ G and a 3-cochain α of K over U(1), which satisfies the generalized Frobenius condition. The Hamiltonian is invariant under the topological renormalization group flow (via Pachner moves). Each solution {K, α} to the generalized Frobenius condition specifies a gapped boundary condition. We derive a closed-form formula of the ground state degeneracy of the model on a three-cylinder, which can be naturally generalized to three-spaces with more boundaries. We also derive the explicit ground-state wavefunction of the model on a three-ball. The ground state degeneracy and ground-state wavefunction are both presented solely in terms of the input data of the model, namely, {G, ω, K, α}.
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References
X.G. Wen, Vacuum Degeneracy of Chiral Spin States in Compactified Space, Phys. Rev. B 40 (1989) 7387 [INSPIRE].
X.G. Wen, F. Wilczek and A. Zee, Chiral Spin States and Superconductivity, Phys. Rev. B 39 (1989) 11413 [INSPIRE].
X.G. Wen, Topological Order in Rigid States, Int. J. Mod. Phys. B 4 (1990) 239 [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].
A.Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [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].
A. Kitaev, Anyons in an exactly solved model and beyond, Annals Phys. 321 (2006) 2 [INSPIRE].
X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry-protected topological orders in interacting bosonic systems, Science 338 (2012) 1604.
M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86 (2012) 115109 [arXiv:1202.3120] [INSPIRE].
L.-Y. Hung and Y. Wan, String-Net Models with Z N Fusion Algebra, Phys. Rev. B 86 (2012) 235132 [arXiv:1207.6169] [INSPIRE].
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].
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].
A. Mesaros and Y. Ran, Classification of symmetry enriched topological phases with exactly solvable models, Phys. Rev. B 87 (2013) 155115 [arXiv:1212.0835] [INSPIRE].
C.-H. Lin and M. Levin, Generalizations and limitations of string-net models, Phys. Rev. B 89 (2014) 195130 [arXiv:1402.4081] [INSPIRE].
L. Kong and X.-G. Wen, Braided fusion categories, gravitational anomalies and the mathematical framework for topological orders in any dimensions, arXiv:1405.5858 [INSPIRE].
T. Lan, A Classification of (2+1)D Topological Phases with Symmetries, Ph.D. Thesis (2018) [arXiv:1801.01210] [INSPIRE].
K. Walker and Z. Wang, (3+1)-TQFTs and Topological Insulators, arXiv:1104.2632 [INSPIRE].
C. Wang and M. Levin, Braiding statistics of loop excitations in three dimensions, Phys. Rev. Lett. 113 (2014) 080403 [arXiv:1403.7437] [INSPIRE].
S. Jiang, A. Mesaros and Y. Ran, Generalized Modular Transformations in (3+1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding, Phys. Rev. X 4 (2014) 031048 [arXiv:1404.1062] [INSPIRE].
J. Wang and X.-G. Wen, Non-Abelian string and particle braiding in topological order: Modular SL(3, ℤ) representation and (3+1)-dimensional twisted gauge theory, Phys. Rev. B 91 (2015) 035134 [arXiv:1404.7854] [INSPIRE].
Y. Wan, J.C. Wang and H. He, Twisted Gauge Theory Model of Topological Phases in Three Dimensions, Phys. Rev. B 92 (2015) 045101 [arXiv:1409.3216] [INSPIRE].
C. Wang and M. Levin, Topological invariants for gauge theories and symmetry-protected topological phases, Phys. Rev. B 91 (2015) 165119 [arXiv:1412.1781] [INSPIRE].
A. Bullivant, M. Calçada, Z. Kádár, P. Martin and J.F. Martins, Topological phases from higher gauge symmetry in 3+1 dimensions, Phys. Rev. B 95 (2017) 155118 [arXiv:1606.06639] [INSPIRE].
A. Bullivant, M. Calcada, Z. Kádár, J.F. Martins and P. Martin, Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry, arXiv:1702.00868 [INSPIRE].
T. Lan, L. Kong and X.-G. Wen, A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons, Phys. Rev. X 8 (2018) 021074 [arXiv:1704.04221].
C. Delcamp, Excitation basis for (3+1)d topological phases, JHEP 12 (2017) 128 [arXiv:1709.04924] [INSPIRE].
C. Delcamp and A. Tiwari, From gauge to higher gauge models of topological phases, arXiv:1802.10104 [INSPIRE].
T. Lan and X.-G. Wen, A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions, arXiv:1801.08530 [INSPIRE].
M. Cheng, N. Tantivasadakarn and C. Wang, Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions, Phys. Rev. X 8 (2018) 011054.
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].
V. Kalmeyer and R.B. Laughlin, Equivalence of the resonating valence bond and fractional quantum Hall states, Phys. Rev. Lett. 59 (1987) 2095 [INSPIRE].
N. Read and S. Sachdev, Large-N expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 (1991) 1773 [INSPIRE].
X.-G. Wen, Topological orders and Chern-Simons theory in strongly correlated quantum liquid, Int. J. Mod. Phys. B 5 (1991) 1641 [INSPIRE].
R. Moessner and S.L. Sondhi, Resonating Valence Bond Phase in the Triangular Lattice Quantum Dimer Model, Phys. Rev. Lett. 86 (2001) 1881 [INSPIRE].
K. von Klitzing, G. Dorda and M. Pepper, New method for high accuracy determination of the fine structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45 (1980) 494 [INSPIRE].
D.C. Tsui, H.L. Stormer and A.C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48 (1982) 1559 [INSPIRE].
R.B. Laughlin, Anomalous quantum Hall effect: An Incompressible quantum fluid with fractionallycharged excitations, Phys. Rev. Lett. 50 (1983) 1395 [INSPIRE].
R. Tao and Y.-S. Wu, Gauge invariance and fractional quantum Hall effect, Phys. Rev. B 30 (1984) 1097 [INSPIRE].
G.W. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360 (1991) 362 [INSPIRE].
X.G. Wen, NonAbelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991) 802 [INSPIRE].
R. Willett, J.P. Eisenstein, H.L. Stormer, D.C. Tsui, A.C. Gossard and J.H. English, Observation of an even-denominator quantum number in the fractional quantum Hall effect, Phys. Rev. Lett. 59 (1987) 1776 [INSPIRE].
I.P. Radu, J.B. Miller, C.M. Marcus, M.A. Kastner, L.N. Pfeiffer and K.W. West, Quasi-Particle Properties from Tunneling in the Formula Fractional Quantum Hall State, Science 320 (2008) 899 [INSPIRE].
B. Dittrich, (3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces, JHEP 05 (2017) 123 [arXiv:1701.02037] [INSPIRE].
C. Delcamp and B. Dittrich, Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases, arXiv:1806.00456 [INSPIRE].
A. Kitaev and L. Kong, Models for Gapped Boundaries and Domain Walls, Commun. Math. Phys. 313 (2012) 351.
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].
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].
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].
L. Kong, X.-G. Wen and H. Zheng, Boundary-bulk relation in topological orders, Nucl. Phys. B 922 (2017) 62 [arXiv:1702.00673] [INSPIRE].
J. Wang and X.-G. Wen, Boundary Degeneracy of Topological Order, Phys. Rev. B 91 (2015) 125124 [arXiv:1212.4863] [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].
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].
C. Wang, C.-H. Lin and M. Levin, Bulk-Boundary Correspondence for Three-Dimensional Symmetry-Protected Topological Phases, Phys. Rev. X 6 (2016) 021015.
S.X. Cui, Higher Categories and Topological Quantum Field Theories, arXiv:1610.07628 [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions, Phys. Rev. X 8 (2018) 031048 [arXiv:1705.06728] [INSPIRE].
L. Fidkowski and A. Vishwanath, Realizing anomalous anyonic symmetries at the surfaces of three-dimensional gauge theories, Phys. Rev. B 96 (2017) 045131 [arXiv:1511.01502] [INSPIRE].
D.J. Williamson and Z. Wang, Hamiltonian models for topological phases of matter in three spatial dimensions, Annals Phys. 377 (2017) 311 [arXiv:1606.07144] [INSPIRE].
Y. Hu, Y. Wan and Y.-S. Wu, Boundary Hamiltonian theory for gapped topological orders, Chin. Phys. Lett. 34 (2017) 077103 [arXiv:1706.00650] [INSPIRE].
Y. Hu, Z.-X. Luo, R. Pankovich, Y. Wan and Y.-S. Wu, Boundary Hamiltonian theory for gapped topological phases on an open surface, JHEP 01 (2018) 134 [arXiv:1706.03329] [INSPIRE].
U. Pachner, Bistellare Äquivalenz kombinatorischer Mannigfaltigkeiten, Arch. Math. 30 (1978) 89.
U. Pachner, Konstruktionsmethoden und das kominatorische Homoomorphieproblem für Triangulationen semilinearer Mannigfaltigkeiten, Ahb. Math. Sem. Univ. Hamburg 57 (1987) 69.
Y. Hu, Y. Wan and Y.-S. Wu, From effective Hamiltonian to anomaly inflow in topological orders with boundaries, JHEP 08 (2018) 092 [arXiv:1706.09782] [INSPIRE].
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Wang, H., Li, Y., Hu, Y. et al. Gapped boundary theory of the twisted gauge theory model of three-dimensional topological orders. J. High Energ. Phys. 2018, 114 (2018). https://doi.org/10.1007/JHEP10(2018)114
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DOI: https://doi.org/10.1007/JHEP10(2018)114