Abstract
We study boundary states for Dirac fermions in d = 1 + 1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (−1)F , which are characterised by the existence of an unpaired Majorana zero mode.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Wang and T. Senthil, Boson topological insulators: A window into highly entangled quantum phases, Phys. Rev. B 87 (2013) 235122 [arXiv:1302.6234] [INSPIRE].
B. Han, A. Tiwari, C.-T. Hsieh and S. Ryu, Boundary conformal field theory and symmetry protected topological phases in 2 + 1 dimensions, Phys. Rev. B 96 (2017) 125105 [arXiv:1704.01193] [INSPIRE].
K. Jensen, E. Shaverin and A. Yarom, ’t Hooft anomalies and boundaries, JHEP 01 (2018) 085 [arXiv:1710.07299] [INSPIRE].
H. Lückock, Mixed boundary conditions in quantum field theory, J. Math. Phys. 32 (1991) 1755 [INSPIRE].
J. Polchinski, Monopole Catalysis: The Fermion Rotor System, Nucl. Phys. B 242 (1984) 345 [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
C. Bachas, I. Brunner and D. Roggenkamp, A worldsheet extension of O(d, d : Z ), JHEP 10 (2012) 039 [arXiv:1205.4647] [INSPIRE].
A. Sen, SO(32) spinors of type-I and other solitons on brane-antibrane pair, JHEP 09 (1998) 023 [hep-th/9808141] [INSPIRE].
E. Witten, D-branes and k-theory, JHEP 12 (1998) 019 [hep-th/9810188] [INSPIRE].
R. Dijkgraaf and E. Witten, Developments in Topological Gravity, Int. J. Mod. Phys. A 33 (2018) 1830029 [arXiv:1804.03275] [INSPIRE].
J. Kaidi, J. Parra-Martinez, Y. Tachikawa and A. Debray, Topological Superconductors on Superstring Worldsheets, SciPost Phys. 9 (2020) 10 [arXiv:1911.11780] [INSPIRE].
F.D.M. Haldane, Stability of Chiral Luttinger Liquids and Abelian Quantum Hall States, Phys. Rev. Lett. 74 (1995) 2090 [cond-mat/9501007] [INSPIRE].
A. Kapustin and N. Saulina, Topological boundary conditions in abelian Chern-Simons theory, Nucl. Phys. B 845 (2011) 393 [arXiv:1008.0654] [INSPIRE].
J. Wang and X.-G. Wen, Boundary Degeneracy of Topological Order, Phys. Rev. B 91 (2015) 125124 [arXiv:1212.4863] [INSPIRE].
M. Levin, Protected edge modes without symmetry, Phys. Rev. X 3 (2013) 021009 [arXiv:1301.7355] [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
M.R. Gaberdiel, D-branes from conformal field theory, Fortsch. Phys. 50 (2002) 783 [hep-th/0201113] [INSPIRE].
N. Ishibashi, The Boundary and Crosscap States in Conformal Field Theories, Mod. Phys. Lett. A 4 (1989) 251 [INSPIRE].
G.Y. Cho, K. Shiozaki, S. Ryu and A.W.W. Ludwig, Relationship between Symmetry Protected Topological Phases and Boundary Conformal Field Theories via the Entanglement Spectrum, J. Phys. A 50 (2017) 304002 [arXiv:1606.06402] [INSPIRE].
J. Lou, C. Shen and L.-Y. Hung, Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I, JHEP 04 (2019) 017 [arXiv:1901.08238] [INSPIRE].
C. Shen, J. Lou and L.-Y. Hung, Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part II. Cutting through the boundary, JHEP 11 (2019) 168 [arXiv:1908.07700] [INSPIRE].
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].
D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [INSPIRE].
M.R. Gaberdiel and A. Recknagel, Conformal boundary states for free bosons and fermions, JHEP 11 (2001) 016 [hep-th/0108238] [INSPIRE].
M.R. Gaberdiel, A. Recknagel and G.M.T. Watts, The Conformal boundary states for SU(2) at level 1, Nucl. Phys. B 626 (2002) 344 [hep-th/0108102] [INSPIRE].
I. Affleck and J. Sagi, Monopole catalyzed baryon decay: A Boundary conformal field theory approach, Nucl. Phys. B 417 (1994) 374 [hep-th/9311056] [INSPIRE].
M.B. Green and M. Gutperle, Symmetry breaking at enhanced symmetry points, Nucl. Phys. B 460 (1996) 77 [hep-th/9509171] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997) [DOI] [INSPIRE].
A. Kapustin, Ground-state degeneracy for abelian anyons in the presence of gapped boundaries, Phys. Rev. B 89 (2014) 125307 [arXiv:1306.4254] [INSPIRE].
C.G. Callan Jr., Monopole Catalysis of Baryon Decay, Nucl. Phys. B 212 (1983) 391 [INSPIRE].
H. Liebeck and A. Osborne, The generation of all rational orthogonal matrices, Am. Math. Mon. 98 (1991) 131.
J. von Delft and H. Schoeller, Bosonization for beginners: Refermionization for experts, Annalen Phys. 7 (1998) 225 [cond-mat/9805275] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1912.01602
Rights and permissions
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.
About this article
Cite this article
Smith, P.B., Tong, D. Boundary states for chiral symmetries in two dimensions. J. High Energ. Phys. 2020, 18 (2020). https://doi.org/10.1007/JHEP09(2020)018
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2020)018