Abstract
Boundary conditions for a massless Dirac fermion in 2+1 dimensions where the space is a half-plane are discussed in detail. It is argued that linear boundary conditions that leave the Hamiltonian Hermitian generically break C P and T symmetries as well as Lorentz and conformal symmetry. We show that there is essentially one special case where a single species of fermion has C PT and the full Poincare and conformal symmetry of the boundary. We show that, with doubled fermions, there is a second special case which respects C PT but still violates Lorentz and conformal symmetry. This second special case is essentially the unique boundary condition where the Dirac operator has fermion zero mode edge states. We discuss how the edge states lead to exotic representations of scale, phase and translation symmetries and how imposing a symmetry requirement leads to edge ferromagnetism of the system. We prove that the exotic ferromagnetic representations are indeed carried by the ground states of the system perturbed by a class of interaction Hamiltonians which includes the non-relativistic Coulomb interaction.
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References
S. Dutta and S.K. Pati, Novel properties of graphene nanoribbons: a review, J. Mater. Chem. 20 (2010) 8207.
H. Wang et al., Graphene nanoribbons for quantum electronics, Nature Rev. Phys. 3 (2021) 791.
R. Jackiw and C. Rebbi, Solitons with Fermion Number 1/2, Phys. Rev. D 13 (1976) 3398 [INSPIRE].
A.J. Niemi and G.W. Semenoff, Fermion Number Fractionization in Quantum Field Theory, Phys. Rept. 135 (1986) 99 [INSPIRE].
G.W. Semenoff and F. Zhou, Magnetic Catalysis and Quantum Hall Ferromagnetism in Weakly Coupled Graphene, JHEP 07 (2011) 037 [arXiv:1104.4714] [INSPIRE].
T.O. Wehling, A.M. Black-Schaffer and A.V. Balatsky, Dirac materials, Adv. Phys. 63 (2014) 1 [arXiv:1405.5774] [INSPIRE].
G.W. Semenoff, Condensed Matter Simulation of a Three-dimensional Anomaly, Phys. Rev. Lett. 53 (1984) 2449 [INSPIRE].
A.R. Akhmerov and C.W.J. Beenakker, Boundary conditions for dirac fermions on a terminated honeycomb lattice, Phys. Rev. B 77 (2008) 085423.
J.A.M. van Ostaay, A.R. Akhmerov, C.W.J. Beenakker and M. Wimmer, Dirac boundary condition at the reconstructed zigzag edge of graphene, Phys. Rev. B 84 (2011) 195434.
M. Fujita, K. Wakabayashi, K. Nakada and K. Kusakabe, Peculiar localized state at zigzag graphite edge, J. Phys. Soc. Jap. 65 (1996) 1920.
K. Nakada, M. Fujita, G. Dresselhaus and M.S. Dresselhaus, Edge state in graphene ribbons: Nanometer size effect and edge shape dependence, Phys. Rev. B 54 (1996) 17954.
Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada and H. Fukuyama, Scanning tunneling microscopy and spectroscopy of the electronic local density of states of graphite surfaces near monoatomic step edges, Phys. Rev. B 73 (2006) 085421.
Y. Kobayashi, K. ichi Fukui, T. Enoki and K. Kusakabe, Edge state on hydrogen-terminated graphite edges investigated by scanning tunneling microscopy, Phys. Rev. B 73 (2006) 125415.
Z. Liu, K. Suenaga, P.J.F. Harris and S. Iijima, Open and closed edges of graphene layers, Phys. Rev. Lett. 102 (2009) 015501.
K. Suenaga and M. Koshino, Atom-by-atom spectroscopy at graphene edge, Nature 468 (2010) 1088.
J. Jung, T. Pereg-Barnea and A.H. MacDonald, Theory of interedge superexchange in zigzag edge magnetism, Phys. Rev. Lett. 102 (2009) 227205.
J. Jung and A.H. MacDonald, Carrier density and magnetism in graphene zigzag nanoribbons, Phys. Rev. B 79 (2009) 235433.
Y.-W. Son, M.L. Cohen and S.G. Louie, Energy gaps in graphene nanoribbons, Phys. Rev. Lett. 97 (2006) 216803.
Y.-W. Son, M.L. Cohen and S.G. Louie, Half-metallic graphene nanoribbons, Nature 444 (2006) 347.
T. Hikihara, X. Hu, H.-H. Lin and C.-Y. Mou, Ground-state properties of nanographite systems with zigzag edges, Phys. Rev. B 68 (2003) 035432.
S. Dutta, S. Lakshmi and S.K. Pati, Electron-electron interactions on the edge states of graphene: A many-body configuration interaction study, Phys. Rev. B 77 (2008) 073412.
H. Feldner, Z.Y. Meng, A. Honecker, D. Cabra, S. Wessel and F.F. Assaad, Magnetism of finite graphene samples: Mean-field theory compared with exact diagonalization and quantum monte carlo simulations, Phys. Rev. B 81 (2010) 115416.
H. Karimi and I. Affleck, Towards a rigorous proof of magnetism on the edges of graphene nanoribbons, Phys. Rev. B 86 (2012) 115446.
Z. Shi and I. Affleck, Effect of long-range interaction on graphene edge magnetism, Phys. Rev. B 95 (2017) 195420.
D.E. Sheehy and J. Schmalian, Quantum critical scaling in graphene, Phys. Rev. Lett. 99 (2007) 226803.
D.T. Son, Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction, Phys. Rev. B 75 (2007) 235423 [cond-mat/0701501] [INSPIRE].
N. Andrei et al., Boundary and Defect CFT: Open Problems and Applications, J. Phys. A 53 (2020) 453002 [arXiv:1810.05697] [INSPIRE].
Y. Sato, Free energy and defect C -theorem in free fermion, JHEP 05 (2021) 202 [arXiv:2102.11468] [INSPIRE].
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Biswas, S., Semenoff, G.W. Massless fermions on a half-space: the curious case of 2+1-dimensions. J. High Energ. Phys. 2022, 45 (2022). https://doi.org/10.1007/JHEP10(2022)045
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DOI: https://doi.org/10.1007/JHEP10(2022)045