Abstract
In this paper we study a simple gravity model dual to a 2 + 1-dimensional system with a boundary at finite charge density and temperature. In our naive AdS/BCF T extension of a well known AdS/CFT system a non-zero charge density must be supported by a magnetic field. As a result, the Hall conductivity is a constant inversely proportional to the coefficients of pertinent topological terms. Since the direct conductivity vanishes, such behaviors resemble that of a quantum Hall system with Fermi energy in the gap between the Landau levels. We further analyze the properties stemming from our holographic approach to a quantum Hall system. We find that at low temperatures the thermal and electric conductivities are related through the Wiedemann-Franz law, so that every charge conductance mode carries precisely one quantum of the heat conductance. From the computation of the edge currents we learn that the naive holographic model is dual to a gapless system if tensionless RS branes are used in the AdS/BCFT construction. To reconcile this result with the expected quantum Hall behavior we conclude that gravity solutions with tensionless RS branes must be unstable, calling for a search of more general solutions. We briefly discuss the expected features of more realistic holographic setups.
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T. Takayanagi, Holographic dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [INSPIRE].
S.S. Gubser, Breaking an abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
H. Liu, J. McGreevy and D. Vegh, Non-Fermi liquids from holography, Phys. Rev. D 83 (2011) 065029 [arXiv:0903.2477] [INSPIRE].
M. Fujita, M. Kaminski and A. Karch, SL(2, \( \mathbb{Z} \)) Action on AdS/BCFT and Hall conductivities, JHEP 07 (2012) 150 [arXiv:1204.0012] [INSPIRE].
R. Laughlin, Quantized Hall conductivity in two-dimensions, Phys. Rev. B 23 (1981) 5632 [INSPIRE].
R. Laughlin, Anomalous quantum Hall effect: an incompressible quantum fluid with fractionallycharged excitations, Phys. Rev. Lett. 50 (1983) 1395 [INSPIRE].
J. Jain, Composite fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63 (1989) 199 [INSPIRE].
J. Jain, Theory of the fractional quantum Hall effect, Phys. Rev. B 41 (1990) 7653 [INSPIRE].
R.E. Prange and and S.M. Girvin, The quantum Hall effect, 2nd edition, Springer-Verlag, Berlin Germany (1990).
X-G. Wen, Quantum field theory of many-body systems, Oxford University Press, Oxford U.K. (2004).
R. Jackiw, Fractional charge and zero modes for planar systems in a magnetic field, Phys. Rev. D 29 (1984) 2375 [Erratum ibid. D 33 (1986) 2500] [INSPIRE].
R. Jackiw, J. Avron, R. Seiler and B. Simon, Quantization of the Hall conductance for general multiparticle Schrödinger hamiltonians, Phys. Rev. Lett. 54 (1985) 259.
G.W. Semenoff and P. Sodano, Nonabelian adiabatic phases and the fractional quantum Hall effect, Phys. Rev. Lett. 57 (1986) 1195 [INSPIRE].
G.W. Semenoff, P. Sodano and Y.-S. Wu, Renormalization of the statistics parameter in three-dimensional electrodynamics, Phys. Rev. Lett. 62 (1989) 715 [INSPIRE].
A.P. Polychronakos, Topological mass quantization and parity violation in (2 + 1)-dimensional QED, Nucl. Phys. B 281 (1987) 241 [INSPIRE].
A.P. Polychronakos, On the quantization of the coefficient of the abelian Chern-Simons term, Phys. Lett. B 241 (1990) 37 [INSPIRE].
N. Bralic, C. Fosco and F. Schaposnik, On the quantization of the Abelian Chern-Simons coefficient at finite temperature, Phys. Lett. B 383 (1996) 199 [hep-th/9509110] [INSPIRE].
C.L. Kane and M.P.A. Fisher, Quantized thermal transport in the fractional quantum Hall effect, Phys. Rev. B 55 (1997) 15832 [cond-mat/9603118].
B.I. Halperin, Quantized Hall conductance, current carrying edge states and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25 (1982) 2185 [INSPIRE].
S.A. Hartnoll and P. Kovtun, Hall conductivity from dyonic black holes, Phys. Rev. D 76 (2007) 066001 [arXiv:0704.1160] [INSPIRE].
S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76 (2007) 144502 [arXiv:0706.3215] [INSPIRE].
J.L. Davis, P. Kraus and A. Shah, Gravity dual of a quantum Hall plateau transition, JHEP 11 (2008) 020 [arXiv:0809.1876] [INSPIRE].
M. Fujita, W. Li, S. Ryu and T. Takayanagi, Fractional quantum Hall effect via holography: Chern-Simons, edge states and hierarchy, JHEP 06 (2009) 066 [arXiv:0901.0924] [INSPIRE].
O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Quantum Hall effect in a holographic model, JHEP 10 (2010) 063 [arXiv:1003.4965] [INSPIRE].
O. Aharony, D. Marolf and M. Rangamani, Conformal field theories in Anti-de Sitter space, JHEP 02 (2011) 041 [arXiv:1011.6144] [INSPIRE].
M. Nozaki, T. Takayanagi and T. Ugajin, Central charges for BCFTs and holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].
M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
G. Hayward, Gravitational action for space-times with nonsmooth boundaries, Phys. Rev. D 47 (1993) 3275 [INSPIRE].
S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
C. Herzog and D. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [INSPIRE].
N.R. Cooper, B. I. Halperin and I.M. Ruzin, Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field, Phys. Rev. B 55 (1997) 2344 [cond-mat/9607001].
T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].
O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Striped instability of a holographic Fermi-like liquid, JHEP 10 (2011) 034 [arXiv:1106.3883] [INSPIRE].
D.B. Chklovskii, B.I. Shklovskii and L.I. Glazman, Electrostatics of edge channels, Phys. Rev. B 46 (1992) 4026.
N.B. Zhitenev, R.J. Haug, K. von Klitzing and K. Eberl, Time-resolved measurements of transport in edge channels, Phys. Rev. Lett. 71 (1993) 2292.
S.W. Hwang, D.C. Tsui and M. Shayegan, Experimental evidence for finite-width edge channels in integer and fractional quantum Hall effects, Phys. Rev. B 48 (1993) 8161.
A.H. MacDonald, Edge states in the fractional-quantum-Hall-effect regime, Phys. Rev. Lett. 64 (1990) 220 [INSPIRE].
X.G. Wen, Gapless boundary excitations in the quantum Hall states and in the chiral spin states, Phys. Rev. B 43 (1991) 11025 [INSPIRE].
X.G. Wen, Electrodynamical properties of gapless edge excitations in the fractional quantum Hall states, Phys. Rev. Lett. 64 (1990) 2206 [INSPIRE].
X.G. Wen, Edge transport properties of the fractional quantum Hall states and weak-impurity scattering of a one-dimensional charge-density wave, Phys. Rev. B 44 (1991) 5708.
M.D. Johnson and A.H. MacDonald, Composite edges in the ν = 2/3 fractional quantum Hall effect, Phys. Rev. Lett. 67 (1991) 2060.
V. Venkatachalam, S. Hart, L. Pfeiffer, K. West and A. Yacoby, Local thermometry of neutral modes on the quantum Hall edge, Nat. Phys. 8 (2012) 676 [arXiv:1202.6681].
D. Melnikov, E. Orazi and P. Sodano, On the stability of the black hole solutions in AdS/BCFT models, to appear.
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ArXiv ePrint: 1211.1416
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Melnikov, D., Orazi, E. & Sodano, P. On the AdS/BCFT approach to quantum Hall systems. J. High Energ. Phys. 2013, 116 (2013). https://doi.org/10.1007/JHEP05(2013)116
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DOI: https://doi.org/10.1007/JHEP05(2013)116