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On the AdS/BCFT Approach to Quantum Hall Systems

Dmitry Melnikov, Emanuele Orazi, Pasquale Sodano

TL;DR

This work develops a holographic AdS/BCFT framework to model a 2+1D boundary system at finite temperature and charge density, using a charged AdS4 black hole with RS branes and Neumann boundary conditions. The analysis shows that a nonzero charge density must be supported by an external magnetic field, yielding a Hall conductivity $\sigma_H$ fixed by topological terms through $\sigma_H = \rho/B = - c_1/c_2$, while the longitudinal conductivity vanishes. Transport calculations reveal a Wiedemann–Franz law with a Lorenz number $\mathcal{L}=\pi^2/3 (k_B/e)^2$ at low $T$, consistent with noninteracting electrons in the transverse channel, and edge currents arise only if the RS branes carry nonzero tension. However, in the tensionless-brane sector the model is gapless and lacks edge currents, implying that tensionful RS-brane backgrounds are needed for a realistic quantum Hall description; the authors argue these solutions may be unstable and call for more general holographic constructions that incorporate a bulk gap and edge modes. The results illuminate how AdS/BCFT topological terms influence Hall responses and guide future holographic QHE models toward including a geometric gap and disorder effects.

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/BCFT 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.

On the AdS/BCFT Approach to Quantum Hall Systems

TL;DR

This work develops a holographic AdS/BCFT framework to model a 2+1D boundary system at finite temperature and charge density, using a charged AdS4 black hole with RS branes and Neumann boundary conditions. The analysis shows that a nonzero charge density must be supported by an external magnetic field, yielding a Hall conductivity fixed by topological terms through , while the longitudinal conductivity vanishes. Transport calculations reveal a Wiedemann–Franz law with a Lorenz number at low , consistent with noninteracting electrons in the transverse channel, and edge currents arise only if the RS branes carry nonzero tension. However, in the tensionless-brane sector the model is gapless and lacks edge currents, implying that tensionful RS-brane backgrounds are needed for a realistic quantum Hall description; the authors argue these solutions may be unstable and call for more general holographic constructions that incorporate a bulk gap and edge modes. The results illuminate how AdS/BCFT topological terms influence Hall responses and guide future holographic QHE models toward including a geometric gap and disorder effects.

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/BCFT 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.

Paper Structure

This paper contains 7 sections, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup of the AdS/BCFT holographic model
  3. The model of a half-plane with a finite charge density
  4. Electric and thermal conductivities
  5. Edge modes and geometric gap
  6. Summary and concluding remarks
  7. Acknowledgements

Figures (2)

  • Figure 1: (a) AdS/BCFT geometry of the half-plane $M$. $N$ denotes the part of the bulk space bounded by $M$, horizon of the black hole at $z_h$ and RS brane $Q$. (b) Dyonic black hole solution ($\theta=\pi/2$).
  • Figure 2: AdS/BCFT geometry in the case of zero, ($\theta=\pi/2$) (a), finite negative, $0<\theta<\pi/2$ (b) and minimal tension, $\theta=0$ (c). (b) and (c) can be interpreted as a position dependent and constant gap in the dual theory respectively. More realistic configurations are expected to have shape (d).