On the AdS/BCFT approach to quantum Hall systems

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.

Bending branes for DCFT in two dimensions

“…In general however, this field requires its own junction conditions. Interestingly, AdS/BCFT like setups with Chern-Simons fields in the bulk were studied before in [24,42]. Building upon the results of this work, we intend to revisit all of these questions in a future publication [64].…”

“…A further interesting question will be to evaluate the on-shell action and examining the thermodynamic behaviour of these solutions. Quantum Hall effect: Furthermore, we would like to shortly comment on the AdS/BCFT model of the quantum Hall effect presented in [24] 24 . There (see especially figure 2(d) of [24]), it was argued that a realistic model of the quantum Hall effect would likely include a brane that is anchored to the boundary twice similar to Q 1 in figure 3.…”

“…Quantum Hall effect: Furthermore, we would like to shortly comment on the AdS/BCFT model of the quantum Hall effect presented in [24] 24 . There (see especially figure 2(d) of [24]), it was argued that a realistic model of the quantum Hall effect would likely include a brane that is anchored to the boundary twice similar to Q 1 in figure 3. From our results in section 5, it is now clear that in 2 + 1-dimensions the brane can be forced to show such a behaviour by choosing its matter content to satisfy both weak and strong energy condition everywhere.…”

“…Gravity duals of DCFT in which the dual of the defect remains localised were recently used also for holographic models of the quantum Hall effect. Recent examples include both bottom-up models [24,25] and top-down models involving brane constructions [26,27]. For example, these models display incompressible states with quantised Hall conductivity.…”

Hall viscosity in a strongly coupled magnetized plasma

“…Although theories with holographic duals can be microscopically very different to a typical condensed system, their long wavelength behavior can be similar and be used as a playing ground to understand better the properties of FQH states. Indeed, there have been many proposals at describing QH states using holography [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Although some of these models describe fractional filling fractions and features like transitions between Hall plateaus, the situation is not entirely satisfactory when the holographic models are examined in detail.…”