Duality and modular symmetry in the quantum Hall effect from Lifshitz holography

Abstract

The temperature dependence of quantum Hall conductivities is studied in the context of the AdS/CMT paradigm using a model with a bulk theory consisting of (3+1)-dimensional Einstein-Maxwell action coupled to a dilaton and an axion, with a negative cosmological constant. We consider a solution which has a Lifshitz like geometry with a dyonic black-brane in the bulk. There is an Sl(2,R) action in the bulk corresponding to electromagnetic duality, which maps between classical solutions, and is broken to Sl(2,Z) by Dirac quantisation of dyons. This bulk Sl(2,Z) action translates to an action of the modular group on the 2-dimensional transverse conductivities. The temperature dependence of the infra-red conductivities is then linked to modular forms via gradient flow and the resulting flow diagrams show remarkable agreement with existing experimental data on the temperature flow of both integral and fractional quantum Hall conductivities.

On the hydrodynamics of (2 + 1)-dimensional strongly coupled relativistic theories in an external magnetic field

“…These kinds of excitations are the only long lived modes expected to survive at strong coupling, where the notion of "quasi-particle" ceases to exist. Consequently, hydrodynamic effective field theories have been applied, among other materials, to the study of graphene [3,4], quantum hall systems [5][6][7][8][9][10][11][12], bad metals [13][14][15][16][17][18][19][20], Wigner solids [21] and high temperature superconductors [22][23][24][25][26][27][28][29][30][31]. Regarding the latter materials, the community interested in the hydrodynamic approach has broadly attempted to reproduce the transport properties of the so called strange metal phase.…”

“…A particularly famous example is bulk SL(2, Z) duality [44][45][46] which in the boundary theory effectively swaps electric charges for magnetic charges and/or attaches units of magnetic flux to charged particles. The outcome of such operations is an infinite family of related longitudinal and Hall conductivities [11,[47][48][49][50][51][52]. Similarly, by combining certain SL(2, Z) operations in the presence of background charge and magnetic field one can generate anyons [53,54].…”

Novel semi-circle law and Hall sliding in a strongly interacting electron liquid

“…Magnetic fields and the quantum Hall effect have also been investigated in a number of other topdown[23][24][25][26][27][28] and bottom-up holographic models[29][30][31][32][33].…”

Novel semi-circle law and Hall sliding in a strongly interacting electron liquid

“…Magnetic fields and the quantum Hall effect have also been investigated in a number of other topdown[20][21][22][23][24][25] and bottom-up holographic models[26][27][28][29][30].…”