Abstract

We consider charge transport properties of $2+1$ dimensional conformal field theories at nonzero temperature. For theories with only Abelian U(1) charges, we describe the action of particle-vortex duality on the hydrodynamic-to-collisionless crossover function: this leads to powerful functional constraints for self-dual theories. For $\mathcal{N}=8$ supersymmetric, $\mathrm{SU}(N)$ Yang-Mills theory at the conformal fixed point, exact hydrodynamic-to-collisionless crossover functions of the SO(8) R-currents can be obtained in the large $N$ limit by applying the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to M theory. In the gravity theory, fluctuating currents are mapped to fluctuating gauge fields in the background of a black hole in $3+1$ dimensional anti-de Sitter space. The electromagnetic self-duality of the $3+1$ dimensional theory implies that the correlators of the R-currents obey a functional constraint similar to that found from particle-vortex duality in $2+1$ dimensional Abelian theories. Thus the $2+1$ dimensional, superconformal Yang Mills theory obeys a ``holographic self-duality'' in the large $N$ limit, and perhaps more generally.

Highlights

  • The quantum phase transitions of two dimensional systems have been the focus of much study in the condensed matter community

  • Examples are (i ) the superfluidinsulator transition in the boson Hubbard model at integer filling [9–11], which is described by the φ4 field theory with O(2) symmetry, and so is controlled by the Wilson-Fisher fixed point in D = 2+1; (ii ) the spin-gap paramagnet to Neel order transition of coupled spin dimers/ladders/layers which is described by the O(3) φ4 field theory [12, 13]; and (iii ) the ‘deconfined’ critical point of a S = 1/2 antiferromagnet between a Neel and a valence bond solid state [14, 15], which is described by the CP1 model with a non-compact U(1) gauge field [16]

  • In all these cases the critical point is described by a relativistic conformal field theory (CFT)

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Summary

INTRODUCTION

The quantum phase transitions of two (spatial) dimensional systems have been the focus of much study in the condensed matter community. More relevant is the self-dual field theory proposed recently by Motrunich and Vishwanath [16], and we discuss its charge transport properties below It was subsequently pointed out [29–31] that the Kab are not the d.c. conductivities observed at small but non-zero temperature. We have considered a similar connection at non-zero temperature for the N =8 SCFT, and shown that it is “holographically self dual” in the large N limit; combined with the non-Abelian SO(8) symmetry (which implies a single K), the constraints for the current correlators are stronger than those for Abelian theories.

Conserved currents
Vortices and duality
THE M2-BRANE THEORY
Current-current correlators
Transverse channel
Longitudinal channel
Conductivity
Electric-magnetic duality
Full spectral functions
CONCLUSIONS
Full Text
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