Quantum critical transport, duality, and M theory

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.

Figure 1

Phase diagram [16] of the easy-plane noncompact $\mathbb{C}{\mathbb{P}}^1$ model (Eq. (2.1)) in 2 spatial dimensions as a function of the coupling $s$ and temperature $T$. The quantum critical point is at $s=s_c$, $T=0$. The finite $T$ correlations of the CFT describe the shaded quantum critical region; the boundary of the shaded region is a crossover into a different physical region, not a phase transition. The full lines are Kosterlitz-Thouless (KT) phase transitions. The KT line for $s<s_c$ describes the disappearance of quasi-long-range $xy$ order of $\overrightarrow{N}$. The KT transition for $s>s_c$ describes the deconfinement of $z$ quanta which are logarithmically bound by the Coulomb interaction in the low temperature phase into particle-antiparticle pairs. The phase diagram can also be described in terms of the dual $w$ theory in Eq. (2.11). Duality interchanges the two sides of $s=s_c$ ($T$ remains invariant under duality), and the $z$ Coulomb phase is interpreted as a $w$ Higgs phase and vice versa.

Figure 2

Imaginary part of the retarded function $C_{yy}(\omega ,k)$, plotted in units of ($-\chi$), as a function of dimensionless frequency $w\equiv 3\omega /(4\pi T)$, for several values of dimensionless momentum $q\equiv 3k/(4\pi T)$. Curves from left to right correspond to $q=0$, 0.5, 1.0, 2.0, 3.0. Left: $\mathrm{Im}{C}_{yy}(w,q)$, Right: $\mathrm{Im}{C}_{yy}(w,q)/w$.

Figure 3

Imaginary part of the retarded function $C_{tt}(w,q)/q^2$, plotted in units of ($-\chi$), as a function of dimensionless frequency $w\equiv 3\omega /(4\pi T)$, for several values of dimensionless momentum $q\equiv 3k/(4\pi T)$. Curves from left to right correspond to $q=0.2$, 0.5, 1.0 (left panel), and $q=1.0$, 2.0, 3.0, 4.0 (right panel). The dashed curves are plots of Eq. (3.37) divided by $k^2$.

Figure 4

The position of the peak of the spectral function in Fig. 3. The dashed line is $w=q$.