Multipoint correlators of conformal field theories: implications for quantum critical transport

TL;DR

This work computes the zero-temperature three-point correlator $\langle TJJ\rangle$ in 2+1D CFTs by both a large-$N_F$ field-theory approach and a holographic AdS$_4$ method, and fixes the bulk four-derivative Weyl coupling $\gamma$ by exact matching. It shows that free scalars and fermions saturate the holographic bound $|\gamma|\le 1/12$ with $\gamma_s=-1/12$ and $\gamma_f=+1/12$, while a topological current carries an intermediate value $\gamma_t=(N_s-N_f)/(12(N_s+N_f))$. The coefficient $\gamma$ controls the frequency dependence of the conductivity and other quantum-critical transport properties, enabling a physical interpretation in terms of particle-like versus vortex-like transport at quantum phase transitions. Moreover, a position-space analysis of energy flux confirms the relation $\mathcal{A} = -4 d (d-1) \gamma$ between the energy-flux anisotropy and the holographic coupling, thereby providing a cohesive link between CFT data and bulk couplings and offering predictions for nonzero-temperature transport in condensed-matter systems.

Abstract

We compute three-point correlators between the stress-energy tensor and conserved currents of conformal field theories (CFTs) in 2+1 dimensions. We first compute the correlators in the large-flavor-number expansion of conformal gauge theories and then do the computation using holography. In the holographic approach, the correlators are computed from an effective action on 3+1 dimensional anti-de Sitter space (AdS_4) proposed by Myers et al., and depend upon the co-efficient, γ, of a four-derivative term in the action. We find a precise match between the CFT and the holographic results, thus fixing the values of γ. The CFTs of free fermions and bosons take the values γ=1/12,-1/12 respectively, and so saturate the bound |γ| <= 1/12 obtained earlier from the holographic theory; the correlator of the conserved gauge flux of U(1) gauge theories takes intermediate values of γ. The value of γalso controls the frequency dependence of the conductivity, and other properties of quantum-critical transport at non-zero temperatures. Our results for the values of γlead to an appealing physical interpretation of particle-like or vortex-like transport near quantum phase transitions of interest in condensed matter physics.This paper includes appendices reviewing key features of the AdS/CFT correspondence for condensed matter physicists.

Figures (6)

• Figure 1: Illustration of the AdS-CFT correspondence in the context of quantum critical transport at finite temperatures. The present paper is concerned with the upper blue arrow: we fix couplings by matching correlators of the CFT to those of the gravity theory. The bottom blue arrow is addressed in Refs. Myers:2010pk and quasinormal, which computed the relevant conductivities and quasi-normal modes of the gravity dual for general values of the couplings in Eq. (\ref{['Smyers']}).
• Figure 2: Correlators (with helicity projections) that fix the numerical values of the couplings in the holographic action specified by Eqs. (\ref{['Smyers']}) and (\ref{['heaction']}). These correlators are evaluated in the present paper in the boundary conformal field theory.
• Figure 3: One-loop triangle diagrams for the scalar contribution to $\langle T J J \rangle$. The top corner of the respective triangles are (momentum-dependent) stress-tensor vertices while the bottom two corners represent current vertices.
• Figure 4: Momentum structure of the stress tensor (top) and current vertex (bottom) after contracting with transverse and traceless polarization tensors.
• Figure 5: Feynman diagrams contributing to the 3-point correlator of $J_t$. The full lines are the bosonic or fermionic matter fields, and the zigzag line is the $a_i$ propagator.