arXiv:1307.6863 · Physical Review B

Conserved current correlators of conformal field theories in 2+1 dimensions

Yejin Huh, Philipp Strack, Subir Sachdev A

Published Oct 7, 2013• 31 citations

Abstract

We compute current correlators of the $CP$${}^{N\ensuremath{-}1}$ field theory in 2+1 dimensions, both at the critical point and in the phase with spontaneously broken $\mathrm{SU}(N)$ symmetry. Universal constants are obtained to next-to-leading order in the $1/N$ expansion. Implications are noted for quantum critical points of antiferromagnets and their vicinity.

DOI: 10.1103/physrevb.88.155109

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arXiv:1307.6863 · Physical Review B

Conserved current correlators of conformal field theories in 2+1 dimensions

Yejin Huh, Philipp Strack, Subir Sachdev A

Published Oct 7, 2013• 31 citations

arXiv HTML PDF

Abstract

We compute current correlators of the

CP$${}^{N\ensuremath{-}1}

field theory in 2+1 dimensions, both at the critical point and in the phase with spontaneously broken

\mathrm{SU}(N)

symmetry. Universal constants are obtained to next-to-leading order in the

1/N

expansion. Implications are noted for quantum critical points of antiferromagnets and their vicinity.

DOI: 10.1103/physrevb.88.155109

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We compute three-point correlators between the stress-energy tensor and the 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 perform the computation using holography. In the holographic approach, the correlators are computed from an effective action on (3+1)-dimensional anti-de Sitter space ( ${\mathrm{AdS}}_{4}$ ) and depend upon the coefficient $\ensuremath{\gamma}$ 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 $\ensuremath{\gamma}$. The CFTs of free fermions and bosons take the values $\ensuremath{\gamma}=1/12,\ensuremath{-}1/12$, respectively, and so saturate the bound $|\ensuremath{\gamma}|\ensuremath{\le}1/12$ obtained earlier from the holographic theory; the correlator of the conserved gauge flux of $U(1)$ gauge theories takes intermediate values of $\ensuremath{\gamma}$. The value of $\ensuremath{\gamma}$ also controls the frequency dependence of the conductivity and other properties of quantum critical transport at nonzero temperatures. Our results for the values of $\ensuremath{\gamma}$ lead to an appealing physical interpretation of particlelike or vortexlike 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.We compute three-point correlators between the stress-energy tensor and the 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 perform the computation using holography. In the holographic approach, the correlators are computed from an effective action on (3+1)-dimensional anti-de Sitter space (${\mathrm{AdS}}_{4}$) and depend upon the coefficient $\ensuremath{\gamma}$ 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 $\ensuremath{\gamma}$. The CFTs of free fermions and bosons take the values $\ensuremath{\gamma}=1/12,\ensuremath{-}1/12$, respectively, and so saturate the bound $|\ensuremath{\gamma}|\ensuremath{\le}1/12$ obtained earlier from the holographic theory; the correlator of the conserved gauge flux of $U(1)$ gauge theories takes intermediate values of $\ensuremath{\gamma}$. The value of $\ensuremath{\gamma}$ also controls the frequency dependence of the conductivity and other properties of quantum critical transport at nonzero temperatures. Our results for the values of $\ensuremath{\gamma}$ lead to an appealing physical interpretation of particlelike or vortexlike 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.

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We compute current correlators of the

CP$${}^{N\ensuremath{-}1}

field theory in 2+1 dimensions, both at the critical point and in the phase with spontaneously broken

\mathrm{SU}(N)

symmetry. Universal constants are obtained to next-to-leading order in the

1/N

expansion. Implications are noted for quantum critical points of antiferromagnets and their vicinity.

{\mathrm{AdS}}_{4}

) and depend upon the coefficient $\ensuremath{\gamma}$ 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 $\ensuremath{\gamma}$. The CFTs of free fermions and bosons take the values $\ensuremath{\gamma}=1/12,\ensuremath{-}1/12$, respectively, and so saturate the bound $|\ensuremath{\gamma}|\ensuremath{\le}1/12$ obtained earlier from the holographic theory; the correlator of the conserved gauge flux of

U(1)

gauge theories takes intermediate values of $\ensuremath{\gamma}$. The value of $\ensuremath{\gamma}$ also controls the frequency dependence of the conductivity and other properties of quantum critical transport at nonzero temperatures. Our results for the values of $\ensuremath{\gamma}$ lead to an appealing physical interpretation of particlelike or vortexlike 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.

Deconfined quantum criticality of two-dimensional

SU(2)

quantum antiferromagnets featuring a transition from an antiferromagnetically ordered ground state to a so-called valence-bond solid state, is governed by a non-compact CP

^1

model with a Maxwell term in 2+1 spacetime dimensions. We introduce a new perspective on deconfined quantum criticality within a field-theoretic framework based on an expansion in powers of

ε=4-d

for fixed number

N

of complex matter fields. We show that in the allegedly weak first-order transition regime, a so-called conformal phase transition leads to a genuine deconfined quantum critical point. In such a transition, the gap vanishes when the critical point is approached from above and diverges when it is approached from below. We also find that the spin stiffness has a universal jump at the critical point.

{]}^{\mathit{z}}

[\ensuremath{\kappa}(k)${]}^{2\mathrm{\ensuremath{-}}\mathit{d}}$, which reduces to the universal conductance in 2+1 dimensions. We find that the ratio \ensuremath{\surd}\ensuremath{\kappa}\ensuremath{\chi} /\ensuremath{\sigma} always approaches a constant as we approach the critical point. We also determine exponents of many scaling relations, using the scaling properties of the current.

The deconfined quantum critical point of a two-dimensional

SU(N)

antiferromagnet is governed by an Abelian Higgs model in

d=2+1

spacetime dimensions featuring

N

complex scalar fields. In this context, we derive for $2\ensuremath{\le}d\ensuremath{\le}4$ an exact formula for the central charge of the

U(1)

current in terms of the gauge coupling at quantum criticality and compare it with the corresponding result obtained using gauge-gravity duality. There is a remarkable similarity precisely for

d=2+1

. In this case the amplitude of the current correlation function has the same form as predicted by the gauge-gravity duality. We also compare finite temperature results for the charge susceptibility in the large

N

limit with the result predicted by the gauge-gravity duality. Our results suggest that condensed matter systems at quantum criticality may provide interesting quantitative tests of the gauge-gravity duality even in the absence of supersymmetry.