# Stress tensor and current correlators of interacting conformal field theories in 2+1 dimensions: fermionic Dirac matter coupled to U(1) gauge field

Open Access

Regular Article - Theoretical PhysicsFirst Online:

Received:

Revised:

Accepted:

- 284 Downloads
- 8 Citations

## Abstract

We compute the central charge *C* _{ T } and universal conductivity *C* _{ J } of *N* _{ F } fermions coupled to a *U* (1) gauge field up to next-to-leading order in the 1*/N* _{ F } expansion. We discuss implications of these precision computations as a diagnostic for response and entanglement properties of interacting conformal field theories for strongly correlated condensed matter phases and conformal quantum electrodynamics in 2 + 1 dimensions.

## Keywords

Conformal and W Symmetry 1/N Expansion Download
to read the full article text

## Notes

### **Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

- [1]K.G. Wilson and M.E. Fisher,
*Critical exponents in*3*.*99*dimensions*,*Phys. Rev. Lett.***28**(1972) 240 [INSPIRE].ADSCrossRefGoogle Scholar - [2]R. Abe,
*Critical exponent η up to*1*/N*^{2}*for the three-dimensional system with short-range interaction*,*Prog. Theor. Phys.***49**(1973) 6.Google Scholar - [3]A.C. Petkou,
*C*_{T}*and C*_{J}*up to next-to-leading order in*1*/N in the conformally invariant O*(*N*)*vector model for*2*< d <*4,*Phys. Lett.***B 359**(1995) 101 [hep-th/9506116] [INSPIRE].ADSCrossRefGoogle Scholar - [4]S. El-Showk et al.,
*Solving the*3*D Ising model with the conformal bootstrap*,*Phys. Rev.***D 86**(2012) 025022 [arXiv:1203.6064] [INSPIRE].ADSGoogle Scholar - [5]S. El-Showk et al.,
*Solving the*3*D Ising model with the conformal bootstrap II. c-minimization and precise critical exponents*,*J. Stat. Phys.***157**(2014) 869 [arXiv:1403.4545] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar - [6]M.C. Cha, et al.,
*Universal conductivity of two-dimensional films at the superconductor-insulator transition*,*Phys. Rev.***B 44**(1991) 6883.ADSCrossRefGoogle Scholar - [7]R. Fazio and D. Zappala,
*ϵ expansion of the conductivity at the superconductor-Mott-insulator transitions*,*Phys. Rev.***B 53**(1996) R8885 [cond-mat/9511004].ADSGoogle Scholar - [8]S. Chakravarty, B.I. Halperin and D.R. Nelson,
*Two-dimensional quantum Heisenberg antiferromagnet at low temperatures*,*Phys. Rev.***B 39**(1989) 2344 [INSPIRE].ADSCrossRefGoogle Scholar - [9]A.V. Chubukov, S. Sachdev and J. Ye,
*Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state*,*Phys. Rev.***B 49**(1994) 11919 [INSPIRE].ADSCrossRefGoogle Scholar - [10]R.K. Kaul and S. Sachdev,
*Quantum criticality of*U(1)*gauge theories with fermionic and bosonic matter in two spatial dimensions*,*Phys. Rev.***B 77**(2008) 155105 [arXiv:0801.0723] [INSPIRE].ADSCrossRefGoogle Scholar - [11]W. Chen, G.W. Semenoff and Y.-S. Wu,
*Two loop analysis of nonAbelian Chern-Simons theory*,*Phys. Rev.***D 46**(1992) 5521 [hep-th/9209005] [INSPIRE].ADSMathSciNetGoogle Scholar - [12]W. Chen, M.P.A. Fisher and Y.-S. Wu,
*Mott transition in an anyon gas*,*Phys. Rev.***B 48**(1993) 13749 [cond-mat/9301037] [INSPIRE].ADSCrossRefGoogle Scholar - [13]T. Senthil et al.,
*Deconfined quantum criticality*,*Science***303**(2004) 1490.ADSCrossRefGoogle Scholar - [14]A.W. Sandvik,
*Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions*,*Phys. Rev. Lett.***98**(2007) 227202 [cond-mat/0611343].ADSCrossRefGoogle Scholar - [15]Y. Huh, P. Strack and S. Sachdev,
*Vector boson excitations near deconfined quantum critical points*,*Phys. Rev. Lett.***111**(2013) 166401 [arXiv:1307.6860] [INSPIRE].ADSCrossRefGoogle Scholar - [16]W. Rantner and X.-G. Wen,
*Spin correlations in the algebraic spin liquid: implications for high-T*_{c}*superconductors*,*Phys. Rev.***B 66**(2002) 144501 [INSPIRE].ADSCrossRefGoogle Scholar - [17]M. Franz, Z. Tesanovic and O. Vafek,
*QED*_{3}*theory of pairing pseudogap in cuprates. 1. From D wave superconductor to antiferromagnet via*‘*algebraic*’*Fermi liquid*,*Phys. Rev.***B 66**(2002) 054535 [cond-mat/0203333] [INSPIRE].ADSCrossRefGoogle Scholar - [18]M. Franz, T. Pereg-Barnea, D.E. Sheehy and Z. Tesanovic,
*Gauge invariant response functions in algebraic Fermi liquids*,*Phys. Rev.***B 68**(2003) 024508 [cond-mat/0211119] [INSPIRE].ADSCrossRefGoogle Scholar - [19]R.K. Kaul, Y.B. Kaim, S. Sachdev and T. Senthil,
*Algebraic charge liquids*,*Nature Phys.***4**(2007) 28.ADSCrossRefGoogle Scholar - [20]J. Cardy,
*Conformal field theory and statistical mechanics*, arXiv:0807.3472. - [21]A.M. Polyakov,
*Gauge fields and strings*, Harwood Academic, Chur, Switzerland (1987).Google Scholar - [22]S. Coleman,
*Aspects of symmetry*, Cambridge University Press, Cambridge U.K. (1988).Google Scholar - [23]T.W. Appelquist, M.J. Bowick, D. Karabali and L.C.R. Wijewardhana,
*Spontaneous chiral symmetry breaking in three-dimensional QED*,*Phys. Rev.***D 33**(1986) 3704 [INSPIRE].ADSGoogle Scholar - [24]T. Appelquist, D. Nash and L.C.R. Wijewardhana,
*Critical behavior in*(2 + 1)*-dimensional QED*,*Phys. Rev. Lett.***60**(1988) 2575 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [25]D. Nash,
*Higher order corrections in*(2 + 1)*-dimensional QED*,*Phys. Rev. Lett.***62**(1989) 3024 [INSPIRE].ADSCrossRefGoogle Scholar - [26]D.T. Son,
*Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction*,*Phys. Rev.***B 75**(2007) 235423 [cond-mat/0701501].ADSCrossRefGoogle Scholar - [27]V. Juricic, O. Vafek and I.F. Herbut,
*Conductivity of interacting massless Dirac particles in graphene: collisionless regime*,*Phys. Rev.***B 82**(2010) 235402 [arXiv:1009.3269] [INSPIRE].ADSCrossRefGoogle Scholar - [28]I.F. Herbut and V. Mastropietro,
*Universal conductivity of graphene in the ultrarelativistic regime*,*Phys. Rev.***B 87**(2013) 205445 [arXiv:1304.1988] [INSPIRE].ADSCrossRefGoogle Scholar - [29]A.V. Kotikov and S. Teber,
*Two-loop fermion self-energy in reduced quantum electrodynamics and application to the ultra-relativistic limit of graphene*,*Phys. Rev.***D 89**(2014) 065038 [arXiv:1312.2430] [INSPIRE].ADSGoogle Scholar - [30]E. Barnes, E.H. Hwang, R. Throckmorton and S. Das Sarma,
*Effective field theory, three-loop perturbative expansion and their experimental implications in graphene many-body effects*,*Phys. Rev.***B 89**(2014) 235431 [arXiv:1401.7011] [INSPIRE].ADSCrossRefGoogle Scholar - [31]J. Braun, H. Gies, L. Janssen and D. Roscher,
*Phase structure of many-flavor QED*_{3},*Phys. Rev.***D 90**(2014) 036002 [arXiv:1404.1362] [INSPIRE].ADSGoogle Scholar - [32]Y. Huh, P. Strack and S. Sachdev,
*Conserved current correlators of conformal field theories in*2 + 1*dimensions*,*Phys. Rev.***B 88**(2013) 155109 [arXiv:1307.6863] [INSPIRE].ADSCrossRefGoogle Scholar - [33]J. Cardy,
*The ubiquitous*‘*c*’*: from the Stefan-Boltzmann law to quantum information*,*J. Stat. Mech.*(2010) P10004.Google Scholar - [34]E. Perlmutter,
*A universal feature of CFT Rényi entropy*,*JHEP***03**(2014) 117 [arXiv:1308.1083] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [35]R.K. Kaul, R.G. Melko and A.W. Sandvik,
*Bridging lattice-scale physics and continuum field theory with quantum Monte Carlo simulations*,*Annu. Rev. Cond. Mat. Phys.***4**(2013) 179 [arXiv:1204.5405].ADSCrossRefGoogle Scholar - [36]S.J. Hathrell,
*Trace anomalies and*λ*ϕ*^{4}*theory in curved space*,*Annals Phys.***139**(1982) 136 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [37]S.J. Hathrell,
*Trace anomalies and QED in curved space*,*Annals Phys.***142**(1982) 34 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [38]I. Jack and H. Osborn,
*Background field calculations in curved spacetime: I. General application and application to scalar fields*,*Nucl. Phys.***B 234**(1984) 331.ADSCrossRefGoogle Scholar - [39]I. Jack,
*Background field calculations in curved space-time. 3. Application to a general gauge theory coupled to fermions and scalars*,*Nucl. Phys.***B 253**(1985) 323 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [40]A. Cappelli, D. Friedan and J.I. Latorre,
*C theorem and spectral representation*,*Nucl. Phys.***B 352**(1991) 616 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [41]H. Osborn and A.C. Petkou,
*Implications of conformal invariance in field theories for general dimensions*,*Annals Phys.***231**(1994) 311 [hep-th/9307010] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [42]A. Petkou,
*Conserved currents, consistency relations and operator product expansions in the conformally invariant O*(*N*)*vector model*,*Annals Phys.***249**(1996) 180 [hep-th/9410093] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [43]M.F. Zoller and K.G. Chetyrkin,
*OPE of the energy-momentum tensor correlator in massless QCD*,*JHEP***12**(2012) 119 [arXiv:1209.1516] [INSPIRE].ADSCrossRefGoogle Scholar - [44]D. Chowdhury, S. Raju, S. Sachdev, A. Singh and P. Strack,
*Multipoint correlators of conformal field theories: implications for quantum critical transport*,*Phys. Rev.***B 87**(2013) 085138 [arXiv:1210.5247] [INSPIRE].ADSCrossRefGoogle Scholar - [45]J.M. Maldacena and G.L. Pimentel,
*On graviton non-gaussianities during inflation*,*JHEP***09**(2011) 045 [arXiv:1104.2846] [INSPIRE].ADSCrossRefMATHGoogle Scholar - [46]I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi,
*Rényi entropies for free field theories*,*JHEP***04**(2012) 074 [arXiv:1111.6290] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [47]A. Dymarsky, Z. Komargodski, A. Schwimmer and S. Theisen,
*On scale and conformal invariance in four dimensions*, arXiv:1309.2921. - [48]A. Bzowski and K. Skenderis,
*Comments on scale and conformal invariance in four dimensions*, arXiv:1402.3208. - [49]R.C. Myers and A. Sinha,
*Holographic c-theorems in arbitrary dimensions*,*JHEP***01**(2011) 125 [arXiv:1011.5819] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [50]I.R. Klebanov, S.S. Pufu and B.R. Safdi,
*F-theorem without supersymmetry*,*JHEP***10**(2011) 038 [arXiv:1105.4598] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [51]T. Appelquist, A.G. Cohen and M. Schmaltz,
*A new constraint on strongly coupled gauge theories*,*Phys. Rev.***D 60**(1999) 045003 [hep-th/9901109] [INSPIRE].ADSGoogle Scholar - [52]J.L. Cardy,
*Anisotropic corrections to correlation functions in finite size systems*,*Nucl. Phys.***B 290**(1987) 355 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [53]
*Tools and tables for quantum field theory calculations*, http://www.feyncalc.org/. - [54]A. Bzowski, P. McFadden and K. Skenderis,
*Holographic predictions for cosmological 3-point functions*,*JHEP***03**(2012) 091 [arXiv:1112.1967] [INSPIRE].ADSCrossRefMATHGoogle Scholar - [55]A.I. Davydychev,
*A simple formula for reducing Feynman diagrams to scalar integrals*,*Phys. Lett.***B 263**(1991) 107 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [56]A.I. Davydychev,
*Recursive algorithm for evaluating vertex-type Feynman integrals*,*J. Phys.***A 25**(1992) 5587.ADSMathSciNetMATHGoogle Scholar - [57]O. Aharony, G. Gur-Ari and R. Yacoby,
*Correlation functions of large-N Chern-Simons-Matter theories and bosonization in three dimensions*,*JHEP***12**(2012) 028 [arXiv:1207.4593] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [58]G. Gur-Ari and R. Yacoby,
*Correlators of large-N fermionic Chern-Simons vector models*,*JHEP***02**(2013) 150 [arXiv:1211.1866] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [59]O. Aharony et al.,
*The thermal free energy in large N Chern-Simons-matter theories*,*JHEP***03**(2013) 121.ADSCrossRefGoogle Scholar

## Copyright information

© The Author(s) 2015