# Magnetic catalysis and quantum Hall ferromagnetism in weakly coupled graphene

## Abstract

We study the realization in a model of graphene of the phenomenon whereby the tendency of gauge-field mediated interactions to break chiral symmetry spontaneously is greatly enhanced in an external magnetic field. We prove that, in the weak coupling limit, and where the electron-electron interaction satisfies certain mild conditions, the ground state of charge neutral graphene in an external magnetic field is a quantum Hall ferromagnet which spontaneously breaks the emergent U(4) symmetry to U(2) × U(2). We argue that, due to a residual CP symmetry, the quantum Hall ferromagnet order parameter is given exactly by the leading order in perturbation theory. On the other hand, the chiral condensate which is the order parameter for chiral symmetry breaking generically obtains contributions at all orders. We compute the leading correction to the chiral condensate. We argue that the ensuing fermion spectrum resembles that of massive fermions with a vanishing U(4)-valued chemical potential. We discuss the realization of parity and charge conjugation symmetries and argue that, in the context of our model, the charge neutral quantum Hall state in graphene is a bulk insulator, with vanishing longitudinal conductivity due to a charge gap and Hall conductivity vanishing due to a residual discrete particle-hole symmetry.

### Keywords

Field Theories in Lower Dimensions Spontaneous Symmetry Breaking### References

- [1]G.W. Semenoff,
*Condensed matter simulation of a three-dimensional anomaly*,*Phys. Rev. Lett.***53**(1984) 2449 [SPIRES].MathSciNetADSCrossRefGoogle Scholar - [2]K.S. Novoselov et al. ,
*Electric field effect in atomically thin carbon films*,*Science***306**(2004) 666.ADSCrossRefGoogle Scholar - [3]K.S. Novoselov et al.,
*Two-dimensional gas of massless Dirac fermions in graphene*,*Nature***438**(2005) 197 [cond-mat/0509330] [SPIRES].ADSCrossRefGoogle Scholar - [4]K. S. Novoselov et al.,
*Two-dimensional atomic crystals*,*PNAS***102**(2005) 10451.ADSCrossRefGoogle Scholar - [5]A.K. Geim and K.S. Novoselov,
*The rise of graphene*,*Nat. Mater.***6**(2007) 183.ADSCrossRefGoogle Scholar - [6]
- [7]S.Y. Zhou et al. ,
*Substrate-induced bandgap opening in epitaxial graphene*,*Nat. Mater.***6**(2007) 770.ADSCrossRefGoogle Scholar - [8]M.I. Katsnelson,
*Graphene: carbon in two dimensions*,*Materials Today***10**(2007) 20.CrossRefGoogle Scholar - [9]Y. Zhang, Y.-W. Tan, H.L. Stormer and P. Kim,
*Experimental observation of the quantum Hall effect and Berry’s phase in graphene*,*Nature***438**(2005) 201.ADSCrossRefGoogle Scholar - [10]V.P. Gusynin and S.G. Sharapov,
*Unconventional integer quantum Hall effect in graphene*,*Phys. Rev. Lett.***95**(2005) 146801 [cond-mat/0506575] [SPIRES].ADSCrossRefGoogle Scholar - [11]N.M.R. Peres, F. Guinea, A.H. Castro Neto,
*Electronic properties of disordered two-dimensional carbon*,*Phys. Rev. Lett.***73**(2006) 125411 [cond-mat/0512091].ADSGoogle Scholar - [12]D.V. Khveshchenko,
*Magnetic-field-induced insulating behavior in highly oriented pyrolitic graphite*,*Phys. Rev. Lett.***87**(2001) 206401 [SPIRES].ADSCrossRefGoogle Scholar - [13]E.V. Gorbar, V.P. Gusynin, V.A. Miransky and I.A. Shovkovy,
*Magnetic field driven metal-insulator phase transition in planar systems*,*Phys. Rev.***B 66**(2002) 045108 [cond-mat/0202422] [SPIRES].ADSGoogle Scholar - [14]I.F. Herbut,
*Interactions and phase transitions on graphene’s honeycomb lattice*,*Phys. Rev. Lett.***97**(2006) 146401 [cond-mat/0606195] [SPIRES].ADSCrossRefGoogle Scholar - [15]D.T. Son,
*critical point in graphene approached in the limit of infinitely strong Coulomb interaction*,*Phys. Rev.***B 75**(2007) 235423 [cond-mat/0701501].ADSGoogle Scholar - [16]J.E. Drut and D.T. Son,
*Renormalization group flow of quartic perturbations in graphene: Strong coupling and large-N limits*,*Phys. Rev.***B 77**(2008) 075115 [arXiv:0710.1315] [SPIRES].ADSGoogle Scholar - [17]
- [18]I.F. Herbut, V. Juricic and B. Roy,
*Theory of interacting electrons on the honeycomb lattice*,*Phys. Rev.***B 79**(2009) 085116 [arXiv:0811.0610] [SPIRES].ADSGoogle Scholar - [19]I.F. Herbut, V. Juricic and O. Vafek,
*Relativistic Mott criticality in graphene*,*Phys. Rev.***B 80**(2009) 075432 [arXiv:0904.1019] [SPIRES].ADSGoogle Scholar - [20]V. Juricic, I.F. Herbut and G.W. Semenoff,
*Coulomb interaction at the metal-insulator critical point in graphene*,*Phys. Rev.***B 80**(2009) 081405 [arXiv:0906.3513] [SPIRES].ADSGoogle Scholar - [21]W. Armour, S. Hands and C. Strouthos,
*Monte Carlo simulation of the semimetal-insulator phase transition in monolayer graphene*,*Phys. Rev.***B 81**(2010) 125105 [arXiv:0910.5646] [SPIRES].ADSGoogle Scholar - [22]W. Armour, S. Hands and C. Strouthos,
*Lattice simulations near the semimetal-insulator phase transition of graphene*, arXiv:0908.0118 [SPIRES]. - [23]S. Hands and C. Strouthos,
*Quantum phase transition in a graphene model*,*J. Phys. Conf. Ser.***150**(2009) 042191 [arXiv:0808.2720] [SPIRES].ADSCrossRefGoogle Scholar - [24]S. Hands and C. Strouthos,
*Quantum critical behaviour in a graphene-like model*,*Phys. Rev.***B 78**(2008) 165423 [arXiv:0806.4877] [SPIRES].ADSGoogle Scholar - [25]J.E. Drut and T.A. Lahde,
*Critical exponents of the semimetal-insulator transition in graphene: a Monte Carlo study*,*Phys. Rev.***B 79**(2009) 241405 [arXiv:0905.1320] [SPIRES].ADSGoogle Scholar - [26]J.E. Drut, T.A. Lahde and L. Suoranta,
*First-order chiral transition in the compact lattice theory of graphene and the case for improved actions*, arXiv:1002.1273 [SPIRES]. - [27]
- [28]
- [29]J.E. Drut, T.A. Lahde and E. Tolo,
*Signatures of a gap in the conductivity of graphene*, arXiv:1005.5089 [SPIRES]. - [30]J.E. Drut, T.A. Lahde and E. Tolo,
*Graphene: from materials science to particle physics*,*PoS*LATTICE2010 (2010) 006 [arXiv:1011.0643] [SPIRES]. - [31]Y. Zhang et al.,
*Landau-level splitting in graphene in high magnetic fields*,*Phys. Rev. Lett.***96**(2006) 136806 [cond-mat/0602649].ADSCrossRefGoogle Scholar - [32]D.A. Abanin et al.,
*Dissipative quantum Hall effect in graphene near the Dirac point*,*Phys. Rev. Lett.***98**(2007) 196806 [cond-mat/0702125].ADSCrossRefGoogle Scholar - [33]Z. Jiang et al.,
*Quantum Hall states near the charge-neutral Dirac point in graphene*,*Phys. Rev. Lett.***99**(2007) 106802 [arXiv:0705.1102].ADSCrossRefGoogle Scholar - [34]A.J.M. Giesbers et al.,
*Quantum-Hall activation gaps in graphene*,*Phys. Rev. Lett.***99**(2007) 206803 [arXiv:1009.5485].ADSCrossRefGoogle Scholar - [35]J.G. Checkelsky, L. Li and N.P. Ong,
*Zero-energy state in graphene in a high magnetic field*,*Phys. Rev. Lett.***100**(2008) 206801 [arXiv:0708.1959].ADSCrossRefGoogle Scholar - [36]J.G. Checkelsky, L. Li, N.P. Ong,
*Divergent resistance at the Dirac point in graphene: evidence for a transition in a high magnetic field*,*Phys. Rev.***B 79**(2009) 115434 [arXiv:0808.0906].ADSGoogle Scholar - [37]L. Zhang et al.,
*Breakdown of the N*= 0*quantum Hall state in graphene: two insulating regimes*,*Phys. Rev.***B 80**(2009) 241412 [arXiv:0904.1996].ADSGoogle Scholar - [38]A.J.M. Giesbers et al.,
*Gap opening in the zeroth Landau level of graphene*,*Phys. Rev.***B 80**(2009) 201403(R) [arXiv:0904.0948].ADSGoogle Scholar - [39]Xu Du, I. Skachko, F. Duerr, A. Luican, E.Y. Andrei,
*Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene*,*Nature***462**(2009) 192.ADSCrossRefGoogle Scholar - [40]K.I. Bolotin et al.,
*Observation of the fractional quantum Hall effect in graphene*,*Nature***462**(2009) 196.ADSCrossRefGoogle Scholar - [41]D.A. Abanin et al.,
*Fractional quantum Hall effect in suspended graphene: transport coefficients and electron interaction strength*,*Phys. Rev.***B 81**(2010) 115410 [arXiv:0912.1134].ADSGoogle Scholar - [42]C.R. Dean et al.,
*Multicomponent fractional quantum Hall effect in graphene*, arXiv:1010.1179. - [43]F. Ghahari et al.,
*Measurement of the*ν = 1/3*fractional quantum Hall energy gap in suspended graphene*,*Phys. Rev. Lett.***106**(2011) 046801.ADSCrossRefGoogle Scholar - [44]K.G. Klimenko,
*Three-dimensional Gross-Neveu model in an external magnetic field*,*Theor. Math. Phys.***89**(1992) 1161 [SPIRES].MathSciNetCrossRefGoogle Scholar - [45]K.G. Klimenko,
*Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field*,*Theor. Math. Phys.***90**(1992) 1 [SPIRES].MathSciNetCrossRefGoogle Scholar - [46]V.P. Gusynin, V.A. Miransky and I.A. Shovkovy,
*Catalysis of dynamical flavor symmetry breaking by a magnetic field in (2 + 1)-dimensions*,*Phys. Rev. Lett.***73**(1994) 3499 [hep-ph/9405262] [SPIRES].ADSCrossRefGoogle Scholar - [47]V.P. Gusynin, V.A. Miransky and I.A. Shovkovy,
*Dynamical flavor symmetry breaking by a magnetic field in*(2 + 1)*-dimensions*,*Phys. Rev.***D 52**(1995) 4718 [hep-th/9407168] [SPIRES].ADSGoogle Scholar - [48]V.P. Gusynin, V.A. Miransky and I.A. Shovkovy,
*Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field*,*Nucl. Phys.***B 462**(1996) 249 [hep-ph/9509320] [SPIRES].ADSCrossRefGoogle Scholar - [49]V.P. Gusynin, V.A. Miransky, S.G. Sharapov and I.A. Shovkovy,
*Excitonic gap, phase transition and quantum Hall effect in graphene*,*Phys. Rev.***B 74**(2006) 195429 [cond-mat/0605348] [SPIRES].ADSGoogle Scholar - [50]I.F. Herbut,
*Pseudomagnetic catalysis of the time-reversal symmetry breaking in graphene*,*Phys. Rev.***B 78**(2008) 205433 [arXiv:0804.3594].ADSGoogle Scholar - [51]E.V. Gorbar, V.P. Gusynin, V.A. Miransky and I.A. Shovkovy,
*Dynamics in the quantum Hall effect and the phase diagram of graphene*,*Phys. Rev.***B 78**(2008) 085437 [arXiv:0806.0846] [SPIRES].ADSGoogle Scholar - [52]M. Ezawa,
*Intrinsic Zeeman effect in graphene*,*J. Phys. Soc. Jpn.***76**(2007) 094701 [SPIRES].ADSCrossRefGoogle Scholar - [53]G.W. Semenoff, I.A. Shovkovy and L.C.R. Wijewardhana,
*Phase transition induced by a magnetic field*,*Mod. Phys. Lett.***A 13**(1998) 1143 [hep-ph/9803371] [SPIRES].ADSCrossRefGoogle Scholar - [54]V.P. Gusynin, S.G. Sharapov and J.P. Carbotte,
*AC conductivity of graphene: from tight-binding model to*2 + 1*-dimensional quantum electrodynamics*,*Int. J. Mod. Phys.***B 21**(2007) 4611 [arXiv:0706.3016] [SPIRES].ADSGoogle Scholar - [55]K. Nomura, A.H. MacDonald,
*Quantum Hall ferromagnetism in graphene*,*Phys. Rev. Lett.***96**(2006) 256602 [cond-mat/0604113].ADSCrossRefGoogle Scholar - [56]S.M. Girvin and A.H. MacDonald,
*Multicomponent quantum Hall systems: the sum of their parts and more*, in*Perspectives in Quantum Hall Effects*, S. Das Sarma and A. Pinczuk eds., John Wiley and Soons, New York U.S.A. (1997).Google Scholar - [57]K. Yang, S. Das Sarma and A.H. MacDonald,
*Collective modes and skyrmion excitations in graphene*SU(4)*quantum Hall ferromagnets*,*Phys. Rev.***B 74**(2006) 075423.ADSGoogle Scholar - [58]A.J. Niemi and G.W. Semenoff,
*Axial anomaly induced fermion fractionization and effective gauge theory actions in odd dimensional space-times*,*Phys. Rev. Lett.***51**(1983) 2077 [SPIRES].MathSciNetADSCrossRefGoogle Scholar - [59]A.N. Redlich,
*Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions*,*Phys. Rev.***D 29**(1984) 2366 [SPIRES].MathSciNetADSGoogle Scholar - [60]A.N. Redlich,
*Gauge noninvariance and parity nonconservation of three-dimensional fermions*,*Phys. Rev. Lett.***52**(1984) 18 [SPIRES].MathSciNetADSCrossRefGoogle Scholar - [61]C. Vafa and E. Witten,
*Restrictions on symmetry breaking in vector-like gauge theories*,*Nucl. Phys.***B 234**(1984) 173 [SPIRES].MathSciNetADSCrossRefGoogle Scholar - [62]S. Ryu, C. Mudry, C.-Y. Hou, C. Chamon,
*Masses in graphenelike two-dimensional electronic systems: topological defects in order parameters and their fractional exchange statistics*,*Phys. Rev.***B 80**(2009) 205319.ADSGoogle Scholar - [63]A. Tanaka, X. Hu,
*Many-body spin Berry phases emerging from the*π*-flux state: competition between antiferromagnetism and the valence-bond-solid state*,*Phys. Rev. Lett.***95**(2005) 036402.ADSCrossRefGoogle Scholar - [64]A. Tanaka, X. Hu,
*Effective field theory with a*θ*-vacua structure for two-dimensional spin systems*,*Phys. Rev.***B 74**(2006) 140407.ADSGoogle Scholar - [65]P. Ghaemi, S. Ryu, D.-H. Lee,
*The quantum valley Hall effect in proximity-induced superconducting graphene: an experimental window for deconfined quantum criticality*, arXiv:0704.2234. - [66]I.F. Herbut,
*Zero-energy states and fragmentation of spin in the easy- plane antiferromagnet on honeycomb lattice*,*Phys. Rev. Lett.***99**(2007) 206404 [arXiv:0704.2234] [SPIRES].ADSCrossRefGoogle Scholar - [67]J. Gonzalez, F. Guinea and M.A.H. Vozmediano,
*NonFermi liquid behavior of electrons in the half filled honeycomb lattice (A Renormalization group approach)*,*Nucl. Phys.***B 424**(1994) 595 [hep-th/9311105] [SPIRES].ADSCrossRefGoogle Scholar - [68]J. Gonzalez, F. Guinea and M.A.H. Vozmediano,
*Marginal-Fermi-liquid behavior from two-dimensional Coulomb interaction*,*Phys. Rev.***B 59**(1999) 2474(R) [cond-mat/0302164]. - [69]H. Leal and D.V. Khveshchenko,
*Excitonic instability in two-dimensional degenerate semimetals*,*Nucl. Phys.***B 687**(2004) 323 [cond-mat/0302164] [SPIRES].ADSGoogle Scholar - [70]D.V. Khveshchenko,
*Coulomb-interacting Dirac fermions in disordered graphene*,*Phys. Rev.***B 74**(2006) 161402 [cond-mat/0612651].ADSGoogle Scholar - [71]E.G. Mishchenko,
*Effect of electron-electron interactions on the conductivity of clean graphene*,*Phys. Rev. Lett.***98**(2007) 216801 [cond-mat/0604601].ADSCrossRefGoogle Scholar - [72]D.E. Sheehy, J. Schmalian,
*Quantum critical scaling in graphene*,*Phys. Rev. Lett.***99**(2007) 226803.ADSCrossRefGoogle Scholar - [73]O. Vafek, M.J. Case,
*Renormalization group approach to two-dimensional Coulomb interacting Dirac fermions with random gauge potential*,*Phys. Rev.***B 77**(2008) 033410 [arXiv:1103.6285].ADSGoogle Scholar - [74]J. Alicea, M.P.A. Fisher,
*Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes*,*Phys. Rev.***B 74**(2006) 075422 [SPIRES].ADSGoogle Scholar - [75]
- [76]R. Jackiw and C. Rebbi,
*Solitons with fermion number*1/2,*Phys. Rev.***D 13**(1976) 3398 [SPIRES].MathSciNetADSGoogle Scholar - [77]A.J. Niemi and G.W. Semenoff,
*Fermion number fractionization in quantum field theory*,*Phys. Rept.***135**(1986) 99 [SPIRES].MathSciNetADSCrossRefGoogle Scholar - [78]S.R. Coleman and B.R. Hill,
*No more corrections to the topological mass term in QED in three-dimensions*,*Phys. Lett.***B 159**(1985) 184 [SPIRES].ADSGoogle Scholar - [79]G.W. Semenoff, P. Sodano and Y.-S. Wu,
*Renormalization of the statistics parameter in three-dimensional electrodynamics*,*Phys. Rev. Lett.***62**(1989) 715 [SPIRES].ADSCrossRefGoogle Scholar