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Chiral (exciton condensation) transition in graphene. Baruch Rosenstein Nat. Chiao Tung University, Hsinchu Collaborators: Hsien-Chong Kao (Nat. Taiwan Normal University, Taipei) Meir Lewkowicz (Ariel University, Israel), Tsofar Maniv (Technion, Israel) KITPC, Beijing , August 10, 2012 Outline 1. Noninteracting electrons in graphene : relativistic massless fermions in 2+1. 2. Finite conductivity without either carriers or impurities. Screening. 3. Effect of large electric coupling: how can graphene become an insulator? 4. Dynamical chiral symmetry breaking. 5. Chiral universality classes. 6. Understanding a conductor with long range Coulomb interactions. 1. MASSLESS RELATIVISTIC FERMIONS IN GRAPHENE Carbon atoms in graphene are arranged in honeycomb 2D crystal by the covalent bonds between nearest neighbours. E. Andrei et al, Nature Nano 3, 491(08) Single graphene sheet on STM It can be viewed as a prototype of various 0D-3D structures based on the similar covalent C bonds. Novoselov et al, Proc. Nat. Acad. Sci. USA 102 10451 (05) Geim et al, Nature Mat. 6, 183 (07) Graphene shows metallic behaviour even at neutrality point where resistivity is maximal. Quantum Hall effect at room temperature clearly shows high mobility. Inhomogeneities or interaction, definitely present are somehow almost invisible in transport. Honeycomb has two hexagonal sublattices. Two representations of unit cell: Two atoms 1/3 each of 6 atoms = 2 atoms Tight binding model Hˆ As Bs ˆ a rn aˆrn sites 1,2,3 s , c.c. Spectrum consists of two bands with Fermi surface pinched right between them: not an obvious band insulator or a metal. Fermi surface is located exactly at two non - equivalent points of the Brillouin zone with sufficient little group to support a two dimensional representation K K KK, K K Around K the Hamiltonian is that of the left and right two component Weyl spinors: H vg σ p σ x , y Wallace, PR71 622 (1947) 3 a c vg 2 300 The Dirac point in band structure was pointed out very early by Wallace: electrons in graphene should behave like a 2D analogue of relativistic massless particles (“charged neutrinos”). The L,R Weyl spinors can be combined into two 4Dirac spinor described by L 0 t 1 x 2 y s s that possesses the Z 2 chiral (sublattice) symmetry s 5 s ; 5 i 0 1 2 3 2. FINITE CONDUCTIVITY WITHOUT EITHER CHARGES OR INPURITIES. SCREENING. Nonzero pseudo Ohmic conductivity There is resistivity at Dirac point, but resistivity is finite Geim et al, Nature Mat. 6, 183 (07) SiO2 Yacoby et al, Nature (08) Si graphene Samples on substrate generally correspond to either p or n doped graphene. Recently developed suspended samples are clearly undoped and exhibit the minimal resistivity at zero temperature. e2 2 h E. Andrei et al, Nature Nano 3, 491(08) Optical frequencies conductivity was measured to accuracy of 1%. And agrees with the theoretical value Geim, Novoselov et al, Science 102 10451 (08) Explanation: Schwinger’s electron – hole pair creation by electric field The basic picture of the pseudo – diffusive resistivity in pure graphene is the creation of the electron – hole pairs by electric field. The pairs carry current that can be further increased by reorientation of moving particles E e2 2 h Linear response Dynamical approach to linear response Let’s try to understand qualitatively how massless fermions react on electric field by just switching a homogeneous electric field and observing the creation of electron - hole pairs and the induced charge motion. Lewkowicz, B.R., PRL102, 106802 (09) hk ei k e 2 2 2 h h ' h ' hk 2 k sin 2 2 BZ k * k k * k e2 t 2 2 h t / One indeed observes that the current stabilizes at finite value. In DC this requires the use of tight binding model rather than its Dirac continuum theory. In addition to the finite term, there is also a linear acceleration term that vanishes due to cancellation of part near the Dirac points and far away . It can be presented as a full derivative 2 Et J k / k 4k 4 k Et d Sk 0 4 k dk y * y div k Im hk hk / k ; k Im hk*hk '' / k Kao, Lewkowicz, B.R., PRB81, 041416 (2010) Therefore there is no perfect scale separation when massless fermions are involved. This is also the source of parity anomaly . K K The 2D vacuum polarization 0,q q 0 Is too weak to exponentially screen the 3D Coulomb with 2D Fourier transform 0 q 4 2 V q q 2 2 The linearity of the current 3 vF eEt 2 1 ... dependence on electric field is 64 t quite accidental. It is not an Ohmic dissipative systems. For example B.R., H. C. Kao, Lewkowicz, the third order “correction” is Kornijenko, PRB81, R041416 (2010) This means that linear response breaks down at time scale tnl / eEv g Dynamical approach to nonlinear transport B.R, Lewkowicz, PRL 77, (09) ; B.R., Kao, Lewkowicz, PRB 81, R041416 (10) 2.5 J E 2.0 E0 E 2 6 E0 2 7 E0 / ea pair creation rate conductivity 2 E E 2 8 E0 1.5 1.0 0.5 1010V / m 0.0 0 20 40 60 80 100 120 time in units of t Stays there till a crossover time tnl / eEvg to a linearly increasing Schwinger’s regime J t 2 2 1/ 2 g ev eE 3/ 2 t At each k x one solves, using the WKB approximation, a tunneling problem similar to that in the Landau – Zener transition. Gavrilov, Gitman, PRD53, 7162 (1986) The interband transition probability at large times is N t ky k y eEt 2 kx k y k y eEt 4 eEt / 2 k 0 2 1 eEt 2 y vF k x2 kx exp eE eE 1 1/ 2 3/ 2 2 vF eE t vF Schwinger, PR82, 664 (1951) Electric field is screened very effectively beyond linear response. Therefore let us assume that the Schwinger, PR (1962)interaction is effectively screened and becomes local When density of created pairs rises, the recombination process is amplified due to Coulomb forces. The system crosses over to a true dissipative regime due to radiation friction. The neutral electron – hole plasma state is achieved when density approaches 12 2 N pl 10 cm Vandecasteele et al, PRB 82, 045416 (08) 3. IS GRAPHENE MORE METALLIC OR INSULATING? Electron – electron interactions are strong e2 1 H int r r ' 2 r ,r ' r r' r r rn cˆnAs † cˆnAs r rn 3 cˆnBs † cˆnBs n r 0 r s † r s r They break the (pseudo) relativistic invariance and are still 3D, namely less long range in 2D than that of the 2+1 dimensional massless QED. Coulomb interactions naively are strong due to coupling constant of order 1 2 e c 300 g QED vF vF 137 Just rescale the field in 2 e H vF † r 2 1 † r 0 r ' r,r ' 0 r r' † If unscreened, in graphene this would lead to a strongly coupled electronic system. A point - like charge in graphene would even supercritical phenomena (fall to the center). Z 1 137 Z supercritical regime Z Z 1 137 Positron emission Although supercriticality phenomena were never observed (perhaps due to nonlinear screening discussed above), the interactions should be strong enough to break chiral symmetry. 4. CHIRAL SYMMETRY BREAKING in GRAPHENE Exciton condensation If experience with relativistic interacting fermions is of any value, the attractive force that strong should be enough to create the electron – hole pairs condensate that would also break the chiral symmetry spontaneously, while the spin (flavour) rotation symmetry would remain unbroken. s s 0 This would make quasiparticle massive and the phase insulating. Nothing of that was observed in graphene as yet, but there might be experimental reasons (excuses). 1. In graphene on substrate there are metallic puddles that screen reducing the coupling and in addition dielectric constant. 2. In suspended samples electrostatics is nontrivial and one worries about capacitances near Yaish (12) suspended contacts. Moreover in another two phenomena where interactions are crucial 1. FQHE although observed for 1/ 3 is weaker than expected 2. DC, AC conductivity nearly coincides with the noninteracting one. Simulations Lattice simulation using staggered fermions indeed demonstrated formation of the condensate S d 2 rdt s vF t iA0 0 s 2 1 2 2 d rdzdt A0 r, z, t 2e Simulation predicted that the critical coupling is below g g 1/ 4 g Drut, Lande, PRB 79, 79167425 (09) Critical exponent however is turned out to be different from the Ising’s and points towards relativistic chiral universality class c ; m ; c ; m 1 m N f 2 : 1, 2.26 0.06 N f 4 : 1, 2.62 0.11 Drut, Lahde, PRB 23, 345601 (09) Critical exponent points to the chiral universality class: a relativistic model with local (screened) effective interaction Critical exponent however is turned out to be different from the Ising’s and points towards relativistic chiral universality class c ; m ; c ; m 1 m N f 2 : 1, 2.26 0.06 N f 4 : 1, 2.62 0.11 Drut, Lahde, PRB 23, 345601 (09) Critical exponent points to the chiral universality class: a relativistic model with local (screened) effective interaction F d x v g1 3 s s s s 2 g 2 s s 2 ... RG flow calculated by various means, Dyson – Schwinger equations truncation Khveshchenko, Veal , PRL 87, 246801 (01); Khveshchenko, Veal , Nucl. Phys. B687, 323 (04) … Perturbation theory in couplings, 1/N, various expansions Drut, Son, PRB 77, 075115 (08),…. The situation is still not entirely settled, but apparently the Coulomb interaction is “irrelevant” and at criticality the interaction becomes local and relativistically invariant, hence belongs to a chiral universality class. 5. UNIVERSALITY REVISITED Universality of transitions A second order phase transition is generally well described phenomenologically if one identifies: a. The order parameter field i ( x ) b. Symmetry group G and its spontaneous breaking pattern. L.D. Landau ( 1937 ) Rest of degrees of freedom are “irrelevant” sufficiently close to the critical temperature Tc. Later, the “relevant” part, namely the symmetry breaking pattern and dimensionality was termed “the universality class”. Its critical properties (“exponents”) are described by a renormalizable FT (CFT via its “deformations”). Wilson, Fisher, …1960s a g 1 2 2 F d x (i ) ii (ii ) 2 4 2 3 The CFT appears at strong coupling on the critical line a g a ac ; 0.33 UV Various aspects of physics of transitions of interacting 2D relativistic fermions were discussed in eighties is connection with nodal quasi-particles and a weak one to graphite Chiral universality classes It turns out that chiral phase transitions involving Dirac point are qualitatively different from the scalar ones due to presence of massless excitations in the ordered phase. s s g s s 2 F d x 2N f 3 gc g Unbroken chiral symmetry, Massless fermions Broken chiral symmetry, Massive fermions How the massless fermions create a strongly coupled fixed point It is convenient to consider the 1/N expansions (despite the fact that other “nonperturbative” methods achieve the same). g N g N N g N g r g g 2 g 3 2 ... gc g g 1 g g r1 g 1 g c1 From weakly coupled FP to chiral FP The four Fermi theory is similar to fixed length scalar models (sigma models). gc UV Gat, Kovner, B.R. Nucl. Phys. B385, 76 ( 1992 ) g This way one can classify chiral phase transitions according to both the symmetry breaking pattern and the number of massless fermions. Can this be called “topological” nowdays? Some critical exponents were calculated analytically and numerically (A. Li talk on Wednesday) The flow to the chiral CFT can be simulated on lattice using the Yukawa model. 6. UNDERSTANDING A CONDUCTOR WITH LONG RANGE COULOMB INTERACTIONS Early conflicting results A first step in estimating the corrections to conductivity would be a perturbative calculation of corrections, especially when experimental value almost coincides with the free electron result. First attempt in which sharp momentum cutoff was used gave a disappointingly large correction indicating that interactions are not weak e2 1 C g ... 2 h C 1 25 0.51 12 2 Herbut, Juricic, Vafek, PRL100 , 046403 (08) However it was shown that Ward identities were not satisfied. When the interaction and current were modified one obtains a very small correction consistent with experiment C 2 19 0.012 12 2 Mischenko, EurPL100 , 046403 (08); Sheehy, Schmalian, PRB80, 193411 (09) Despite this certain identities were not obeyed and a variant of dimensional regularization was applied with yet different answer C 3 22 0.25 12 2 Juricic,Vafek, Herbut, PRB82 , 235402 (10) New calculations appear, most consistent with the prefered small value MacDonald…Vignale…, PRB84, 045429 (11) As was noticed in calculation with noninteracting fermions, there is no complete scale separation in the model due to anomaly related zero modes. It is interesting to note that the results always differ by 1 C 4 And one might suspect that “incorrect UV regularization brings in “topological” term. Only calculation with more physical regularization will clarify this. Calculation on the lattice To resolve the ambiguity we have performed the calculation within the tight binding model and obtained the third result. Hˆ s s ˆ ˆ a U t a rn rn , rn c.c. n, , s 1 cˆns † cˆnsV rn rm cˆms † cˆms ... 2 n ,m e U n , t exp i c A r s , t ds 0 n 1 The current density operator is defined in such a way 1 J r, t H c A r, t That the Ward identities are satisfied for the lattice model and one does not have to guess corrections for the continuum operators. Results B.R., Maniv, Lewkowicz, arxiv cond mat (12) 1. For a local, quasilocal and even long range potential 1 V r ; 1 r The correction to DC conductivity is zero MacDonald…Vignale…, PRB84, 045429 (11) 2. For 1 0 there is an IR divergence and the perturbation theory is invalid. 3. For the Coulomb interaction one obtains the third value. 25 3 C 0.25 2 2 Interactions are not accidentally exceptionally small. One still have to look for explanation of the small interaction correction in experiment elswhere. Conclusions 1. Graphene undergoes the chiral symmetry breaking transition, however several theoretical questions remain: a. is the critical value of coupling larger than that of “ideal”=suspended graphene? b. Relativistic Z 2 ,N f 2 chiral universality class is the strongest candidate, but more simulations or analytic calculations are needed to assertain this. 2. The chirally unbroken quasi – Ohmic phase is by now better understood, still has several puzzles like why the conductivity is so close to the noninteracting one. 3. The insulating chiral symmetry broken phase still awaits discovery. It might be a laboratory of massless strongly coupled QED.