Phase transitions in strong QED3

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1 Phase transitions in strong QED3 Christian S. Fischer Justus Liebig Universität Gießen SFB November 2012 Christian Fischer (University of Gießen) Phase transitions in strong QED3 1 / 32

2 Overview 1.Introduction to QED3 2.QED3 and high Tc superconductors 3.QED3 and graphene Christian Fischer (University of Gießen) Phase transitions in strong QED3 2 / 32

3 QCD phase diagram Interesting open questions: Details of phase transitions Existence and location of critical point Properties of quarks and gluons in different phases Consequences for astrophysics Christian Fischer (University of Gießen) Phase transitions in strong QED3 3 / 32

4 QCD phase diagram Quarks de-confined and (almost) massless Quarks confined and massive Interesting open questions: Details of phase transitions Existence and location of critical point Properties of quarks and gluons in different phases Consequences for astrophysics Christian Fischer (University of Gießen) Phase transitions in strong QED3 3 / 32

5 Strong QFTs: QCD vs QED3 QCD non-abelian Abelian QED3 dynamical generation of scale scale set by coupling α = N f e 2 /8 asymptotically free asymptotically free α(p 2 ) N f =2 N f = p 2 Confinement and DχSB Confinement and DχSB Christian Fischer (University of Gießen) Phase transitions in strong QED3 4 / 32

6 Properties of QCD: Dynamical mass generation Dynamical quark masses via weak and strong force Yoichiro Nambu, Nobel prize 2008 [MeV/c 2 ] u d s c b t Mweak [MeV/c 2 ] Mstrong Mtotal [MeV/c 2 ] S 1 (p) =[ip/ + M(p 2 )]/Z f (p 2 ) Christian Fischer (University of Gießen) Phase transitions in strong QED3 5 / 32

7 Properties of QCD: Dynamical mass generation Dynamical quark masses via weak and strong force Yoichiro Nambu, Nobel prize 2008 [MeV/c 2 ] u d s c b t Mweak [MeV/c 2 ] Input parameters in Nf=2+1 QCD Mstrong Mtotal [MeV/c 2 ] S 1 (p) =[ip/ + M(p 2 )]/Z f (p 2 ) Christian Fischer (University of Gießen) Phase transitions in strong QED3 5 / 32

8 Properties of QED3: Chiral Symmetry S = d 3 x Nf ΨiD/ Ψ 1 4 F µνf µν Four component spinors Clifford algebra {γ µ, γ ν } =2δ µν, µ,ν =0..2 Generators for chiral symmetry: γ 3, γ 5, [γ 3, γ 5 ] U(2N f ) Chiral symmetry breaking U(2N f ) U(1) U(1) SU(N f ) SU(N f ) Christian Fischer (University of Gießen) Phase transitions in strong QED3 6 / 32

9 Properties of QCD: Confinement Quark confinement Millenium-Prize (1 Mio Dollar) Clay Mathematics Institute baryons, mesons (and glueballs?) linear rising potential related to center symmetry Bali, Phys. Rept 343 (2001) Jeff Greensite, Lecture Notes in Physics 821 (2011) 1. Christian Fischer (University of Gießen) Phase transitions in strong QED3 7 / 32

10 Properties of QED3: Confinement Logarithmically rising potential: V (r) d 2 k 1 k 2 eikr ln(r) massless one boson exchange geometrical confinement dressed Polyakov loop does NOT show confinement (similar to NJL model...) Christian Fischer (University of Gießen) Phase transitions in strong QED3 8 / 32

11 Overview 1.Introduction to QED3 2.QED3 and high Tc superconductors 3.QED3 and graphene Christian Fischer (University of Gießen) Phase transitions in strong QED3 9 / 32

12 High-temperature superconductors superconducting CuO2-planes doping is important normal state is insulator order parameter has d-wave symmetry Christian Fischer (University of Gießen) Phase transitions in strong QED3 10 / 32

13 Fermi surface Schematic Fermi surface: Ding et al. PRB 54 R9678 (1996) Gap has nodes on Fermi surface: d-wave symmetry Measured via ARPES experiments Damascelli,Hussain, Shen, Rev. Mod. Phys. 75 (2003) Christian Fischer (University of Gießen) Phase transitions in strong QED3 11 / 32

14 Disperson relation Effective BCS-like Hamiltonian: 2d-Quasiparticles k 2 k + 2 k Linear expansion around nodes: k = v f q k = v q Anisotropy! Two neutral spin 1/2 quasiparticles combined in four-spinors interacting with topological excitations of the gap function Christian Fischer (University of Gießen) Phase transitions in strong QED3 12 / 32

15 Strong QED3: Eff. theory for superconductors Christian Fischer (University of Gießen) Phase transitions in strong QED3 13 / 32

16 Lattice QCD vs. DSE/FRG: Complementary! Fodor, Karsch, Phillipsen... } relevant Models: PNJL, PQM Technically easier Exploratory for QED3 Weise, Schaefer,... Christian Fischer (University of Gießen) Phase transitions in strong QED3 14 / 32

17 DSEs of QED3 in Landau gauge Transverse photon D µν (p 2 )= δ µν p µp ν p 2 1 Z 3 + Π(p 2 ) Quark propagator Quark-photon vertex: S(p) = ip/ A(p2 ) + B(p 2 ) p 2 A 2 + B 2 k µ Γ µ (p, q) =S 1 (p) S 1 (q) Ball, Chiu, PRD 22 (1980) 2542 Curtis, Pennington, PRD 42 (1990) Christian Fischer (University of Gießen) Phase transitions in strong QED3 15 / 32

18 Analytic solutions: PT vs deep infrared large momenta (PT): Π(p 2 )= α p A(p 2 ) 1 B =0 small momenta: Deconfined Symmetric phase: almost-conformal infrared behaviour with running coupling: α(p 2 )=α/(p(1 + Π(p)) (p 2 ) κ κ = N f N 2 f + O(1/N 3 f ) Christian Fischer (University of Gießen) Phase transitions in strong QED3 16 / 32

19 Numerical solutions Christian Fischer (University of Gießen) Phase transitions in strong QED3 17 / 32

20 Running coupling broken phase: IR scale set by generated fermion mass Christian Fischer (University of Gießen) Phase transitions in strong QED3 18 / 32

21 Phase transition: Miransky scaling Christian Fischer (University of Gießen) Phase transitions in strong QED3 19 / 32

22 Finite volume: DSEs on a torus critical Nf much smaller on lattice - why? separation of scales: dynamical mass << α - volume effects? Put DSEs on torus with (anti-)periodic boundary conditions Integrals become Matsubara sums d 4 p 2π L 4 j 1,j 2,j 3,j 4 = 2π L 4 j,l Formalism well known from QCD... C.F., Alkofer and Reinhardt, PRD 65 (2002) C.F., Gruter and Alkofer, Annals Phys.321 (2006) 1918 C.F. and Pennington, PRD 73 (2006) C.F., Maas, Pawlowski and von Smekal, Annals Phys. 322 (2007) 2916 Luecker, C.F. and Williams, PRD 81 (2010) Christian Fischer (University of Gießen) Phase transitions in strong QED3 20 / 32

23 Volume effects: results Continuum: Lattice: N c f = N c f 1.5 } Volume effect! Christian Fischer (University of Gießen) Phase transitions in strong QED3 21 / 32

24 Anisotropy Recall: high temperature superconductors governed by (large) anisotropy k = v f q k = v q Define metric-like quantity and modify Lagrangian accordingly S = d 3 x Nf Ψiγ µ gµν ( ν + ia ν )Ψ 1 4 F µνf µν Christian Fischer (University of Gießen) Phase transitions in strong QED3 22 / 32

25 Modified critical Nf Bonnet, C.F. and Williams, PRB 84 (2011) Assume isotropic volume effects: N c f (V = ) 3N c f (V ) vf > c anticipated N c f (V = ) > 2 Experiment: YBa 2 Cu 3 O 7 : v f v = 14 Bi 2 Sr 2 CaCu 2 O 8 : v f v = 19 v f v = 10 Chiao et al., PRB 62 (2000) 3554 Christian Fischer (University of Gießen) Phase transitions in strong QED3 23 / 32

26 Finite Temperature: beyond Miransky scaling Universal power law corrections to Miransky scaling: Two generic cases for critical exponent: Braun, CF, Gies, PRD 84, (2011) Braun, Gies, JHEP 1005 (2010) 060 Christian Fischer (University of Gießen) Phase transitions in strong QED3 24 / 32

27 Finite T and scaling in anisotropic QED3 v f = v =0.8 v f = v Scaling observed Critical exponent is controlled by anisotropy Physical case numerically not yet accessible Bonnet and CF, PLB 718, (2012) 532 Christian Fischer (University of Gießen) Phase transitions in strong QED3 25 / 32

28 Overview 1.Introduction to QED3 2.QED3 and high Tc superconductors 3.QED3 and graphene Christian Fischer (University of Gießen) Phase transitions in strong QED3 26 / 32

29 Graphene [Wallace, 1947; Semenoff, 1984] Christian Fischer (University of Gießen) Phase transitions in strong QED3 27 / 32

30 Quantum critical point Christian Fischer (University of Gießen) Phase transitions in strong QED3 28 / 32

31 Quantum critical point Gamayun et al, 2007, 2010; Drut, Lähde, 2009; Son,2007 Christian Fischer (University of Gießen) Phase transitions in strong QED3 28 / 32

32 QED3 as an effective theory for graphene Herbut, PRL 97 (2006) Christian Fischer (University of Gießen) Phase transitions in strong QED3 29 / 32

33 DSE for Fermions!"!" # S 1 (p 0, p) =p 0 γ 0 p/ A (p 0, p) B(p 0, p) Vector dressing function A renormalizes fermi velocity v f (p) =v f A(p) Solve DSEs with bare vertex and one-loop photon D(q 0, q) = 2π q + Π(q 0, q), Π(q 0, q) = πe2 N f 4ε q 2 2 v 2 F q2 q 2 0 Christian Fischer (University of Gießen) Phase transitions in strong QED3 30 / 32

34 DSE for Fermions!"!" # S 1 (p 0, p) =p 0 γ 0 p/ A (p 0, p) B(p 0, p) Vector dressing function A renormalizes fermi velocity v f (p) =v f A(p) Gonzalez, Vozmediano et al,1994 Solve DSEs with bare vertex and one-loop photon D(q 0, q) = 2π q + Π(q 0, q), Π(q 0, q) = πe2 N f 4ε q 2 2 v 2 F q2 q 2 0 Christian Fischer (University of Gießen) Phase transitions in strong QED3 30 / 32

35 Results Analytic result A k 3.0 in symmeric phase: v F (p) =v F 1+f 1 (α)ln Λ p + f 2(α) k Symmetric phase: infrared divergence experiment?! Critical coupling: α c 2.7 larger than value 2.19 for suspended graphene! Christian Fischer (University of Gießen) Phase transitions in strong QED3 31 / 32

36 Results Analytic result A k 3.0 in symmeric phase: v F (p) =v F 1+f 1 (α)ln Λ p + f 2(α) k 3 Symmetric phase: infrared divergence experiment?! Critical coupling: α c 2.7 larger than value 2.19 for suspended graphene! F (106 m s 1 ) ν n (10 10 cm 2 ) Popovici, CF, von Smekal, in preparation Elias, Gorbachev, Mayorov et al, Nature Physics, 2011 Christian Fischer (University of Gießen) Phase transitions in strong QED3 31 / 32

37 Summary QED3 Analytic and numerical solutions from DSEs Transition at Nfc with Miransky scaling Large volume effects: extremely difficult for lattice Anisotropies taken into account Nfc > 2: at zero temperature direct transition from dsc to AF instantaneous QED3 large effects due to running fermion velocity critical coupling too large: suspended graphene remains semi-metalic Christian Fischer (University of Gießen) Phase transitions in strong QED3 32 / 32