Magnetic Catalysis and Confinement in QED3

Transcription

1 Alfredo Raya IFM-UMSNH XQCD-2011, San Carlos, Sonora, Mexico

2 In collaboration with: Alejandro Ayala, ICN-UNAM Adnan Bashir, IFM-UMSNH Angel Sánchez, UTEP

3 1 Motivation 2 QED3 3 Magnetic Catalysis 4 Phase splitting?

4 QCD Phase Diagram

5 QCD Phase Diagram Mizher, Chernodub, Fraga, PRD82, (2010)

6 QCD Phase Diagram Ayala, Bashir, Sánchez, AR, J. Phys. G37, (2010).

7 QED3: Toy model of QCD High-T QCD QCD3 Large N, Abelianization: QCD3 QED3 Asymptotically free Super-renormalizable Dimensionful coupling Exhibits Chiral Symmetry Breaking and Confinement If it happens in QED3, it also happens in QCD

8 QED3: Applications in Condensed Matter Rich structure Anyons Chern-Simons term (topological mass to photons) Additional fermion mass terms

9 QED3: Applications in Condensed Matter Rich structure Anyons Chern-Simons term (topological mass to photons) Additional fermion mass terms Applications High T c Superconductivity Quantum Hall Effect Graphene Topological Insulators

10 QED3: Applications in Condensed Matter Rich structure Anyons Chern-Simons term (topological mass to photons) Additional fermion mass terms Applications High T c Superconductivity Quantum Hall Effect Graphene Topological Insulators Lagrangian L = ψ i D ψ 1 4 F µνf µν 1 2ξ ( µa µ ) 2

11 Schwinger-Dyson Equations 1 = 1

12 Schwinger-Dyson Equations 1 1 =

13 Dynamical Chiral Symmetry Breaking in Rainbow Approximation M p

14 Dynamical Chiral Symmetry Breaking in Rainbow Approximation M p

15 Confinement Static potential (r ) V (r) = e2 8π G(0) ln(e2 r) + cte + O(1/r) Quenched, G(0) = 1 Massless fermions in loops, G(0) = 0 Massive fermions in loops, G(0) 0

16 Confinement Axiom of reflexion positivity Define (t) = d 2 x d 3 k (2π) 3 eik x σ s (k), σ s (k) = F (k)m(k) k 2 + M 2 (k). Free particle, F (k) = 1 and M(k) = m, Rainbow solution (t) = 1 2 e mt 0

17 Confinement 2 Log t

18 Confinement Oscillatory behavior, (t) = 1 2 e m 1t cos(m 2 t + δ) 0 Corresponds to a pair of complex conjugate mass poles m = m 1 ± im 2 Position of the first dip (inverse) order parameter for confinement

19 Gap Equation Start from Σ(x, x ) = ie 2 γ µ G(x, x )γ ν D µν (x x ). In Ritus formalism d 3 xd 3 x E l p(x)σ(x, x )E l p (x ) = ie 2 d 3 xd 3 x E l p(x)g(x, x )γ ν D µν (x x )E l p (x ) with and D µν (x x ) = d 3 q (2π) 3 e iq (x x ) D µν (q) G(x, x ) = dpe l p(x)π(l)g l (p)e l p(x ),

20 Gap Equation Ritus eigenfunctions (γ Π) 2 E p = p 2 E p,

21 Gap Equation Ritus eigenfunctions (γ Π) 2 E p = p 2 E p, Orthogonality dze p (z)e p (z) = Iδ(p p ), dpe p (z)e p (z ) = Iδ(z z ), with E p (z) = γ 0 E p(z)γ 0.

22 Gap Equation Ritus eigenfunctions (γ Π) 2 E p = p 2 E p, Orthogonality dze p (z)e p (z) = Iδ(p p ), dpe p (z)e p (z ) = Iδ(z z ), with E p (z) = γ 0 E p(z)γ 0. Quantization (γ Π)E p = E p (γ p) with p µ = (p 0, 0, k), where p 2 = p 2 0 k

23 Gap Equation Equation to solve { d δ ll Σ l (p)π(l) = ie 2 3 q (2π) 3 Dµν (q)e ˆq Π(l)γ νg l (p q)γ µδ ll + σ + σ ˆ (σ)γ ν Π(l + σ)g l+σ (p q)γ µ ˆ (σ)δ ll [ ˆ (σ)γ ν Π(l + σ)g l+σ (p q)γ µ ˆ ( σ)δ l,l+2σ +Π(l)γ νg l (p q)γ µ ˆ (σ)δ l,l +σ ]} + ˆ (σ)π(l + σ)γ νg l+σ (p q)γ µδ l,l+σ with D µν (q) = g µν /q 2, ˆ (σ) = I + σγ 1 γ 2 and σ = ±1.

24 Lowest Landau level In the lowest Landau level (LLL), gap equation decouples: dq M(p q) M(p) = α 2π (p q) 2 + M 2 (p q) ln 1 + 2eB 0 q 2 In the constant mass approximation CMA, ( ) 1 = 2α 2eB 0 ln πm dyn mdyn 2

25 Constant mass approximation 6 m dyn Α eb 100

26 Constant mass approximation 0.30 m dyn Fit m dyn = alog[beb 0 + c]

27 Constant mass approximation Unfortunately, CMA fails to incorporate confinement (t) = 1 2 e m dynt

28 Full solution M p p

29 Full Solution m dyn Fit m dyn = alog[beb 0 + c]

30 Full Solution 0 2 Log t t

31 Chiral Symmetry and deconfinement transition Vacuum Polarization Intense field quenches fermion loops Temperature Numerical challenge Photon mass Relevance in High-T c superconductivity

32 Photon mass Pereg-Barnea, Franz, PRB67, (2003).

33 Photon mass Order parameter for AF and SC phases m g 0 signals coexistence Leads to Chiral symmetry restoration at B = 0 Enters into the propagator as 1 D µν (q) = q 2 + mg 2 g µν Modifies Gap Eq. as dq M(p q) M(p) = α 2π (p q) 2 + M 2 (p q) ln 1 + 2eB 0 q 2 + mg 2

34 Photon mass 0.15 m dyn

35 Photon mass MC exponentially suppressed m dyn = a mg + c e bmg Effective chiral symmetry restoration

36 Confinement Yukawa potential V (r) K 0 (m g r) Asymptotically, ln(r) m g 0 V (r) e r r m g

37 Confinement 5 Log t

38 Phase splitting 1 Order Parameter

39 Conclusions (preliminary) CMA overestimates the value of m dyn

40 Conclusions (preliminary) CMA overestimates the value of m dyn Incapable of describing confinement

41 Conclusions (preliminary) CMA overestimates the value of m dyn Incapable of describing confinement Confinement requires knowledge of full momentum dependence of the mass function

42 Conclusions (preliminary) CMA overestimates the value of m dyn Incapable of describing confinement Confinement requires knowledge of full momentum dependence of the mass function Intense magnetic field shortens confinement radius

43 Conclusions (preliminary) CMA overestimates the value of m dyn Incapable of describing confinement Confinement requires knowledge of full momentum dependence of the mass function Intense magnetic field shortens confinement radius Photon mass effectively screens charges

44 Conclusions (preliminary) CMA overestimates the value of m dyn Incapable of describing confinement Confinement requires knowledge of full momentum dependence of the mass function Intense magnetic field shortens confinement radius Photon mass effectively screens charges MC exponentially damped

45 Conclusions (preliminary) CMA overestimates the value of m dyn Incapable of describing confinement Confinement requires knowledge of full momentum dependence of the mass function Intense magnetic field shortens confinement radius Photon mass effectively screens charges MC exponentially damped Evidence of phase splitting?

46 Conclusions (preliminary) CMA overestimates the value of m dyn Incapable of describing confinement Confinement requires knowledge of full momentum dependence of the mass function Intense magnetic field shortens confinement radius Photon mass effectively screens charges MC exponentially damped Evidence of phase splitting? GRACIAS