Magnetic Catalysis and Confinement in QED3

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

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

1 Motivation 2 QED3 3 Magnetic Catalysis 4 Phase splitting?

QCD Phase Diagram

QCD Phase Diagram Mizher, Chernodub, Fraga, PRD82, 105016 (2010)

QCD Phase Diagram Ayala, Bashir, Sánchez, AR, J. Phys. G37, 015001 (2010).

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

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

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

Schwinger-Dyson Equations 1 = 1

Schwinger-Dyson Equations 1 1 =

Dynamical Chiral Symmetry Breaking in Rainbow Approximation 0.12 0.10 0.08 M p 0.06 0.04 0.02 0.00 0.001 0.005 0.010 0.050 0.100 0.500 1.000

Dynamical Chiral Symmetry Breaking in Rainbow Approximation 0.5 0.4 M p 0.3 0.2 0.1 0.0 0.001 0.005 0.010 0.050 0.100 0.500 1.000

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

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

Confinement 2 Log t 4 6 8 10 0 10 20 30 40 0

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

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 ),

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

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.

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

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.

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

Constant mass approximation 6 m dyn 4 2 0 0 Α 5 10 0 50 eb 100

Constant mass approximation 0.30 m dyn 0.25 0.20 0.15 0 1 2 3 4 5 6 Fit m dyn = alog[beb 0 + c]

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

Full solution 0.25 0.20 0.15 M p 0.10 0.05 0.00 10 6 10 4 0.01 1 100 10 4 p

Full Solution 0.20 0.15 m dyn 0.10 0.05 0.00 0 20 40 60 80 0 Fit m dyn = alog[beb 0 + c]

Full Solution 0 2 Log t 4 6 8 10 0 10 20 30 40 50 t

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

Photon mass Pereg-Barnea, Franz, PRB67, 060503 (2003).

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

Photon mass 0.15 m dyn 0.10 0.05 0.00 0 2 4 6 8

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

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

Confinement 5 Log t 10 15 0 50 100 150 0

Phase splitting 1 Order Parameter 2 3 4 5 0 2 4 6 8

Conclusions (preliminary) CMA overestimates the value of m dyn

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

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

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