Fermion Condensate in Lower Dimensions Edward Daniel Reyes Ramirez (ICN-UNAM) Alfredo Raya Montaño (IFM-UMSNH) XII Mexican Workshop on Particles and Fields Mazatlán, México 2009

Motivation Vacuum modification by external fields In QED3 appears a topological mass induced by the Chern-Simons term QED3 and QED2 are important in condensed matter Graphene (massless limit) QED3 and QED2 exhibit confinement like QCD We want to study the effects of an external magnetic field of arbitrary spatial profile

Fermion condensate and current of the vacuum The fermion condensate is an order parameter, when the electrons have zero bare mass, corresponds to a dynamical chiral symmetry breaking. Fermion condensate Fermion current .

Dirac matrices in lower dimensions To satisfy the Clifford algebra, the dimensionality of the Dirac matrices depends of the even or odd parity Massive Schwinger model ((1+1) dimensions) QED3

Dirac Lagrangian in QED (2+1) Inherited Lagrangian: We have two inequivalent representations each one with a Lagrangian of the form. Extended Lagrangian: We combine the two inequivalent representations in one with two fermion species, A and B. Reducible Lagrangian: It uses the Dirac matrices of QED4 and allows to introduce explicitly the second mass term and the Chern-Simons term

Fermion propagator Canonical quantization Schwinger’s proper time method Ritus eigenfunctions .

Fermion propagator The form of the free propagator come from its representation in momentum space In the presence of an electromagnetic field, (g.P) doesn’t commute with the momentum operator. But (g.P)2 commute with the scalar structures compatibles with the properties of QED3

Ritus method en lower dimensions Irreducible representation for (2+1) dimensions with Am=(0,0,W(x)) (1+1) dimensions with Am =(Z(x),0)

Solutions Irreducible representation on (2+1) dimensions Where Fk,p2,si are solutions of Massive Schwinger model ((1+1) dimensions) .

Dirac equation These are the Pauli’s equation of supersymmetric quantum mechanics (SUSY-QM) And we can construct the vector With this, we can find the solutions of the Dirac equation in this form .

Fermion propagator In the basis of the Ritus eigenunctions the propagator is similar to the free propagator with electrons with momentum p that depends of the dimension (2+1) dimensions (1+1) dimensions . Run AAA_PRESSURETEST.mb

Fermion condensate Irreducible representation Extended Reducible Massive Schwinger model .

Uniform field The solutions are in terms of the parabolic cylinder functions

Exponential field The solutions are in terms of the Laguerrre polynomials

Fermion condensate Uniform field Exponential field.

Energy Levels Uniform field Exponential field

Solutions Uniform field Exponential field

Solutions Exponential field

Solutions squared Uniform field Exponential field

Conclusions We have a simple free-like form for the fermion propagator in the presence of a magnetic field in (2+1) dimensions and an electric field in (1+1) dimensions W’(x), both of arbitrary profile in a spatial direction, if we solve We can use the tools of SUSY-QM to solve more complicated potential and find other quantities. We solve explicitly the cases of uniform field and exponential field. In the first case, we recovered the previously reported in the literature.

Conclusions In the case of the exponential field, we found a quantization of the quantum number p2 which result in the generation of Landau sublevels In the case of intense magnetic fields in (2+1) dimensions, the condensate have the same form of the external field Similar conclusion holds for the massive Schwinger model, although for the pair production rate

Fermion Current and Condensate in Lower Dimensions Edward Daniel Reyes Ramirez (ICN-UNAM) Alfredo Raya Montaño (IFM-UMSNH) XII Mexican Workshop on Particles and Fields Mazatlán, México 2009