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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 ChernSimons 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)
Where Fk,p2,si are solutions of
.
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
Fermion condensate
Irreducible
representation
Extended
representation
Reducible
representation
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