1. 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

2. 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

3. 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 .

4. 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

5. 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

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

7. 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

8. 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)

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

10. 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 .

11. 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

12. Fermion condensate Irreducible representation Extended Reducible
Massive
Schwinger model .

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

14. Exponential field
The solutions are in terms of the Laguerrre polynomials

15. Fermion condensate
Uniform field
Exponential field.

16. Energy Levels
Uniform field
Exponential field

17. Solutions
Uniform field
Exponential field

18. Solutions
Exponential field

19. Solutions squared
Uniform field
Exponential field

20. 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.

21. 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

22. 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