Journal of Modern Physics, 2011, 2, 374-378
doi:10.4236/jmp.2011.25046 Published Online May 2011 (http://www.SciRP.org/journal/jmp)
Critical Behavior of a Supersymmetric Extension of the
Ginzburg-Landau Model
Grigoris Panotopoulos
Departament de Fisica Teorica , Universitat de Valencia, Burjassot, Spain; Instituto de Fisica Corpuscular (IFIC),
Institutos de Paterna, Universitat de Va lencia-CSIC, Valencia, Spain
Received January 31, 2011; revised February 26, 2011; accepted March 18, 2011
Abstract
We make a connection between quantum phase transitions in condensed matter systems, and supersymmetric
gauge theories that are of interest in the particle physics literature. In particular, we point out interesting
effects of the supersymmetric quantum electrodynamics upon the critical behavior of the Ginzburg-Landau
model. It is shown that supersymmetry fixes the critical exponents, as well as the Landau-Ginzburg para-
meter, and that the model resides in the type II regime of superconductivity.
Keywords: Superconductivity, Critical Exponents, Supersymmetry
1. Introduction
A very well studied model in the condensed matter
literature is the Ginzburg-Landau (GL) model [1], des-
cribed by the lagrangian of an Abelian Higgs model
224
22
1
=4 Dm F


  (1)
where

is a complex scalar field charged under the
abelian gauge field
A

, with the gauge covariant deri-
vative and field strength
=DieA


 (2)
=
F
AA


 (3)
When 2>0m, the gauge symmetry is exact, and the
model describes a massive complex scalar particle that
interacts with a massless photon. The electric potential
between these scalars has the usual Coulomb form, and
therefore this is referred to as the Coulomb phase. On the
other hand, when 2<0m the gauge symmetry is spon-
taneously broken, and in this Higgs phase the model
describes a massive gauge boson and a massive real sca-
lar field. The nature of the transition between the Higgs
and Coulomb phase has been of great interest to the con-
densed matter community.
The critical fluctuations in the Ginzburg-Landau model
of superconductors were studied in [2], while the fixed
point structure for the GL model was presented in [3].
Furthermore, in previous works the authors have inves-
tigated models in which massless Dirac fermions are
coupled to the Ginzburg-Landau model [4]. The presence
of the Dirac fermions is justified by the fact that effective
microscopic models of strongly correlated electrons
usualy contain them [5]. The critical exponents can be
computed as a function of the number
F
N of the fer-
mions, and for increasing
F
N the models is driven into
the type II regime of superconductivity. In particular, for
the minimum allowed value of the fermion number,
=4
F
N, both values of the  parameter, correspond-
ing to the ‘T’ fixed point and the ‘SC’ fixed point, are
found to be above the mean-field GL value 12, in
contrast to the theoretical [6] and the Monte Carlo num-
bers [7] in the GL model. In this article we point out that
the generalization of the Ginzburg-Landau model to a
supersymmetric one necessarily introduces fermions both
in the matter and gauge supermultiplets, and that the
restrictions imposed by the symmetries of the model
unambigiously determine the critical exponents and the
Landau-Ginzburg parameter, which is found to be in the
type II regime of superconductivity.
Finally, we remind the reader that a) all exactly sol-
vable models show that not all of the critical exponents
are independent. In fact they satisfy certain scaling laws,
supported by all the experimental and numerical results,
and it can be shown that there are only two independent
critical exponents. If we take them to be

and

, the
rest of the critical exponents are given by [8,9]:

G. PANOTOPOULOS 375
=2 D

  (4)

=2
2D




 (5)

=2



(6)
2
=2
D
D




 (7)
where D is the dimension of the system, and b) in the
Landau-Ginzburg theory there are two fundamental length
scales, namely the penetration length

and the cohe-
rence length 0

. The Landau-Ginzburg parameter

is
defined as follows
0



 (8)
and it can be shown that <1 2
 corresponds to type
I superconductors, while 12   corresponds to type
II superconductors.
2. The Supersymmetric Model and Critical
Exponents
Supersymmetric Quantum Electrodynamics (SQED) is an
abelian gauge theory with the following field content [10]:
1) One vector multiplet

,,A





consisting
of the photon and the photino (in the so-called
Wess-Zumino gauge), described by a vector and
a Majorana spinor field.
2) Two chiral multiplets

,
L
L


and


,
R
R


with charges =1
L
Q, =1
R
Q, each consisting
of one Weyl spinor and one scalar field, consti-
tuting the left- and right-handed electron and se-
lectron, the matter fields.
The electron Dirac spinor and the photino Majorana
spinor are given by
=, = .
L
R
i
i







 
 
  


 (9)
The SQED Lagrangian contains kinetic, minimal cou-
pling and mass terms and in addition, due to the super-
symmetry, coupling terms to the photino and quartic terms
in the selectron fields:
 

S
2 2†
††
2
22 22
2
11
=42
2
1
2
QED
LR
LRLLRLL RR
LLRRL R
LFFi
DD iD
eQ PPPP
eQeQm m
 
 

 
 

 
 






(10)
with the gauge covariant derivative and field strength
=,DieQA



 (11)
=.
F
AA



 (12)
It must be noted that the model with just one chiral su-
permultiplet is anomalous, while the inclusion of a second
chiral supermultiplet with opposite electric charge renders
the model anomaly-free, since in this case =0TrQ . The
two Weyl spinors combine to form the Dirac spinor of
the usual spinor electrodynamics in the standard four-
component formalism.
The model contains both bosons and fermions, with
equal masses and degrees of freedom within each multi-
plet. The form of the interactions, as well as the values of
the couplings, are completely determined by the symme-
tries. It is interesting that there is just one coupling con-
stant, namely the electric charge e. We have the usual
types of interaction that one encounters in the usual field
theory, namely quartic interaction for the scalars, Yukawa
coupling, and the gauge (electromagnetic) interaction.
We thus know that the theory is renormalizable. In fact,
here we have just a wave function renormalization both
for the vector and the chiral multiplets due to supersym-
metry [11], and furthermore the beta function for the elec-
tric charge is determined by the photon self-energy and
wave-function renormalization due to gauge invariance
[12].
The investigation regarding the critical behavior is ac-
cording to the following program: a) Perform a one-loop
analysis to compute the relevant counterterms that elimi-
nate the unwanted divergencies, b) determine the beta-
function for the electric charge

e  , as well as the
anomalous dimensions for the scalars ,
m


, c) find the
fixed points from the condition

*=0e
 , and finally d)
compute the critical exponents ,

 using the well-
known formulas [4,13]
*
=2


(13)

*
1
=21 m

 (14)
where the anomalous dimensions, as well as the beta
function are given by [14]

=

 


 (15)
ln
1
=2
Z





 (16)
=
mm
m




 (17)
with

the renormalization mass scale, and 2 =4πe 
the fine-structure constant. Note that our definitions for the
anomalous dimensions are slightly different than [4,13].

G. PANOTOPOULOS 376
We start from the photon self energy, that will allows
to compute the electric charge beta function



 . The
relevant loop-diagrams are shown in Figure 1. We have
the same diagrams as in the usual spinor and scalar
electrodynamics. The electric charge beta function has a
contribution from a Dirac spinor and a contribution from
two complex scalars. At one loop and using dimensional
regularization [15] (the space-time dimension 4 
=4D

, then take the limit 0

 and isolate the
divergent part 1/

) one obtains the result

2
,= π

   (18)
Next we turn to the scalar field self-energy. The
relevant diagrams can be shown in Figure 2. We have
the three usual diagrams from scalar electrodynamics, plus
a new one with the Yukawa coupling with the Dirac
electron and the photino Majorana fermion. For the sca-
lar field wave-function renormalization we find the result
(in the Lorentz gauge)
2
2
5
=1 8π
e
Z


 (19)
Now it is a straightforward algebraic task to compute
the anomalous dimensions and then the critical exponents.
We thus obtain our final results (for =3D or =1

)
=2.5

 (20)
(a)

(b)

(c)
Figure 1. Feynman diagrams for the photon self-energy
with the usual spinor and scalar electrodynamics interac-
tion vertices.

L

L


L

L

L

(a) (b)

L

L



L

L


(c) (d)
Figure 2. Feynman diagrams for the scalar self-energy with
(a) the quartic, (b) the single photon, (c) the two-photon,
and (d) the Yukawa interaction vertices.
1
=0.17
6
  (21)
Our results for


e  and


agree with the
corresponding formulas of [16] at one loop. In [4] there
are
F
N massless fermions, and two coupling constants
with two different beta functions. The authors in [4] have
found two fixed points (tricritical and superconducting),
and that the number of massless fermions must be at
least four. The

critical exponent is always negative,
while the

critical exponent is always positive, and for
both exponents the absolute value is a number around 0.5
when
F
N is small. In our supersymmetric version of
the model, there is just one massive fermion, since super-
symmetry requires that there are equal number of
fermionic and bosonic degrees of freedom, and with the
same masses. There is only one coupling constant, namely
the electric charge e, and thus just one beta function,
and a single infrared stable fixed point. Despite this, there
is also here a quartic self-interaction potential for the
scalar field, where the coupling is fixed by supersym-
metry, and it is given in terms of the electric charge.
Furthermore, we find also a negative

critical ex-
ponent and a positive

exponent, with values not too
different from the ones obtained in [4] for small
F
N.
3. Supersymmetry Breaking and the

Parameter
So far we have not seen any superpartners yet, and thus
supersymmetry must be broken. In this section we shall
discuss spontaneous breaking of supersymmetry, follow-
ing [17], within the framework of the Fayet-Iliopoulos
mechanism [18]. In an abelian


1U supersymmetric
gauge theory an extra term is allowed by the symmetries,
the so called Fayet-Iliopoulos term, D

, where D is
the auxiliary field in the vector supermultiplet, and

is

G. PANOTOPOULOS 377
a new parameter with mass dimension two. If 12 ,,
F
FD
are the auxiliary fields in the off-shell formulation of the
supersymmetric theory, the scalar potential is given by
2* *
112 2
1
=2DFFFF (22)
and the auxiliary fields satisfy the following equations of
motion
*
12
=0FmA (23)
*
21
=0FmA (24)

**
1122=0
2
e
DAAAA    (25)
where now the scalar fields are denoted by 12 ,
A
A in-
stead of ,
L
R

. Supersymmetry is broken since there is
no solution that leaves =0. Upon substitution the
scalar potential takes the form

2*2*
112 2
2
2* *
1122
=22 2
1
8
ee
mAAmAA
eAA AA
 

 




(26)
We can see that there are two possibilities, namely that
2>2me

or 2<2me

. In the first case the 1=
A
2
0=
A
minimizes the potential, the form of which is
shown in Figure 3(a). The supersymmetry is sponta-
bneously broken but the gauge symmetry is exact. The
theory describes two complex scalar fields with masses
2
2
e
m

 and 2
2
e
m

. The rest of the fields, namely
the photon

, the photino

, and the two fermions
12
,

retain their masses. In particular, the photino is the
massless goldstino. In the second case the 1=
A
2
0=
A
no longer minimizes the potential, the form of which is
shown in Figure 3(b). This time both the supersymmetry
and the gauge symmetry are broken simultaneously. The
minimum corresponds to 12 =0, =
A
Av, where the
vacuum expectation value v is determined from
22
2=0
42
ev e
m





(27)
This model describes a vector field and a scalar field
of mass 2e

, a complex scalar field with mass
2
2m, a massless goldstino, and two spinor fields with
mass 2
2
e
m

. The Landau-Ginzburg parameter
therefore is easily computed to be
1.41
==1 2
s
v
m
m
  (28)
which is larger than 12, and we thus have a type II
(a)
(b)
Figure 3. (a) The scalar potential versus 2
A
in the
2>2me

case (in arbitrary units). Supersymme try is spon-
taneously broken, but the

1U gauge symmetry is exact.
(b) As in (a) but in the 2<2me
 case. Here, both super-
symmetry and


1U gauge symmetry are spontaneously
broken.
superconductor. It is interesting to see again that our
value of the Ginzburg parameter is comparable to the
value obtained in [4] at the superconducting fixed point
and for =4
F
N. Therefore, we conclude that super-
symmetry provides the kind of lagrangian studied in [4],
and that the values of the Ginzburg parameter and of the
critical exponents are similar to the ones obtained in [4],
without a second coupling constant

for the scalar
quartic self-interaction, and without many fermions.
4. Conclusions
We have proposed and analyzed a supersymmetric
extension of the Landau-Ginzburg theory, which is es-
sentially the supersymmetric version of quantum electro-
dynamics. The model describes the interaction of a Dirac
fermion and two complex scalar fields with the photon
and its superpartner, the photino, which is a Majorana

G. PANOTOPOULOS 378
fermion. All the couplings in the model are given in
terms of the electric charge. It is interesting that there is a
quartic self-interaction coupling for the scalar fields even
in the absence of a coupling

. Within the one-loop
renormalization program we give the expression for the
wave-function renormalization, and according to the
standard prescription we compute the critical exponents
,

 from the beta function and the anomalous dimen-
sions. Finally, we have discussed spontaneous supersym-
metry breaking a la Fayet-Iliopoulos mechanism. There
is a case in which both supersymmetry and gauge sym-
metry can be broken at the same time. The photon ac-
quires a non-vanishing mass, and the Landau-Ginzburg
parameter is computed. We find that its value corre-
sponds to type II superconductors. Our values of the
Ginzburg parameter and of the critical exponents are
similar to the ones obtained in [4], without many fer-
mions and without the introduction of a second coupling
constant for the scalar quartic self-interaction.
5. Acknowledgments
The author acknowledges financial support from FPA
2008-02878 and Generalitat Valenciana under the grant
PROMETEO/2008/004.
6. References
[1] V. L. Ginzburg and L. D. Landau, “On the Theory of
Superconductivity,” Zhurnal Eksperimental’noi i Teo-
reticheskoi Fiziki, Vol. 20, 1950, pp. 1064-1082.
[2] B. I. Halperin, T. C. Lubensky and S. K. Ma, “First-
Order Phase Transitions in Superconductors and
Smectic-A Liquid Crystals,” Physical Review Letters,
Vol. 32, 1974, pp. 292-295. J. H. Chen, T. C. Lubensky
and D. R. Nelson, “Crossover near Fluctuation-Induced
First-Order Phase Transitions in Superconductors,”
Physical Review B, Vol. 17, No. 11, 1978, pp. 4274-4286.
doi:10.1103/PhysRevB.17.4274
[3] I. D. Lawrie, “On the Phase Transitions in Abelian Higgs
Models,” Nuclear Physics B, Vol. 200, No. 1, 1982, pp.
1-19. doi:10.1016/0550-3213(82)90055-4
I. F. Herbut and Z. Tesanovic, “Critical Fluctuations in
Superconductors and the Magnetic Field Penetration
Depth,” Physical Review Letters, Vol. 76, No. 24, 1996,
pp. 4588-4591. doi:10.1103/PhysRevLett.76.4588
[4] H. Kleinert and F. S. Nogueira, “Critical Behavior of the
Ginzburg-Landau Model Coupled to Massless Dirac Fer-
mions,” Physical Review B, Vol. 66, No. 1, 2002, p.
012504. doi:10.1103/PhysRevB.66.012504
[5] J. B. Marston, “U(1) Gauge Theory of the Heisenberg
Antiferromagnet,” Physical Review Letters, Vol. 61, No.
17, 1988, pp. 1914-1917.
doi:10.1103/PhysRevLett.61.1914
J. B. Marston and I. Affleck, “Large-n Limit of the Hub-
bard-Heisenberg Model,” Physical Review B, Vol. 39, No.
16, 1989, pp. 11538-11558.
doi:10.1103/PhysRevB.39.11538
D. H. Kim and P. A. Lee, “Theory of Spin Excitations in
Undoped and Underdoped Cuprates,” Annals of Physics,
Vol. 272, No. 1, 1999, pp. 130-164.
doi:10.1006/aphy.1998.5888
[6] H. Kleinert, “Disorder Version of the Abelian Higgs
Model and the Order of the Superconductive Phase Tran-
sition,” Lette re Al Nuov o Ciment o, Vol. 35, No. 13, 1982,
pp. 405-412. doi:10.1007/BF02754760
[7] S. Mo, J. Hove and A. Sudbo, “Order of the
Metal-to-Superconductor Transition,” Physical Review B,
Vol. 65, No. 10, 2002, p. 104501.
doi:10.1103/PhysRevB.65.104501
[8] M. Le Bellac, “Quantum and Statistical Field Theory,”
Oxford University Press, Oxford, 1992.
[9] H. Kleinert and V. Schulte-Frohlinde, “Critical Phenom-
ena in 4

-Theory,” World Scientific, Singapore, 2001.
http://www.physik.fu-berlin.de/kleinert/b8
[10] W. Hollik, E. Kraus and D. Stockinger, “Renormalization
and Symmetry Conditions in Supersymmetric QED,”
European Physical Journal C, Vol. 11, No. 2, 1999, pp.
365-381. doi:10.1007/s100529900216
[11] S. Ferrara and O. Piguet, “Perturbation Theory and Re-
normalization of Supersymmetric Yang-Mills Theories,”
Nuclear Physics B, Vol. 93, No. 2, 1975, pp. 261-302.
doi:10.1016/0550-3213(75)90573-8
[12] J. D. Bjorken and S. D. Drell, “Relativistic Quantum
Mechanics,” McGraw-Hill, New York, 1998.
[13] F. S. Nogueira and H. Kleinert, “Field Theoretical Ap-
proaches to the Superconducting Phase Transition,” The
Smithsonian/NASA Astrophysics D ata System.
[14] M. E. Peskin and D. V. Schroeder, “An Introduction to
Quantum Field Theory (Frontiers in Physics),” Westview
Press, New York, 1995.
M. Srednicki, “Quantum Field Theory,” Cambridge Uni-
versity Press, Cambridge, 2007.
[15] G. ‘t Hooft and M. J. G. Veltman, “Regularization and
Renormalization of Gauge Fields,” Nuclear Physics B,
Vol. 44, No. 1, 1972, pp. 189-213.
doi:10.1016/0550-3213(72)90279-9
[16] M. E. Machacek and M. T. Vaughn, “Two-Loop Renor-
malization Group Equations in a General Quantum Field
Theory: (I). Wave Function Renormalization,” Nuclear
Physics B, Vol. 222, No. 1, 1983, pp. 83-103.
doi:10.1016/0550-3213(83)90610-7
[17] J. Wess and J. Bagger, “Supersymmetry and Supergrav-
ity,” Princeton University Press, New Jersey, 1992.
[18] P. Fayet and J. Iliopoulos, “Spontaneously Broken Super-
gauge Symmetries and Goldstone Spinors,” Physics Let-
ters B, Vol. 51, No. 5, 1974, pp. 461-464.
doi:10.1016/0370-2693(74)90310-4