Critical Behavior of a Supersymmetric Extension of the Ginzburg-Landau Model

doi:10.4236/jmp.2011.25046

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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

E-mail: [email protected]

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

Dm F







 (1)

where



is a complex scalar field charged under the

abelian gauge field



, with the gauge covariant deri-

vative and field strength

=DieA





 (2)





 (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

N of the fer-

mions, and for increasing

N the models is driven into

the type II regime of superconductivity. In particular, for

the minimum allowed value of the fermion number,

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)











 (5)









(6)









 (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



defined as follows







 (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







and









with charges =1

Q, =1

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

=, = .













 

 



 





 (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:

 



2†

††

22 22

=42

QED

LRLLRLL RR

LLRRL R

LFFi

DD iD

eQ PPPP

eQeQm m

 



 

 



 

 













(10)

with the gauge covariant derivative and field strength

=,DieQA







 (11)







 (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





, as well as the

anomalous dimensions for the scalars ,





, c) find the

fixed points from the condition



*=0e



, and finally d)

compute the critical exponents ,





using the well-

known formulas [4,13]





(13)



=21 m



 (14)

where the anomalous dimensions, as well as the beta

function are given by [14]





 





 (15)









 (16)









 (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



,= 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)

=1 8π





 (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.







(a) (b)













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.

=0.17



 (21)

Our results for







and





agree with the

corresponding formulas of [16] at one loop. In [4] there

are

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

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

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



G. PANOTOPOULOS

377

a new parameter with mass dimension two. If 12

are the auxiliary fields in the off-shell formulation of the

supersymmetric theory, the scalar potential is given by

2* *

112 2

=2DFFFF (22)

and the auxiliary fields satisfy the following equations of

motion

=0FmA (23)

=0FmA (24)



1122=0

DAAAA



  (25)

where now the scalar fields are denoted by 12

A in-

stead of ,



. Supersymmetry is broken since there is

no solution that leaves =0. Upon substitution the

scalar potential takes the form



2*2*

112 2

2* *

1122

=22 2

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=

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



 and 2



. The rest of the fields, namely

the photon



, the photino



, and the two fermions



retain their masses. In particular, the photino is the

massless goldstino. In the second case the 1=

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, =

Av, where the

vacuum expectation value v is determined from

2=0

ev e











(27)

This model describes a vector field and a scalar field

of mass 2e



, a complex scalar field with mass

2m, a massless goldstino, and two spinor fields with

mass 2



. The Landau-Ginzburg parameter

therefore is easily computed to be

1.41

==1 2



 (28)

which is larger than 12, and we thus have a type II

(a)

(b)

Figure 3. (a) The scalar potential versus 2

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

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.

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