The supersymmetric Kosterlitz-Thouless phase transition

Nuclear Physics B270 [FSI6] (1986) 519-535 © North-Holland Publishing Company

THE SUPERSYMMETRIC KOSTERLITZ-THOULESS PHASE TRANSITION* Mauro M. DORIA* and YU-LIFA *~

Department of Physics, University of Florida, Gainesville, FL 32611, USA Received 29 October 1984 (Revised 2 December 1985) The Coulomb gas is constructed in superspace. The critical temperature of the KosterlitzThouless phase transition is obtained in mean-field approximation. We argue that close to this transition the gas becomes supersymmetric and that the supersymmetric sine-Gordon model undergoes the same transition.

I. Introduction

The ideas of Kosterlitz and Thouless [1] (KT) about the nature of phase transition of two-dimensional systems with a continuous symmetry have proven to be very fruitful**. They have been successfully applied to a variety of situations, namely thin films of 4He, Josephson-junction arrays and melting experiments [2]. The main difficulty encountered in characterizing the critical behavior of such systems was that in two dimensions continuous global symmetries cannot break down [3]. Then KT realized the importance of the topological configurations, namely the vortices. They are collective modes that still respect the symmetries of the theory. Energy considerations favor neutrality, i.e. an equal number of vortices and antivortices. The interaction between two of them is of the Coulomb type, thus depending on the logarithm of the distance. Therefore to understand the phase transition that those systems undergo, KT looked at the statistical problem of a neutral gas of interacting vortices and antivortices, the Coulomb gas. The Coulomb gas has had several other interesting theoretical applications. Coleman [4] discovered that the quantum sine-Gordon field theory in Minkowski space, L = l(a~b)2+ ( m / o t 2) cOS ol~, becomes ill-defined when the coupling constant c~ is larger than ~ . The theory has no ground state because the energy per unit volume is unbounded from below according to a variational calculation by that author. Later, others [5] realized that the sine-Gordon theory and the grand partition * Work supported in part by the US Department of Energy under contract no. DE-AS-05-81-ER40008. ** The KT definition of the two-body Coulomb energy, U = -2q~qe In r is slightly different from the one used in this paper. Present address: Los Alamos Natl. Lab., CNLS MS-B258, Los Alamos, NM 87545, USA. t, Present address: Southwest-China Teacher's College, Chongging, People's Republic of China. 519

(t550-3213/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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M.M. Doria, Yu-L([a / Kosterlitz Thouless phase transition

function for the Coulomb gas are in fact equivalent in euclidean space. The point a = 8x/~ corresponds exactly to the KT critical temperature of the Coulomb gas. Another interesting application of the Coulomb gas has been in the understanding of the critical properties of several two-dimensional lattice models [6]. In fact the Coulomb gas can be seen as a representation that many of these models admit exactly [7]. Recently [8] it has been discovered that conformal invariance can lead to a classification of all possible critical exponents for the spatial decay of correlation lengths of two-dimensional theories. The values of these critical exponents are obtained by studying the representation theory for the two infinite-dimensional conformal algebras, Virasoro and R a m o n d - S c h w a r z - N e v e u (RSN) [9], the latter being a supersymmetric generalization of the first. The KT Coulomb gas has its critical behavior (r/=~) falling in the Virasoro algebra classification. Although no calculations are performed here in this direction, we believe that this supersymmetric Coulomb gas must be an example of a model falling in the RS N algebra classification. The outline of the paper is as follows: in sect. 2 the vortex interaction is constructed in superspace. The grand partition function for this Coulomb gas in superspace (CGS) is obtained. The theory is supersymmetric invariant up to a short-distance cut-off procedure. By integrating over the anticommuting variables the physical gas is obtained. The partition function becomes a sum over configurations of the kind g2 exp ( - f i E ) . The interaction energy E for vortex-antivortex configurations is exactly the same as in the Coulomb gas studied by KT. The novelty here is the presence of a function Y2 that weights the Boltzmann factor. It solely depends on the position that vortices and antivortices occupy in the plane. Sect. 3 as well as appendices C and D deal with field theories in Minkowski space. The results of the remaining sections and appendices are formulated in euclidean space. Here we show that the functional generator of a field theoretical model is the grand-partition function for the Coulomb gas previously introduced. One can relate the temperature of the gas to the coupling constant of this model, Close to the critical temperature we argue that this model is equivalent to the supersymmetric sine-Gordon theory. In sect. 4 the existence of a phase transition in this Coulomb gas in superspace is discussed. The value of the critical temperature is obtained in the mean-field approximation. Some useful material and detailed calculations are sketched in the appendices. (A) Majorana fermions are defined in euclidean space. Integration for anticommuting variables, Fierz transformations and some useful Taylor expansions are studied here. (B) Complex coordinates are introduced and employed to take the gas in superspace and write it in the euclidean plane. (C) Green functions for the exponentials of a free boson field in Minkowski space are derived. (D) Green functions for free Majorana fermions in Minkowski space are also obtained. The following notation is employed throughout this paper: x, y, z refer to Minkowski space quantities and r, s, t to euclidean space ones.

M.M. Doria, Yu-Lifa / Kosterlitz- Thouless phase transition

521

2. The Coulomb gas in superspace (CGS) The potential energy for two interacting vortices in two dimensions is the same electrostatic energy between two charges q~ and q2 separated by a distance Irl: U(r) = -qlqe In Ir]. The presence of the minus sign accounts for the fact that the interaction between particles of equal charge must be repulsive and for opposite charges attractive. The Coulomb gas is composed by an assembly of vortices of positive and negative sign. The condition of neutrality means that configurations with a surplus of charge do not contribute to the partition function because the hamiltonian is assumed to take a plus infinite value. Now we construct such theory in superspace, thus obtaining a supersymmetric generalization of the C o u l o m b gas. A short-distance cut-off r is also required here. Unfortunately the cut-off procedure employed in this paper destroys the supersymmetric properties of this Coulomb gas. However close to the critical point, we claim that supersymmetry must hold. This is because at the critical temperature large fluctuations (on the dipoles seize) dominate the system and the cut-off ~becomes irrelevant. Superspace is defined by endowing every point in the plane with a variable 0 that transforms as a spinor under 0(2) rotations in the plane (r, 0). A supersymmetric transformation on these coordinates is defined by

r~

r~ +te F~,O,

0' = 0 + e.

(2.1) (2.2)

The F , matrices transform as a vector under O(2), that is, rotations are achieved by employing the unitary matrix U=exp(½iOFs), Fs=-i½e,~FoF,(eOl=+l); t U + F u U = A , ~ F , and so r~,=Au,r,. The F~'s are hermitian and obey { F ~ , F , } = 26uv(/z , v = 0 or 1). In this p a p e r the following Pauli matrix representation is used for them: Fo = o'x, F, = o~v and F5 = oz. The parameter of this supersymmetric transformation is the anticommuting variable e. e and 0 are Majorana spinors in order to guarantee the reality of F . . The definition of a Majorana spinor in this euclidean space is contained in appendix A. Given two points in superspace, say (r i, 0 ~) and (r -~, OJ), one can define a vector connecting them which is invariant under the supersymmetric transformation. This entity is the cornerstone for this supersymmetric generalization of the Coulomb gas: R ~ = r; -

6

+ ioTF.oj.

(2.3)

Under the transformations (2.1) and (2.2) one obtains R ,0' -_ R ~ .iJ

(2.4)

Although the R~' transforms as vectors under 0(2) rotations, they do not form a vector space. The associativity property for vector addition fails. To see this take three distinct points (r ~, 0~), i = 1 to 3 in superspace. Then one immediately obtains 12 "~q R2 ~ # R . + R ; 2 .

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M.M. Doria, Yu-LiJa / Ko~terlitz- Thouless phase transition

NOW define the Coulomb energy of two charges q~ and qj located at (r', 0 ~) and ( # , OJ), respectively, as being V(Rij) =

{o'

5q, qi In ( R i i / r ) 2 , ,

Ir, - ~ l > r

(2.5)

Ir,- r,I < "~'

where (R°) 2= R ~ R ~ . The short-distance cut-ott r avoids divergencies when the particles get very close to each other. From now on we like to take the notation: ( / , 0 ~) and (s i, ~oi) are the coordinates of positive and negative charges, respectively. The grand-canonical partition function is Z=

V a2~ a. (p!)2

dV2p exp [ - / 3 H 2 p ( r ' , 0 ~, s', ~os )],

(2.6)

I'=0 21"

(2.7)

H 2 e ( r ~, 0 ~, s i, w j) = }~ V ( R ii ) , i" 1 2 I"

q,=0, i

dV2e =dv2p

(2.8)

q~=:~cq,

I

dO(FsdO

dw;,l'~ dw,, , 2

dO~l'sdOedw;Fsdoo,

2

2

2

(2.9) (2.10)

dv2p = d2sl • . . d2se d2rl • . • d2rv ,

parameter still to be related to the fugacity. Next, all the anticommuting variables are eliminated through integration. No physical interpretation is attached to them. Then supersymmetry is no longer obvious in the formulation given below A being a

o~ (2P f

Z = p~o2 ~

,dv2eS22p(r'u' s~') exp [-[3h2t'(r" si) ] '

-/3h2,.=/3q2{~ l n } r ' - # ' + ~ i" i

"1"

In } s ' - s ' ' i~j

T

~ln It'-s*'}. i,i

(2.1l) (2.12)

7"

Integration is not performed inside the radius of each vortex. For later purposes the statistical weight is expressed as follows, exp ( - B h 2 p ) = [[l'p:i (It' - # ] ~ / r e )

[[I,.

l I2:j Is' -*'1~/;]..eje

i ,_~,l~lr~]~:/.

~

(2.13)

The function .O (r',, s{,) is better expressed by first introducing the complex variables l ~, 1 <~ 1~ < 2P. Decompose coordinates r ~, s-* into a frame defined by the orthonormal vectors & and ~,, # = r;8o+ r~, and s*= s[,~o+ s;~,. A possible definition for the complex variables is tt

=

i lrl .i , ~ ro+

t .t [So+lSl

1<~ 1 ~ P P
(2.14)

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M.M. Doria, Yu-Lifa / Kosterlitz- Thouless phase transition

their difference defined by t "m= t " - t m. Function ~ is given by ~2p(r~, s{,)= IF2p(r2, s;,)[ 2, 1

F2p(r~, s'~) = E (-1)Mr.,,.,. " M

1 tm3m4

1 tmel'

tin2"°

(2.15)

where the sum is taken over the permutations M : { m ~ , m 2 . . . m2p} of numbers { 1 , 2 , . . . . , 2 P } and ( - 1 ) ~ is its parity. Thus the 2P particles are arranged into P elements of coupled charges. The sum only runs over distinct arrangements of couples. Notice that these pairs are formed regardless of the charge of the two constituents. The variable ~c is the physical fugacity, ( ~ : = e x p [ - / 3 i x ] , with /x the chemical potential). Therefore the parameter A previously introduced is related to the fugacity by ~: : A/3q~. The process of integrating the grand partition function (2.6) over the anticommuting variables and obtaining (2.11) is achieved by first introducing complex coordinates and is sketched in the appendix B. It is important to emphasize that the physical gas is the one defined by (2.11). This means that physical quantities are formulated as averages there, ( O ) = Z - i t~,-~J~o ,. ~

dv2pOS22pexp[-flh2p].

(2.16)

The partition function (2.11) shows the existence of a mathematical symmetry between this supersymmetric Coulomb gas and the K o b a - N i e l s e n amplitude of the R a m o n d - N e v e u - S c h w a r z string model [9].

3. A field theoretical formulation of the Coulomb gas in superspace (CGS)

In this section a field theoretical representation of the CGS valid for all values of temperature is given. We argue that close to the Kosterlitz-Thouless critical temperature this model is a good approximation to the supersymmetric sine-Gordon theory provided that the coupling g is sufficiently small. The field theories discussed in this p a p e r are always in Minkowski space. Therefore contact with statistical mechanics results are valid upon rotation to euclidean space. The supersymmetric sine-Gordon lagrangian is given by 1

2

1 --.

/x.

1

Lss6 =~(cg&) +~017 # , 6 + ~ F

2

+ FV'(fb)-~t#tbV"(O),

V(~h) = - g cos ( a 4 ~ ) / a 2 .

(3.1) (3.2)

g and a are coupling constants, ~p a Majorana fermion, 4, a scalar and F an auxiliary field [10]. The %, matrices obey {y", y"} = 2g ~'" and one defines 3/5 = e ~'"yuy,,/2!, e ol = + I. We use the representation 3,0 = cry, y l = io-~and 35 = or.. The supersymmetric

M.M. Doria, Yu-Lifa / Kosterlitz- Thouless phase transition

524

transformations,

~3fb= go,

(3.3)

~0 = ( F - iOuch3,")e ,

(3.4)

3F = - igy" 3AJ ,

(3.5)

leave the lagrangian invariant up to a total derivative. The field theoretical version of the C G S corresponds to the supersymmetric s i n e - G o r d o n lagrangian with no auxiliary field F, I

2

1 ---

.~

L = 5(&h) +~Oty 0,qJ-~gq~O cos ~d~.

(3.6)

It is not surprising that the equivalence is not to a fully supersymmetric model. After all the supersymmetric invariance o f the C G S is spoiled by the presence of the short-distance cut-off ~-. H o w e v e r close to the critical temperature, we claim that supersymmetry is recovered in the gas. Then one expects this statistical model to a p p r o a c h a fully supersymmetric field theory. This means that at the critical value of the coupling constant a, the two field theories (3.1) and (3.6) must have the same behaviour. Therefore purely based on this a r g u m e n t one can anticipate that the transition occurs for a = 4./~. This is because bosonization rules [10] show that at this value of the coupling the operator ½F2+ FV'(ch) behaves like the kinetic term ½(aq5)2. Let X = ~/~(q~l + iO2) be a Dirac field and t),, qJ2 be i n d e p e n d e n t M a j o r a n a fields. There is only one possible fourth-order non-vanishing operator that can be formed with these M a j o r a n a fermions and that is (f~0t)(q;2~02). Looking at the X field this means that operators ( ~ y ' X ) 2 and (2X) 2 are really the same one. The bosonization rules tell us that some operators d e p e n d i n g on field t" can be reexpressed in terms o f a single b o s o n field q5, (for c~ = 4~/4~, 2X~-~A cos c~oS,~y'1" ~ c~Ae"a,,qS, A being a cut-off-dependent parameter). Since integration over F gives a term proportional to cos 2 acb (see (3.16)) this is like 071")2 and then follows the connection to the kinetic term. For more details on this the reader should consult ref. [10]. N o w consider the functional generator W for this model (3.6). Let us introduce the following notation. Call Lo the free field contribution and write the interacting part in the following form: Lin ,

lira

g+,g

~g

g+L++g_L

,

(3.7)

L i = -~q;t) e ~i~

(3.8)

Hence one can say that

fd[ch, O ] e x p { i f d 2 x [ L o + g + L + + g W=

lira g+,g ~R

f d[cb,q,lexp {i f d2xLo}

- L ]} (3.9)

M.M. Doria,

Y u - L i f a / K o s t e r l i t z - T h o u l e s s p h a s e transition

525

Expanding it in powers of g+, g_ gives W=

lira ~ (ig+)k (ig-)t ~,~ ~gk3=o k! I!

d2x,

.

d2xk

d2yl

.

""

. d2yt

"

f d[qb,O]L+(x,) . . . L+(xk)L-(yl) . . . L-(yl) exp { i f d2x Lo} x

(3.1o)

f d[cb,q']exp {i f d2xLo} In principle the problem is reduced to finding Green functions over free fields. Then eq. (3.10) becomes W=

lim~ g k ~t = o g+,g

(-~ig+)k(-laig)tf k] 1]

d2xl "" " d2xk f

d2yl

"

' " d2y;

X ( e ia4~(x~) . . . e iaO(xk) e -ia4'(y~) . . . e -ia4~(vt))

x (t~(x~)6(xt)...

t~(Xk)6(Xk)t~(yOtO(y~)... ~(Y,)6(Yt)).

(3.11)

The averages over the bosonic and fermionic fields are defined as

f d[cb]Oexp{i f d2x~(O&)2}

- "f d[---~i; x p {i f- d-~ ~i ~aq,-} '

(3.13)

respectively. These Green functions are computed in appendices C and D, respectively. Making use of e q s . ( C . 2 ) a n d ( D . 7 ) , one gets, W=

lim

~°~ [--g+g-( -p ]/(27r)2]e2p " I d2xj )2 t

g+'g ~ g P=O

x

g-l(z~)aQ(z/j)[[[ P i
.

.

d2xp . .

f

.

d2yl .

c2(xi- xj) 2HiP
_ yj)2].2/,~

d2ye

(3.14)

where 12(z~) for a = :t: is defined in eq. (D.8). The connection with the CGS becomes quite transparent by looking at (2.14). Rotating (3.14) to euclidean space can establish a corresponding between the parameters of the theories. In particular one gets the following relation involving the temperature, /3q 2 0 l 2 2 -4rr"

(3.15)

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M.M. Doria, Yu-Lifa / Kosterlitz- Thouless phase transition

NOW we give another argument showing the connection between the two theories (3.l) and (3.6) close to the critical point. Upon integration over the F field, the lagrangian (3.1) becomes 2 I

-

2

1 ---

Lss(~ = :(O(h) + ~ q n y

ta.

,)~th - g½(bqJ cos cr& - ~

g

-

sin- ad,.

(3.16)

For small values of g and sufficiently large values of c~ the last term becomes a small perturbation over the remaining lagrangian, which is no more than the model (3.6) suggests. In the next section we will prove the Kosterlitz-Thouless phase transition occurs exactly in this range of the parameters c~ and g thus rendering the two models approximately equivalent. Hence it happens that the F terms of eq. (3.1) are negligible when compared to ~0V"(
M.M. Doria, Yu-L!fa / Kosterlitz- Thouless phase transition

527

it gives very large contributions to the partition function when the particles are paired. At very low temperature the most relevant configurations are the ones that maximize /22p and exp (-flh2e) simultaneously. Next we revisit the KT argument showing that at very low densities, i.e. when the typical distance r a m o n g dipoles is much bigger than their average size u, the dipoles act as being almost free because the interaction between them is only proportional to (u/r) 2. Corrections to the free dipole a p p r o x i m a t i o n are obviously negligible at very low densities. To see h o w this scenario emerges rearrange (2.5) such that the products are over the dipole self-energy and over the interaction between two dipoles, 1

exp ( - 3 h 2 p ) = [ [ i ; = , [ k

[ p Ir i - rJl21si--sJl21/3q:/2

sk[2/ 2]¢q2/2[,Hs ~;Z sS[2ls'_rSl--5 J

"

(4.1)

Let us say that particles rs and s~ form a dipole o f size lull, r i = si+ ui.

(4.2)

Getting rid o f s ~ and expressing (4.1) as well as (2.15) in terms o f the dipole coordinates r ~ one gets exp (-13h2e) = 1

}ukl2/'r2l

+O(u:/r2).

(4.3)

1

The neglected terms O(u2/r 2) are o f the type lukl2/lr'q 2, k = i or j, and u ~. uJlr°12, r ij =

r i __ r

j.

N o w the g22e function has corrections to the free dipole gas only at order u2/r: [111:

F2e = 1/ulu2... ue +O(u2/t°2) ,

(4.4)

where u k stands for the complex coordinate, u k = Uok + iu~ and t 'J has been defined after eq. (2.14). It follows that g22p = l/[ul121u212.., luP[2+O(u2/r2).

(4.5)

The dipole arrangement can be seen as some kind o f saddle-point a p p r o x i m a t i o n to the C G S too since no terms linear in u/r a p p e a r a r o u n d it. Joining (4.3) and (4.5) one sees that the function S22p enhances the free dipole picture. Thus the critical temperature in this gas is expected to be higher than the KT value for the usual C o u l o m b gas. The phase transition takes place when the dipole configuration breaks down. The physical entity used to determine the critical temperature is the mean square separation between the positive and negative particles that makes up a pair. The hypothesis that the gas has very low density (u/r small) leads to a picture o f almost free dipoles. Therefore c o m p u t i n g the mean square separation for * This hypothesis has been explicitly checked up to six particles in the gas.

M.M. Doria, Yu-Lifa / Kosterlitz- Thouless phase transition

528

a single dipole will give a reliable information about the transition,

(u 2)

(4.6)

"

f d~r d2s /22(r, s) exp (-/3q 2 In ( I r - s l ) / ~ ) r

Introducing the relative coordinate u = r - s , [22 = l/[u[ ~ and cancelling the volume factor that appears both in the numerator and denominator, one gets (u 2) = r2~q ~/ ~ q 2 - 2,

(4.7)

this result is being valid only for /3q2> 2. Hence this expression leads on to the conclusion that a phase transition must be occurring at the temperature /3cq2-2,

(4.8)

which is twice the critical temperature of the usual C o u l o m b gas. A further improvement to this free dipole approximation follows by introducing a dielectric constant field e ( u ) , that describes the effects caused by the remaining dipoles over a particular one. For this, one initially couples the C o u l o m b gas to an external electric field and then proceeds to study the susceptibility X. Microscopically, a local contribution dx is given by the polarizability p ( u ) o f a dipole times the local density o f such pairs, d n (u). Macroscopically, the susceptibility is measured through the dielectric constant and hence it follows that d e ( u ) -- 2n-p(u) d n ( u ) .

(4.9)

We want to have the C G S coupled to an external electric field in a way that the model can still be formulated in superspace. This means that the only breaking of supersymmetry comes from the presence of the cut-off r. Thus the coupling to the external electric field in superspace is k i h ' = - - -q ~Z ~ E.[r~,-s.-iOk'l~uwt].

(4.10)

P k.l

This contribution to the statistical weight is exp(-/3h')=exp

x

/3q

[

(£"rk--E-s

l + i13q/~*

0 k*

k) l + i ~ q E

)/]}

~ o ~*

P

Ok

.

to;

P

(4.11)

The notation in the above expression is the following: 0 ~ and w; are the upper c o m p o n e n t s of the c o r r e s p o n d i n g 0 ( 2 ) M a j o r a n a spinors of (3.9) and /~ = Eo+ iE~. Integrating over the antisymmetric variables one obtains that the C o u l o m b gas (2.11 ) has two sources of couplings to the electric field. One is the expected exponential

M.M. Doria, Yu-Lifa / Kosterlitz- Thouless phase transition

529

factor of (4.11). The other induces a modification of the function /~2P, now g22p(r i, s s, E), due to the Ok, w ~ dependent terms of (4.11). Instead of giving the general formula for g]2P, we write the first two cases and then draw some general conclusions. Recalling the notation (2.16), one has 1

1

1

F2

f12

1

1

/~*

(4.12)

q ' 1

1

1

1

1

(4.13)

Fn= t12t34 t13 ~'24 ~'13 124 ~- tl--~ ~-3-1- ~'14 /23 , 1

1 1

/~*

T'j

2 tu

Pq

P=2.

(4.14)

When writing F4 we assumed that particles labeled 1 and 2 have opposite charge to 3 and 4. Similarly for F2 particles 1 and 2 have opposite signs. Notice that factors 1/t o describing particles of equal sign do not get an electric field correction. This is the case for 1/t 12 and 1/t 34 in the first term of (4.13). For arbitrary P, it is equally true that F2p is linear in /~*/P. Eqs. (4.11) and (4.14) also tell us that in the thermodynamic limit, when P is large, the electric field dependency of S22p becomes very small. This means that the standard coupling to the electric field violates supersymmetry only in the order lIP. If this electric field dependency of -O2e is entirely neglected one can follow the identical steps of KT and arrive at the same conclusions [1]. The dielectric constant has a singularity that confirms the critical temperature value of (4.10). We briefly repeat their mean-field calculations to check that their procedure can be generalized to superspace. Consider the polarization that a dipole of size u = lul feels in the presence of an external electric field. Take the average over all possible orientations in space and only the contribution from the area [u, u + d u ] ,

p(~) = qo(~. ~)/a~l~:o

u du

(dO/2rr)S22(u)E. uO exp ( - f l U ( u , E)/oE)IE~ o

=

c~.~

u du J o

,

(4.15)

(dO/2~r)S22(u) exp ( - ~ U ( u , 0))

where /~. u = u cos 0 and

exp(-flU(u,E))=exp{[-~qZ/e(u)]ln(u/r)+~qE.u}.

(4.16)

One can check that p(u) is proportional to the highest spinor component of the following superspace function, 0 a--E-exp {[-flq2/2e(R2)] In ( R 2 / r :)

+~q~R.}~.R./E]E=

o.

(4.17)

530

M.M. Doria, Yu-L(fil / Kosterlitz- Thouless phase transition

This is true under the assumption that the dielectric constant changes very slowly in space and therefore e ( R 2) ~ s ( u 2) according to (A.7). Employing the KT argument we say that the density of dipoles in this area [u, u + d u ] is approximately given by dn(u)=(~/'r)

2

dOduugl2(u)exp(-fiU(u,O))

(4.18)

in the dilute approximation. Explicit calculation of (4.15) and (4.18) leads to the following behavior for the dielectric constant according to (4.9), d~(u)=27r2~2flq2(u/7) II~,d.,~,,I 2 1 d i n ( u / 7 ) .

(4.19)

Studying the above equation, one finds that the dielectric constant diverges at the critical temperature (4.9).

5. Summary In this paper, we have studied the superspace generalization of the statistical problem of a neutral gas of interacting vortices and antivortices, the Coulomb gas. It turns out that the interaction among them is still the same as the gas studied by KT. The novelty here is the presence of a function depending only on the position that vortices and antivortices take in the plane. This function weighs the Boltzmann factors and can be seen as a suitable modification of the phase space measure to allow supersymmetry. The existence of at least two distinct phases in this gas can be inferred by general considerations taken at extreme temperatures. One looks for vortex configurations that contribute the most to the partition function at a given temperature. Mean field considerations are employed to determine the phase transition point. We claim an equivalence between the supersymmetric sine-Gordon model to the CGS in the neighborhood of the KT phase transition, at least in the limit of diluteness of the gas. In the field-theoretical formulation of the CGS the critical temperature occurs for c~,,= 4-,/4~. We gave arguments supporting the claim that the supersymmetric sine-Gordon and this model behave similarly around this critical value of the coupling. We are led to the conjecture that the Kosterlitz-Tbouless phase transition also takes place in the supersymmetric sine-Gordon theory at the same value of (x¢.= 4x/4~.

Note added In July of 1985, Dr. Yadin Gouldschmidt communicated to one of us (M.M.D.) his desire to perform the renormalization group calculations in order to confirm some of this paper's results - in particular, our prediction that an instability takes place in the supersymmetric sine-Gordon model at a = 4~/4~.

Appendix A Let qJ =(i~';) and )~ : (~,;) be two spinor fields. Their charge conjugated fields ~ = C~*, X~ = CX* in euclidean space have the same current property of Minkowski

M.M. Doria, Yu-Lifa / Kosterlitz-Thoulessphase transition

531

space, namely Xc,-F.oc = - 0 + F . x . It follows that C +F.C = +F f and one also obtains that Xc+ Oc = -O+)( and Xc+Fs0c = +O+I~sx. Notice that the last three relations carry just the opposite sign of their Minkowski counterparts. A self-conjugated or Majorana fermion fulfils the condition X = a'c. In the I ~ representation introduced in sect. 2 one obtains C = a~. Then the Majorana fermion is given by XM = (~,':). Integration over the antisymmetric coordinates is defined by the well-known rules, ~dOiOj=~O*dO*=~o and ~ d 0 , 0 * = ~ d 0 i = ~ d 0 * = 0 . Recalling that ½d0+Fsd0= dO* dO, it is a straightforward exercise to check that

f dO+ FsdOo+FsO=2 2 I dO~ Fs 2

d01 .

.

.

(A.1)

d0+p F5 dO2PO.lO#2.. " O~pO2p.. " 0 2 0 1 2

==

1.

(A.2)

The Fierz transformation is given by

I ® I = ½ [ I x I + F . x F . + F s x Fs],

(A.3)

where I is the identity matrix and A ® B, A x B mean in components A~bB,,a, A~a&~, respectively. For practical calculations the identities

I~F~ = 6,~ + iGt3Fs,

F.Fs = -is~¢Vt3

(A.4)

are very handy too. Now for the expansion of a superfield in its components consider two points in superspace, namely (r, 0), (,, w) and define the vector u = r - s . The supersymmetric invariant R 2 = R~,R., R . = r~, - s. + iO+Fuw is the superfield

el) =- R 2 = A q- O+X -? I O+FsOG,

(A.5)

where A = u 2, x = 2iuuF.~o and G = 2w+Fs~o. The general form for an arbitrary function of superfield 4~ is

v(4,) =

v ( a ) + v ' ( a )

0 + x+~O ' + F s O [ V' ( A ) G + ~ ' V "( A ) x ~+l s x ] ,

(A.6)

obtained by Taylor expanding v(q~). Adapting it to our special values of A, X and G, one gets

F( R 2) = F( u 2) + 2 iO+I~wu, F~)( u 2) + O+F,&o+Fsw[F't'(u 2) +

u2F(I)(u2)]

,

(A.7)

where F(")(x 2) = d"F(y)/dy"l~,=x~, As an example, it follows that

• + F~OT~ u .,L+ut 0 + rsoro + Fs~o411u[" ~/e2 =-I"l q- tO

(A.8)

Appendix B The supersymmetric invariant R 2= R,.R~, R~. = r~-s.+iO+Furo that connects two points in superspace, namely (v, 0) and (s, w) can be written in terms of the

M.M. Doria, Yu-LiJa / Kosterlitz-Thouless phase transition

532

complex coordinates T = R~ + iR2 and T* = R* - JR*: R 2 = TT*,

T = t - T+2iO*to*,

T * = t * - ?*+2i0w.

(B.I)

In this expression 0 and to as well as their complex conjugates are the components of the 0(2) spinors defined in each point of the plane 0 = (o,) and to = (,,%). Variables t, ? are defined in the same spirit of (2.16), i.e. t = ro+ ir~ and ?= So+ is~. The anticommutative properties of 0 and to imply that an arbitrary function of T has a very simple form,

f ( T) = f ( t - 7) + 2iO* to *f'( t - ?).

(B.2)

In order to have the partition function (2.8) in the form (2.13), first express the statistical weight in terms of complex variables: exp (-/3H2e) = I1 i
i
i
Repeated use of (B.2) renders the following expressions, 2P

I1

]3qfl:/ 2

13q q j 2

= (i~)"o*, O,* 02* ... O*p

i
,, t.. i
x

]

[~M ( - 1 ) Mq,n, qm2 qm3qm4. . . qt.%p-'":" . . . qm2e]~ ~ t ''m4 jj+'''.

(B.4)

M corresponds to a summation over permutations of the integers from 1 to 2P as already described after eq. (2.15), Integration over the anticommuting variables [see (A.2)] and formula (B.4) are enough to go from (2.8) to (2.11). The remaining terms in (B.4) don't contribute upon integration of 0i variables since they are of lower order in these variables.

Appendix C In this appendix the Green functions for the boson field~ are obtained. This result is well-known in the literature and the reader is encouraged to read Coleman's paper [4]. Here we just present a heuristic derivation of it. Green functions should be obtained in euclidean space and then rotated back to Minkowski space [11]. Consider the path integral over the classical current J ( r ) ,

f d[49]expfd2r[-½(O4a)2+J~] -exp{-f

d2rd2r'½J(r)G(r-r')J(r')},

(C.I)

Id[~]expfd2r[-½(3~)2] where 3 2 G ( r - s ) = ~ 2 ( r - s ) and thus G ( r - s ) = (In c[r-sl)/2rr.

For the current

533

M . M . Doria, Y u - L i f a / K o s t e r l i t z - Thouless p h a s e transition

given by J ( r ) = ia[~m= k l 62(r - rm) - Z , =I , 32(r - r,)] one obtains

I

d[~b] [I,~=1 exp (ic~4)(rm)) H.=I exp (-ia~b(s.)) exp

=

--r,)

-

d2r½(od~) 2

l~m
II'. c2(r.,

[lq

(I)

(c.2)

- s.)2]

In principle (C.1) leads to divergent self-interactions of the kind exp In 0 but if one assumes the existence of a cut-oil e in the propagator they don't appear at all. Now to transpose (C.2) to Minkowski space, we rotate the spatial coordinates of vectors r, r,, and s,, (ro=X0, r, = - i x O . Next another condition has to be imposed upon (C.2). In Minkowski space the free field theory has the continuous symmetry tb - ~b+ A, A a parameter. Imposing this symmetry on the Green functions, exp ia4J exp iaa exp ia4,, immediately leads to the condition that k = 1. As it is well known, no continuous symmetry can be broken in two dimensions [12]. Hence one finally obtains that in Minkowski space, (ei~4~(x~) . . . e iC~4aIXk) e-iC,4~(yl ) . . .

=

1

[fl'
e-i'~'b(YJ ))

- x,) [Illi, j

IIl
(C.3)

In the Coulomb gas the condition k = l corresponds to neutrality.

Appendix D In this appendix the Green functions involving Majorana fermions are obtained. The charge conjugated field 4'c-- Cq~T, qSc= - 4 ' TC-~ enjoys the following properties in two dimension (Minkowski space), ~cy~Xc=-)?y~'4J; qTcXc=)?@ and ~cYSXc = -)?3,54,. The C matrix fulfills Cy~'C -1= _ y , , v and in the particular representation employed in this paper C = try,. A self-conjugated or Majorana fermion satisfies 4,c = C47 v. For a spinor 4, = (~), this condition means 4,1 = -i4,* and 4,2 = + i 6 " . Now the propagator for the Majorana fermion is just twice the one for a Dirac fermion, (4,~(x)~t~(y)) = iSM(X--Y)~t3,

(D.1)

where ½i7~'G, SM(X --y) = 62(x --y) and S~a(x - y ) = (1 - 7ri) y~(x - y ) J ( x _y)2. The average symbol in (D.1) has been previously introduced in (3.13). Using the y , representation introduced in sect. 3 together with the Majorana condition for 4, leads to {4,l(X)4,,(y)) = 1/ rrz + ,

(D.2)

534

M.M. Doria, Yu-Lija / Kosterlitz- Thouless phase transition

(th2(x)th2(y)) = 1/ rrz

,

(D.3)

(~,,(x)4,:(y)) = 0.

(D.4)

where

z" = ( x ° - y ° ) ± ( x ' - y ~ ) . Here we c o m p u t e the Green function ( f ( z ~ ) t h ( z ~ ) . . . tfi(ZN)O(ZN)), involving only M a j o r a n a fermions in N points. First recall that in our g a m m a matrix representation CO = 2iO~ t)2. Next Wick's theorem is used to d e c o m p o s e the N - p o i n t Green function into a p r o d u c t of two-point Green functions. Since there is no propagation between the upper (01) and the lower (qs2) c o m p o n e n t , we can treat the two blocks independently. For N even, one obtains ( ~ ( z,)tb( z,) . . . f(ZN)th(ZN)) = ( 2i ) N (--1) N/2(thi( Zi) . . . thi( ZN ))

X(thz(Z , ) . . . O2(ZN)).

(D.5)

Then Wick's theorem says that

(O,(z,)~,,(z~)... 4,,(zN)> = E (-l)°(¢,(zo,)4',(z,2)> o x (thi(Zq~)th~(Zq,))... (O,(z,., ,)O,(Zq~ )),

(D.6)

for i = 1 and 2. O = { q ~ , . . . qN} is a sum over all the permutations that lead to distinct arrangements of two-point Green functions and ( - 1 ) ° is the parity of the permutation. Notice that (D.6) requires N to be even otherwise the Green function vanishes. One finally obtains that (~(-71)[//(ZI)

. • • ~(ZN)~I(ZN)

) = a N,2P

1

a

n(zo)=E(-1)?z, Q

1

z~ qiq2

, . .

~(ZO)~(Zij),

(D.7)

1

z,

q3q4

q2p

,

(D.8)

Iq21"

where a means + or - and zi~ = z,~ - z , ,± z~ = z~0 ± z i 1.

References [1] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181 [2] J.M. Kosterlitz and D.J. Thouless, Two-dimensional physics; progress in low temperature physics, VIIB (1978), ed. D.F. Brewer (North-Holland); R.A. Webb, R.F. Voss, G. Grinstein and P.M. Horn, Phys. Rev. Lett. 51 (1983) 690 [3] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133 [4] S. Coleman, Phys. Rev. l i D (1975) 2088 [5] J. Frohlick, Comm. Math. Phys. 47 (1976) 233; S. Samuel, Phys. Rev. 18D (1978) 1916

M.M. Doria, Yu-Lifa / Kosterlitz- Thouless phase transition [6] B. Nienhuis, J. Stat. Phys., 34 (1984) 731; M. den Nijs, Phys. Rev. B27 (1983) 1674 [7] L.P. Kadanoff, J. Phys. All (1978) 1399 [8] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333; D. Friedan, Z. Qiu and S. Shenker, Phys. Rev, Lett. 52 (1984) 1575 [9] P.H. Frampton, Dual resonance models, Frontiers in physics (Benjamin/Cummings, 1974) [10] E. Witten, Nucl. Phys. B142 (1978) 285 [11] E.S. Abers and B.W. Lee, Phys. Reports 9 (1973) 1 [12] S. Coleman, Comm. Math. Phys. 31 (1973) 259

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