PHYSICAL REVIEW B VOLUME 47, NUMBER 9 1 MARCH 1993-I

Effec of vortex bending in the phase transition mediated by vortex ringsin three dimensions and its relation to the A. transition in liquid helium

Alejandro F. Ramirez* and Fernando LundDepartamento de Fs'sica, Facultad de Ciencias FI'sicas y Matematicas, UniUersidad de Chile,

Casilla 487-3, Santiago, Chile

Rodrigo HernandezDepartamento de Ingeniena Mecanica, Facultad de Ciencias FI'sicas y Matematicas,

Universidad de Chile, Casilla 2777, Santiago, Chile(Received 6 August 1992)

We compute the critical behavior of a dilute gas of circular and elliptical vortex rings that can be bent.It is confirmed that this system presents a phase transition, and its critical exponents do not differ greatlyfrom those of planar rings. Implications concerning the X transition in liquid helium are discussed.

Recently' it has become apparent that vortex fila-ments qualitatively describe the A, transition in liquidhelium, as conjectured many years ago by Feynman andOnsager. The physics is much the same as in theKosterlitz-Thouless mechanism that drives the superAuidtransition in two dimensions. These computations haverelied on simplifying the phase space from the (infinite-dimensional) space of all curves in three dimensions tothe (six-dimensional) space of circles in three dimensions.This vortex loop approach still keeps the many-body as-pects of the problem through the interaction amongloops. In Ref. 1 it was shown that enlarging the phasespace to include elliptical shapes confirms that there is aphase transition and that its qualitative behavior remainsunchanged. The quantitative agreement with the mea-sured critical exponents for the X transition is, however,not improved. The question remains: What happens ifthe phase space is enlarged to include more generalshapes?

The purpose of this paper is to present results on thecritical behavior of a dilute gas of circular and ellipticalvortex rings that are allowed to bend (Fig. 1).

For circular rings, their phase space is eight dimen-sional and they are parametrized as

X(cr)=(R coso, R sino, (a/2)R cos2o), 0(c.r (2~ .

(1)Their projection on the x-y plane are circles and their

projection on the x-z plane are parabola segments. Thereare then two additional degrees of freedom with respectto circular vortices: orientation and curvature. Thelatter is measured by the parameter 0, , and is assumed tobe small: 0&a &a «1. In principle the critical behav-ior of the system depends on the parameter a, the max-imum allowed bending, but as we shall see, the criticalexponent turns out to be quite insensitive to it.

Following Ref. 1, we consider very thin vortices havingan interaction energy given by

p2 X'H = f do'IX'I ln +O(1)4~ 7

II= p p2

4

2

2+ R ln —+R 2C+ +2 2 2

where we have kept terms to order a . C is a quantitythat depends on what happens inside the vortex core andwhose exact value will not inAuence the critical behaviorof our system. We shall take C =0.464. '

Since we consider bent vortices with small curvature,the length scale is fixed by R, and the screening of theself-energy of a large vortex due to the presence of manysmall vortices is described by a scale-dependent "diamag-netic constant" e(R), which is a function only of radiusR. The screened energy is then, following Ref. 1,

U(R, a)=p+ j, (4+a )lnpI 2 R'8 r e(R')

where I is the circulation, p the superAuid density, and w

a small cutofF radius. Here O(1) terms remain finite whenIX' /ri co.

For a vortex parametrized as in Eq. (1)

IX'I =RI 1+(a /2) sin 2o. j,

and, using the fact that a is small, the line integral in theexpression for the filament energy is readily performed toyield

+(4C+4+2a +Ca )

FIG. 1. Cseometry for a bent vortex ring. (2)

5465 1993 The American Physical Society

5466 BRIEF REPORTS 47

where p is the energy needed to form a vortex of radius ~.To have a closed system of equations we now need

another relation between the diamagnetic constant e andthe energy U. This is provided by the definition of e as alinear-response function given by a sum over states up toscale R of the polarizability of a single vortex times aBoltzmann factor. The polarizability of a vortex, definedby

2

Pp )XX X'der

Then

D(X, T~X)=(1+a )dR

aI1Cl

g=f (1+a )

R 7(6)

Rotations should be performed with respect to the"center of mass" of the system. In our case this meanswith respect to the origin since

is, for a vortex parametrized as in Eq. (1),

q = —(~/12)/3pI R (3)

This is the same as for a circular ring.The next step is to compute the "magnetic susceptibili-

ty"

(X)=(2~) ' f X(o.)do. =(0,0,0) .0

We consider first infinitesimal rotations around the prin-ciple vortex axis (Fig. 1). If we parametrize them by anangle P they can be represented as

1 d1( 0Td~= —dQ 1 0

0 0 1

where V is the volume of the system, i labels the vortexstates, q, is the polarizability of a vortex in the ith state,and n; is the number of vortices in that state.

The counting of states is delicate and will be dealt within some detail. We shall consider two states as differentwhen they are separated by a distance greater than orequal to r. Now, the separation d(cr) produced by aninfinitesimal transformation T between a fixed vortexpoint X(o. ) and the transformed vortex TX is

~

[X(o.)—TX(o. )] P, X'(cr )

~

CT

/X'(a)/

and thus, we can define the separation between vortex Xand TX as

Then

ancl

D(X, Tdg) =aR d1(

g= f a—dP.0 7

T = 0 cosO sinO

0 —sinO cosO

Since this is proportional to a, the term in a in Eq. (6)will not contribute.

The rotation about the x axis (Fig. 1) can be written as

D (X, TX ) = max d (o ) (4)

so that the number of states related to an infinitesimaltransformation T is D (X, TX ) /r.

We next distinguish three different degrees of freedomseparating the sum over states as g, = g, g~ g„, wheret stands for translations, p for parameters, and r for rota-tions. The energy of a vortex does not depend on posi-tion so that the translation degrees of freedom contributeV/~ to the susceptibility. As for the parametric degreesof freedom, we treat separately n and R. An infinitesimalvariation of a is represented by

T X(o ) =(R cosa, R sincr, [(a+da)R cos2o ]/2) .

Then, from (4) we obtain

D(X, T X)=R da/2

aR do.0 2V

An infinitesimal variation of R is represented by

Tz X(o ) =((R +dR) coso, (R +dR) sincr,

(a/2)(R +dR) cos2o. ) .

where O is the polar angle. Then

D(X, TdaX)=R dO

ancl

—dO.0

so that

aIlCl

D(T&X, Td&T~X)= sinOR dP

g = f sinO —dP0 7

Putting all this together we get the following expres-sion for the "magnetic susceptibility" to leading order in

We now consider infinitesimal azimuthal rotations for afixed O. They are given by

1 dP 0

Td~= —dp 1 00 0 1

47 BRIEF REPORTS 5467

8I

y(R, a)= PpI I f ae ~ ' ' 'dadR' .6 7 0

80—

(10)

The integration over a can be done explicitly. Usinge(R)= 1 —4~y(R) and differentiating with respect to R,we obtain the scaling equations for a dilute gas of bentvortex rings,

8

P~ ~ R —Pw(R)1

—Pa v(R))

2 4

dR 3V(R)

20

d W(R) = Pr' 1„R +C+1dR 2e(R )

d V(R) pI RdR 8e(R)

FIG. 2. Solutions to the scaling equations (11). There is acritical point at w, —1, . . . , 6 where the asymptotic behavior ofthe solutions changes abruptly.

which have to be solved numerically with initial condi-tions e(r)=1, W(r)=p, and V(r)=0. The superfluiddensity is given by the inverse of the diamagnetic con-stant at large scales, ' (p, )

' =e( oo ) so that the presenceof a phase transition will be signaled by an abrupt changein the asymptotic behavior of that quantity. Rescalingvariables to

l = ln(R /w),

E( 1 ) =e(1)/PpI

u (l) =PV(l),(v (l) =p8'(l),

t is the reduced temperature, t =(1—T/T, ), and T, isthe critical temperature. For values corresponding toliquid helium (po =0.137 g/cm, ~- 1 A, I =9.97 X 10cm /s), we get T, —1 K which is in the right ball park,remembering that the critical temperature is not auniversal quantity and depends on the values of the core-dependent parameters.

The next step is to find a critical exponent v. To dothis we solve the scaling equations (12) for differentvalues of w (0) in the vicinity of (v, with initial conditions

(v(0)=w, /(1 —t) .

The resulting c.( oo ) as a function of reduced temperaturet was adjusted with a linear least-squares fit to the curve

transforms the system (11) to the dimensionless form,lnE( oo, t)= —v lnt +const, (13)

d E(l) 1 4 9! —w(!)1

—a U(l)

dl 3u ( I )

d(v (1) 1

dl 2s(l)du (I) 1

e (l+C+2),dl 8E l

(12)

in the reduced temperature interval t &[10,10 ], fora wide range of maximum allowed bending curvatures a.The resulting fits are shown in Fig. 3 and the numericalresults are summarized in Table I. We see that, for therange of parameters studied,

with initial conditions e(0)=1/(ppI v), (u(0)=p(M, andv(0)=0. Different initial conditions thus correspond todifferent temperatures. Choosing p = 1.39pt ~ as in Ref.1, these initial conditions become e(0) =1.39/(v(0) andv (0)=0.

A heuristic argument suggesting that this set of equa-tions does indeed describe a phase transition goes as fol-lows: If v(l) grows faster than exp(I) the derivative ofs(1) vanishes and E(1) is a constant when f~ oo. On theother hand, if E(l) grows faster than exp(l), the derivativeof v(l) vanishes and v(l) is a constant when l~ oo. Nu-merical solution confirms this expectation (Fig. 2). Thereis a transition at (Table I)

m, —1, . . . , 6,depending on the maximum allowed bending, where thesuperfluid density p=E( oo )

' vanishes like

p —t

4,4

2.2

0.0—22 —20 —18 —16 —14 —12 —10 —8 —6 —4

FIG. 3. Computation of the critical exponents through rela-tion (13) for various allowed limits to the vortex curvature forcircular rings that can be bent. For a wide range of maximumbending curvature a, we get a very good fit to v-0. 46.

5468 BRIEF REPORTS 47

TABLE I. Critical behavior of a dilute gas of circular vortexloops for difTerent values of the maximum allowed curvature a.w, is the transition point and v the resulting critical exponent.

TABLE II. Critical behavior of a dilute gas of elliptical vor-tex loops for difTerent values of the maximum allowed curvaturea and eccentricity E. Symbols as in Table I.

0.0010.0050.1

0.5

wc

1.0561.7464.0815.884

0.4570.4600.4680.466

0.50.30.1

0.0010.50.001

0.50.30.1

0.0010.0010.5

wc

6.0825.1463.3413.4413.3131.325

0.4110.4100.4060.3980.4080.402

v-0. 46 .

U(R, a) =@+ f4 ~ e(R')2 I

2+f + ln2 "7

2

+2 C+1+f +2

It is also possible to study the critical behavior for vor-tex rings with more complex geometries. As an examplewe consider elliptical vortices bent in an arbitrary direc-tion. They are parametrized as follows:

X(o ) =(R coso, R (1+f) sino, (a/2)R cos2(o +y) ),O~o- ~2~ .

There are two additional degrees of freedom which arethe eccentricity f, 0 &f & E & 1, and y which is a mea-sure of the relative orientation between the direction ofelliptical deformation (the ellipse major axis) and thedirection defined by the vortex points of maximum bend-ing. This time the phase space is ten dimensional and theeffective potential is given by

e'(l +C+2),dl 8E( l )

'(l +C+2),dl 4E(l)

3 =2[a +f +(a +f +2a f cos4y)'~ ] .

These equations have to be solved with initial conditionsE(0)=1.39/to (0), U(0)=0, and z(0)=0, and again it isfound that there is a transition at which the superAuiddensity p vanishes like t . The critical exponent v wasnumerically calculated for different values of maximumbending a and maximum eccentricity E using the reducedtemperature interval t H [ 10,10 3 ]. The results aresummarized in Table II, and it is seen that for a widerange of parameters we get

v-0. 40 .

2

+ f+ C2

A calculation analogous to the one that was just car-ried out for circles yields the following scaling equations,to first order in a and f:

4 11I —w(t)

dl 3=—~e e

X f f f A' ae f'e 'df dad)0 0 0

'(l +C+1)dl 2E(l)

To conclude, we have shown that the result that circu-lar vortex rings drive a phase transition that is qualita-tively similar to the k transition in liquid helium is robustin the sense that it does not go away if the phase space isenlarged, and the critical behavior does not qualitativelychange. The experimental value is, however, v-0. 67 andit is not clear that the agreement can be quantitativelyimproved using this approach.

The work of A.F.R. and R.H. was supported byCONICYT and the Departamento de Postitulo y Post-grado, U. de Chile, and that of F.L. by FONDECYTGrant No. 91-1265 and DTI Grant No. E-2854-9244.

*Present address: Courant Institute of Mathematical Sciences,New York University, 261 Mercer Street, New York, NY10012.

F. Lund, A. Reisenegger, and C. Utreras, Phys. Rev. B 41, 155(1990).

2G. Williams, Phys. Rev. Lett. 59, 1926 (1987); 61, 1142(E)(1988); in Excitations in 2D and 3D Quantum Fluids, edited

by A. G. F. Wyatt and H. J. Lauter (Plenum, New York,1991).

S. R. Shenoy, Phys. Rev. B 40, 5056 (1989);42, 8595 (1990).4L, Onsager, Nuovo Cimento Suppl. 6, 249 (1949); R. P. Feyn-

mann, in Progress in Low Temperature Physics, edited by C. J.Gorter (North-Holland, Amsterdam, 1955), Vol. 1, p. 17.

5J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1131 (1973).