a. holz- three-dimensional rotator model and vortex plasma

8/3/2019 A. Holz- Three-Dimensional Rotator Model and Vortex Plasma

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THREE-DIMENSIONAL ROTATOR MODELAND VORTEX PLASMA (*)

A. HOLZ

Institut fr Theoretische Physik, Freie Universitt Berlin, 1 Berlin 33, Germany

(Reu le 18 mai 1978, revise le 17 juillet 1978, accepte le 24 aofit 1978)

Rsum. 2014 On tablit une relation triangulaire entre les modles statistiques suivants : le modle

des excitations discrtes, le modle des rotateurs gnraliss et le plasma de tourbillons. On montreque la fonction de corrlation du modle rotateur en dimension D ~ 2 dcrot exponentiellement haute temprature.

Abstract. 2014 We establish a triangular relation between discrete excitation models, generalizedrotator models and generalized vortex plasma models. The high temperature correlation functionfor the planar rotator models in dimensions D ~ 2 is shown to decrease exponentially.

LE JOURNAL DE PHYSIQUE - LETTRES

Classification

PhysicsAbstracts

In a recent letter Knops [1] gave an exact relationbetween solid-on-solid interfacial models and gene-ralized planar rotator (GPR) models in two dimen-sions (2D). In particular he showed that the discreteGaussian (DG) interfacial model [2] maps onto theVillain [3] rotator model and which corresponds tothe 2D plasma model [3], as does the DG model [2].Similar methods have been developed by Jose et al. [4].Based on these ideas a similar triangular relationfor the corresponding 3D models will be established.We consider on a 3D simple cubic (sc) lattice

with N lattice sites the Hamiltonian

where n(ijkl) = 0, :t 1, ..., and V(x) is a positivefunction of x which will be specified later. Eachelementary square of the sc lattice is labeled by i, j, k, 1 >, where i, j, k, I are taken consecutivelycounterlockwise around the square face. Each face

carries only two labels i, j, k, I > and I, k, j, isgiving the face two orientations. Correspondingly werequire

The set { ~j~ } is thus defined on a face centredcubic (fcc) lattice. Furthermore we require that

holds for each point i on the dual lattice (dsc) tothe sc lattice. Here n~ ~~kl > = n(ijkl) holds for thesix neighbouring outward oriented faces to thepoint i otherwise it vanishes.Accordingly the discrete

excitation (DE) model defined by equations (1),(2) and (3) has 2 N degrees of freedom. Using theintegral representation for equation (3)

the partition function of the problem can be written

where i, j > indicates summation over nearestneighbours of the dsc lattice, and

This is a GPR model with nearest neighbour interac-tion and which, in the case of V(n) = n2, correspondsto the Villain [3] model for the 3D rotator model.A further representation of the DE model similarto the one obtained in [2] for the DG model willnow be derived. Because equation (3) implies thatthe set {~~ } forms a divergenceless field, it may

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019780039019033100
http://www.edpsciences.org/http://dx.doi.org/10.1051/jphyslet:019780039019033100http://dx.doi.org/10.1051/jphyslet:019780039019033100http://www.edpsciences.org/


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be obtained via a rot-operation [4] from a vectorfield { mi } defined on the sc lattice. Using the follow-ing representation

M~234~ =~2-~+~ "~4-(~4-~+~-~) ~

~6721~=~-~+~-~-(~-~+~-~).(7)

"4561>=~"~+~~~"(~-~4+~6-~)

for the three faces illustrated in figure 1 and the rulesstated earlier one confirms that equation (7) satisfiesequation (3). Using in addition the condition

I V t~

FiG. 1. - Relation between subindices used in equation (7) andsquare face orientations.Arrows point inside first quadrant.

where 8~ is a nearest neighbour vector, one confirmsthat to the 2 N independent variables of the set{ n(ijkl)} correspond 2 N independent variables of

the set { mi }.A unique mapping between these twosets of variables is therefore provided if the trans-formation matrixA constructed from equation (7)

has rank 2 N. Here xn and yrn are 3 N vectors andA

is a 3 N, 3 N matrix. Using Fourier transforms andfor sake of simplicity a long wave length approxima-tion one gets

where (Y,,, Y,7, Y,7) are normal coordinates referedto the orthogonal trihedron with e; = qjl q I, andq assumes N values in the first Brillouin zone of thesc lattice. It follows from equation (10) that rank

FIG. 2. - Planar

partof some

loopconformation.

C1and

C2are losed loops. The loop parts C3 and C4 enter the plane at cir-cular drawn points and coalesce to C5. Construction satisfiesequation (8). Vortex strengths are Ti == 0"2 = ~4 = 1/2, ~3 = 1,65 = 3/2. Dashed drawn circuits indicate sense of integration in

equation (12).

[A] = 2 N. All sets {mai} c { y } satisfying equa-tion (8) can now be constructed as an oriented andinterconnected loop system (Fig. 2) obeying Kir-chhoffs law in the form of equation (8) and corres-ponding uniquely via equation (9) to a set { n~ i jk1) }with integer valued entries as required for the deri-vation ofequation (5). The question if all sets { n~ i jkl> }which satisfy equation (3) and which can beconstructed in an obvious fashion are obtained inthis way is of different nature and will not be consi-dered further. Let me point out that Villain [5] hasalready studied the relation between Hamiltoniansexpressed in terms of sets of variables { qp } and {~ },which are essentially equivalent to the sets { n~~~k~> }and { m~ } respectively although he does not introducethe DE model explicitely. The difference in the twoprocedures to construct the set { n~ ~~kl ~ } from theset { mi } and the analogous construction in [5] hastwo reasons. First the vij are bond variables whereasthe mi are variables on the sc lattice and second inthe present method both sets of variables used areconfined from the outset to 2 N degrees of freedom.


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L-333THREE-DIMENSIONAL ROTATOR MODELAND VORTEX PLASMA

The constructions of sets of { mi } satisfying equa-tion (8) and which involves the oriented loops pre-sented in figure 2 allows to associate with each part Ciof the loop system a vortex strength (Ji defined by

which can be interpreted as a current j( i,j) associatedwith the bond i, j >.Now we can express the partition function in the

form

From the generalized vortex loop (GVL) models defined by equation (11) only the one obtained from V(n)=n2leads to a Coulomb vortex loop (CVL) model. One obtains after standard manipulations taking into accountthat L mi = 0 holds for each closed loop

i

where 6~ _ (0, + 1, + 2, ...)/2, and integration isperformed along the conformations of the loopsystem extending along the edges of the sc lattice,and at each vertex Kirchhoffs law in the form of

equation (8) has to be satisfied. In equation (12) thediscrete summation over the m~ has been approximatedby a continuous integration which involves the aias defined above. Because these quantities are definedwith respect to bonds they can assume half integervalues. The interaction in equation (12) is the Cou-lomb interaction known from the interaction of

Amperian currents but is of apposite sign. Equa-tion (12) represents the grand canonical ensemblepartition function of a vortex plasma with couplingconstant

and chemical potential

for the loop bordering a square face on the sc lattice.Further relations can be established by expanding

equation (5) for fiDE JDE > 1 and using n = 0, 1only. This leads to the planar rotator PR model

with

It follows from equations (15) and (16) that the hightemperature CVL- and PR-model can be mapped

onto each other for #PR JPR 1 leading to

and

For low and intermediate temperatures one usesBerezinskiis [6] representation of equation (15) toderive the following Hamiltonian

A _2 r

Here the first term represents spin wave excitations,the second term represents the vortex plasma on thesc lattice with (Ji = 0, 1, ... and the anharmonicterms couple spin waves and vortices such that

is satisfied for all dsc lattice sites. For PPR JPR >> 11

equation (19) is characterized by the coupling cons-tant and chemical potential given in equation (17)with 9(1/#PR JpR) ~ 0. For increasing temperaturethe third term in equation (19) can be taken intoaccount by means of a wave function renormalizationof the spin propagator [7]

yielding the renormalized expressions given in equa-tion (17). It apparently follows from this and theexact high

temperaturelimit

equation(18) that the

PR model equation (15) can be studied over thewhole temperature range by means of equation (19)where the last term is taken into account over equa-tion (20) and 9(/PPR JPR) can be calculated via aperturbative procedure. Apparently therefore no


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dimensional change of the spin propagator in thedisordered phase takes place. It should, however, bepointed out that the plasma model defined by equa-tion (19) allows integer valued vortex strengths,whereas the plasma model defined by equation (12)

allows also half integer vortex strengths. The topo-logical conformations of the vortex loops in thelatter model are, however, severely constrained dueto Kirchhoffs law in the form of equation (8), whereasthe usual Kirchhoffs law as applied to equation (19)allows more configurations. For instance the ele-mentary vortex loop of the CVL-model equation (12)borders four square faces implying a chemical potentialfour times the value given in equation (14) whereasthe elementary loop to equation (19) borders onesquare face.Accordingly there is still a certain diffe-rence between the two models.

The most interesting case, however, arises forV(n) = n2, n = 0, 1, which allows only n~i~k~~=o, 1 excitations. Equation (1) and its constraintsdefine this time a vortex loop problem on the sclattice with an excluded volume effect as interactionand vortex strengths ~ _ 1. It is obvious that thisholds for all hypercubic lattices with D > 2 whenthe ~...~ are defined with respect to the (D - 1)dimensional elementary hypercube faces. Equation (5)gives in that case

The PR model equation (15) can now be studied forthe range

as follows from equation (21).A lower bound forthe PT in the DE model follows from the Peierls

type argument [8] namely that below a certain tempe-rature calculated within a random walk approxima-tion no infinitely extended loop can be present,

Comparison of equations (22) and (23) shows thatthe PT in the PR model for D = 2 cannot be studiedvia the present DE model, but only its high tempe-rature properties. For D > 3 this is, perhaps, possibleand has recently been done by Helfrich [9] whoderived the present representation of the PR modelby means of a different procedure. The correlationfunctions for the PR model are easily derived bysubstituting the right hand side of equation (3) fori = 0, r by + 1, and - 1 respectively. This leads to

where all string conformations attached to i = 0,r and all loop conformations are summed over.Equation (24) holds for~ ~ given by equation (16)under the restriction equation (22) and for D > 2.It can be studied by the methods of polymer physicsand leads for PDE JDE > 1 to exponential decay ofcorrelations.

Finally we would like to point out with respect to

the GVL model that due to the constraints imposedby the requirements of the DE model, to permitn = 0, 1 excitations only, onto the set { ml } viaequation (9) vortex strengths and loop conformationsof the GVL model are severely constrained. Thismay invalidate the conclusions reached below equa-tion (20) because the PR model is obtained from theGPR model with n = 0, :t 1 excitations only.

Interesting discussions with W. Helfrich are kindlyacknowledged.

References

[1] KNOPS, H. J. F., Phys. Rev. Lett. 39 (1977) 766.[2] CHUI, S. T. and WEEKS, J. D., Phys. Rev. B 14 (1976) 4978.[3] VILLAIN, J., J. Physique 36 (1975) 581.[4] JOS, J. V., KADANOFF, L. P., KIRKPATRICK, S., NELSON, D. R.,

Phys. Rev. B 16 (1977) 1217.[5] VILLAIN, J., J. Phys. C : Solid State Phys. 11 (1978) 745.

[6] BEREZINSKII, V. L., Sov. Phys. JETP 34 (1972) 610.[7] HOLZ,A., to be published in the Proceedings of the 13. IUPAP

Statphys. Conference, Hafa 1977.[8] GALLAVOTTI, G., Nuovo Cimento 2 (1972) 133.[9] HELFRICH, W., to be published in J. Physique.

a. holz- three-dimensional rotator model and vortex plasma

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