IC/83/211

v :. • " '• >

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

AH EQUIVALENCE BETWEEN THE DISCRETE GAUSSIAN MODEL AND A GENERALIZED SINE GORDON THEORY ON A LATTICE

G. Baskaran

and INTERNATIONAL Neelima Gupte ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1983 MIRAMARE-TRIESTE

The discrete Gaussian (DG) model was introduced by Chui and Weeks as IC/83/211 an approximation to the Solid on Solid model in the interfacial roughening transition

problem. They also showed that this model maps onto the two dimensional neutral International Atomic Energy Agency Coulomb plasma. There exists an equivalence of this model to the Sine-Gordon (SG) and theory in two dimensions through the Coulomb plasma and massive Luttinger model United Nations Educational Scientific and Cultural Organization In this note we establish the equivalence between the DG model and a generalized INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SG theory defined on an Euclidean lattice. The equivalence is obvious in some sense,

because in both cases the integer values of the field variable are preferred

However, we give a precise mathematical connection between the two models. AN EQUIVALENCE BETWEEN THE DISCRETE GAUSSIAN MODEL AND A GENEflALIZED

SINE GORDON THEORY ON A LATTICE* EQUIVALENCE

G. Baskaran The DG model is defined by the interaction energy

International Centre for Theoretical Physics, Trieste, Italy T

coupling between the two sites i and j. The partition function is given by and

Neelima Gupte

International Centre for Theoretical Physics, Trieste, Italy. 2 - Z **f[-lk)Z7*i<*-»i?-]

ABSTflACT (2)

We demonstrate an equivalence between the statistical mechanics of the where J = £J discrete Gaussian model and a generalized Sine-Gordon theory on an Euclidean 0 j ij lattice in arbitrary dimensions. The connection is obtained by a simple We use the identity transformation of the partition function and is non perturbative in nature. - 2 *>• *$ *f -t 2.1 \ ) f <* MIRAMARE-TRIESTE

November 1983 7T* (M if (3) * To be submitted for publication.

to rewrite the partition function as

~ 2 - (4)

This is the generating function of a scalar theory with periodic self coupling in

an Euclidean lattice. We call this a generalized Sine Gordon theory.

When J (^>^1, the dominant Fourier coefficient is for n = 1 and its

harmonics are exponentially small. Thus the interaction energy is vhere H is the total number of lattice sites and M is a constant. .z Notice that the periodic function inside the product resembles the Villain * 7' (10) approximation to the term exp(cos -A). Defining to = —— we can write Z as

From this we can find the phenomenological coefficient y used by Chui

and Weeks in their study of the dynamics of roughening transition (where they 2 approximate the DO nodel by the SG model). This is

(ID

wherewe have used the definition of the Jacobi theta function, Notice also that the self coupling term is exponentially small compared to the Gaussian term, which maJtes the model closer to the massless gaussian f (7) theory and hence to the low temperature phase of the XY model. As we go to very low temperature/, that is the massive phase, the harmonics

To make connection with the Sine-Gordon theory on an Euclidean lattice, we use the become important. In fact, for a given J fc , the harmonics are important and go 12 as ~ for ii < (J g/??r ) . Other higher order terms are exponentially small, Fourier expansion. n u One may also view the present result as if the Villain approximation to

Mine Gordon theory produces the DGM. Thus our work complements the work of Knops

(8) who shows that the DG model maps exactly onto the Villain approximation of the & XT model in two dimensions. and write the partition function a:; ACKNOWLEDGMENTS REFERENCES

The authors are grateful to Professor Abdus Salam, the International 1) S.T. Chui and J.D. Weeks, Phys, Rev. B14, 4978 (1976).

Atomic Energy Agency and UNESCO for hospitality at the International Centre for 2) J. Solyom, Adv. Phys. 28, 201 (1978).

Theoretical Physics, Trieste. They would also like to thank Professors M.P. Tosi 3) S.T. Chui and J.D. Weeks, Phys. Rev. Lett. k0_, 733 (1978). and N.H. March for advice and encouragement. 4) j. Villain, J. Phys. (Paris) 36, 581 (1975).

One of the authors (G.B.) would like to thank Professors Paolo Budinich 5} H.J.F. Knops, Phys. Rev. Lett. 39, 766 (1977). and Erio Tosatti for hospitality at SISSA.