Volume 134B, number 3,4 PHYSICS LETTERS 12 January 1984

STOCHASTIC QUANTIZATION A N D MEAN FIELD APPROXIMATION

R. JENGO International Centre for Theoretical Physics, Trieste, Italy International School for Advanced Studies (SISSA), Trieste, Italy and INFN, Sezione di Trieste, Italy

and

N. PARGA 1 International Centre for Theoretical Physics, Trieste, Italy

Received 14 October 1983

In the context of the stochastic quantization we propose factorized approximate solutions for the Fokker-Planck equation for the X Y and Z N spin systems in D dimensions. The resulting differential equation for a factor can be solved and it is found to give in the limit of t --, ~ the mean field or, in the more general case, the Bethe-Peierls approximation.

The stochastic quantization procedure [ 1 ] has received much at tention lately [2,3] as a general the- oretical frame to study the thermodynamics of field theory, and also because it is a useful one to deal with numerical simulations [4,5]. In addition, it can be conceptually compared with the Monte Carlo compu- tations where the thermodynamical equilibrium is reached in the limit of an infinite number of itera- tions. It is interesting therefore to see how results or approximation schemes which are usual in the stan- dard part i t ion function formulation are obtained in the stochastic formalism (see for instance ref. [6] on the derivation of the reduced models).

As is well known one introduces an extra time vari- able t and an evolution equation containing random forces. One can use a probabil i ty distribution which is time dependent and obeys the Fokke r -P l anck equa- tion, giving in the t -~ oo limit the equilibrium proba- bility distribution [7]. The variables of this distribu- tion are of course the field variables on the lattice, on the sites or on the links, depending on the theory at hand.

1 Permanent address: Centro Atomico Bariloche and Instituto Balseiro, Universidad de Cuyo, Bariloche, Argentina.

0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

In this letter we study the evolution equation for spin systems in the simplifying assumption that at least from a given time on the probabil i ty distribution factorizes, each factor depending only on a single site variable or on a group of variables of a cluster of neighbouring sites.

Our result is that, with the assumption of factori- zation, the Fokke r -P l anck equation for a factor gives in the t ~ ~o limit a probabil i ty distribution which co- incides with the mean field one in the case of site by site factorization, or more in general with the B e t h e - Peierls distribution [8] when factorization in clusters of sites is assumed.

Let us notice that the assumption of a factorized probabil i ty distribution is not a priori equivalent to a mean field-like approximation, since the latter also requires a particular functional form of each factor. We study explicitly the examples of the X Y model and the ZN (clock) spin systems in D dimensions. In the first case the stochastic formalism can be written down immediately for the unconstrained variables (the phases). For the ZN models the stochastic quan- tization is not completely straightforward. As noticed by Parisi [4], it is necessary to convert the discrete spin variables into continuous degrees of freedom

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which will obey a Langevin equation. We first discuss the X Y model which illustrates the

procedure in a simple way. We then discuss the Ising model in some detail, generalizing afterwards to the ZN spin model. Finally we extend the discussion to the case where a factorization in clusters is assumed.

In the stochastic version of the X Y model the prob- ability distribution satisfies the Fokke r -P lanck equa- tion

_ 6 2 aP({0), t) ~r77g~2P({0), t) at n LaOn

+a0- ' (11

where t is the auxiliary time to be taken to infinity [ I ] , On is the phase of the spin defined on the site n of a hypercubical lattice and {0} denotes a spin con- figuration. H is the X Y model hamiltonian given by

H = -/3 ~.. c o s ( 0 / - 0 / ) , (2) t ,]

where the sum extends only over nearest neighbours. As we said before we approximate the solution of

eq. (1) by a factorized expression

P({O), t) ~- ]-1 P(On, t ) , (3) n

where each factor is normalized

i P(O n, t) = 1 . (4 )

In order to obtain a differential equation for P(On, t) we replace (3) in the Fokker -P lanck equation and integrate over all the On's except one (say n = k). The left-hand side of eq. (1) then becomes:

aP({0}, t) cq ~

at --Tf

_ aP(Ok, t) at

(s)

When we do the same on the right-hand side of eq. (1) the only term which contributes in the sum over all lattice sites is the one with n = k. Then we end up with the differential equation

aP(Ok, t ) /at = a 2P(Ok, t)/a 02 (6)

O ,dO ) where k' refers to the nearest neighbours to k.

This can also be written as

aP(Ok, O/at = a 2p(Ok, t)1602

+ (J(6/6Ok)[P(Ok, t)(A sin Ok B cos Ok)] , (7)

which can be interpreted as the stochastic quantiza- tion of a single degree of freedom with an effective hamiltonian

h(Ok) = -13(A cos 0 k + B sin Ok). (8)

The solution ofeq . (7) for t ~ o~ is then [1,7]

P(Ok, t) ~ N exp[ h(Ok)] . (9)

A and B are to be computed self-consistently, i.e. they are given by

A = 2D(cos Ok}h, B = 2D(sin Ok)h , (10)

where the averages are taken with the weight given in (9). We recognize in (9), together with the prescrip- tion of eq. (10), the mean field probabil i ty distribu- tion. Actually both equations yield the same result and only the modulus of the external field M = (A 2 + B2) 1/2 is determined and given by M =

1 I(2Df3M)/IO(2DclM) where I0,1 are Bessel functions. It is not immediate to carry out this computat ion

for the lsing model. As we mentioned before, since the spin variables are discrete it is not possible to write down a Langevin equation for them. Instead it is nec- essary to perform a gaussian transformation

Z = ~ exp(ll3VikSiSk) ( l l a ) {Si=+I)

+ ~ In ch(l~ V/k ~k)l ( 1 1 b) i !

and in terms of the continuous field ~ the magnetiza-

tion M = (Si) is given by

M = (th [3Vik~k). (12)

Here and in eqs. (11) there is an explicit sum over

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repeated indices. Now the Langevin equation for the scalar field is

~i = -~ Vik [¢k -- th(/3 Vkl¢l)] + rli(t), (13)

where rli(t) is a gaussian random force satisfying

(rli(t) nk(t')) = 26ik5 (t - t ' ) . (14)

The matrix V cannot be taken just as a nearest neighbour interaction, because in this case it is not positive definite and the gaussian transformation is not defined. A way out of this problem is to define Vii to be a for i = ], 1 for i, ] nearest neighbours and zero otherwise [9]. The piece agq is irrelevant since S 2= 1.

From eq. (13) we see that it is convenient to work with the linear combination

Ck = Vkl q~l (15)

which satisfies the equation

~i = - a / 3 ¢ i + a 2 [3 th/3 ¢i +/3 ~ (-¢i' + 2a th/3 ¢i')

+/3i,~,i,,th(3¢i,,+arli(t)+ ~i, rli ,(t). (16)

Here we wrote explicitly the form of V; i ' runs over the nearest neighbours of i and similarly i" runs over the nearest neighbours of i'. Consequently ¢i appears not only in the first two terms of the equa- tion but also in the double sum.

We can now write the Fokker-Planck equation for the probability distribution P ( ( ¢}, t). Let us no- tice that since in the Langevin equation the random force appears not only on the site i but also on its neighbours the diffusion matrix present in the Fok- ker-Planck equation will not be diagonal [ 10]. How- ever since we are proposing a factorized probability distribution the differential equation we obtain for the factor P(¢k, t), once we integrate eq. (16) over all the ¢i 's except ¢k, is simply

~P( ¢k, t)/~t = (2D + a 2) 6 2P( ¢k, t)/6 ¢2

+ t3(~/~ ¢~){P(¢~, t)[c+ a¢~

- (2D+ a 2) th(/3¢k)]}. (17)

C is a constant to be determined self-consistently and given formally by

C= 2D(¢ ) h - 2D(2D+ 2 a - 1)(th/3¢)h. (18)

Here the averages are taken with respect to the proba- bility distribution which is the solution of eq. (I 7) at large times

P( ¢k, t) ----+ N e x p [ - h ( ¢ k ) ] , (19) t ~

where h is the effective action

h ( ¢ k ) = [~C/(2D + a2)] ¢k + [/3a/2(2D + a2)] ¢2

- In ch(/3 Ck)- (20)

It is easy to find that

C = 2D(a - 1) th(t3C/a), (21)

and besides from eq. (12)

M = -th(t3C/a). (22)

Combining these two equations M is determined self- consistently by

M = th [2D/3M(a - 1)/a] . (23)

Notice that since we treat the path integral (1 lb) in an approximate way by assuming a factorized solution of the Fokker-Planck equation the equivalence of it with the Ising model expression (t la) is, in general, lost and indeed the arbitrary constant a appears in the final result eq. (23). However if in eq. (1 lb) we take the limit a -+ oo the original expression (1 la) is recov- ered since we get for every site exp a~l~i[ (1¢ i1 /2- 1) which is maximum for 4~i = + 1 and then inch t3 Vik ~bk

{3(Pi Vik~k ,1. Therefore any approximation in this limit will be an approximation for the Ising model. Taking the limit in eq. (23) we obtain the mean field equation.

What we did can easily be generalized for the case of the ZN spin model. The hamiltonian is

H = - ½ ~S i Vik Sk , (24)

where S i = (COS 27rsi/N, sin 2~rsi/N), si = O, ..., N - 1. By performing the gaussian transformation (as before Vik contains a diagonal term a6 ik) w e can write the partition function

z = f ( ~i d~') exp(--½~VikC~i "dPk)

× I-I ~ exp(/3 Vik¢~ k • S i ) , (25) i s i

4:1 Notice that in this limit qJi -->acki.

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Volume 134B, number 3,4 PHYSICS LETTERS 12 January 1984

and in terms of dO the magnetization M is

M = (Si)

= ~Zsi Si exp(t3 VikdOk " Si!; ( •si-----exp (fl--~Vik dO~-k 7 g ) / " (26)

It is easily seen that in the limit a -+ ~ the continuous fielddO = (p cos 0, O sin 0) becomes a ZN variable p = 1, 0 = 27rn/N.

Proceeding like before and assuming a factorized probabili ty distribution in terms of

Ok = VkldOl (27)

we get the equation

3P( Ok, t)/Ot = (2D + a 2) 62P(~k, 0 / 6 0 2

+ (2D+a2)(g/6dlk)[P(dtk, t) Sh(dlk) /60k], (28)

where the effective hamiltonian is

h(~k) = [JC" Ok/(2D + a 2) + ~ 0 2 / 2 ( 2 D + a 2)

- In ~ ] exp(/30k" S ) . (29) $

As before the vector C is self-consistently determined by

c= 2D(0>~

, , INs S exp(/3 0" S ) \ - 2D(2D + 28 - l ) \ ~;s exp(/3 0" S) / h " (30)

This gives the equation for the magnetization

M = :~s S exp [-2D/3(1 - a) M . S/a] (31) 2; s exp [ -2D~(1 - a) M . S/a] "

As we discussed previously we take the limit a ~ oo. We then obtain the mean field result.

Finally, one can include short range correlated fluctuations by assuming, for instance, for the X Y model that

e ( (0} , t) -~ ]q P((0}c, t ) , (32) C

where c denotes a cluster of neighbouring sites. From the Fokke r -P lanck equation we get an equation for P((0}c, t) which gives for t ~ "~

P({0}c , t) ~ N exp [ - h ((0}c) ] , (33)

where the effective hamiltonian is given by

h((O)c) = ~ 7__2, cos(Oi- Oj) if,j)

- / 3 ~ (Au) cos 0(/) + Bff) sin 0 q ) ) . (34) (1)

In this formula (i, j) means a pair of neighbouring sites belonging both to the cluster, (/) denotes the sites which have n /4 :0 neighbours outside the cluster. Aj, 17/have to be detemlined self-consistently and are ex- pressed as expectation values

AU) = n/(cos Off)) h , B(/) = n/(sin O(i))h . (35)

This gives the Bethe-Peierls approximation [8].

N.P. would like to thank G. Parisi for a useful dis- cussion. He also thanks Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitali ty at the International Centre for Theoret- ical Physics, Trieste, and in particular for an award which made possible his participation in the Carg~se Summer Institute (1983).

References

[1] G. Parisi and Wu Yongshi, Sci. Sin. 24 (1981) 483. [2] T. Fukai, H. Nakazato, 1. Ohba, J. Okano and Y. Yama-

naka, Waseda University preprint WU-HEP-82-7 (1982); J.D. Breit, S. Gupta and A. Zaks, IAS Princeton preprint (March 1983); P.H. Damgaard and K. Tsokos, preprint MD-TP-218 (June 1983).

[3] G. Aldazabal, E. Dagotto, A. Gonzalez-Arroyo and N. Paxga, Phys. Lett. 125B (1983) 305; G. Aldazabal, A. Gonzalez-Arroyo and N. Parga, Bari- loche preprint (1983); G. Aldazabal and N. Parga, ICTP Trieste preprint IC/ 83/155.

[4] G. Parisi, Nucl. Phys. B180 [FS2] (1981) 378. [5] G. Parisi, Nucl. Phys. B205 [FS5] (1982) 337;

M. Falcioni et al., Nucl. Phys. B215 [FS7] (1983) 265; A. Guha and S.-C. Lee, State University of New York preprint ITP-SB-83-33 (1983).

[6] J. Alfaro and B. Sakita, Phys. Lett. 121B (1983) 339. [7] E. Floratos and J. lliopoulos, LPTENS Preprint 82/31

(1982); B. Sakita, City College of the City University of New York preprint CCNY-HEP-83/14 (1983).

[8] K. FIuang, Statistical mechanics (Wiley, New York, 1963).

[9] D.J. Amit, Field theory, tile renormalization group and critical phenomena (McGraw-Hill, New York, 1978).

[10] R. Graham, Z. Phys. B26 (1977) 397.

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