Zn spin systems in two dimensions

Nuclear Physics B205 [FS5] (1982) 414--432 © North-Holland Publishing Company

THE P H A S E S T R U C T U R E OF SU(n)/Z,,xSU(n)/Z,, SPIN SYSTEMS IN TWO DIMENSIONS N.D. HARI DASS and A. PATKOS 1

The Niels Bohr Institute, University of Copenhagen DK-2100 Copenhagen ~, Denmark F. DE,~K

Department of Atomic Physics, E6tv~s University VIII. Puskin u. 5-7, Budapest, Hungary Received 22 January 1982 The ground state of the two-dimensional chiral adjoint spin systems is studied using the most general nearest-neighbor ansatz for the eigenfunctions of the transfer matrix. The parameters entering this ansatz are determined variationally and the resulting phase structure is in close agreement with the results of Monte Carlo simulations.

1. Introduction

Two-dimensional models of global S U ( n ) × S U ( n ) symmetry are thought [1] to share important features with four-dimensional SU(n) gauge systems. They are asymptotically free [2] and are expected to be disordered for any non-zero value of the coupling constant. The lattice regularization of these models has so far been investigated with the action in the fundamental representation only. Extended studies have been performed in the weak [3] and strong coupling regimes [4]. A better understanding of the continuum limit requires careful investigation of other models belonging to the same universality class. In particular, important practical procedures like the continuation of the strong coupling series towards weak coupling or the early continuum behavior might depend essentially on non-universal characteristics of the model. Simultaneously with the gauge theories [5] a detailed study of variant actions of non-abelian spin models in two dimensions was started. A strong coupling calculation of the mass gap in the adjoint 0(3) model [6] indicated a possible first-order transition at finite coupling. A candidate for the mechanism underlying this transition is the condensation of non-abelian vortices similar to what happens in the XY-model [7]. SU(n) × SU(n) models also admit Z , vortices and therefore a parallel suggestion may be applicable to these latter models too. 1 On leave from Dept. of Atomic Physics, E6tv6s University, Hungary. 414

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Subsequent Monte Carlo studies [8] excluded the above possibility for the adjoint 0(3) model and also the investigation of the SU(2)/Z2 x SU(2)/Z2 model led to negative results [9]. Very recently, clear evidence for a first-order transition in the SU(3)/Z3 x SU(3)/Z3 model was reported by Kogut, Snow and Stone [10]. They also presented results showing that the condensation of Z3 vortices accompanies the transition. Even in those cases where no transition is observed the effect of these topologically sensitive objects is a very strong disordering which shifts the applicability of the perturbative renormalization behavior into the very small coupling region. The failure of the strong coupling series, the inapplicability of the asymptotically free perturbation theory in the transition range and also the surprising difference between the adjoint chiral SU(3) and SU(2) symmetric models demands the application in the intermediate range of some truly non-perturbative techniques. The application of the mean field approximation in 4d gauge theories has proved to be quite useful [11]. As mean field theory is expected to be reliable only when the dimension of space-time is large, its applicability to two-dimensional theories appears doubtful. In any case the M e r m i n - W a g n e r - C o l e m a n theorem [12] makes the applicability of any homogeneous background configuration very questionable. In this paper we propose a simple variational ansatz for the symmetric ground state, which takes into account nearest-neighbor correlations in an optimal way. In virtue of this latter it is sensitive to first-order transitions in systems with nearest neighbor interaction. It will be shown that this ansatz distinguishes nicely between the SU(2) and SU(3) cases in agreement with the numerical simulations. Namely, a smooth joining of the weak and strong solutions of the variational problem is observed in the first case, excluding a sizeable discontinuity in the internal energy. In the latter model, however, a sharp crossing occurs in the range 3/3 = (2.7-3.1) depending on the approximation used in the solution of the variational problem. The discontinuity of the internal energy is of the same order as observed in the Monte Carlo data. Our ansatz differs in some respect from the standard variational approach discussed currently in the literature [13]. Therefore a more detailed description of the idea is given in sects. 2 and 3. (Some earlier applications of our approach are listed in ref. [14].) Sect. 2 contains a presentation particularly appropriate to the high-temperature regime. The parallel weak coupling (low-temperature) behavior is derived in sect. 3. The general formulae of these sections are applied to the SU(2)/Z2 x SU(2)/Z2 model in sect. 4. There the eventual degeneracy of several higher excited states is also investigated, which would yield infinite correlation length and correspondingly a higher order phase transition. The SU(3) case is treated in sect. 5, where it is shown that the minima of the weak and the strong coupling solutions are distinct from each other and they cross close to the transition point determined numerically (3/3 Mc = 2.97 [10]). We discuss the possible mechanism of these transitions and the case of higher SU(n) groups in sect. 6.

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2. The variational problem in the character basis

We study the partition function of the models defined by Z = f DUx exp (BS) ,

S = ~ h ( U x U x++ t ) ,

,1

links

(2.1)

where Ux is an element of the SU(n) group (also denotes the fundamental representation of the SU(n)× SU(n) group). The lattice vector l connects the nearest neighbor sites in the square lattice. For the S U ( n ) / Z , × S U ( n ) / Z , model we have h (U) = xA(U) = [xf(U)[ 2 - 1

(2.2)

(XA(f) denotes the character in the adjoint (fundamental) representation).. The transfer matrix technique [15] will be used for the evaluation of Z. In the thermodynamic limit one has Z

=•0

N

r ,

where Nr is the number of rows of the lattice and A0 stands for the highest eigenvalue of the transfer matrix defined symmetrically in the following way: I O U i exp {½/3 ~ [h(UiUi++l ) + h(

gigi++x)] 2r-/~ ~ / h ( g i g + )} ~ ( { g i } ) = h ~({ gi}) (2.3)

({Ui} denote the variables in a given row and {V~} are those in the subsequent one). We now write the most general trial functional for the ground state of eq. (2.3) which is symmetric under SU(n)× SU(n) transformations and positive everywhere (as is obligatory for the ground-state wave functional of any quantum mechanical system). In addition the trial wave functional treats the neighboring pairs independently. Such an ansatz is qs0({ Ui}) = H exp ( f ( U i U i + + l ) ) , i

(2.4)

where f is a scalar function of its argument. Being a class function each factor of (2.4) can be represented by its character expansion e [(UIU~*l) = ~ arXr(UiUi++l ) ,

(2.5)

r

where Xr is the character of the representation r. Characters obey the orthonormality relation I d U X r (VU+)Xr (UW) ,

d r l~rr,Xr( V W )

(dr is the dimension of r). The coefficients ar will be treated as variational parameters. The expansion (2.5) is particularly useful in the high-temperature region

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(small fl), where it is known that the ground state is dominated by the closest representations to those appearing in the interaction function h. Using (2.4) in (2.3) the following lower bound is reached for h0: h o ~ f DU/D Vi exp {~/[f(UiUi++l )+f( gi v i++l )+ 21~(h ( g i s ; 1 )

+h(ViVi++l))+.h(UiV+)]}/~DUiexp{2~f(UiUi+,)} (2.6) Both the numerator and the denominator are interpreted as the partition functions of some (quasi)- one-dimensional systems of the same symmetry as earlier and with only nearest-neighbor interactions again. Eq. (2.6) is evaluated by applying the transfer matrix to the numerator (the denominator can be evaluated exactly):

f dX~ dX2 exp

(X~X~) + lfl (h (X~ Y~ ) + h (X2 Y~ )) + f(X~ Y~ ) +f(X2Y~ )}$(Xl, X2) = e$( Yb II2).

(2.7)

Using the character expansion

e 2:(v) = F. brXr(U),

(2.8)

r

and the orthonormality relation of the characters one evaluates the denominator directly: f DUi e2Ej(uiui+l) Nc-~OC, b~ o

(2.9)

(Nc is the number of columns of the lattice, bl is the coefficient of the character of the trivial representation in (2.8)). The partition function Z is then bounded from below by

Z ~>max (eo(fl, a,)] NoN"

(2.10)

where e0 is the highest eigenvalue of (2.7). The integral eigenvalue problem (2.7) is transformed into a matrix diagonalization if one uses the character expansion of its kernel in terms of the expression: exp (f(U) + lflh (U)) = X BrXr ( U ) .

(2.11)

r

Also the eigenfunction belonging to the highest eigenvalue eo should depend only

418

on Tr

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XIX~ ; therefore,

one writes 6 0 ( X l , X2) ~---~'. Cr)(r(XlX2 r

).

(2.12)

Using (2.11) and (2.12) in (2.7) together with the orthonormality relation one arrives at

~r' {I d X e ~ h ( X ) X r ( X ) X r ' ( X + ) ~ } Cr'=eOCr"

(2.13)

The character expansion of the Boltzmann factor in (2.1) contains only selfconjugate combinations of the representations. Therefore an analogue of the reality of the quantum mechanical ground state can be stated by ar = as

(2.14)

(where ~ denotes the representation conjugate to r). This argument can be repeated for the numerator system and (2.12) too, modifying (2.13) in the following manner: (i) one has to sum over each conjugate pair of representations only once, (ii) Xr should be replaced in (2.13) and (2.11) by 14~(xr(U) + x~(U)), Kr(U) = (Xr(U),

r # ~ representations, r = ~ representations.

(2.15)

One solves (2.13) numerically by selecting some important set of representations r. The strong coupling series of the matrix elements allow a systematic truncation. One associates the first direction in the {cr} space with the trivial representation; c2 is chosen equal to Cro appearing next in the strong coupling expansion of the Boltzmann factor and so on. For a given (not too high)/3 we could reach a numerical convergence of the value of e0 by keeping a moderate number of the representations (and consequently of at's). Therefore we conclude that the representation (2.13) is useful in the small/3 region. In the weak coupling regime all matrix elements in (2.13) become of about the same order of magnitude; therefore, one has to turn to a different representation which will be presented in the next section.

3. The weak coupling solution of the variational problem In sect. 4 numerical evidence will be presented that the results of the most general ansatz (2.4) in the case of the group SU(2)/Z2 × SU(2)/Zz can be reproduced very accurately by the one-parameter trial functional ~o = l-I exp i

{ah(UiUi~l)}.

(3.1)

H e r e we notice two general features of it, making the above assertion very plausible.

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First, it reproduces the O(fl 4) term of the strong coupling expansion of Z correctly for any n. Second, if one writes (2.4) in the form

IIro=l~i exp{~r arXr(giui++l) } , and in the low-temperature region expands Xr around the frozen configuration U~ = L one gets in the leading approximation a gaussian of width (Yr ar½dr) -1/2. This form is reproduced by (3.1) also; therefore, it will adequately represent the most general form, if the strong-weak crossover region is not too wide. Among the tremendous technical simplifications arising from its use, the possibility of an easy general treatment of its weak coupling solution is one of the most important. On substituting (3.1) the symmetrical form of (2.7) becomes [ dU

f dW e(tr/2)E~"(uw+)+xA(wv)ld)o(U)=eod~(V),

(3.2)

where/3' =/3 + 2~ and U = X~X;, W = X1 Y[, V = Y1 Y~ are introduced. We are going to derive a series for e0 in the 13 ~ oo limit: eo = e~0°)+ 1 e(o-" +. • •.

(3.3)

13

For this purpose the most convenient parameterization of the group elements is U = exp (iA),

A = AT,

(3.4)

where T are the generators of the group normalized according to t r v T / rt : ~5ii. We need the expansion of Xa with T 4 accuracy for the computation of e0 up to e(0 1):

XA(ei2 e -gf ) = n 2 - 1 - l n ( Z - Y)2 + I ( Z -

y)4

+ ~2n TrF ( 2 4 "4- ~4 + 6 2 2 i7.2_ 42I~3 _ 423 ~ ) .

(3.5) The cartesian parametrization of the Haar measure reads [16]

sin ½ATA ~ C* d(n 2 1)A(1 - ~ T r C* d("2-1)a I "aet" } - ~ A

(ATA) 2)

= C* d("2-1~A (1 - ~nA2),

(3.6)

where TA are the generators in the adjoint representation. The constant C* ensures that the weak coupling asymptotics of integrals of the type

f dU exp (xxA(U)) evaluated with the normalized measure (S d U = 1) agree with their value obtained

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by evaluating it in the algebraic basis (3.4)*. Thus we require

C*fdA

1, . sin~ATa [oet 1--7~ ~ A I A [ e xr~(~') = f d/x(~i) e x~A~'~')

x~oo,

(3.7)

where on the right-hand side Weyl's parameterization of the group integral 1s applied [17]. It is sufficient to compute the integrals in (3.7) with gaussian accuracy in order to fix the value of C*: 1 C~U(2 ) =

1 C~U(3 )

877-2,

(3.8)

x/37r (47r) 4"

We can apply to (3.2) the results of the standard perturbation theory if its kernel is defined symmetric with the measure ~_~ dA. For this we have to rescale the eigenfunctions: q~(A) = det sin ½ATA 1/2 ~b(A).

(3.9)

Then the kernel of (3.2) is written up to the required order as K(A1, A3) = Ko(A1, A3) + K - I ( A 1 , A3) = C ' 2 1 dA2 exp {(n 2 - 1)(/3 +/3')-1n/3(A21 +A~) -¼n/3'[(A 1 -A2)2 + (A2-A3)2]} X{ 1

1 2 2 --4-8F/(A 1 --t-A 3 +

2 1 4 2A2) + 3~/3 (A 41 +A3)

+ 3~/3'[(A 1 - A:) 4 + (A2 - A3) 4] +~n/3 Tr (A~ + * ~ ) + ~ n / 3 ' Tr [~4 +.2.4+2.~4+6(,{~ + * ~ ) ~ - 4 ( ~ ^3

A

+ A3) e{3 - 4(A 3 + A3)A2]},

(3.10) where Ko is the 0(/3 °) part of the kernel (the pure exponential), while K 1 is 0(/3 1). The leading order equation is therefore of the form

C.2 f dAldA2exp{(n2_l)(/3

1 2 +A3) 2 +/3 ,) - ~n/3(A1

-¼n/3'[(A1-A2)2+(A2-A3)2]}O(°)(A~) = E(o°)~(°)(A3)

.

(3.11)

The 0(/3 1) correction to e(o°) is calculated through the familiar formula E (0- 1 , -- ( ( ~ ( 0 ) i g _ l i~(oO) )



(3.12)

* The importance of C* was realized independentlyby B. Lautrup in his study of first-order transitions in gauge systems.

N.D. Hari Dass et al. / SU(n)/Z, x SU(n)/Zn spin systems

42l

The denominator with the same accuracy is given by the expression I d U e 2 ° ' x A ( V ) = C * f d A ( 1 - ~ n A 2 + l - - 4 ' l g a . z i ± ~na Tr A4) e2"("2-~) ~"'*2e =_d(o) + l d ( 1 ) .

(3.13)

/3

Using (3.11), (3.12), and (3.13) with the appropriate accuracy in (2.10) the expression of the free energy is given by -F=max

ln~T05+~(e(-~-y

~T65] + O

.

(3.14)

The solution of (3.11) is a simple gaussian

4700)(a) = exp (-aA21, 6=~n/341+r, .

y-=~=1+2~, . n2--1.

e~oo' = C .2 e2(n2 a)(~+-)/Z ] d (°)= C*

(3.15)

. (n2--1)/2

~

~v(1 +½v +,q--7~)/

2(n2--1)~{

e

~

(

\n/3]

O/

77"

'

~ (n2-1)/2

~n/3(~-1i]

"

The maximalization of (3.14) to leading order leads to ')/max =

3,

O~max = / 3 ,

q~(o)= exp (-~n/3A2),

- F = 2 ( n 2 - 1)/3 +ln C * +

n2-1 2

16 ln-27n/3 "

(3.16)

The above value of a [accurate to O(/3°)] makes it possible to calculate any local expectation value (like internal energy) from a one-dimensional system of effective coupling 2/3, when 13 >> 1. This means that the larger coherence length of the two-dimensional system is imitated by the one-dimensional one by approaching its critical point twice as fast. One understands that F calculated from (3.16) becomes exact for /3 >> 1 by the following argument. Moving each "vertical" bond to its neighboring "horizontal" one we get an assembly of one-dimensional chains of coupling 2/3. The estimate for Z one gets from this special bond-moving transformation is known to be an upper bound [18], while our variational estimate is a lower one. Therefore the asymptotic effective coupling 2/3 corresponds to a saturation of the inequality. The knowledge of (~(0) enables us to calculate (3.11). It is simple to understand, that the correction to the amax =/3 relation should not be taken into account in the matrix elements of (3.12) to the present order. The results of the straightforward

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calculation which uses the general expression for the quadratic Casimir operators of SU(n) in the fundamental and adjoint representations is the following: 5 ( n 2 - 1)

-AF = - -

(3.17)

96/3

F r o m the point of view of first-order transitions the quantity of central importance is the density of the interaction function: f D U ~ exp [2a(/3) Y~XA(U~U,+I )]XA(UioUi+o+l )

1 dF


i

-

(3.18)

f DU, exp(2a([3) ~ X g ( U ~ U + a ) )

2d/3

It is a continuous function of a (or more generally of at's); therefore, the only possibility for observing a discontinuity is to find a discontinuous change in a . . . . In the next two sections the n = 2, 3 cases will be investigated to see whether the strong coupling representation joins the weak coupling results in a smooth way or not. The latter possibility corresponds to F(o~) having two distinct minima a l ~ a2, which eventually cross each other for some/3c. In case of SU(2) a l = a2 seems to be the case within our computing accuracy, while for SU(3) we observe a clear evidence for eel ~ O~2,

4. The SU(2)IZ2 X SU(2)Z2 model This model is the same as the R P 3 system studied numerically by D u a n e and G r e e n [9]. The relationship between their definition of the relevant quantities and ours is the following: / 3 D o = 8/3,

u p o.(8/3) = 21- ~ u ( / 3 ) ,

where U(/3)--= (X3(/3))+ 1. Our fundamental conclusion, as well as that of ref. [9], is the absence of the first-order transition suggested by ref. [6]. The representations of SU(2) will be labeled by their respective dimensions (d). The relevant coefficients in the character expansion (2.8) and (2.11) are found by explicit integration:

Bd

=

1 | .2,, ~ 30 d~

E

d"=l d " = d ( m o d 2)

sin 2

et~/zad"(IId-d"l/2(fl)--I(d+d")/2(fl)),

I bl = ~ d,d'

sin ~0 ad" sm ~a1 ~ et3/z+t3cos ½(p sm ~o d"= 1 sin ~

dU

d+d'--I Y. xf(U)xl(U)adad '= f =ld-d']+ l

(4.1)

a 2. d=l

(4.2)

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The integral written out explicitly on the left-hand side of (2.13) is calculated in the same way as (4.1) and we arrive at the following form for the n u m e r a t o r ' s eigenvalue problem: BdBa'

a'=l

dd'

e° [Ila-a'l/2(2~)-I~a+a')/2(2B)]ca, = eoca.

(4.3)

d'=d(mod 2)

Both bl and e0 depend quadratically on at's; therefore, one fixes the value of one p a r a m e t e r to an arbitrary value (al -~ 1 was chosen). It is clear that eq. (4.3) contains two decoupled channels: d = odd or even. It has been checked that 80 comes from the d = o d d channel, as is expected because the character expansion of the Boltzmann factor with adjoint action gets contribution only from integer spin representations. In this case c~ = Ca=l, c2 = C d = 3 , • • • , C j = C d = 2 ] + 1 , . . . is the ordering dictated by the strong coupling series. The numerical convergence of the estimate for the free energy was reached in the interval/~ = (0.1-1.4) with a truncation dmax = 9 (4 free parameters) and with dmax--- 15 for /~ = (1.4-2.3). All parameters ad(~) increase smoothly in the range

.l: IO

-

- - R : O

---

R=4

-'/// ,

0

1

I

2

3

Fig. l. The estimate of the energy densities of the variational ground state (2.5) (ri = 0) and of two excited states (~ = 2, 4) (4.9) for the group SU(2)/Z2 × SU(2)/Z2 ( f = - F + 2/9 - I n 2).

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considered; the maximal value as a function of d at given/3 shifts gradually f r o m d = 1 (/3 = 0) to d = 5 (/3 = 2.3). T h e resulting u p p e r b o u n d for the free e n e r g y is plotted in fig. 1. Numerical differentiation yields the internal e n e r g y and the specific heat according to the f o r m u l a e 1 dF U = - ~ d-~+ 1,

= _/32 d U

C

dfl

(4.4)

(fig. 2). N o evidence can be f o u n d in their b e h a v i o r for a transition of first order. T h e calculation was next r e p e a t e d with the ansatz (3.1). T h e f o r m u l a e c o r r e s p o n d ing to (4.1) and (4.2) are bl

= e2~ (I0(4a) --I1(4a)) ,

Bd = e~+O/2(I(d-1)/2(/3 + 2c~) --I(d+1)/2(/3 + 2 a ) ) ,

d = odd.

(4.5)

T h e n u m e r a t o r ' s eigenvalue p r o b l e m was solved with dmax = 15 in the full /3-(0.1-2.3) range (and also for s o m e values close to/3 = 5). T h e am,x = ct(/3) function resulting f r o m the maximization is displayed in fig. 3. It increases m o n o t o n i c a l l y f r o m the a = ½/3 (/3 ~ 0 ) regime to the a =/3 + c o n s t (/3 >> 1) behavior, with welllocalized t u r n o v e r in the/3 = (0.5-1.0) interval. T h e crossover region is correlated with the position of the p e a k in the specific heat (/3peak-- 0.85). T h e estimate of the free e n e r g y is c o m p a r e d to that of the ansatz (2.4) in table 1. A very g o o d a g r e e m e n t is seen suggesting that (3.1) might be the best trial function in the class considered, but we did not elaborate on this conjecture further. T h e truncation of the eigenvalue p r o b l e m (2.13) at some dmax means the omission of matrix elements of O(/3dma'). H e n c e the truncation works best in the strong 4

I

U'CI3

i

/ F

iI I iI

/ /I i/I

II

I

o

i

2

Fig. 2. The internal energy density and the specific heat of the SU(2)/Z2 x SU(2)/Z2 obtained by numerical derivation of the free energy.

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425

/

2

/ 1

2

6 Fig. 3. The variational parameter a as a function of/3 (G = SU(2)/Z2 × SU(2)/Z2). The straight lines represent the strong and weak coupling asymptotia.

c o u p l i n g r e g i m e . T h e i m p o r t a n t q u e s t i o n is w h e t h e r this a p p r o x i m a t i o n grows into the t r u e w e a k c o u p l i n g a s y m p t o t i c s c o n t i n u o u s l y o r not. O u r best a r g u m e n t for an affirmative a n s w e r is to s h o w that t h e a(/3) f u n c t i o n f o u n d n u m e r i c a l l y a b o v e r e p r o d u c e s t h e t r u e a s y m p t o t i c b e h a v i o r of the p r e v i o u s section. W i t h i n o u r n u m e r i cal accuracy, anom =/3 + k ,

k =-0.04+0.01,

if/3 > 1.

(4.6)

U s i n g (4.6) in (3.18) o n e easily gets, with h e l p of t h e l a r g e - x a s y m p t o t i c s of t h e m o d i f i e d Bessel f u n c t i o n s I ( x ) , (X3) =

3 3 In {e2'~lm[I0(4c~ ( / 3 ) - I1(4ce))]} = 3 - ~ - ~ + ~ B ~ (2k - 2 ) + ' " .

(4.7)

TABLE 1

Comparison of the estimates for the free energy density of the SU(2)/Z2 × SU(2)/Z2 model coming from eqs. (2.4) and (3.1), respectively /3 0.1 0.4 0.7 1.0 1.3 1.6 1.9 2.2

- F (multiparameter) + 2/3 0.2103 0.9864 2.0867 3.7262 5.6706 7.7295 9.8506 12.0179

- F (1 parameter)+2/3 0.2102 0.9866 2.0868 3.7246 5.6670 7.7288 9.8517 12.0182

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The integration of (4.7) yields the following continuation of our strong coupling calculation towards weak coupling -Fweak = const + 6/3 - 3 In/3 -~ 0.154 + 0.015 + . . . ,

(4.8)

/3 while the true asymptotics (3.16)+(3.17) yields for the coefficient of/3-1, 0.156. Therefore the discontinuity in a and consequently in U is ruled out up to about 1~(o accuracy. A n o t h e r illustration of the smoothness of the transition is provided by fig. 4, where (Fweak-Fstrong)/Fstrong is displayed. One can ask whether the crossover reflects a higher order transition. This eventuality was tested by computing the energy density of some excited states of the form

~exc=I~i(+gifl(gigi+l)),

(4.9)

where ~)~ U; --- U~'1°1 . . . U7 ~°" and fl is some invariant function of its argument.

F~- Fst Fst

J

I

G = SU 2 / Z 2 x S U 2 / Z 2

0.1

0.01

0.001

J

I

,

I

2

13 >

Fig. 4. The difference of the 0(/3 -1) accurate weak coupling and the strong coupling estimates of the free energy normalized by -Fstro.g (G = SU(2)/Z2 × SU(2)/Z2).

N.D. Hari Dass et at / S U ( n ) / Z ,

x

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427

Using the scalar product f l-I d U i ~ ~ ' ~ ' ~ # ~ ~ " ' # ' " " ~ i

,

one gets the following estimate for the corresponding eigenvalue A 1: ~ l q ~ [ d U ~ d V ~ ( T r U ~ V ~ + )~f l ( V~V,+, + )fl (U~U~++I)]T({U}I{ V}) A1 =

2aN° ~ l-Ii [dUifl(UiUi++l )2]

,

(4.10)

where T({U}, {V}) denotes the kernel of the transfer matrix. We have considered the ti = 2, 4 cases because these are the first states contributing to the chiral vector-vector (vector under both the left and right transformations) correlation function. The parameters to be varied were the coefficients of the character expansion of fl. The numerator and the denominator of (4.10) was treated in full analogy with the t~ = 0 (ground state) case described above. The resulting estimates for In Aa are displayed also in fig. 1. No signal for any degeneracy, correlated with the specific heat peak is observed. However, the correlation length (if we assume that geese(ri = 2) dominates the above correlation function) increases rapidly (at/3 = 5.2, st = 19.6); therefore, a transition for/3 > 5 can not be excluded. These conclusions are very similar to those drawn by Sinclair in the adjoint 0(3) model [8]. 5. The SU(3)/Z3 x S U ( 3 ) / Z 3 model

The study of this system was accomplished with the help of the one-parameter trial function (3.1) in view of its attractive features illustrated in the previous sections. However, even its use does not allow the character expansions (2.11) and (2.8) to take a convenient closed form for the coefficients br and Br. Therefore, we used in the strong coupling regime the power series of these quantities in/3. For a semiquantitative comparison with the numerical simulation [10] we contented ourselves with an accuracy 0(/36). The first method for calculating (2.6) was to evaluate the respective hightemperature series both of the numerator and the denominator. Using the irreducible decomposition of the product of two characters in SU(3) one easily gets bl = f d U e x p ( x x s ( u ) ) = l . ~ x

, 1 2 ,t1~ X 3 ,t1g X 4 ,t ~ X4 5 ,t l ~29 X 6-tO(x7),

(5.1)

where 2(8 is the character of the adjoint representation. The expression of the numerator to 0(/3 6) is given exclusively in terms of B1 and Bs: Z

tor= BI(/3)N¢BI(/3,)2No[

+ 1 N { B S ( / 3 ) " ~2 { B 8(/3, )'~2 [~ +(B8(/3')'~2'1]

(5.2)

428

N.D. Hari Dass et al. / SU(n)/Zn x SU(n)/Zn spin systems

(/3' = 1/3 + a). H e r e BI(X)

x2(1 + 2 x '[- 8x2)-~- O ( x 6 ) .

\BI(X)]

= bx(x) ,

(5.3)

Substituting (5.1) into (2.9) and using it together with the exponentiated form of (5.2) in eq. (2.6) one arrives after subsequent power expansions at a polynomial of sixth degree for In h0, which was maximized with respect to a. The solution of the e x t r e m u m condition for a was found as a power series in/3: amax(/3)=l/3-[-64-/3

1 3

3 4 355

"[-64/3

"[- 6--Z~').- /3

5

-~t-O(/36)

•

(5.4)

With the help of (5.4) the following series is obtained for the variational estimate of the free energy: -F=/3

2 , 2~3--

83 ~ 4

t~p .~p

, 21~5-

t~p

1155

6 . ~/~7\

~-6~/3 +utlJ ).

(5.5)

In fig. 5 we display the 0(/34) and 0(/36) estimates of the free energy (F(s4) and F(s6), respectively) compared to the O(/3 °) and 0(/3 -1 ) weak couphng • approximations • • ( ~ , ( 0 ) r , ( - 1) • w, 1 w ). A very clear crossing is observed. The positions of the crossing points are scattered around the Monte Carlo transition point. The best agreement appears by "pairing" the 0(/3 o) weak with the 0(/34 ) strong and the 0(/3 -1 ) weak with the 0(/36 ) strong coupling curves. This suggests that the simultaneous i m p r o v e m e n t of both kinds of asymptotic expansions is necessary for a quantitative treatment. Therefore, a fair error estimate of our determination of the transition point is the distance of the crossing of the "worst" pairs (see the 3/3 axis in fig. 5). The crossing points A and B yield the following quantitative characteristics of the transition

/35

2.9 3 '

/3~

2.92 3


(Axs) B = 3.23,

A a A ---- 0.35,

Aa a = 0.47.

' (5.6)

The discontinuity of (Xs) found in ref. [1] equals 1.96. One has to emphasize that the internal energies calculated from the asymptotic solutions are about 25% higher than the Monte Carlo estimates, therefore one has to consider the above data only as qualitative indicators. A n o t h e r way to evaluate (2.6) is to calculate the n u m e r a t o r ' s partition function by solving the corresponding transfer matrix eigenvalue problem in the character basis as we did in sect. 4 for S U ( 2 ) / Z 2 × S U ( 2 ) / Z > In the analogue of (2.13) we kept those matrix elements which are different from zero t o /36 order. The matrix elements connecting the representations !, 8_, 10, 27, 35, 64 should be retained and the expressions necessary for the matrix of eq. (2.13) are given in

N.D. Hari Dass et al. / S U ( n ) / Z . × S U ( n ) / Z . spin systems

429

-F

G = S U 3 / Z3×SU3/Z/

/ //

(0/

-~

C~MC 3

Z

313

Fig. 5. The free energy of the SU(3)/Z3 × SU(3)/Z3 system. The way the different curves are obtained is explained in the text.

tables 2 and 3 with the appropriate accuracy. The denominator is given with b l ( 2 a ) taken from (5.1). The resulting free-energy estimate crosses the weak coupling solution in a very similar m a n n e r as earlier (the dashed line in fig. 5). It is worth noticing that the internal energy on the strong coupling side is now much closer to the values found numerically. The striking difference between the SU(2) and SU(3) cases (the order of magnitude difference in Ac~ in particular) allows us to state unconditionally: the variational ansatz (3.1) signals the first-order transition of the S W ( 3 ) / Z 3 × S U ( 3 ) / Z 3 system in the correct place and provides a satisfactory description of the internal energy on both sides of it.

TABLE

2

Series of the coefficients in the character expansion of exp [xxs(U)]

bl b8 blo b27 b35 b64

1

x

x2/2!

1 0 0 0 0 0

0 1 0 0 0 0

1 2 x~1 0 0

x3/3! 2 8 4x/T 6 2"/21

x4/4!

x5/5!

x6/6!

8 32 20,f233 15",~12

32 145 lOOx/~ 180 100~,/~ 94

145 702 52Sx/2999 630~660

430

N.D. Hari Dass et al. / S U ( n ) / Z n × S U ( n ) / Z , spin systems TABLE 3 Series for Trs = I d U exp (xxs(U))xr(U)xs(U) X2

X3

~.

Tl1=1+--+2--+8 •

2

3!

~

.

2

+32

3

5.

X6

+145

4

6.1'

Xt

T t 's = x + 2 X +2!8' X + 33! 2 x + 1 4 54!~ 2

3

4

322

T1 lo = 42(x-- + 4 x-- + 20 x - - ~ • \2 3! 4U' 3 T1'35 = 2x/2 ~.I'.

X4

~ .T

X3 T1"64 = ~ . '

2

3

T88= l + 2 x + 8 X + 3 2 x + 1 4 5 '

X3

T127=--+6 ~+33 ' 2

2!

2

3!

X3

T8,27 = x + 6 x--+ 33 2! 3! ' Tlo,to=l+x+67.~,

4

-(

x 4!

x

2

x

3

T8 lo = ",/2 x + 4 ~7, + 2 0 ~.~ '

'

z!

--X 2

T8.35 = 2x/2 ~ . , Tlo.27=x/2 x+5~.~

J!;

'

X2 T8'64 = 2.1 '

,

Tlo,35 = x ,

T10,64 = 0 ,

X2

T27,27 = 1 + 2 x + 10 ~ . ,

T27,35= "/~x,

T27.64 = x ,

T35,35 = 1 ,

T35,64 = 0 ,

T64,64 = 1 .

6.

Summary

We have proposed a variational description of the thermodynamics of twodimensional spin models with S U ( n ) / Z ~ × SU(n )/Z~ symmetry (n = 2, 3). Its appealing feature is that the ground state is characterized by a single parameter, o~. The crossover between the weak and strong coupling regimes can be summarized through the change of the effective inverse t e m p e r a t u r e of the one dimensional system with the same symmetry, which reproduces the thermodynamics of the original model [/3e~=2a(/3)]. For fl<-flc, Be~/3, while for fl>/3c the effective inverse t e m p e r a t u r e is doubled. The crossover means a radical deviation from the one-dimensional behavior and the turnover is well-localized in /3. In this crossover region we have to see the intrinsic two-dimensional properties of the internal energy, if there is any, before the essentially one-dimensional large-/3 behavior sets in again. If the increase of /3en is not fast enough, the transition between the two regimes becomes discontinuous and the above possibility is probably lost. However, one has to keep in mind that in our approach the difference between the n = 2 and n = 3 cases is just quantitative (the bound for A U~ U). From our investigation we conclude that the ground state at low t e m p e r a t u r e (down to the crossover region) is given in terms of spin wave type excitations (the

N.D. Hari Dass et al. / S U ( n ) / Z . x SU(n)/Zn spin systems

431

gaussian approximation of the ansatz). For low /3 the d o m i n a n t configurations become sensitive to the global group structure (the characters "feel" at which particular group element we are). However, our successful description shows no need for selecting any particular set of these topologically sensitive objects (vortices, etc.). We believe that any type of these semilocal fields would suffer the same sudden change in its expectation value as the Z3 vortices, observed in ref. [10]. An independent combination for S U ( 2 ) / Z 2 x S U ( 2 ) / Z 2 can be sign Tr . sign Tr U / + I U i4+ 2 • sign Tr U/+2 U + , where all 3 U's are taken along the same axis. Finally we comment shortly about the existence of first-order transitions for n > 3. The generalization of (5.4) and (5.5) starts as

UiUi+14-

1 3 O~max(/3) = 1 / 3 + ~ A ~ +" ' "' -Fstrong(~)=~2"f-2~3jr-" . . . This should be combined with the weak coupling expansion. For this we have to know the expression of C* (whose appearance is also a novel feature of the present paper). At large n its leading term (O(n2)) can be extracted from the U(n) value /~adjoint*

'--u(.)

-

1

1

l(-fi,

(4~') "2/2 (2¢r) "/2 n / l l0] ~

i!

X "

Cu(n) was evaluated by using the method for matrix integration discussed by Mehta, and Zuber and Itzykson [19]. Using the asymptotic expression of the last factor [20] we have up to 0(/3 -2) corrections 4 4 5 2 --Fw~ak=n2(2/3+lln27/3 3 ~9-g-~+ o(/3- )) +O(n) It is well known that the kth order weak coupling corrections behave as /3-kn2k (planar contributions). However, if their sum makes any thermodynamic sense, it cannot increase faster than n 2, because the free energy density should be proportional to the number of degrees of freedom per site. Therefore, tentatively we just represent this sum by the O(/3 ~) term, which does not violate this behavior. Then the strong weak crossover occurs at the root of the equation 2/3~+~In

4 27/3c

4 + 5 + (1~ 3 96/3c O \ n ] = O '

giving /3c= 1.1583. It is remarkable that it is very close to the place, where the crossover happens in the n = 2 , 3 cases. For finite n we notice that 2 c ~ ( / 3 ) - / 3 - O ( 1 / d 2 ) ; that is, the take-off from the one-dimensional character becomes more difficult as n increases. Therefore, we expect the existence of first-order transitions in all adjoint models with n > 3.

432

N.D. Hari Dass et al. / SU(n)/Zn x S U ( n ) / Z , spin systems

T h e a u t h o r s b e n e f i t e d from discussions with P. Cvitanovi6, B, L a u t r u p a n d J.B. Z u b e r . T h e idea of this i n v e s t i g a t i o n was greatly s t i m u l a t e d by a lecture of J.B. Kogut. I n the c o m p u t e r work the help of B. L a u t r u p a n d B. Nilsson a n d the financial s u p p o r t of the D a n i s h R e s e a r c h C o u n c i l is a c k n o w l e d g e d . N o t e a d d e d in proof. Caselle, Gliozzi a n d M e g n a in a r e c e n t analysis of the high

t e m p e r a t u r e series for the mass gap of S U ( 2 ) / Z 2 x SU(2)Z2 spin m o d e l s in two d i m e n s i o n s [21] have persistently f o u n d a zero a t / 3 = 0.85 at the p o i n t w h e r e the sharp p e a k in the specific heat occurs according to the p r e s e n t paper. F u r t h e r study is necessary to clarify the p r o b l e m of higher o r d e r t r a n s i t i o n s at finite/3.

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