correlations in inhomogeneous ising models

Z. Phys. B - Condensed Matter 47, 341-352 (1982) Condensed Zeitschrift Matter far Physik B

�9 Springer-Verlag 1982

Correlations in Inhomogeneous Ising Models I. General Methods, the "Fully-Frustrated Square Lattice" and the "Chessboard" Model*

W.F. Wolff and J. Zittartz

Institut Rir Theoretische Physik, Universit~it zu K61n, Federal Republic of Germany

Received April 28, 1982

We study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant either in horizontal or in diagonal direction, i.e. the models have a layered structure. By using transfer matrix techniques the spin-spin correlations are calculated parallel to the layering and are expressed as Toeplitz determinants. After working out the general methods we discuss two special examples in detail: the "fully frustrated square lattice" (FFS) and the "chessboard" model, both having no phase transition. At zero temperature correlations in the chessboard model decay exponentially, while in the FFS model one has algebraic decay with a critical index t/=1, i.e. T=0 is a critical point. At finite temperature we find exponential decay in both models with a correlation length determined by the excitation gap in the fermion spectrum. Due to frustration correlations may develop on oscillatory structure and spins separated by an odd diagonal distance are totally uncorrelated at all temperatures.

I. Introduction

Recently we have studied inhomogeneous Ising models on a square lattice [1, 2]. These models describe Ising spins interacting via nearest-neighbour couplings varying both in strength and sign and are considered to be relevant for spin-glasses where one expects a random distribution of couplings. In order to apply the transfer-matrix-method for an exact calculation of the partition function and thus the thermodynamics, the models had to be restricted to a layered structure. In [1] we have considered horizontally layered models, while the diagonally layered models in [2] are even more general, as they allow an easy extension also to other planar lattices, for instance the triangular and honey-comb lattice. Horizontally layered Ising models have been considered in many earlier papers [3-8], usually with the restriction on only ferromagnetic couplings, i.e. couplings of the same sign. However, it has become

* Work performed within the research program of the Sonder- forschungsbereich 125 Aachen-Jtflich-K61n

clear from the work of Toulouse [9] that the con- cept of frustration is most relevant for spin glasses. This then leads quite naturally to study inhomogeneous Ising models in which both ferro- and antiferromagnetic couplings compete with one another. Some typical frustration models, which also appear as special cases of the models in [1, 2] have been considered in [10-14]. This paper is the first in a series in which we investigate spin-spin correlations in frustrated Ising models. The work on correlations in the homogeneous Ising model, which, of course, starts with Onsager's magnetization formula [15] and its calculation by Yang [16], is well described in the book of McCoy and Wu [17]. Correlations for layered models without frustration have been considered in [6], within the continuum approximation in [7], while correlations in frustration models have already been investigated in [11, 18-23]. In Sect. II we develop the general method to calculate the two-spin correlation function for layered models by using the transfer-matrix formalism [24]

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342 W.F. Wolff and J, Zittartz: Correlations in Inhomogeneous Ising Models. I.

and applying techniques developed in [1, 2]. As this is necessarily rather formal and technical, we first give a brief survey and point out basic formulas of Sect. II. These are needed later on in Sects. III and IV as well as in subsequent papers where we consider special frustration models. In Subsect. II.a the transfer matrix techniques of [1, 2] are applied to the evaluation of the spin correlation f(r) as a function of distance defined by (2.2). It is shown that f(r) can be written as a block Toeplitz determinant (2.13) which in all our applications can be further reduced to a usual Toeplitz determinant (2.23) (Subsect. II.b). Subsection II.c then summarizes the methods to calculate the Toep- litz determinants and thus the correlation f(r). As in the case of the homogeneous Ising model E17] we have to distinguish the three cases T < T~, T > T~, and T=T~. Below the transition, whenever we encounter a finite T~, we obtain from the asymptotic behaviour (2.25) the order parameter or spontaneous magnetization. Above the transition we expect exponential decay of correlations given by the general formula (2.31). At T~ we shall find that f(r) is always related to the Ising correlation (2.36). Then we turn to special cases. In Sect. III we consider the "fully frustrated square lattice model" (FFS) whose thermodynamics has been explored before [10, 1]. This model remains disordered down to zero temperature because of its large degree of frustration, but T = 0 is a critical point in the sense that the correlations show algebraic decay with critical exponent t/=�89 [19]. We shall calculate the correlation in the whole temperature range explicitly both in horizontal and in diagonal direction of the lattice. This is done in several Subsects. III.a-III.e. At T = 0 we recover the results of [19]. Most remarkable is the fact that for all temperatures the correlations differ quantitatively for odd and even distances due to frustration such that spins separated by an odd diagonal distance are even completely uncorrelated. In Sect. IV we calculate the correlations of the "chessboard model" whose thermodynamics has been obtained before in [12, 2]. This model again has no transition at finite temperatures, but in con- trast to the FFS-model it does not even become critical at T = 0 , as the correlations decay exponentially with finite correlation length ~. This behaviour must be attributed to the large groundstate degeneracy (the largest among comparable models El, 2]) and the effectiveness of frustration; according to E21~ the model is "superfrustrated". Section V contains a short summary and an outlook on further results which will be published in subsequent papers.

II. General Formulation

a) Transfer Matrix Method

Layered inhomogeneous Ising models are defined by the Hamiltonian

~n=-~EKj ~ ~#'+~j Y #~'] (2.1) j h .n .n , v.n.n.

where the sums inside the brackets denote horizontal and vertical nearest neighbour pairs of spins, # = + 1, with couplings J = fl- 1 K and J = fl- 1/~, respectively. The external j-sum extends over horizontal rows [1] or over diagonal rows of the square lattice [2] with the periodic condition Kj=Kj+v. This means that we are considering layers of finite width v which are then repeated periodically. It is convenient for these models to calculate spin- spin correlations in the direction of homogeneity, i.e. in horizontal rows for the horizontally layered models and in diagonal rows for the diagonally layered models. Using the technique of transfer matrices as in [ l, 2] one can write the correlation between two spins separated by r steps in both cases as

3 3z T Traceaoa r t 1... T~) M f ( r ) - (#o#r>- Trace (T 1 ... T~) M (2.2)

w h e r e 0 -3 is the third Pauli matrix with eigenvalues # = + 1. The transfer matrices T; ~ T(Kj, Kj) describe the transfer from row to row (or from diagonal to diagonal) starting with the row (or diagonal) in which we have placed the two spins at the sites denoted by 0 and r. As described in [1, 2] the

i quantities Tj are given in terms of Pauli matrices a n ( i= 1,2, 3) at sites n and are represented in terms of fermion operators cn, c 2 by the Jordan-Wigner representation. Fourier transforming to momentum operators cq, cq + (0< q < re) all relevant quantities can then be expressed in terms of

3 _ 1 _ i ( c ~ c+_q + c ~ c _ q ) ; " C q - - C ~ C q + C + q C q - - l ; _ _ "Cq--

2 _ - 1 3 (2.3) 27q - - ~17q ~ q .

In the thermodynamic limit the z ~ only act in the subspace of zero and double occupancy of the two fermion states q and - q and then they have the usual properties of Pauli spin matrices. To calculate the partition function which is just the denominator in (2.2) one only needs the largest eigen-

v

value of the layer transfer matrix Y = [ [ Tj; to j = l

calculate expressions like (2.2), however, more infor- mation is necessary. As discussed in [1, 2] it is important to go over to a symmetrized form:

3 - - ~ = IF[ T~(q). (2.4) q > O

W.F. Wolff and J. Zittartz: Correlations in Inhomogeneous Ising Models. I. 343

The 2 x 2 matrix T~ is given by

~ ( q ) = D . ~ (q) . e ~ ' "~. V (q) - ~ (2.5)

with the determinant

D = [Det T~(q)] ~, (2.6)

the unitary matrix

i z -- ~ a(q) ~q U(q)=e , (2.7)

and the fermion excitation energies E(q). In the thermodynamic limit ( M ~ oo) the trace in (2.2) reduces to the contribution from the state of largest eigenvalue such that the partition function is

Z = T r a c e J ~ ~t ~ 1-I (@olT~(q)MlTJo), (2.8) q>O

and [7~0) is given by

I '/~o ) = U I 0 ) , z,~ [0) = I0) (all q), (2.9)

with U - I - I u(q). q>0

To symmetrize also the numerator expression in (2.2) one may have to apply a similarity transformation to the operator o 03 G3 (this will be seen explicitly in the applications in Sects. III and IV). With some nonsingular matrix V, which is usually the product of some of the transfer matrices Tj in (2.2), one gets

VaoG V , (2.10)

and the correlation (2.2) reduces to

f (r)=(OlU-: (V%G3 3 V - 1) U [ 0 ) (2.11)

in the thermodynamic limit. This quantity is a free fermion average of a product of fermion operators [23]. In the Jordan-Wigner

3 is given by representation a~o-~

3 3 6oG =(co + -Co)(C~[ +q)(c] - c 0 ... (G + +G), (2.12)

involving all sites along the row (or diagonal) between the sites 0 and r. By using Wick's theorem the fermion average in (2.11) can be written as the Pfaf- fian Pf(~r of a 2 r x 2 r antisymmetric matrix sr [23]. Furthermore one has Det(sC)=[Pf(~r z [17], and thus we finally obtain the result that

f(r) 2 = [Pf(sd)] 2 = Det (sr ~. (2.13)

The matrix d

(i ~ :ior l) d = A: A 2 - (2.14)

r - - 1

is a block Toeplitz matrix (i.e. Am,,-Am_,) where each block A, is a 2 x 2 matrix

( v. u,) (2.15) A n = __u_ n V n

with the two characteristic fermion averages

u . = ( ( c ; ~ - + ck)(ck + .+ 1 + q + . + 1))

v . = ( ( c ~ - c ~ ) ( c L . - c~ + . ) )

= ((c~- +ck)(c~+,+G+,,)), n.i=O. (2.16)

' (2.3), Fourier transforming, using the quantities rq and observing that all averages have q ~ - q symmetry one shows that

i dq -i , 2 - _ ~ G e q (zq), Un=

u,= i dq e_iq(,+l)((z3 +iz~)), (2.17) _ ~

and explicitly

(z~q)=folg(q) -1 V(q)z~oV(q) -1 U(q)lO). (2.18)

These formulas are the starting point for the calculation of correlation functions in each particular case; (2.13) expresses the square of fir) as a block Toeplitz determinant whose elements are given by (2.14)-(2.18).

b) Reduction to a Toeplitz Determinant

In all applications in this and subsequent papers the 2 x 2 matrices A,, (2.15) can be diagonalized simul- taneously, i.e. independently of n and q. With a suitable quantity F one has

A n = i o "2 . e ~r'l. 4 . - e- Ir , l , (2.19)

a i are usual Pauli matrices, and A, is diagonal:

2) It is then easy to see that the full matrix sd (2.14) can be written as the same product (2.19) of matrices

+ 2 1 in which the blocks ia 2 and e-2r~ appear along the matrix diagonal with zeros otherwise and where the matrix ~7 corresponds to ~4 (A,---,,4,). The determinant of this matrix product then obviously reduces to

Det (d ) = Det (~) = Det (a)~. Det (a~r')r, (2.21)

as the determinants of the matrices with _+ F cancel and Det( ia2)=l . Furthermore one can rearrange

344 W.F. Wolff and J. Zittartz: Correlations in Inhomogeneous Ising Models. I.

rows and columns in Det(s~) to perform the last step in (2.21) where z~ is now the r x r matrix

li0 am a 1)�9 al a~ (2.22) Z 2 5 ~ ".

r - 1 a o

and t r . denotes the transposed. As both have the same determinant, we finally obtain the correlation

f(r) = Det (c@ (2.23)

as a usual Toeplitz determinant of order r. It should be remarked that the sign in (2.23) must be checked in each individual case.

c) Evaluation of Toeplitz Determinants

The calculation of the correlation f(r) is by (2.23) reduced to the evaluation of a Toeplitz determinant with entries a, given by

an= i dq _~q,-, , _ ~ 2~ e atq). (2.24)

As in the case of the homogeneous Ising model we then distinguish three different cases and we proceed as in 1,17].

(i) T < T~: If the particular model has a phase transition at some finite temperature Tc, then below T~ the quantity ln(5(q)) is continuously differentiable and periodic in q with period 2re. Under these con- ditions Szeg6's theorem 1-25] (Chap. X of 1-17]) can be used for the asymptotic evaluation of the correlation. For r-~ 0o one obtains

f(r)~-exp (rgo + ~ang, g- , )

where

(2.25)

dq g. = j -~ ~ - e -~q" In a(q). (2.26)

Usually we have go=0 and f(r) approaches a con- stant (for r~oo) , which is just the square of the spontaneous magnetization re(T) or the local order parameter. It will then be interesting to study the behaviour of re(T) in both limits T ~ 0 as also T~T~.

ii) T > T~: In the disordered high-temperature phase above T~ we expect that the correlation shows an exponential decay. For the explicit calculation we follow the treatment in Chap. XI of [17]. The quan-

tity 8(q) in (2.24) can now be written as

fi(q) = e- ~q 1'P(r �9 Q(( - 1 ) ] - 1, ~ = e iq, (2.27)

such that P(() and Q(() are both analytic for 141< 1, and continuous and nonzero for ICf < 1. Q is norma- lized, Q(0) = 1. First one considers the determinant of the matrix (:), with elements

b.= :,~ ~ e-'~"[e '~ a(q)] =-a._ 1 . (2.28)

SzegS's theorem discussed before can then be applied to Det (:)~ with the result

D~ = lim Det (~)r = exp n g n g _ n (2.29) r ~ oo n 1

where now

g'.: i dq e-iq"ln[eiqa(q)] _~ 2~

(2.30)

(with g0 = 0 in all applications). The correlation f(r) (2.23) is then related to (2.29) via the solution h(r) of Wiener-Hopf sum equations [17] and is ultimately given to leading order by the expression:

f(r)~-(-1)r.D~.h(r), r ~ oo (2.31)

where

h(r)= ~ ff~.. f f -x . Q(~)-l .p(~- 1). (2.32) Ir f z m

This contour integral around the unit circle in the ~- plane is usually determined solely by the singularity z o which is closest to 141=1 which then leads to exponential decay with Jzol r. In this context we shall frequently encounter Euler's beta-function

1 . . . . . . . r(1/2)r(r+l) (~]~ !au.ut~-u) ~= r(r+~) ~ - U : '

r ~ oo (2.33)

with the indicated asymptotic behaviour. To summarize the case T>T~ we have the correlation f(r) given by (2.31) involving the quantities D~o (2.29), (2.30) and h(r) (2.32). We shall see in the particular models of Sects. III and IV and in the subsequent papers that besides the exponential decay we may also encounter quite interesting oscillatory behaviour for r ~ o o . The cases of even or odd distance r may differ quantitatively, and we may also observe that the periods of oscillations change from being commensurate with

W.F. Wolff and J, Zittartz: Correlations in [nhomogeneous Ising Models. I. 345

the lattice to being incommensurate and depending on temperature (paper III of this series).

iii) T = T j This case, which also includes T~=0, is characterized by the fact that the periodic function h(q) of (2.24) is discontinuous somewhere in the in- terval - rc < q_<_ re. However, all cases can be treated and the correlation f(r) can be expressed in terms of the particular Toeplitz determinant of the matrix s ~ which appears in the homogeneous Ising model at its T~ [17]. With its elements

1 o _ ( 2 . 3 4 )

a. ~(n+~)

one obtains explicitly

-- f l F(m)2 fls" (r)= Det (~~ 1 s189189 (2.35)

with the asymptotoc expansion (r ~ oo)

a

1 2 3 5 6 7 8 9

1 2

u~

% o%

#o ~

L

3 L 5 6 7 8 9

f~s. (r) ~- A/r +, A = 0.6450..., (2.36)

and the critical exponent t/=�88 This ends our general treatment of correlations in layered models. In the next two sections we shall treat special models by using the general results and formulas of this section.

III. The Fully-Frustrated Square Lattice Model (FFS)

a) Model and Symmetries

b Fig. la and b. FFS-model in version (a) and version (b). Thick bonds: antiferromagnetic couplings - K . Thin bonds: ferromagnetic coupiings +K. All squares are frustrated. Diagonal and horizontal correlations as indicated

The FFS-model, whose thermodynamics has been discussed in [10, 1] and whose correlations at T = 0 have been determined in [19] by a mapping to a special 8-vertex model, is a most typical frustration model in which all couplings K u between sites i,j have equal strength ([Kul=K), but may differ in sign. It is characterized by the fact that all plaquettes of the square lattice are frustrated [9]. Among the many coupling distributions which real- ize this particular distribution of frustrations we shall consider two particular (regular) distributions. As shown in Fig. 1 the FFS-model can both be realized as a diagonally layered model with period v = 2 (version (a)) and as a horizontally layered model with period v=2 (version (b)). Thick bonds denote negative (antiferromagnetic) couplings and thin bonds denote positive (ferromagnetic) couplings. For models where all couplings have equal strength the Hamiltonian (2.1) is invariant under the "local gauge transformations"

K u ~ siKus ~ (3.1 a)

#i -~ si#~ (3.1 b)

for any choice of si= _+1. As (3.1b) has no effect on the partition function, we see that the thermodynamics is the same for all those distributions of couplings which can be related to one another by (3.1a). As far as the correlation (2.2) is concerned, it is also obvious that the transformation (3.1) at most may change the sign, if only one of the two spins #0 and #,. changes sign under (3.1b). In our particular case it is easily seen that version (a) and version (b) in Fig. 1 are related to one another by turning over all spins (st= - 1 in (3.1)) in columns numbered 3, 4, 7, 8, 11, 12, .~. while leaving all spins unchanged (s~= 1) in columns numbered 1, 2 ,5 ,6 ,9 , 10, ... In the following we shall consider correlations f(r) (2.2) in horizontal, vertical, and diagonal direction, denoted by an upper index h, v, or d, both in versions


(a) and (b) (lower index a or b), furthermore in version (b) we distinguish correlations in horizontal rows with positive and with negative couplings (lower index + or - ) . Applying the above transformation to the diagonal correlations and also using simple lattice symmetry arguments, we then derive the following important symmetry relations:

f~(r)=fbd(r)----0, r odd (3.2a) r

fad(r) = ( - 1)?-f~(r), r even. (3.2b)

The result (3.2a) is particularly remarkable as it means that spins separated by an odd number of diagonal steps are completely uncorrelated in the FFS-model at any temperature (a diagonal step, of course, corresponds to a horizontal plus a vertical step along the lattice bonds). This is a striking example of the effect of frustration or the competition of interactions with opposite sign. For horizontal and vertical correlations we obtain from the same transformation the relations:

f~(r) = fb~ (r) (3.3a) P

fah(r) = ( - 1)Tfbh(r), r even, (3.3b)

and a similar relation for an odd distance. Finally one can demonstrate the equivalence of horizontal and vertical direction in version (b) of Fig. 1, corresponding to a 90 ~ rotation of the lattice. The appropriate transformation is the one where in odd numbered columns we turn over all the spins which are adjacent to the horizontal negative couplings while leaving all other spins unchanged. This implies the relations

f{(r) =f~ + (r)= ( - 1)'fb h, _ (r). , (3.4)

In the following we shall therefore calculate just two correlations, namely the diagonal correlation f~(r) in version (a) and the horizontal correlation f~h, + (r) in version (b) in rows with positive couplings. All other correlations of interest can then be obtained via the relations (3.2)-(3.4).

b) The Diagonal Case

The FFS-model as a diagonally layered model (version (a) in Fig. 1) with period v = 2 is characterized by the two transfer matrices

T~,2~ T+_-T(+K,K)= l~ T+(q) (3 .5) q > O

which have been determined in [2]. Explicitly we have

T_+ (q) =D_+. U+. e E~ ~ U ; 1 (3.6)

with

D_+ e E ~ =211 +S 2 +R+_ z3], (3.7a)

-~-"~ ~ (3.7b) g 4 _ ~ C 2

R§ iq, R+_>O (3.7c)

and we use the abbreviations

S=s inh2K, e = 1 +S-2=coth2(2K) . (3.8)

A suitable symmetrized T~(q) corresponding to the product T+. T appearing in (2.2) is then

-• +a)~ T = T~2. T+ . ~ = D . e 2 .eE*~

i

�9 e ?-(a- +~)*~ (3 .9)

E(q) is the fermion excitation energy for diagonal transfer across the v=2 layer. Using (3.6)-(3.8) simple, but tedious algebra leads explicitly to

D=8S2isinq[, coshE(q)=~]sinq] -1,

sin a. sinh E_ = sin q (e2 _ sin 2 q)- ~. (3.10)

From this we can obtain the partition function according to (2.8) as

Z = [ l-[ D'e~] ~ (3.11) q > 0

and then the free energy per site as

fl F = - ~f dq

-ln {8 cosh2 2K (1 + [1 - sin2q, tanh4 2K]~)}, (3.12)

which formula is equivalent to expressions derived in [2, 10]. F is analytic in temperature and thus there is no phase transition. From (3.10) we can also deduce the usual relation between correlation length and the gap A=minE(q) of the fermion excitation energies. The minimum is at q = re/2 and we obtain

~ - 1 =A --ln z o a (3.13)

where z o is defined as

z o =(c~ +(c~ 2 - 1)+) - 1 __< 1. (3.14)

Actually (3.13) is somewhat subtle. E(q) refers to fermion energies for layers of width v=2; thus the gap per diagonal should be �89 But excitations in the subspace of zero plus double occupancy require


two fermions to be excited and therefore we get 2.A/2=A as the relevant excitation gap to be iden- tified with the inverse correlation length. This will be explicitly confirmed later on. Equation (3.13) also indicates that Tc=O will be the critical point of the FFS-model, as e ( T = 0 ) = 1 and thus ~ ( T = 0 ) = oo. The diagonal correlation f~(r) is now obtained from (2.2) as

aoa ~ (T+. T_)M f f ( r ) _ T r a c e 3 3 Trace T y (3.15)

To symmetrize also the numerator and to write f(r) as in (2.11) we have to perform a similarity transformation V according to (2.10) which by (3.9) is obviously

V = T_ ~ = I ] T_ (q)~-. q>0 (3.16)

Then the analysis of Sect. II goes through. The quantities (2.18) can be calculated explicitly with the help of (3.6), (3.7), (3.10) and are given by

(~q2)_ - i s i n q (z~ + izq l )_ c~ .

sin 2 q ' l/c~ 2 - sin 2 q (3.17)

From (2.17) we then obtain the block matrix (2.15)

dq e ~q~ A ,=

_,~J 2re ]//a2_sin2 q

�9 {i sin q - i~ sin q. a 1 + ie cos q. a 2} (3.18)

with the Pauli matrices cr ~ in standard notation. Diagonalization according to (2.19) is achieved with

tanh F = ~- 1 (3.19)

leading to (2.20), the a, being defined as in (2.24) with

c ~ c o s q - i ~ l . s i n q a(q) (3.20)

1/~ 2 - sin 2 q

The correlation is then given as the Toeplitz determinant (2.23)�9 From (3.20) it is easily seen that

a ,=0 , n even, (3.21)

quite generally which is sufficient to show that

fad (r) = Det (~)r = 0, r odd, (3.22)

confirming the result (3.2a), now by explicit calculation.

c) Temperature Dependence of Diagonal Correlations

For all T > 0 , i.e. c~>l (3.8) or z o < l (3.14), the quantity ~ (3.20) can be written as

{ 1 2 a(q)=e -'q \ 1+z2~_2] , ~ = e ~*. (3.23)

Obviously a is thus of the form (2.27), expected for T > T~, with

Q (4) = [P({)] - 1 = (1 + z z ~2)�89 (3.24)

where here, and henceforth, the positive root is taken for positive radicand. The quantities g, (2.30) are easily calculated,

g = _ c o s ( ~ n t . _ _ = z ~ - g - , , go =0, (3.25) \z / n

such that (2.29) leads to

4 ~ D~ =(1 -Zo) ~. (3.26)

To determine f(r) asymptotically from (2.31) we still need the contour integral h(r) (2.32). Substituting (3.24) into (2.32), transforming ~=iu, and contracting the contour 1{1=1 to the cut extending from - z o to z0, we obtain the integral

h(r)=rc-le'g r duur(z2u2)--~(1 __ZoR2 2 , -~) . (3.27) - z o

This shows again that h(r), and thus f(r), is zero for odd r because the integrand is odd. For even r we substitute u--+ z o - u and we have

1 .Sduu~(l_u)-~(l+u)-~(1 4 2,-~ -ZoU ) �9 (3.28)

0

As r ~ o% the essential contribution to the integral comes from u = l and thus we can set u = l in the last two factors of the integrand to leading order. The remaining integral is (2.33). Using its asymptotic form and collecting all factors we then obtain the correlation from (2.31) for T > 0 as

f f (r) ~ cos r z; , r ~ oQ (3.29)

which has a surprisingly simple structure. As expected the exponential decay

z~ = e -~/r c-~ ~- 1 = in z o 1 (3.30)

leads to the predicted correlation length (3.13).

348 W.F . W o l f f a n d J. Z i t t a r t z : C o r r e l a t i o n s in I n h o m o g e n e o u s Is ing Models . I.

At T=0 , i.e. e = 1, gt(q) in (3.20) reduces to

a(q) = sign (cos q), (3.31)

from which we calculate the matrix elements (2.24):

2 [2([n[-1)] . (3.32) a n = ~ cos

Now we form the Toeplitz determinant of (2.22) with these a,. Observing that the a, vanish for even n, we can interchange rows and columns in the determinant in such a way that, apart from a factor

cos . r , the resulting determinant is just the

square of the Toeplitz determinant found in the homogeneous Ising model at its critical point, with the replacement r ~ r/2. As this is given by (2.35), we have the exact result

or in symmetrized form

L T;-). rl. v.(+)* 2 , (3.36)

where T2 (-+) describes the interaction in horizontal rows with couplings + K and T 1 transfers from row to row with vertical couplings +K. Explicitly we have [1]

i 2 i 2 --~-q~q +9g(~3 ~qVq Tt(q)=e2I~; T(z+-)(q)=e ~ e . . . . . ~e ~ (3.37)

with the dual coupling S- 1 =sinh 2L (3.8). The symmetrized transfer matrix ~ = 1-I T~(q) is then readily determined as ~ > o

i :z "

T~(q) = e -2(q+~)~. e ~ . e ~2 (q+")~ (3.38)

with

I = F(m- �89189 =-f~

( 3 . 3 3 )

which function will be denoted as fo(r) for future applications. Our result coincides with that of For- gacs [19] who obtained the T = 0 correlation for even distances r by mapping the FFS-model to a special 8-vertex model. The asymptotic expansion for fo(r) follows from (2.36)

fo(r)~-cos (2 r)2~ A2/r~, (3.34)

thus we have the critical exponent t/=�89 which in [18] has also been obtained for the triangular anti- ferromagnet at T = 0. The diagonal correlation f f(r) in version (a) is thus an oscillating function with period 4 (because of the

factor c o s ( 2 r ) both in (g.29) and (3.gg)), i.e. com-

mensurate with the lattice. We mention this because in a subsequent paper we shall also find oscillating correlations which are incommensurate with the lattice. Finally we see that in version (b) the correlation fbd(r) is always positive, as (3.2b) implies: f~(r) =lff(r)l.

d) The Horizontal Case

The FFS-model as a horizontally layered model (version (b) in Fig. 2) with period v=2 has been considered in the first paper of [1]. The total transfer matrix J - f o r the transfer across the v = 2-layer is n O W

~- = T2 (+). T 1 �9 Tz (-). Tz (3.35)

cosh E(q) = 1 + 2. (sin 2 q + S - 2),

sinh E. sin a = 2. sin q(S- 1 cosq - e). (3.39)

Again one could obtain the partition function (2.8) from (3.38), in a form somewhat different from, but equivalent to formula (3.12). The horizontal correlation f~ + (r) in a row of positive couplings K (version (b) in Fig. 1) is now obtained from (2.2) in the same way as (3.15) was obtained in Subsect, b). However, as T2 (+)

= H (+) T~ (q) commutes with 3 3 %o-r, no similarity

q > 0 transformation (2.10) is necessary, i.e. V--1 in the present case. Again the analysis of Sect. II goes through. We obtain

(z 2) = O, (r~ + iv 1 ) = e i (q+a) . (3.40)

Using this in (2.15), (2.17) we see that the block - i ~ 2 A , is already diagonal. The correlation f(r) is then the Toeplitz determinant (2.23) with elements a, being defined by (2.24) with

fi(q)- e i"(q) = e is

[ 1-Zo2@ 2 l + z l ~ ' 1 --Zl~-- 1 ] �89 (3.41)

l--Z1{

~ = e iq. The latter expression has been derived by using (3.39). There are now two characteristic singularities z o and Z t ,

zl = 1 / 1 + c~-lf~; Zo = ] / ~ - ~1/~- 1 - t a n h K (3.42)

which satisfy

z 1 <z o__< 1, (3.43)


and z o = l precisely at T - 0 , i.e. ~=1. Notice that the present z o should not be confused with the z o (3.14) of the diagonal case which has a different K- dependence. At T = 0 ~ is rather given by

f i (q)=- i (s ignq) [- l + z l ~ �9 L1 + z i ~ - i

where in fact z t = ] f 2 - 1 .

1-z~ -~] 1 - z l ~

T = 0 (3.44)

e) Temperature Dependence of Horizontal Correlations

For T>0 , both Zo,~<l, the quantity fi in (3.41) has the form (2.27), expected for T > T~, with

Q(r [P(~)]- ~ [(1 2~2' 1 + z ~ ] ~ = -Zo (3.45)

The quantities ~, (2.30) are easily calculated, go =0,

_ ( - z'Un, n even

g'" = - g - " = ~ z]/n, n odd

from which we obtain (2.29):

z4 ~ 1 - z~ ]~ D~ = [ (1 - o, l~-z~ ] .

(3.46)

(3.47)

We then need the contour integral h(r) (2,32) which in the present case is

d~ ~r-1 h(r)= ~; ~ [(1-z~)(1-?o~-~)1 -~ Ir 1

l[~z~ 1-zt~-l]~ - zl ~ l+Zl~2 i - j �9 (3.48)

The contour can be deformed to the two cuts from - z 0 to -z~ and from z t to z0, which are the four singularities on the real axis inside the unit circle. The integral can then be evaluated asymptotically, as the leading contribution comes solely from ~ _+z o. Omitting details we just state the final result for f(r) which follows from (2.31):

( 2 ) ~ ( ~ ]+~(l+c0~, reven

ki + a ] (~=, r odd. (3.49)

As in the diagonal case (3.29) we have the expected exponential decay, z~ = e -r/C, although with a quantitatively different correlation length. Both, however, go to infinity at T=0. Equation (3.49) again shows the remarkable fact that, due to frustration, spins which are separated by an odd distance are much less correlated than those

which are separated by an even distance. While in the diagonal case these spins were even completely uncorrelated, f i r )=0 , we now have a substantial reduction. For the ratio we obtain

l i m f ( r + l ) = ( c~ ~ 1 r-.~ f(r) . . . . . \ ] ~ 1 z~ ~ i f2 ' (3.50)

the latter value being assumed at T=0. It remains to calculate the horizontal correlation at T = 0 from (3.44). We could follow a similar treatment in [17] to obtain the horizontal correlation from the diagonal one. But rather than going through lengthy calculations, which we have done and which confirm the result (3.51) below, we shall derive the result by a simple scaling argument. At T =0 the correlation length ~ is infinite, the lattice structure will not be seen anymore as r--~ oe and the system becomes isotropic. This means that if r horizontal steps correspond to a distance r, then r diagonal steps correspond to a distance 2~r. As the diagonal correlation i f ( r ) falls off like r -~ for even r, the corresponding fh(r) should be larger by a factor 2 +. For an odd distance r, the preceding argument cannot be used, as f a ( r ) -O. However, now we apply the ratio result (3.50) at T=0. This finally leads to the asymptotic behaviour

2~A 2 (1, r even (3.51) fb~ + (r) ~ - ~ T - l l /2~ , r odd,

and we have taken into account that in version (b) the diagonal fb a is nonoscillating, as stated at the end of Subsect. c).

IV. The Chessboard Model

a) Model and Symmetries

The chessboard model whose thermodynamics has been considered in [12, 2] is characterized by having _+K couplings distributed on a square lattice in such a way that frustrated and nonfrustrated plaquettes form the pattern of a chessboard. Figure 2 shows a particular (regular) coupling distribution for this case. The model has the largest groundstate degeneracy, i.e. rest entropy, among comparable models and has been called "superfrustrated" [21]. In fact, we shall see in the following that frustation is so effective as to even prevent the system from be- coming critical at zero temprature as does the FFS- model of the last section. As is obvious from Fig. 2, the model is a diagonally layered model with period v=4 [2]. The total layer


1 2 3 /* 5 6 7 8 9

•

x %

x

x

x

x

x

"x %

x

% %

x "% x

X [" "... X "". X "..

X ":. X "" �9 %

- f3 x... x "... x

"i

"X. X "" �9 goQq �9 fl

• "X. X %

x N...j

fz Fig. 2. Chessboard model. Bond notation as in Fig. 1. Frustrated plaquettes are crossmarked, f~.2.3 denote the three possible diagonal correlations

4

transfer matrix I~ T~ can be chosen symmetrical 3 = t

~-~= T_. T+ a. T_ (4.1)

where the T+ are given by (3.5)-(3.8). We shall be interested in diagonal correlations only. Figure 2 shows that in principle we may consider three correlations fl,z,3(r). The two f l and fa go through nonfrustrated plaquettes, but have negative or positive horizontal couplings adjacent to the sites of the spins, respectively. The f2 correlation goes through frustated plaquettes. Again we have interesting symmetry relations. By turning over all spins in every second column of Fig. 2 one shifts all horizontal coulings by two horizontal steps according to (3.1). Using also simple lattice symmetry arguments this transformation im- mediately implies

f l (r) = ( - 1)"f3 (r), (4.2)

f2(r) = ( - 1) ~ .fz(r) ~ f2(r)=0, r odd. (4.3)

Again as in the FFS-model we therefore have the remarkable result that spins separated by an odd diagonal distance through frustrated plaquettes are completely uncorrelated at any temperature due to the effect of frustration. Because of (4.2) it is sufficient to calculate only f t and f2- According to (2.2) and with a view to Fig. 2 these quantities can be written as

Trace o'g a~ j - M fl(r) = ~ o - ~ , (4.4)

Trace (T_ a 03 at3 T - 1). ~-~M fz (r) = Trace 9-~ u (4.5)

Thus we have to subject f2 to the similarity transformation (2.10) with V= [ I T_(q), while f l requires none. 0 > o The symmetrical ~ = [-[ T~(q) is written as (2.5),

q > 0

i 2 i

T~(q) = D-e -g("- + a )*~. e E~- e ~-(~- +")~], (4.6)

and the quantities D, a, and E are obtained from (3.6)-(3.8) with some algebra:

D=4S'~ sin2 q, cosh E(q)= l + 2a2/sin2 q, sinh 2E �9 sin a = c~S 2 sin q. (c~ 2 + sin 2 q)- 3. (4.7)

According to (2.8) one could derive partition function and free energy from (4.7), as has been done in [2]. There is no phase transition because the excitation energy never vanishes. This even holds at T =0 as the gap, A =minE(q), is from (4.7)

A =21n Zo 1, Zo =((1 +~2)�89 1, (4.8)

which remains finite at T = 0 with A0=21n(1 +1/2). Again we can predict the correlation length from (4.8) as

- 1 = �89 = In z o 1. (4.9)

The argument is the same as given in Sect. III following formula (3.14). We merely have to take into account that in the present case we have v=4 layers. (4.9) will be confirmed below. As ~ remains finite at T=0 , the system will not become critical at all, i.e. even at T = 0 correlations will decay exponentially.

b) Diagonal Correlations

To evaluate the correlations (4.4), (4.5) the analysis of Sect. II goes through again. The quantities (2.18) are calculated by using (4.7):

( z3+ iz ; )2 / I~-iS2sinq "ei~ ('E3 + i'~1) 1 [ =(~2 a - i s i n q (4.10)

('c~)z | ' +sin2q) ~. _iS2~sinq

J 0

First we consider fl(r). With (zqz)l=0, it is obvious from (2.17), (2.15) that the block -iaZA, is already diagonal. The correlation fl(r) is then the Toeplitz determinant (2.23) where the elements a, are given


by (2.24) in terms of

[_ l-Zo a(q) = e - ~ q ' \ l _ z o ~ - I l + z o ~ ] '

(4.11)

As z o < l (4.8), we see that fi has the form (2.27), expected for T>T~, for all T > 0 where P and Q are given by

Q(~) = [ p ( ~ ) ] - i = ( I -Zo# ]~ (4.12) \1 +ZOO] "

The quantities g, (2.30) are easily calculated, go=0,

g,= _g_=~:Z"o/n, n odd (4.13) ~u, n even

from which we obtain

(remark the surprisingly similar structure with the FFS-model (3.20)). The correlation f2 is then given (2.23) as

f2 (r) = [Det (~)~[ (4.20)

and (~)~ is the matrix (2.22). Here the absolute value must be chosen because one can show that f2(r)>0. From (4.19) one proves again that (4.3) holds, namely that f2( r )=0 for odd r. This follows from the fact that (4.19) implies

a ,=0 , n even, (4.21)

similar to the FFS-case at the end of Subsect. (III.b). Now we can proceed as in the case of f~(r). Instead of (4.11) we write (4.19) as

= ( 1 - 4 V D ~ \t + z~ ] " (4.14)

We are left with the contour integral (2.32) around the unit circle in the ~-plane. Substituting (4.12) and contracting the contour to the cut extending from - z o to z o we get

z0 < ), h(r)=n -1 ~ dx'x ~-1" z~ l + z ~ (4.15)

-zo -x i -ZoX

This integral can be evaluated asymptotically in the same way as in Sect. III. Collecting all factors in the expression (2.31) we have the final result

fl(r)=(-1)rf3(r)~-(-1)~- @)~.z~o, r~oo. (4.16)

Postponing a discussion we now turn to the correlation f2(r). Using (4.10) in (2.17) we obtain the block (2.15)

'~ dq -iq,, 2 A , = j ~ e (c~ +sin2q) -~

{ic~S2sinq-i(~+ S2)sinq-crl +igcosq.a;}. (4.17)

Diagonalization according to (2.19) is now achieved with

tanh F = e S 2- [e + S 2] - ~, (4.18)

leading to (2.20), where the elements a, are defined (2.24) in terms of

fi(q)=(ecosq-il/~ +e2sinq).(e2 +sin2 q)-}, (4.19)

{ (q) = e-"- :-2 ] ' r = e'" (4.22)

which has the form (2.27) with

Q ( ~ ) = [ p ( ~ ) ] - l = ( l _ZO~ .2 2) .~ (4.23)

The quantities ~, (2.30) are easily calculated and D~ (2.29) now becomes

D~ =(1 -Z4o) ~. (4.24)

We are left with the contour integral (2.32) which, similar to (4.15), reduces to

Zo

h(r)=n -1 ~ dx.xr.[(zg-xZ)(1-zgx2)] -~. (4.25) - - z o

This vanishes for odd r, as expected. By asymptotic evaluation as before and collecting all factors in (2.31) we obtain finally the correlation

fz(r)~ cos ( 2 ) @ ) + P ' "Z~, r ---~ o0 . (4.26)

Upon comparison we see that the two correlations f l (4.16) and f2(r) (4.26) become identical asymptotically as r ~ o% if the distance r is even, while f2(r) vanishes for any odd distance due to frustration. Comparing with (3.29) we also see that apart from oscillation factors the diagonal correlations have the same form as in the case of the FFS-model. Equa- tions (4.16), (4.26) confirm the correlation length (4.9), in particular the fact that even at T = 0 we have exponential decay properties.


V. Summary and Outlook References

In the foregoing we have derived the general methods to calculate spin-spin correlat ions in inhomogeneous Ising models which have layered structure, either hor izontal ly or d iagonal ly on the square lattice. We then have used these methods to calculate correlat ions explicitly for the fully-frustrated square lattice model and the chess-board model. In bo th cases there is no phase transit ion at finite temperature, and thus we found exponential decay of the correlat ion for all T > 0 . It is, however, interesting that due to frustrat ion the correlat ions develop oscillating behaviour and that there are completely uncorrelated spins, in our part icular cases spins separated by an odd distance in d i a g o n a l disrection. For the FFS-mode l T = 0 is a critical point, i.e. correlations decay with an algebraic power r - " with t/ =�89 while the chessboard correlat ions decay also exponential ly at T - - 0. We shall cont inue to investigate further frustrat ion models in subsequent papers. It will be shown that even more interesting properties can be observed. For models wi thout phase transi t ion at finite temperatures the T = 0 behaviour distinguishes three cases: (i) even at T = 0 the system is highly disordered with exponential decay of correlat ions (the chess- boa rd model of this paper is the first example), (ii) the system becomes critical at T = 0 , like the FFS- model, (iii) the system is perfectly ordered at T = 0 . In the latter case there is then a first order t ransi t ion at T = 0 . In situations with a finite T~ we shall see that the order parameter may also show quite interesting behaviour due to frustrat ion effects, in part icular near zero temperature. The t ransi t ion at T~ is, however, always of Ising type. As far as critical behaviour at finite T~ is concerned, there is thus noth- ing new to be seen. Final ly we shall see in further applications that the oscillations in the correlat ion functions above T~ become even more interesting than in this paper, because we shall find periods of oscillations both com- mensura te and incommensura te with the lattice, as is already k n o w n for the t r iangular ant i ferromagnet [18] and other models.

We would like to thank P. Hoever for very stimulating dis- cussions.

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W.F. Wolff J. Zittartz Institut ftir Theoretische Physik Universitgt zu K/51n Ziilpicher Strasse 77 D-5000 K61n 41 Federal Republic of Germany

correlations in inhomogeneous ising models

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