Frustrated ising models in two dimensions

Journal of Magnetism and Magnetic North-Holland Publishing Company

INVITED

Materials

38 (1983) 217-224

217

PAPER

FRUSTRATED

ISING

MODELS

IN TWO

DIMENSIONS

*

W.F. WOLFF Institut fiir Theoretische Physik, Vniuersitiit zu Kiln, D - 5000 Cologne 41, Fed. Rep. Germany

A survey of recent theoretical work on frustrated Ising models in two dimensions, which are relevant for spin glasses, is given. We discuss exactly soluble inhomogeneous Ising models of layered structure with nearest-neighbour couplings varying in strength and sign. In particular we discuss the following properties: phase transitions, ground-state properties, degeneracies, and correlations. Finally, we consider a low-concentration expansion and discuss its implications for random bond systems.

1. Introduction Spin glasses [l] are magnetic systems where the interactions are “in competition” with each other due to some disorder in the system, and no conventional long-range ferromagnetic or antiferromagnetic order is established. Classical spin glasses consist of dilute magnetic ions in a nonmagnetic metallic matrix at low concentration, e.g. 1% Fe in Au [2]. The RKKY exchange among the spins strongly oscillates with distance and thus, due to the randomness of distances, some of the interactions with a given spin are ferromagnetic and some are antiferromagnetic. These systems exhibit a transition into a state where the spins are more or less frozen in, and the susceptibility shows a rather sharp cusp. While the nature of this state and of the freezing transition is still debated, some basic concepts have emerged which are generally accepted. Firstly, though the RKKY exchange interaction is fairly extended, models with shortranged interactions are considered to give a realistic description. Secondly, it has become clear from the work of Toulouse [3] that the concept of frustration is most relevant for spin glasses. This concept essentially describes and quantifies the effect of competition between ferromagnetic and antiferromagnetic couplings. Thus one is led quite naturally to study magnetic models with short* Work supported forschungsbereich

within the research program 125, Aachen-Jiilich-Kbln.

0304-8853/83/0000-0000/$03.00

of the Sonder-

0 1983 North-Holland

range interactions containing both disorder and frustration. In this survey we shall report on recent theoretical work on inhomogeneous Ising models on a two-dimensional square lattice, where Ising spins p = + 1 interact via nearest-neighbour couplings varying both in strength and in sign. We shall concentrate on exactly soluble models and analytical results (numerical work has been reviewed by Binder and Kinzel [4]). As the most general case is presumably not solvable we have to restrict ourselves to models with random couplings only in one direction of the lattice, namely to Ising models of layered structure. These models are defined in section 2 where we also briefly describe the method of solution and discuss the occurrence of phase transitions. Ground-state properties and correlations are considered in sections 3 and 4, respectively. In the last section 5, we shall go back to the more general case where all nearest-neighbour couplings are randomly distributed. be discuss the results of a small-concentration expansion recently performed by us and comment on the conclusion about the two-dimensional Ising spin glass.

2. Layered

Ising models

As already mentioned in the introduction we are interested in exactly solvable models containing both disorder and frustration. We then have to restrict ourselves first to two dimensions and sec-

for any choice of s, = + 1. One is therefore introduce the frustration index + = ll sign( K,,) = f 1.

Kv-i DC

:ell and layers I” the HL and I>L

ond to Ising models of layered structure. In these models the nearest-neighbour couplings are homogeneous, i.e. translationally invariant, in one direction of the square lattice but arbitrarily distributed in the orthogonal direction. Two general classes have been considered [5-91: models with horizontal layers (HL) and models with diagonal layers (DL). Each layer consists of v horizontal or diagonal rows with couplings K, = /3J, for the horizontal bonds and ir/ = p< for the vertical bonds with ,j = 1 ,. . .,v (fig. 1). These layers have finite but arbitrary width v and are repeated periodically to make up the whole lattice. Layered Ising models are described by Hamiltonians

.I

.P= PH = - c

c K,pp’ + c i+p’ WI” 1 hnn

1 (1) ,

where the sums inside the square brackets run over horizontal (hnn) and vertical (vnn) nearestneighbour pairs and the external j-sum extends over horizontal or diagonal rows with the periodicity condition K, = K,, “. Thermodynamics is derived from the partition function Z = Tr e-“1 As noticed by Toulouse [3] the Hamiltonian (eq. (1)) is invariant under the following local ‘gauge’ transformations (acting on both spin and interaction variables): I”, + S,ELI3 K,, -+ s,sjK,,

(j adjacent

to i),

(2)

led to

(3)

where the K,,‘s are the four couplings around a plaquette. The thermodynamic properties, i.e. partition function and free energy, of the models defined by eq. (1) are gauge invariant and do not depend on the particular coupling distribution, but rather on the distribution of coupling strengths and of frustration, i.e. many different coupling distributions have the same thermodynamics. For our layered Ising models it follows from the definition (eq. (1)) that the frustrated plaquettes. i.e. plaquettes with $ = - 1, are arranged in horizontal or diagonal rows. Horizontally layered Ising models have been studied in many earlier papers [lo-1.51, however with the restriction on couplings of the same sign. These models thus do not contain frustration. Only after Toulouse introduced the frustration concept some typical frustration models [16-201 have been studied which all appear as special cases of the HL and DL models. The layered structure of our models, i.e. the translational invariance either in horizontal or in diagonal direction of the square lattice, allows the application of the transfer matrix method for an exact calculation of the partition function and thus the thermodynamics. This means that the corresponding transfer operator is diagonalized by using a Jordan-Wigner transformation to fermions and Fourier transforming. The partition function Z is then solely determined by the largest eigenvalue of the transfer operator. Other eigenvalues and the eigenstates are only needed for the correlation functions. In the horizontal case the transfer operator relates the configurations { /_L}of spins in a horizontal row to the configuration {p’} of spins in the next row, while in the diagonal case one considers configurations of spins in diagonal rows. The diagonalization of this operator is well known in the horizontal case [21] and more easily achieved than in the diagonal case [22]. However, the results obtained in the latter case are more general and include all other models as special cases. By letting

W. F. Wolff / Frustrated Ising nlodels in two dimensiotu

couplings go to zero or to infinity in a suitable way one obtains from the DL models [7]: (1) the HL models [5], (2) layered models on the triangular (LT) and honeycomb lattice (LA), (3) layered models on mixed lattices, consisting of arbitrary combinations of rows of triangles, pentagons and hexagons. Layered Ising models are characterized by the global-p + -cc symmetry of the Hamiltonian (1). This symmetry can be spontaneously broken at low temperatures resulting in a phase transition at some finite temperature T,.At the transition the system goes over from the completely disordered, unique high temperature phase to a 2-fold degenerate low temperature phase characterized by an ordering of the spins in complicated domains of ferromagnetic and antiferromagnetic substructures, which depend on the specific distribution of couplings and of frustration. As expected by universality this transition is always of Ising type, i.e. the critical behaviour and critical exponents are the same as in the homogeneous Ising case. A phase transition occurs if there is a singularity in the free energy as a function of temperature. For the DL models the transition temperature is determined by the formula [6] sinh IK,-SK/I I?

j=l cosh(Ki+.@

T= T,

219

of where V* with v = v++ v_ are the numbers horizontal rows with positive and negative horizontal couplings within the period v. This also includes the (m, n) models, where m rows with ferromagnetic couplings are succeeded by n rows with antiferromagnetic couplings such that we have two layers of width m and n without frustration which are separated from each other by a row of frustrated plaquettes. A transition to an ordered state at some finite temperature T,> 0 occurs if one of the conditions (4)-(6) can be satisfied. For the (m, n) models this happens if the two layers have different widths, m # n; otherwise we have no transition, T,= 0, precisely for m = n (or v+= v_). With decreasing temperature first the layers of finite width order internally, i.e. local ordering takes place. As the sum of vertical couplings through the separating frustrated row between successive layers adds up to zero, there would be no interaction between layers, if these would be completely ordered. This means that with decreasing temperature also the effective interaction between layers diminishes at the same time. The global ordering of adjacent layers is therefore aggravated by the internal ordering process and the interaction is maintained only by the disorder of edge spins still present at finite T. This might be called “entropyor dis-

=”

which must be satisfied either for s = 1 or s = - 1 corresponding roughly to predominant antiferromagnetic or ferromagnetic ordering, respectively. For the HL models one obtains [5]

(5) where we could choose all vertical couplings positive, K/ > 0, without loss of generality and L denotes their dual coupling, tanh K, = exp( - 2 L,). Special cases are the pure frustration models with couplings of equal strength, 1K,I= FJI= K, such that only the signs c can vary from row to row. Then all DL models have T, = 0 as eq. (4) cannot be satisfied, except in the case Y = 1, while for HL models we get from eq. (5) the simple formula L=

Iy+--IK

v,+v_

’

0

1

2

3

LW

J Fig. 2. Specific heat of (m, m)-models for different m. m + m corresponds to the Ising model.

5 values of

Fig. 3. Specific heat of (nl. n)-models for the special cahe no =lO. )I = 11. The thin line at T, indicates the logarithmic singularity.

plings. In frustrated systems not all couplings can be satisfied at T = 0, which means that there are spins or clusters of spins with a vanishing net interaction with the rest of the system, i.e. they can flip without cost in energy. This then leads to groundstate degeneracies and possibly to finite rest entropies at T = 0. Pure frustration models with their high symmetry in couplings, namely couplings of equal strength, are the most liable candidates to show this behaviour. In ref. [5] we showed that the rest entropy vanishes if and only if each row of frustrated plaquettes is separated from the next one by at least one row of nonfrustrated plaquettes. Otherwise we have finite rest entropies. For the particular case of (m, n) models we found for the rest entropy per site [5]

sg =

G/T = 0.291,

m=n=l.

(m+n)-‘ln+(J5+1),

ma2,n=l,

I 0, order-coupling”. A global ordering, i.e. a transition, then only takes place if m f n while T, = 0 if m = n. This ordering behaviour is seen in the specific heat curves of (m, n) models shown in figs. 2 and 3 [8]. For models with m = n and no transition the curves (fig. 2) generally show a broad maximum attributed to the internal ordering within the layers. This maximum becomes more pronounced for increasing m and for m + cc approaches asymptotically the case of the homogeneous Ising model with a logarithmic divergence. For models with m # n (fig. 3) and a transition at some finite T, > 0 we again have a broad maximum indicating the internal ordering process. The global ordering, which occurs at much smaller T,, requires a logarithmic singularity indicated by the thin line. The amount of energy needed for this global ordering is, however, much smaller than for the internal layer ordering as the area under the logarithmic peak is hardly visible.

3. Ground-state properties Frustration describes and quantifies the competition between ferro- and antiferromagnetic cou-

m,n>2, (7)

where G is Catalan’s constant. Eq. (7) shows that the occurrence of a phase transition at some finite temperature T, > 0, which happens if m + n, is not simply related to the occurrence of rest entropy. All combinations are possible: transition and no transition on the one hand and presence and absence of rest entropy on the other. However, in ref. [5] we showed the following: if for any two groundstates one can be obtained from the other by a succession of purely local transformations, then the global symmetry p + -p cannot be broken, i.e. then T, = 0. If this is not the case, then there might be a transition or not.

4. Correlations The bulk properties of our layered models can be inferred from the knowledge of the exact free energy and have been described in the foregoing sections. Spin correlation functions are a suitable tool to gain some insight into the microscopic properties, in particular into the details of the ordering process. In refs. [17,23-271 it has already been shown that correlations in frustrated systems

W. F. Wolff /

Frustrured Ising models in two dimensions

show quite interesting behaviour. One usually studies two-spin correlations where for convenience the two spins are separated by r steps in the direction of homogeneity, i.e. parallel to the layering. Here we shall describe the main results obtained in ref. [9]. The transfer matrix technique turns out to be also well-suited for the calculation of correlations. Formally the correlation f(r) = (p0p7) can be written as a block Toeplitz determinant, which in all cases could be reduced to a usual Toeplitz determinant. We then could follow standard treatments described in [28]. For a general discussion it is convenient to distinguish the two cases where (a) the critical temperature is finite, T, > 0, and (b) q = 0, more precisely, no transition at finite temperature. (a) In models with T, > 0 correlations show the usual Ising type critical behaviour. In the disordered high temperature region above T, f(r) decays exponentially with a correlation length 6 diverging as IT- T,I-’ with the Ising exponent v = 1. At T, we have algebraic decay with rm8 and n = l/4. Below T, f(r) approaches m(T)2 asymptotically, where the spontaneous magnetization or local order parameter m (T ) vanishes at T, with exponent /3 = l/8. This general behaviour has to be expected from universality, while the influence of inhomogeneities and frustrations is seen in details. Below T, we may have different local order parameters along layers of spins with different couplings. The ordering is, however, always unique, i.e. ferromagnetic or antiferromagnetic and is determined at T > 0 by the globally dominating couplings. At zero temperature we may, however, encounter typical one-dimensional chain ordering different from the bulk ordering, e.g. an antiferromagnetic ordering in a chain with negative couplings while the rest orders ferromagnetically. Such behaviour causes the local order parameter in the chain to diminish again for T < T,, to go to values smaller than one or even to zero for T + 0, and to jump discontinuously at T = 0. A typical example for such properties is the HL model with period Y = 2 [9b]. One further effect of frustration is the fact that certain spins may be totally uncorrelated, i.e. f(r) = 0, for any temperature T > T,, i.e. in the disordered region. This occurs for instance in the fully-frustrated square

221

lattice model (FFS) and in the chessboard model, discussed in detail in ref. [9a], for spins separated by an odd distance r in diagonal direction. Depending on the details of the couplings the decay of correlations in the disordered region above T, develops in many cases quite interesting oscillatory behaviour superposed on the exponential decay (8) The oscillations may be commensurate with the lattice, in which case 8(T) in eq. (8) takes on values, which are rational fractions of 27~~ e.g. 0 for ferromagnetic alignment and 71 for usual antiferromagnetic alignment or period 2. Such behaviour is seen for instance in the HL model [9b], where in the whole temperature range above T, we either have ferromagnetic or antiferromagnetic correlations depending on the dominating couplings. However, for the “general square lattice” model (GS) [SC] we may have a change from commensurate periods, which we always have for temperatures just above T,, to incommensurate periods. This then occurs at some “disorder temperature” T,, (> T,), where correlations decay as a pure exponential and is just the same behaviour as seen in the triangular antiferromagnet [23]. Above TD we have periods with the wavevector d(T) depending on temperature. This change is seen only in the r + cc limit of the correlation f( r). As the thermodynamic quantities remain analytic, the smooth change at T,, is no phase transition. (b) In the case T, = 0 frustration is so effective that the system cannot order globally at any finite temperature. While for T > 0 we have exponentially decaying correlations as described before, we have to distinguish three cases at T = 0. (i) There is perfect ordering at T = 0, i.e. f(r) = + 1. The ordering is usually one-dimensional chain ordering. As this is expected to be typical for models with layered structure, it is not suprising that the majority of our models with T, = 0 shows this behaviour, i.e. a first order transition at zero temperature. (ii) T = 0 is a critical point, i.e. T, = 0 in the strict sense. In that case the system has a vast groundstate degeneracy and correlations decay with T-T (r + cc). The critical exponent 9, how-

ever, is different from the Ising value a and seems to assume the value TJ= + universally in all investigated cases. The first known case is the triangular antiferromagnet (FFT) [23], the second case if the FFS model [9a]. This also occurs in the general HL model with period v = 2 on a whole one-dimensional manifold in the space of the three couplings of the model [9b]. In the GS model we also find this behaviour, algebraic decay with n = l/2 accompanied by oscillations with both the commensurate periods 3 (as in the FFT) and 4 (as in the FFS) [SC]. (iii) Even at T = 0 the system is disordered. This means that also at T = 0 the correlations decay exponentially with a finite correlation length $( T = 0)c co.That frustration can be so effective and the ground state degeneracy so vast in certain models as to cause this fast exponential decay is perhaps the most surprising result. So far we have found two special models with this behaviour, namely the chessboard model [9a] which is also and this is somewhat surprising - that model with the largest rest entropy per site and the fully frustrated honeycomb model (FFH) solved in [7]. The T = 0 behaviour of the latter model is thus different from the other fully frustrated lattice models in two dimensions, namely the FFS and FFT models [33].

5. Low-concentration

expansion

In the foregoing sections we have considered Ising models with random couplings only in one direction of the lattice. This restriction was sufficient to allow an exact solution. The more general and more realistic case where the couplings also vary randomly in the second direction is presumably not solvable. Thus one is forced to use either numerical methods or approximate treatments. In this section we want to report the results of a low-concentration expansion recently performed by us. We consider an Ising model on a two-dimensional square lattice with nearest-neighbour couplings K,, = /3J,, randomly distributed according to ~W,,)

= (1 -PP@,,

- K) +f44,

- 4.(9)

We shall confine our discussion to the most interesting cases, namely to the diluted ferromagnet with u = 0 [4,29] and to the wellknown *J model with u = - 1 [4]. The diluted ferromagnet does not contain competing interactions and thus no frustration while in the iJ model all couplings have equal strength, i.e. it is purely frustrated. In other words, we have one model which describes only disorder and one model which describes only frustration. The case p = 0 reduces to the homogeneous Ising model with a phase transition at the critical coupling K, = $ ln(J2 + 1) and a logarithmically divergent specific heat. The following question (apart from many others) then immediately arises: do we still have a phase transition for finite p and if this is the case, what is the nature of this transition and what is the critical temperature? The answer to this question is still debated (see ref. [4] for a review), and the exact answer would amount to solving exactly for the configurationally averaged free energy f = (f [ K,,]),,. Numerical analysis [4] and renormalization group treatments [30] suggest a phase diagram as shown in fig. 4. For finite concentration p we still have a transition from a ferromagnetic to a paramagnetic state. The critical temperature Tc(p) diminishes with increasingp until it reaches zero at critical concentrations p, = 0.12 (a = - 1) and p, = 0.5 (a = 0), respectively. In the +J model this can be understood as the breakdown of long-range ferromagnetic order due to the presence of com-

kBT/J +

Fig. 4. Phase diagram of the diluted ferromagnet ((I = 0) and the k J-model (u = - l), respectively. Broken curves correspond to the expansion up to order p2_

W. F. Wolff

/ Fru.s~rrr~ed Ising models in

peting interactions, i.e. of frustration, while in the diluted ferromagnet it is the disappearance of the largest percolating cluster which causes this destruction. The question about the nature of this transition still remains unanswered. However, the solvable case p = 0 suggests an expansion of the averaged free energy f for small concentrations p. For the dilute ferromagnet with a = 0 this has already been done [29], but only to first order p. This then yields the slope of the curve T,(p) near the zero concentration limit p = 0. However, to this order no definite conclusion can be drawn on whether the transition at finite concentration p > 0 is different from the Ising transition at p = 0. We only mention that differences between the (i.e. configurational averages per“quenched” formed on the free energy) and “annealed” (i.e. configurational averages performed on the partition function) models are of order pz or higher [31,32]. It is thus necessary to go higher than to first order p. Up to second order pz this expansion has been performed [33] and it turned out that all relevant quantities could be calculated exactly. The first result is the T,-shift up to order p2, T,(p) = 7’,.(O)- 6,~ - S,p*, where 7’,_(O) is the critical temperature of the homogeneous Ising model, T’(0) = 2/ln(\/l + l), measured in units of k,/J. The coefficients S,, S, can be given as functions of a; in table 1 we give their numerical values for a = 0 and a = - 1. The critical concentrations.p, at which r,(p) vanishes are given by p, = 0.703 for a = 0 and p, = 0.235 for a = 1, respectively. These numbers are larger than the estimates given above, which is not surprising as for larger p higher-order corrections to T,(p) become relevant. The agreement of our T,( p)-formula with the estimated curves however is very good for smaller p. The second result one obtains is the expansion of the averaged free energy pj in the vicinity of

0 -1

3.016

0.306

(16,‘n)(\/Z

7.279

9.982

16/1~ = 5.093

- 1)2 = 0.874

dimensions

223

T,(p), i.e. near the transition. Keeping only the leading terms the singular part of the free energy pf is given up to order p* as Pflsing

t: if*

X

InPI

[

1 +ptln/t[+p2gln2[t[+

...

1,

t -+ 0, (10) where t measures the distance from r,(p). Of course, for p = 0 (eq. (10)) reduces to the familiar Ising singularity. Again the coefficient c can be determined as a function of a and is given in the table for the two cases a = 0 and a = - 1. The free energy expansion (eq. (10) allows us to draw quite remarkable conclusions about the nature of the phase transition for finite concentrations p. Assume that the transition for finitep is governed by a single critical exponent a(p), where a(p) + 0 for p -j 0 to account for the Ising value cr( p = 0) = 0. Then one expects a singular behaviour like #f3j]si”g= d”3’6,

1.

(11)

However, an expansion of eq. (11) in powers of p leads to a coefficient l/6 in thep*-term and not to l/3 as in our expansion (eq. (10)). Quite recently [34] it has been suggested on the basis of an approximate mapping of the Ising model with randomly distributed couplings onto the zero-component Gross-Neveu model that the singular part of the free energy j?f takes the form

Pflsign

3

-

5

%

t* ln[l

- cp ln]t]],

which in the limit t + 0 amounts PfIsing=

Table 1

IWO

-$~t21n]ln]t]],

t+O.

(12) to a singularity

(13)

Comparing the p-expansion of eq. (12) with our expansion (eq. (10)) we see that all terms are reproduced exactly up to order p2. Though this is not a proof we feel that the consistency of the p-expansions is a strong argument for eq. (12) being actually correct.

Acknowledgement I would like to thank Professor J. Zittartz for very stimulating discussions and also for a critical reading of the manuscript.

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