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Partially Frustrated Ising Models in Two Dimensions

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Partially Frustrated Ising Models in Two Dimensions ✦ PHYSICAL REVIEW B 67, 224414 ⑦ 2003 Partially frustrated Ising models in two dimensions Jiansheng Wu and Daniel C. Mattis* Department of Physics, University of Utah, 115 S. 1400 East #201, Salt Lake City, Utah 84112-0830, USA ✦ ⑦ Received 3 March 2003; published 13 June 2003 We examine ordered, periodic, Ising models on a sq lattice at varying levels x of frustration. The thermo- ✺ dynamic singularity of the fully frustrated model (x ✺ 1) is at T 0 while those of partially frustrated lattices ❁ (0✱ x 1) occur at finite Tc . The critical indices in the partially frustrated lattices that we consider—including the logarithmic specific heat—are all identical to those in the ferromagnet (x ✺ 0.) We display exact values of Tc and of ground-state energy and entropy Eo and So ,atx ✺ 1, 2/3, 1/2 , 2/5 ,...,0. ✦ ✶ DOI: 10.1103/PhysRevB.67.224414 PACS number⑦ s : 75.10.Hk, 05.50. q ✁ INTRODUCTION just 2 sites 1 and 2 , it transforms into a regular array in ✁ which the antiferromagnetic AF bonds are all on the left This work discusses the phenomenon of partial vertical riser and all other bonds are ferromagnetic. Among ‘‘frustration.’’ 1 Interest in systems with ‘‘frozen-in’’ random- the many ground states8 of this configuration one finds two ✁ ness, commonly denoted ‘‘spin glasses,’’ has continued un- ferromagnetic states all spins up or all down and a ground- 2 ✄ ✉ ✉ abated since the 1970s. The only statistical systems with state energy E0 ✂ 4 J . Except in its response to an exter- any chance of being solved in closed form are Ising-model nal field, this model is an example of perfect geometrical spin glasses. These come in at least two varieties: site- ✂ frustration i.e., x 1) and not of disorder! Later we shall see centered random gauge glasses that can be analyzed com- that merely specifying the extent of frustration x in the range pletely and sometimes even solved exactly,3 and those with ✆ 0 ✆ x 1isalso generally insufficient to determine whether competing ✻ J interactions that are both random and frus- this model or material can support an ordered phase. trated and generally cannot be solved at all. Toulouse iden- In trying to understand and systematize the distinction tified frustration as the result of competing interactions.4 The between random and geometrically frustrated systems, we unit of frustration is a plaquette in which the product of J’s propose a classification scheme. Type A is representative of around the perimeter has a negative sign.5 Frustration in- pure geometrical frustration, such as the nearest-neighbor creases both the ground-state energy and the ground-state entropy over their values in the ferromagnet. Ramirez6,7 Ising model on a triangular lattice with all equal antiferro- ✁ identified ‘‘geometrical frustration’’ as the disorder inherent magnetic AF bonds in which all plaquettes are frustrated in the correlation functions of what are frustrated but other- (x ✂ 1.) The thermodynamic properties of this model are 9 known: it has no ordered phase at T ✳ 0. Geometrically frus- wise structurally perfectly ordered materials. The prototype is the Ising antiferromagnet on a regular, triangular lattice, trated systems such as this can only sustain disordered about which more will be said later. Other well-known ex- phases at finite T while others frustrated chessboard or frus- 5,6 ✁ amples include ice at 0 °C and the pyrochlores. ✁ Still, in trated hexagonal lattices maintain a finite correlation length 6 the words of Ramirez, ‘‘... there is comparatively little even at T ✂ 0 and thus possess no critical exponents 9 ★ known about ❅ such materials,’’ although, at first blush, geo- whatever. metrically frustrated materials seem to share many properties Type B: the Edwards-Anderson (E-A) model1 on the two- with structurally random glassy materials having random dimensional sq lattice is a prime example of a magnetically bonds and/or fields at every site. amorphous material in which spin-glass behavior is caused ✻ The present work sets out to distinguish between the two. by the randomness. In it, Ising spins Si ✂ 1 are subject to ✻ ✁ ✂ It is introductory and admittedly incomplete, but because our randomly frozen-in nearest-neighbor NN bonds Ji, j J. results are exact they may serve as markers in a field that is The prototype E-A model consists of a sq lattice with a 1 ✆ not fully understood. We show that under certain circum- fraction p ✂ 2 of antiferromagnetic (J 0) bonds located at stances a sharp distinction can be drawn between partially random. This causes half the plaquettes, on average, to be- 1 ✂ ✁ frustrated systems the usual case and the disordered sys- come frustrated, i.e., x 2 . The location of the frustrated tems they superficially resemble. We display exact values of plaquettes is random. In two dimensions this model does not ground-state entropy So , ground-state energy Eo , critical exhibit a phase transition at any finite T although it may in temperature Tc , and quasiparticle dispersion calculated at higher dimensions.✁ some discrete values of x where x is the fraction of frus- The Hamiltonian in the E-A model is H ✄ ❙ ✂ trated plaquettes✁ . We also indicate possible directions for (ij)Ji, jSiS j , the sum being over NN’s. Its partition ✄ ✂ ❜ ✪ future investigations. function is Z Tr ✩ exp H . But it is the free energy, ✄ As a brief example of how much disordered and frustrated F ✂ kBT ln Z and not Z that needs be averaged over the ❜ systems can resemble one another, consider the apparently random variables. In addition to the temperature T ✂ 1/kB random Edwards-Anderson-Ising spin ladder in Fig. 1, one needs consider at least one supplementary material 10 ✻ containing nearest-neighbor bonds Ji, j ✂ J in parameter. This extra parameter is frequently taken to be p, 11 ✁ ✄ ❙ ☎ ✄ H ✂ (i, j)Ji, jSiS j . By gauge transformations S S on the fraction of AF – bonds. ✦ ⑦ ✦ 0163-1829/2003/67⑦ 22 /224414 4 /$20.00 67 224414-1 ©2003 The American Physical Society ✄ JIANSHENG WU AND DANIEL C. MATTIS PHYSICAL REVIEW B 67, 224414 ✂ 2003 ered in other fully frustrated models and may exist here also.14,1 Type C: Regular, homogeneous, systems with partial frozen-in frustration. The Ising versions have finite values of 1 ✺ Tc , hence an ordered low-temperature phase ⑦and 4 in two dimensions✦ . This is the category denoted ‘‘partially frustrated,’’ that is studied below. DISTINCTION BETWEEN TYPES Type-C systems exhibit some geometrical frustration and their ground states are typically degenerate. But unlike type A, they support an ordered phase and unlike type B they are ✦ FIG. 1. Random, or ‘‘ordered but geometrically frustrated?’’ not random ⑦ although the unit cell may be large .Byan Heavy lines are AF bonds, light lines are ferrmagnetic. The appar- obvious gauge symmetry of the sq lattice, or of bipartite ✂ ently disordered ladder on the left turns into an ordered but equally ✺ lattices in general, F(1✁ p) F(p) in the absence of finite frustrated!✄ ladder after two spins are ‘‘flipped’’ by a gauge trans- ✦ external fields. Hence we can ⑦and shall limit our studies to 1 ☎ ✆ formation: S ☎ ✆ S and S S . ❁ 1 1 2 2 ✱ 0 p 2 . 1 At the upper limit of p ✺ 2 one uncovers a fundamental ✦ The fully frustrated Ising model ⑦FFIM , illustrated in Fig. difference between types C and B in two dimensions. Con- 11 1 ✺ ✺ ✦ 2 ⑦a , has p 4 and x 1. Any additional, or any fewer, sider the following type-C model on a sq lattice: all vertical antiferromagnetic bonds necessarily decrease the fraction x bonds are antiferromagnetic ⑦✁ ✦ and all horizontal bonds are of frustrated plaquettes. As illustrated, this model does not ✶ ✦ ferromagnetic ⑦ . Not a single plaquette is frustrated and exhibit any structural randomness. Taking the unit cell to there is a phase transition from disorder to an ordered phase consist of two neighboring columns, this model exhibits as one lowers T below a critical temperature. This is to be translational periodicity—no less so than does the triangular 1 contrasted with the E-A model at p ✺ 2 on the same lattice, lattice with all antiferromagnetic bonds. It is therefore of which is of type B. As we have already noted, in the latter type A. By trivial gauge transformations it can be made to case half the plaquettes are frustrated on average and Tc look perfectly random and seemingly impossible to solve by ✺ 0. So just specifying the fraction p of antiferromagnetic the ordinary methods of statistical mechanics! Yet it has been bonds does not tell us what thermodynamic phase diagram 11 known to be solvable since 1977. can be expected. The FFIM has already been the subject of several inves- But then, neither is the fraction x of frustrated plaquettes tigations, including a mapping onto eight-vertex models, indicative of the thermodynamic properties that are to be 12,13 ✦ renormalization-group ⑦RG studies, etc., that revealed a expected! For example, the thermodynamic behavior of the 1 sort of ‘‘phase transition’’ at T ✺ 0 with a power-law corre- c above-mentioned two-dimensional E-A model (x ✺ 2 ) dif- ❤ 1 ✺ ✺ lation function ⑥ 1/r , exponent . The divergent T 0 2 fers completely from that of the n ✺ 4 model of type C in- paramagnetic susceptibility calculated by Kandel, Ben-Av, 1 ✺ ✦ vestigated below ⑦ in which x 2 also . Unlike the former, ❤ 12 2 ✷ ✺ ✻ and Domany, ① ⑥ L , with 0.507 0.009, confirms the latter has a second-order phase transition and supports an this unusual value of that is shared with the triangular AF ordered low-temperature phase. 13 lattice,the prototype of species A.
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