# Other Exactly Solved Models

There is a number of papers dealing with exactly solved frustrated models published by J. Strecka and coworkers since 2006. These models are essentially decorated Ising models in one or two dimensions.

In Ref. [331] ground-state and finite-temperature properties of the mixed spin-1/2 and spin-S Ising-Heisenberg diamond chains are examined within an exact analytical approach based on the generalized decoration-iteration map. A particular emphasis is laid on the investigation of the effect of geometric frustration, which is generated by the competition between Heisenberg- and Ising-type exchange interactions. It is found that an interplay between the geometric frustration and quantum effects gives rise to several quantum ground states with entangled spin states in addition to some semiclassically ordered ones. Among the most interesting results to emerge from our study, one could mention a rigorous evidence for quantized plateaux in magnetization curves, an appearance of the round minimum in the thermal dependence of susceptibility times temperature data, double-peak zero-field specific heat curves, or an enhanced magneto-caloric effect when the frustration comes into play. The triple-peak specific heat curve is also detected when applying small external field to the system driven by the frustration into the disordered state.

In Ref. [332], the geometric frustration of the spin-1/2 Ising-Heisenberg model on the triangulated kagome "triangles-in- triangles” lattice is investigated within the framework of an exact analytical method based on the generalized star-triangle mapping transformation. Ground-state and finite-temperature phase diagrams are obtained along with other exact results for the partition function, Helmholtz free energy, internal energy, entropy, and specific heat, by establishing a precise mapping relationship to the corresponding spin-1/2 Ising model on the kagome lattice. It is shown that the residual entropy of the disordered spin liquid phase for the quantum Ising-Heisenberg model is significantly lower than for its semiclassical Ising limit (*S*_{0}*/NTk _{B} =* 0.2806 and 0.4752, respectively), which implies that quantum fluctuations partially lift a macroscopic degeneracy of the ground-state manifold in the frustrated regime.

In Ref. [333], spin-1/2 Ising model with a spin-phonon coupling on decorated planar lattices partially amenable to lattice vibrations is examined using the decoration-iteration transformation and harmonic approximation. It is shown that the magneto-elastic coupling gives rise to an effective antiferromagnetic next-nearest- neighbor interaction, which competes with the nearest-neighbor interaction and is responsible for a frustration of decorating spins. A strong enough spin-phonon coupling consequently leads to an appearance of striking partially ordered and partially disordered phase, where a perfect antiferromagnetic alignment of nodal spins is accompanied with a complete disorder of decorating spins.

In the above works, the decorations are local couplings of an Ising spin to decorated Heisenberg spins or to phonons which can be summed up to renormalize the interactions between Ising spins. This leaves the systems solvable by star-triangle transformations, vertex models or other methods. Their results are interesting. We find again in these models striking features shown above in the present chapter for 2D solvable frustrated Ising models. In particular, partially disordered systems, multiple phase transitions and the reentrant phase due to the frustration are shown to exist in the phase diagrams. For details, the reader is referred to Refs. [331- 333].

# Evidence of Partial Disorder and Reentrance in Non-Solvable Frustrated Systems

The previous sections show interesting phenomena due to the frustration. What to be retained is the fact that those phenomena occur around the boundary of two phases of different ground states, namely different symmetries. These phenomena include the following:

- (1) The partial disorder at equilibrium: disorder is not equally shared on all particles as usually the case in non-frustrated systems,
- (2) The reentrance: this occurs around the phase boundary when
*T*increases -> the phase with larger entropy will win at finite*T.*In other words, this is a kind of selection by entropy. - (3) The disorder line: this line occurs in the paramagnetic phase. It separates the pre-ordering zones between two nearby ordered phases.

The partial disorder and the reentrance which occur in exactly solved Ising systems shown above are expected to occur also in models other than the Ising one as well as in some three?dimensional systems. Unfortunately, these systems cannot be exactly solved. One has to use approximations or numerical simulations to study them. This renders difficult the interpretation of the results. Nevertheless, in the light of what has been found in exactly solved systems, we can introduce the necessary ingredients into the model under study if we expect the same phenomenon to occur.

As seen above, the most important ingredient for a partial disorder and a reentrance to occur at low *T* in the Ising model is the existence of a number of free spins in the ground state.

In three dimensions, apart from a particular exactly solved case [155] showing a reentrance, a few Ising systems such as the fully frustrated simple cubic lattice [30, 81], a stacked triangular Ising antiferromagnet [31, 244] and a body-centered cubic (bcc) crystal [17] exhibit a partially disordered phase in the ground state. We believe that reentrance should also exist in the phase space of such systems though evidence is found numerically only for the bcc case [17].

In two dimensions, a few non-Ising models show also evidence of a reentrance. For the q-state Potts model, evidence of a reentrance is found in a recent study of the two-dimensional frustrated Villain lattice [the so-called piled-up domino model) by a numerical transfer matrix calculation [119,120]. It is noted that the reentrance occurs near the fully frustrated situation, i.e. *a _{c}* = Jaf/Jf = -1 [equal antiferromagnetic and ferromagnetic bond strengths), for

*q*between ~ 1.0 and ~ 4. Note that there is no reentrance in the case

*q*= 2. Below [above) this

*q*value, the reentrance occurs above [below) the fully frustrated pointa

_{c}as shown in Figs. 7.17 [120] and 7.18 [120]. For

*q*larger than ~ 4, the reentrance disappears [120].

A frustrated checkerboard lattice with XY spins shows also evidence of a paramagnetic reentrance [45].

In vector spin models such as the Heisenberg and XY models, the frustration is shared by all bonds so that no free spins exist in the ground state. However, one can argue that if there are several kinds of local field in the ground state due to several kinds of interaction, then there is a possibility that a subsystem with weak local field is disordered at low *T* while those of stronger local field stay ordered up to higher temperatures. This conjecture has been verified in a number of recent works on classical [307] and quantum spins

Figure 7.17 Phase diagram for the Potts piled-up-domino model with *q =* 3: periodic boundary conditions (top) and free boundary conditions (bottom). The disorder lines are shown as lines, and the phase boundaries as symbols. The numerical uncertainty is smaller than the size of the symbols.

[285,305]. Consider, for example, Heisenberg spins Sj on a bcc lattice with a unit cell shown in Fig. 7.19 [307].

For convenience, let us call sublattice 1 the sublattice containing the sites at the cube centers and sublattice 2 the other sublattice. The Hamiltonian reads

Figure 7.18 The phase diagram for *q =* 1.5 found using transfer matrices and the phenomenological renormalization group with periodic (top) and free boundary conditions (bottom). The points correspond to finite-size estimates for *T _{c},* whilst the lines correspond to the estimates for the disorder line, (се = /2//1).

where *J2* 1 indicates the sum over the NN spin pairs with exchange coupling/ь while *J2_{2}* *

^{s}limited to the NNN spin pairs belonging to sublattice 2 with exchange coupling /

_{2}. It is easy to see that when

*J*is antiferromagnetic the spin configuration is non- collinear for /2/I/1I < -2/3. In the non-collinear case, one can verify that the local field acting a center spin (sublattice 1) is weaker in magnitude than that acting on a corner spin (sublattice 2). The partial disorder is observed in Fig. 7.20: The sublattice of black spins (sublattice 1) is disordered at a low

_{2}*T.*

The same argument is applied for quantum spins [285]. Consider the bcc crystal as shown in Fig. 7.19, but the sublattices are supposed now to have different spin magnitudes, for example, *S _{A}* = 1/2

Figure 7.19 bcc lattice. Spins are shown by gray and black circles. Interaction between NN (spins numbered 1 and 3) is denoted by *Jx* and that between NNN (spins 1 and 2) by *) _{2}.* Note that there is no interaction between black spins.

Figure 7.20 Monte Carlo results for sublattice magnetizations vs *T* in the case *J* j = —1,/_{2} *=* —1-4: Black squares and black circles are for sublattices

- 1 and 2, respectively. Void circles indicate the total magnetization.
- (sublattice 1) and Sb = 1 (sublattice 2). In addition, one can include NNN interactions in both sublattices, namely
*J*and_{2A}*J*b- The Green’s function technique is then applied for this quantum system [285]. We show in Fig. 7.21 the partial disorder observed in two cases: Sublattice 1 is disordered (Fig. 7.21a) or sublattice_{2} - 2 is disordered (Fig. 7.21b). In each case, one can verify, using the corresponding parameters, that in the ground state the spin of the

Figure 7.21 Sublattice magnetizations vs *T* in the case *S _{A}* = 1/2,

*S*1: (a) Curve

_{B}=*a*(b) is the sublattice-l(2) magnetization in the case /24/I/1I — 0.2 and /2B/I/1I = 0.9 (b) curve

*a*(b) is the sublattice-1(2) magnetization in the case /гл/1У 11 = 2.2 and/

_{2}b/I/iI = 0.1. P is the paramagnetic phase, PO the partial order phase (only one sublattice is ordered), II and III are non collinear spin configuration phases. See text for comments.

disordered sublattice has an energy lower than a spin in the other sublattice.

We show in Fig. 7.22 the specific heat versus *T* for the parameters used in Fig. 7.21. One observes the two peaks corresponding to the two phase transitions associated with the loss of sublattice magnetizations.

The necessary condition for the occurrence of a partial disorder at finite *T* is thus the existence of several kinds of site with different energies in the ground state. This has been so far verified in a number of systems as shown above.

Figure 7.22 Specific heat versus *T* of the parameters used in Fig. 7.21, Sa = 1/2, Sj, = 1: (a) Jza/Ji = 0.2 and *J _{2B}/J_{1} *

*=*0.9 (b)

*J*

_{2A}/J_{l}*=*2.2 and /20/1/1! = 0.1. See the caption of Fig. 7.21 for the meaning of P, PO, II and III. See text for comments.