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Exactly Solved Frustrated Models

Any 2D Ising model with non-crossing interactions can be exactly solved. To avoid the calculation of the partition function one can transform the model into a 16-vertex model or a 32-vertex model. The resulting vertex model is exactly solvable. We have applied this

Kagome lattice

Figure 7.5 Kagome lattice: Diagonal and horizontal bonds are NN antiferromagnetic interactions ]u vertical double lines indicate the NNN interactions ]2.

method to search for the exact solution of several Ising frustrated 2D models with non-crossing interactions shown in Figs. 7.5-7.7.

Details have been given in Ref. [86]. We outline below a simplified formulation of a model for illustration. The aim is to discuss the results. As we will see, these models possess striking phenomena due to the frustration.

Example of the Decimation Method

We take the case of the centered honeycomb lattice with the following Hamiltonian:

where a, = ±1 is an Ising spin at the lattice site /. The first, second, and third sums are performed on the spins interacting via Ji, J2 and J3 bonds, respectively (see Fig. 7.7). The case J2 = /3 = 0 corresponds to the honeycomb lattice, and the case ] = ]2 = Уз to the triangular lattice.

Let a be the central spin of the lattice cell shown in Fig. 7.7. Other spins are numbered from ax to 6. The Boltzmann weight of

Exactly solved dilute centered square lattices

Figure 7.6 Exactly solved dilute centered square lattices: Interactions along diagonal, vertical and horizontal spin pairs are noted by /1( J2, and J3, respectively.

the elementary cell is written as

where Kj = ^ (/ = 1, 2, 3). The partition function reads

where the sum is taken over all spin configurations and the product over all elementary cells of the lattice. One imposes the periodic boundary conditions. The above model is exactly solvable. To that

Centered honeycomb lattice. Spins are numbered for decimation demonstration (see text). Blue, red and black bonds denote interactions J ) 2 and J з, respectively

Figure 7.7 Centered honeycomb lattice. Spins are numbered for decimation demonstration (see text). Blue, red and black bonds denote interactions J ) 2 and J з, respectively.

end, we decimate the central spin of every elementary lattice cell. We finally get a honeycomb Ising model (without centered spins) with multispin interactions.

After decimation of the central spin, namely after summing the values of the central spin a, the Boltzmann weight of an elementary cell reads

We show below that this model is, in fact, a case of the 32-vertex model on the triangular lattice which has an exact solution.

We consider the dual triangular lattice of the honeycomb lattice obtained above [25]. The sites of the dual triangular lattice are at the center of each elementary honeycomb cell with bonds perpendicular to the honeycomb ones, as illustrated in Fig. 7.8.

Let us define the conventional arrow configuration for each site of the dual triangular lattice: If all six spins of the honeycomb cell are parallel, then the arrows, called "standard configuration,” are shown in Fig. 7.9. From this "conventional" configuration, antiparallel spin pairs on the two sides of a triangular lattice bond will have its corresponding arrow change the direction.

The dual triangular lattice, shown by discontinued lines, of the honeycomb lattice

Figure 7.8 The dual triangular lattice, shown by discontinued lines, of the honeycomb lattice.

The conventional definition of the "standard” arrows for NN around a site of the triangular lattice

Figure 7.9 The conventional definition of the "standard” arrows for NN around a site of the triangular lattice: Spins are numbered so the arrows can be recognized in examples shown in Fig. 7.10. Note that the configuration of all down spins has the same arrow configuration. See text.

As examples, two spin configurations on the honeycomb lattice and their corresponding arrow configurations on the triangular lattice are displayed in Fig. 7.10.

Counting all arrow configurations, we obtain 32. To each of these 32 vertices one associates the Boltzmann weight W'(ai, 02, 03, a4, as, 6) given by Eq. (7.10). Let us give explicitly a few of them:

Two examples of spin configurations and their corresponding arrow configurations. To understand, compare with the standard arrows defined in Fig. 7.9. See text

Figure 7.10 Two examples of spin configurations and their corresponding arrow configurations. To understand, compare with the standard arrows defined in Fig. 7.9. See text.

Using the above expressions of the 32-vertex model, one finds the following equation for the critical temperature (see details in Ref. [76]):

The solutions of this equation are given in 7.4.2 below for some special cases.

Following the case studied above, we can study the 2D models shown in Figs. 7.5-7.6: After decimation of the central spin in each square, these models can be transformed into a special case of the 16-vertex model which yields the exact solution for the critical surface (see details in Ref. [86]).

Before showing some results in the space of interaction parameters, let us introduce the definitions of disorder line and reentrant phase.

Disorder Line, Reentrance

It is not the purpose of this review to enter technical details. We would rather like to describe the physical meaning of the disorder line and the reentrance. A full technical review has been given in Ref. [86].

Disorder solutions exist in the paramagnetic region which separate zones of fluctuations of different nature. They are where the short-range pre-ordering correlations change their nature to allow for transitions in the phase diagrams of anisotropic models. They imply constraints on the analytical behavior of the partition function of these models.

To obtain the disorder solution, one makes a certain local decoupling of the degrees of freedom. This yields a dimension reduction: A 2D system then behaves on the disorder line as a ID system. This local decoupling is made by a simple local condition imposed on the Boltzmann weights of the elementary cell [329].

This is very important while interpreting the system behavior: On one side of the disorder line, pre-ordering fluctuations have correlation different from those of the other side. Crossing the line, the system pre-ordering correlation changes. The dimension reduction is often necessary to realize this.

Note that disorder solutions may be used in the study of cellular automata as it has been shown in Ref. [297].

Let us give now a definition for the reentrance. A reentrant phase lies between two ordered phases. For example, at low temperature (Г) the system is in an ordered phase I. Increasing T, it undergoes a transition to a paramagnetic phase R, but if one increases further T, the system enters another ordered phase II before becoming disordered at a higher T. Phase R is thus between two ordered phases I and II. It is called "reentrant paramagnetic phase" or "reentrant phase.”

How physically is it possible? At a first sight, it cannot be possible because the entropy of an ordered phase is smaller than that of an disordered phase so that the disordered phase R cannot exist at lower T than the ordered phase II. In reality, as we will see below, phase II has always a partial disorder which compensates the loss of entropy while going from R to II. The principle that entropy increases with T is thus not violated.

Phase Diagram

Kagomé Lattice

Model with NN and NNN interactions

The Kagome lattice shown in Fig. 7.5 has attracted much attention not only by its great interest in statistical physics but also in real materials [124]. The Kagome Ising model with only NN interaction J i has been solved a long time ago [173]. No phase transition at finite T when /i is antiferromagnetic. Taking into account the NNN interaction J2, we have solved [16] this model by transforming it into a 16-vertex model which satisfies the free-fermion condition. The equation of the critical surface is

We are interested in the region near the phase boundaiy between two phases IV (partially disordered) and I (ferromagnetic) in Fig. 7.11 (top). We show in Fig. 7.11 (bottom) the small region near the boundary a =J2/Ji = — 1 which has the reentrant paramagnetic phase and a disorder line.

We note that only near the phase boundary such a reentrant phase and a disorder line can exist.

Generalized Kagomé lattice

If we suppose that all interactions J b J2 and /3 in the model shown in Fig. 7.5 are different, the phase diagram becomes very rich [67]. For instance, the reentrance can occur in an infinite regionpav

Top: Each color represents the ground-state configuration in the space С/ьУг) where +, — and x denote up, down and free spins, respectively. Bottom

Figure 7.11 Top: Each color represents the ground-state configuration in the space С/ьУг) where +, — and x denote up, down and free spins, respectively. Bottom: Phase diagram in the space (a = J2/Jь T) with )! > 0. Г is in the unit of Ji/кц. Solid lines are critical lines, dashed line is the disorder line. P, F and X stand for paramagnetic, ferromagnetic and partially disordered phases, respectively. The inset shows schematically the enlarged region near the critical value )г/] = — 1.

of interaction parameters and several reentrant phases can occur for a given set of interactions when T varies.

The Hamiltonian reads

where ]J2 and /3, respectively.

(a) Generalized Kagome lattice

Figure 7.12 (a) Generalized Kagome lattice: J, J2 and /3 denote the

diagonal, vertical and horizontal bonds, respectively, (b) The ground-state phase diagram in the space (a = /2//1, P = Js/Ji)- Each phase is displayed by a color with up, down and free spins denoted by +, — and 0, respectively. I, II, III and F indicate the three partially disordered phases and the ferromagnetic phase, respectively.

The phase diagram at temperature T = 0 is shown in Fig. 7.12 in the space [a=J2/]i, P=Js/]), supposing /i>0. The spin configuration of each phase is indicated. The three partially disordered phases (I, II, and III) have free central spins. With У1 < 0, it suffices to reverse the central spin in the F phase of Fig. 7.12. In addition, the permutation of J2 and /3 will not change the system, because it is equivalent to а я/2 rotation of the lattice.

We examine now the temperature effect. We have seen above that a partially disordered phase lies next to the ferromagnetic phase in the ground state gives rise to the reentrance phenomenon. We expect, therefore, similar phenomena near the phase boundary in the present model. As it turns out, we find below a new and richer behavior of the phase diagram.

We use the decimation of central spins described in Ref. [86], we get then a checkerboard Ising model with multispin interactions. This corresponds to a symmetric 16-vertex model which satisfies the free-fermion condition [123, 336, 366]. The critical temperature

Phase diagram in the

Figure 7.13 Phase diagram in the (p = J3/J u Г) space for negative a = )2/J i- (a) a = —0.25; (b) a = —0.8. Partially disordered phases of type I and II and F are defined in Fig. 7.12. The disorder lines are shown by dotted lines.

is the solution of the following equation:

Note the invariance of Eq. (7.18) with respect to changing K ->■ -Ki and interchanging K2 and K3. Let us show just the solution near the phase boundary in the plane {ft = J3/)i, T) for two values of « = Ji/Jilt is interesting to note that in the interval 0 > a > —1, there exist three critical lines. Two of them have a common horizontal asymptote as p tends to infinity. They limit a reentrant paramagnetic phase between the F phase and the partially disordered phase I for p between p2 and infinite p (see Fig. 7.13). Such an infinite reentrance has never been found before in other models. With decreasing a, p2 tends to zero and the F phase is reduced (comparing Figs. 7.13a and 7.13b). Fora < — 1, the F phase and the reentrance no longer exist.

We note that for —1 < a < 0, the model possesses two disorder lines (see equations in Ref. [67]) starting from a point near the phase boundaiy p = —1 for a close to zero; this point position moves to ft = 0 as a tends to -1 (see Fig. 7.13).

Centered Honeycomb Lattice

We use the decimation of the central spin of each elementary cell as shown in paragraph 7.3.1. After the decimation, we obtain a model equivalent to a special case of the 32-vertex model [299] on a triangular lattice which satisfies the free-fermion condition. The general treatment has been given in Ref. [76]. Here we show the result of the case where K2 = K2. Equation (7.15) is reduced to

When K2 = 0, Eq. (7.15) gives the critical line

When K3 = 0, we observe a reentrant phase. The critical lines are given by

The phase diagram obtained from Eqs. (7.21) and (7.22) near the phase boundaiy a = -0.5 is displayed in Fig. 7.14. One observes here that the reentrant zone goes down to T = 0 at the boundary a = -0.5 separating the GS phases II and III (see Fig. 7.14b).

Note that phase II has the antiferromagnetic ordering on the hexagon and the central spin free to flip, while phase III is the ordered phase where the central spin is parallel to 4 diagonal spins (see Fig. 2 of Ref. [76]). Therefore, if-0.6 < a < —0.5 (reentrant region, see Fig. 7.14b), when one increases T from T = 0, ones goes across successively the ordered phase III, the narrow paramagnetic

Centered honeycomb lattice

Figure 7.14 Centered honeycomb lattice: (a) Phase diagram in the space (/f1; K2), discontinued line is the asymptote; (b) Reentrance in the space (T, a = K2/Ki). I, II, III phases denote paramagnetic, partially disordered and ordered phases, respectively.

reentrant phase and the partially disordered phase II. Two remarks are in order: (i) The reentrant phase occurs here between an ordered phase and a partially disordered phase. However, as will be seen below, we discover in the three-center square lattice, reentrance can occur between two partially disordered phase; (ii) In any case, we find reentrance between phases when and only when there are free spins in the ground state. The entropy of the high-Г partially disordered phase is higher than that of the low-Г one. The second thermodynamic principle is not violated.

It is noted that the present honeycomb model does not possess a disorder solution with a reduction of dimension as the Kagome lattice shown earlier.

Centered Square Lattices

In this paragraph, we study several centered square Ising models by mapping them onto 8-vertex models that satisfy the free-fermion condition. The exact solution is then obtained for each case. Let us anticipate that in some cases, for a given set of parameters, up to five transitions have been observed with varying temperature. In addition, there are two reentrant paramagnetic phases going to infinity in the space of interaction parameters, and there are two additional reentrant phases found, each in a small zone of the phase space [68, 77].

Ground-state phase diagram in the space

Figure 7.15 Ground-state phase diagram in the space (a = J2/J1, b = J3//1) for (a) three-center square lattice; (b) two-adjacent center case; (c) and one-center case. Phase boundaries are indicated by heavy lines. Each phase is numbered and the spin configuration is displayed (+, —, and о are up, down, and free spins, respectively).

We consider the dilute centered square lattices shown in Fig. 7.6. The Hamiltonian of these models reads

where <7, is an Ising spin at the lattice site The sums are performed over the spin pairs interacting by /ь J2 and /3 bonds (diagonal, vertical and horizontal bonds, respectively).

Figure 7.15 shows the ground-state phase diagrams of the models displayed in Figs. 7.6a, 7.6b and 7.6d, where a = J2/J1 and b = у3/y 1. The spin configurations in different phases are also displayed. The model in Fig. 7.15a has six phases (numbered from

Three-center model

Figure 7.16 Three-center model: phase diagram in the space (T, a = У2/У1) for several values of b = JzU‘- (a) b = -1.25, (b) b = -0.75, (c) b = —0.25, (d) b = 0.75. Reentrant regions indicated by discontinued lines are enlarged in the insets. A number indicates the corresponding spin configuration shown in Fig. 7.15a. P is paramagnetic phase.

I to VI), five of which (I, II, IV, V and VI) are partially disordered (at least one centered spin being free), the model in Fig. 7.15b has five phases, three of which (I, IV, and V) are partially disordered, and the model in Fig. 7.15c has seven phases with three partially disordered ones (I, VI and VII).

It is interesting to note that each model shown in Fig. 7.6 possesses the reentrance along most of the phase boundary lines when the temperature is turned on. This striking feature of the centered square Ising lattices has not been observed in other known models.

Let us show in Fig. 7.16 the results of the three-center model of Fig. 7.6a, in the space (a = J2/J1, T) for typical values of b = Jz/Ji-

For b < —1, there are two reentrances as seen in Fig. 7.16a for b = -1.25. The phase diagram is shown using the same numbers of corresponding ground state configurations of Fig. 7.15. Note that the centered spins disordered at Г = 0 in phases I, II and VI (Fig. 7.15a) remain so at all T. Note also that the reentrance occurs always at a phase boundary. This point is emphasized in this chapter through various shown models.

For -1 < b < -0.5, there are three reentrant paramagnetic phases as shown in Fig. 7.16b, two of them on the positive a are so narrow while a goes to infinity. Note that the critical lines in these regions have horizontal asymptotes. For a large value of a, one has five transitions with decreasing T: paramagnetic phase-partially disordered phase I-first reentrant paramagnetic phase-partially disordered phase II-second reentrant paramagnetic phase-ferromagnetic phase (see Fig. 7.16b). To our knowledge, a model that exhibits such five phase transitions with two reentrances has never been found before.

For -0.5 < b < 0, another reentrance is found for я < —1 as seen in the inset of Fig. 7.16c. With increasing b, the ferromagnetic phase III in the phase diagram becomes large, reducing phases I and

II. At b = 0, only the ferromagnetic phase remains.

For positive b, we have two reentrances for a < 0, ending at a = —2 and a = -1 when T = 0 as seen in Fig. 7.16d.

In conclusion, we summarize that in the three-center square lattice model shown in Fig. 7.6a, we found two reentrant phases occurring on the temperature scale at a given set of interaction parameters. A new feature found here is that a reentrant phase can occur between two partially disordered phases, unlike in other models such as the Kagome Ising lattice where a reentrant phase occurs between an ordered phase and a partially disordered phase.

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