II: Magnetism of Thin Films
Exactly Solved Frustrated Models in Two Dimensions
In this chapter, we show properties of exactly solved frustrated two- dimensional (2D) spin systems. We would like to emphasize the physics near a phase boundary where interesting phenomena can occur due to competing interactions of the two phases around the boundaiy. Two-dimensional systems are, in fact, the limiting case of thin films with a monolayer. We give examples of frustrated 2D Ising systems that we can exactly solve by transforming them into vertex models. We show that these simple systems contain already most of the striking features of frustrated systems such as the high degeneracy of the ground state (GS), many phases in the GS phase diagram in the space of interaction parameters, the reentrance occurring near the boundaries of these phases, the disorder lines in the paramagnetic phase and the partial disorder coexisting with the order at equilibrium.
We discuss a number examples of thin films and 3D systems where we see some phenomena observed in the above exactly found solutions such as the reentrance and the partial disorder at equilibrium.
The results shown in this chapter are taken from Refs. [16,67,68, 76,77,153,285,305,307].
Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep
Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.
ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com
Extensive investigations on materials have been carried out during the past three decades. This is due to an enormous number of industrial applications which drastically change our life style. The progress in experimental techniques, the advance on theoretical understanding and the development of high-precision simulation methods together with the rapid increase of computer power have made possible the rapid development in materials science. Today, it is difficult to predict what will be discovered in this research area in 10 years.
The purpose of this chapter is to recall important results found by solving exactly several frustrated 2D Ising systems which help understand recent results in other non-solvable models such as frustrated thin films studied in the following chapters of the present book. The physics of frustrated spin systems at low dimensions, 2D systems and magnetic thin films, attracts an enormous number of investigations during the past decades due to many industrial applications. We would like to connect these results, published over a long period of time, on a line of thoughts: physics near a phase boundary. A boundary between two phases of different orderings is determined as a compromise of competing interactions each of which favors one kind of ordering. The frustration is thus minimum on the boundary (see reviews on many aspects of frustrated spin systems in Ref. ). When an external parameter varies, this boundary changes and we will see in this review that many interesting phenomena occur in the boundary region. We will concentrate ourselves in the search for interesting physics near the phase boundaries in various frustrated spin systems in this review.
In the 1970s, statistical physics with the Renormalization Group analysis has greatly contributed to the understanding of the phase transition from an ordered phase to a disordered phase [88, 380]. We will show methods to study the phase transition in magnetic thin films where surface effects when combined with frustration effects give rise to many new phenomena. Physical properties of solid surfaces, thin films and superlattices have been intensively studied due to their many applications [18, 22, 36, 87,113,134, 374].
Section 7.2 is devoted to the definition of the frustration and to the determination of the ground states of a few popular models. We show in Section 7.3 the decimation method and how to find the disorder line and the reentrance phase. Section 7.4 shows several exactly solved models with their striking properties. Section 7.5 shows some other 2D exactly solved decorated models. As seen, many interesting phenomena such as partial disorder, reentrance, disorder lines and multiple phase transitions are exactly uncovered. Only exact mathematical techniques can allow us to reveal such beautiful phenomena which occur around the boundary separating two phase of different ground-state orderings. These exact results permit to understand similar behaviors in systems that cannot be solved, some of these systems are shown and discussed in Sections
7.6 and 7.7.
Since the 1980s, frustrated spin systems have been subjects of intensive studies . The word "frustration" has been introduced to describe the fact that a spin cannot find an orientation to fully satisfy all interactions with its neighbors, namely the energy of a bond is not the lowest one [344, 348]. This will be seen below for Ising spins where at least one among the bond with the neighbors is not satisfied. For vector spins, frustration is shared by all spins so that all bonds are only partially satisfied, i.e., the energy of each bond is not minimum. Frustration results either from the competing interactions or from the lattice geometry such as the triangular lattice with antiferromagnetic nearest-neighbor (NN) interaction, the face-centered cubic (FCC) antiferromagnet and the antiferromagnetic hexagonal-close-packed (HCP) lattice (see ).
Note that real magnetic materials have complicated interactions and there are large families of frustrated systems such as the heavy lanthanides metals (holmium, terbium and dysprosium) [381, 382], helical MnSi , pyrochore antiferromagnets , and spinice materials . Exact solutions on simpler systems may help understand qualitatively real materials. Besides, exact results can be used to validate approximations.
We recall in the following some basic arguments leading to the definition of the frustration.
The interaction energy of two spins S, and Sy interacting with each other by J is written as E = —) (S, ■ Sy). If J is ferromagnetic (У > 0) then the minimum of E is — ] corresponding to S, parallel to Sy. If у is antiferromagnetic (J < 0), E is minimum when S, is antiparallel to Sy. One sees that in a crystal with NN ferromagnetic interaction, the ground state (GS) of the system is the configuration where all spins are parallel: The interaction of eveiy pair of spins is ''fully" satisfied, namely the bond energy is equal to — J. This is true for any lattice structure. If J is antiferromagnetic, the GS depends on the lattice structure: (i) For lattices containing no elementary triangles, i.e., bipartite lattices (such as square lattice, simple cubic lattices, ...) in the GS, each spin is antiparallel to its neighbors, i.e., every bond is fully satisfied, its energy is equal to — |y |; (ii) for lattices containing elementary triangles such as the triangular lattice, the FCC lattice and the HCP lattice, one cannot construct a GS where all bonds are fully satisfied (see Fig. 7.1). The GS does not correspond to the minimum interaction energy of every spin pair: The system is frustrated.
Let us formally define the frustration. We consider an elementary lattice cell which is a polygon formed by faces called "plaquettes.” For example, the elementary cell of the simple cubic lattice is a cube with six square plaquettes, the elementary cell of the FCC lattice is a tetrahedron formed by four triangular plaquettes. According to the definition of Toulouse  the plaquette is frustrated if the parameter P defined below is negative
where y,j is the interaction between two NN spins of the plaquette and the product is performed over all y,-(y around the plaquette.
We show two examples of frustrated plaquettes in Fig. 7.1, a triangle with three antiferromagnetic bonds and a square with three ferromagnetic bonds and one antiferromagnetic bond. P is negative
Figure 7.1 Two frustrated cells are shown. The thin (heavy) lines denote the ferromagnetic (antiferromagnetic) bonds. Up and down spins are shown by green and red circles, respectively. Question marks indicate undetermined spin orientation. Choosing an orientation for the spin marked by the question mark will leave one of its bonds unsatisfied (frustrated bond with positive energy).
in both cases. If one tries to put Ising spins on those plaquettes, at least one of the bonds around the plaquette will not be satisfied. For vector spins, we show below that the frustration is equally shared by all bonds so that in the GS, each bond is only partially satisfied.
One sees that for the triangular plaquette, the degeneracy is three, and for the square plaquette it is four. Therefore, the degeneracy of an infinite lattice for these cases is infinite, unlike the non-frustrated case.
The frustrated triangular lattice with NN interacting Ising spins was studied in 1950 .
Non-Collinear Ground-State Spin Configurations
For vector spins, non-collinear configurations due to competing interactions were found in 1959 independently by Yoshimori , Villain  and Kaplan .
We emphasize that the frustration may be due to the competition between a Heisenberg exchange model which favors a collinear spin configuration and the Dzyaloshinskii-Moriya interaction E = -D • (S, л S;) [99,240] which favors the perpendicular configuration. We will return to this interaction in Chapters 13,14 and 15.
We show below how to determine the GS of some frustrated systems and discuss some of their properties.
We consider the plaquettes shown in Fig. 7.1 with X Y spins. The GS configuration corresponds to the minimum of the energy E of the plaquette. In the case of the triangular plaquette, suppose that spin S, (/' = 1, 2, 3) of amplitude S makes an angle 0, with the Ox axis. One has
where ] < 0 (antiferromagnetic). Minimizing E with respect to 3 angles 6j, we find the solution в — 02 = #2 - #з = #з — #i = 2,т/3. One can also write
J is negative, the minimum thus corresponds to Si + S2 + S3 = 0 which gives the 120° structure. This is true also for the Heisenberg spins.
For the frustrated square plaquette, we suppose that the ferromagnetic bonds are J and the antiferromagnetic bond is —J) connecting the spins Si and S4 (see Fig. 7.2). The energy minimization gives
If the antiferromagnetic interaction is —>jJ (>/ > 0), the angles are 
and |014| = 3|0|, where cos= cos#, — cos6j. This solution exists if | cos 01 < 1, namely /? > r]c = 1/3. One recovers when i] = 1, в = п/4, $i4 = Зя/4.
The GS spin configurations of the frustrated triangular and square lattices are displayed in Fig. 7.2 with XY spins. We see that the frustration is shared by all bonds: The energy of each bond is —0.5] for the triangular lattice, and - /2 for the square lattice. Thus, the bond energy in both cases is not fully satisfied, namely not equal to —], as we said above when defining the frustration.
At this stage, we note that the GS found above have a twofold degeneracy resulting from the equivalence of clockwise and
Figure 7.2 Ground state of XY spins on frustrated triangular and square cells: non-collinear spin arrangements. The thin lines denote the ferromagnetic interaction, the thick line is the antiferromagnetic one.
counter-clockwise turning angle (noted by + and - in Fig. 7.3) between adjacent spins on a plaquette in Fig. 7.2. Therefore, the symmetry of these spin systems is of Ising type 0(1), in addition to the symmetry S0(2) due to the invariance by global spin rotation in the plane.
From the GS symmetry, one expects that the respective breaking of 0(1) and 50(2) symmetries would behave, respectively, as the 2D Ising universality class and the Kosterlitz-Thouless transition .
Figure 7.3 Triangular antiferromagnet with XY spins: The left (right) chirality is indicated by + (—). See text.
However, the question of whether the two phase transitions would occur at the same temperature and the nature of their universality remains an open question [27,45].
Let us determine the GS of a helimagnet. Consider the simplest case: a chain of Heisenberg spins with ferromagnetic interaction У x(> 0) between NN and antiferromagnetic interaction /2(<0) between NNN. The interaction energy is
where one has supposed that the angle between NN spins is в. The first solution is
sin# = 0 —> в = 0 which is the ferromagnetic solution and the second one is
This solution is possible when -l
Note that the two frequently studied frustrated spin systems are the FCC and HCP antiferromagnets. These two magnets are constructed by stacking tetrahedra with four frustrated triangular faces. Frustration by the lattice structure such as these cases are called "geometry frustration.” Another 3D popular model which has been extensively studied since 1984 is the system of stacked antiferromagnetic triangular lattices (satl). The phase transition of this system with XY and Heisenberg spins was a controversial subject for more than 20 years. The controversy was ended with our works: The reader is referred to Refs. [187, 250, 251] for the history.
Figure 7.4 Example of a helimagnetic configuration using e = I/2I//1 > £c = 1/4 C/i > 0,J2 < 0), namely в = 2л/3. Left: 3D view. Right: top view.
In short, we found that in known 3D frustrated spin systems (FCC, HCP, satl, helimagnets,...) with Ising, XY or Heisenberg spins, the transition is of first order [83,152].
Another subject which has been much studied since the 1980s is the phenomenon called "order by disorder": We have seen that the ground state of frustrated spin systems is highly degenerate and often infinitely degenerate (entropy not zero at temperature T = 0). However, it has been shown in many cases that when T is turned on the system chooses a state which has the largest entropy, namely the system chooses its order by the largest disorder. We call this phenomenon "order by disorder” or "order by entropic selection” (see references cited in Section Ш.В of Ref. ).
We will not discuss these subjects in this review, which is devoted to low-dimensional frustrated spin systems.