No. 7] Proc. Jpn. Acad., Ser. B 86 (2010) 643
Review Phase transition in spin systems with various types of fluctuations
† By Seiji MIYASHITA*1,*2,
(Communicated by Jun KONDO, M.J.A.)
Abstract: Various types ordering processes in systems with large fluctuation are overviewed. Generally, the so-called order–disorder phase transition takes place in competition between the interaction causing the system be ordered and the entropy causing a random disturbance. Nature of the phase transition strongly depends on the type of fluctuation which is determined by the structure of the order parameter of the system. As to the critical property of phase transitions, the concept “universality of the critical phenomena” is well established. However, we still find variety of features of ordering processes. In this article, we study effects of various mechanisms which bring large fluctuation in the system, e.g., continuous symmetry of the spin in low dimensions, contradictions among interactions (frustration), randomness of the lattice, quantum fluctuations, and a long range interaction in off-lattice systems.
Keywords: phase transition, critical phenomena, frustration, quantum effect, randomness, long-range interacting system
1. Introduction cal ensemble, all the thermodynamic properties are The phase transition is one of the most derived from the partition function which consists significant phenomena in the nature.1) The most of analytic functions of the temperature, i.e. Ei=kBT familiar phase transition is the boiling phenomenon e , where Ei is an energy of a state i, T is of the water. The state of the water changes at the the temperature, and kB is the Boltzmann con- boiling temperature (’100°C), i.e., from the liquid to stant. the gas. The microscopic description of the system This singular property was one of the main does not change at all at this point. Namely, the topics of the statistical physics in the early twenty interaction among molecules is given by a normal century.2) L. Onsager finally succeeded to obtain the non-singular form which does not depend on the exact form of the free energy in the thermodynamic temperature. However, the macroscopic property limit, and showed a divergence of the specific heat.3) shows the singular dependence on the temperature. This work first demonstrated the intrinsic difference This dependence was first explained by the van der between the so-called microscopic state and the Waals equation of state. This equation takes the macroscopic state. interaction between molecules into the equation of Since then, natures of phase transitions have state of the ideal gas (Boyle–Charles law). Thus, it been extensively studied, and various properties have became clear that the interaction is important for been clarified. One of the most significant properties the phase transition. But it was still mystery that is the universality of the criticality. This indicates the singular behavior of phase transition can be that the critical property of systems only depends on explained by the statistical mechanics. In the canoni- the so-called relevant characteristics, such as the spatial dimensionality of the system, symmetry of the *1 Department of Physics, The University of Tokyo, Tokyo, order parameter, and the range of interaction. The Japan. concept of the universality has been supported by the *2 CREST, JST, Saitama, Japan. 4) † idea of the renormalization group. However, in the Correspondence should be addressed: S. Miyashita, two dimensions, it has turned out that infinite Department of Physics, The University of Tokyo, 7-3-1 Hongo, ff Bunkyo-ku, Tokyo 113-0033, Japan (e-mail: [email protected]. di erent types of critical properties exist in the u-tokyo.ac.jp). studies on the exactly solvable models,5),6) and also doi: 10.2183/pjab.86.643 ©2010 The Japan Academy 644 S. MIYASHITA [Vol. 86, on the conformal field theory.7) There the concept of property is characterized by the singularity at a “relevance” of quantities became rather unclear. critical point TC. For example, the susceptibility In three dimensions, so far we knew little on the diverges as variety of the criticality. At present, the relevant 2 2 hM i hMi quantities in three dimensions are only the dimen- ¼ /jT TCj ; ½4 sionality of spin and the range of interaction. Most kBT models which show the second-order phase transi- and the spontaneous order parameter mS appears as tions in the two dimensions with different critical hMi exponents exhibit the first order phase transition. For mS ¼ lim lim /jTC Tj ; ½5 H!0 N!1 N example, the case of the three-state Potts model in three dimension was very marginal,8),9) but it has which represents the symmetry breaking of the turned out to have a weak first-order transition.10) system. The specificheatC shows a singular Thus, the approach to classify the second order phase dependence as transition is not very powerful in three dimensions. hH2i hHi2 There may be various unknown types of ordered C ¼ 2 /jT TCj : ½6 states in three dimensions, and a new concept for kBT classification would be necessary. Here, the exponents ,, - and . are called “critical The phase transition is understood as competi- exponents”. tion between the interaction which causes the order As an example, we show the critical properties of and the thermal fluctuation (the entropy) which the two-dimensional Ising model on the square lattice causes random configurations. Beside this fundamen- which is the most well-known and a prototype system tal mechanism of phase transition, ordering processes for the phase transition: X X show various aspects depending on the structure of H ¼ J H ; ½7 the order parameter of the system. In this paper, Ising i j i ij i we study several examples of ordering processes in h i systems with some mechanisms for large fluctuations. where < = ±1 and hiji denotes all the nearest There are various types of models, such as spin neighbor pairs on the lattice. In Fig. 1, we depict models with Ising, XY and Heisenberg spins, Potts the specific heat obtained by Onsager3) by the thin model, and the clock model, etc. They are expressed curves. There, we see the divergencepffiffiffi at the critical in the following form point kBTC=J ¼ð2= lnð1 þ 2Þ’2:269 . X X In the figure, we also show the temperature H 0 ¼ JijXiXj H Xi; ½1 dependence of the spontaneous magnetization ob- hiji tained by C. N. Yang11) where Xi represents a local quantities such as the spin at the position of i, and H is an external field conjugate to the order parameter. Here hiji denotes the interacting pairsP and Jij is the coupling constant 0 3 for the pair, andP denotes the sum for the order 0 parameter M ¼ Xi. We assume that the local quantities are on lattice points which are fixed except
in the last section. B 2
Critical property is studied by investigating the C/k order parameter
H s TrMe m 1 hMi¼ ; ½2 Tre H and its fluctuation 0 2 H 0 2 4 2 2 TrM e 2 hM i hMi ¼ hMi ; ½3 kBT/J Tre H Fig. 1. Temperature dependences of the specific heat (thin line) which corresponds to the response to the conjugate and of the spontaneous magnetization of the ferromagnetic two- field H, i.e., ¼ dM=dH. Type of the critical dimensional Ising model [7] with H =0. No. 7] Phase transition in spin systems with various types of fluctuations 645
rather straightforward. However, when the system has additional sources of fluctuation, the phase transition shows various interesting aspects of the ordering. In this paper, we study characteristics of phase transitions of systems in which the fluctuation is enhanced by various reasons, such as the contin- uous symmetry of the order parameter which causes the so-called infrared divergence of fluctuation in low dimensions (d = 1 and 2), contradictions among the interactions (frustration), randomness of the lattice, and quantum fluctuation. We also study the effect of the long range interaction induced by an elastic interaction in off-lattice systems.
Fig. 2. A snapshot of equilibrium spin configuration obtained by 2. XY model (a continuous spin symmetry) a Monte Carlo method near the critical point T = 2.3/J and In the Ising model, the boundary of the two H =0(cf. TC=J 2:269 ). The solid and open circles denote phases is given by a domain wall. On the other hand, < = 1 and < = −1, respectively. i i when the spin has a continuous symmetry, e.g., the 1=8 XY model with S ¼ðcos ; sin Þ: 1 i Xi i ms ¼ 1 4 : ½8 sinh 2K HXY ¼ J cosð i jÞ; ½12 ij The free energy of the Ising model has been obtained h i by various methods,3),5),12)–14) and a new field of the the direction of the spin can change smoothly, so-called “exactly solvable model” has been devel- and the domain wall is not formed. Thus, the oped.15) fluctuation easily occurs, and it causes the so-called In Fig. 2, we plot a typical spin configuration infrared divergence of the fluctuation, and the long near the critical temperature where we find large range order can not appear in two dimensions.17) domains. The linear dimension of the domain Although the long range order does not exist in corresponds to the correlation length 9 the two-dimensional XY model, it has been known 1 that the system exhibits a peculiar critical phenom- rij= hSiSji e for large rij; ½9 enon which is called “Kosterlitz–Thouless transi- rd 2þ tion”.18) where rij is the distance between the sites i and j, and At low temperatures, this system can be 2 is the exponent so-called anomalous dimension. The approximatedP by a harmonic system, H ’ 2 divergence of the correlation length is expressed as J=2 hijið i jÞ , where the periodicity of the angle 3 is not relevant. The spin correlation function decays jT T j ½10 C by a power law19) with 8 = 1. It is known that the exponent 2 is 1/4 for hS S i r ; ½13 the present model. At the critical point, fluctuation of i j ij the order parameter becomes large which results in with a temperature dependent exponent 2 = kBT/4J the divergence of @. Although the susceptibility has in the harmonic approximation. In contrast, the not yet been obtained analytically, it has been correlation function decays exponentially at high established that it diverges with the exponent . = temperatures, where the periodicity of the potential 7/4 of Eq. [4]. This divergence of @ corresponds to cos(3i − 3j) is relevant. This difference is explained the correlation length 9. The exponent 8 is related to by using the picture of vortex-pair association. The . and 2 as vortex is characterized by the vortex number n defined by ¼ð2 Þ : ½11 I This is an example of the scaling relations among the d ¼ 2n : ½14 critical exponents.16) For the Ising ferromagnet, the ordered state is a Because n can be any integer, this type of vortex is simple ferromagnetic state and ordering process is called Z vortex. But vortices with large n cause high 646 S. MIYASHITA [Vol. 86, energies, and thus the vortices with n = ±1 are important. We call them ‘±’ vortex, respectively. The vortex represents a topological defect in config- urations of the system with O(1) symmetry repre- senting the periodicity of the angle. Appearance of single vortex indicates the relevance of the periodicity, and thus it is a symbol of the high temperature phase. On the other hand, in the low temperature phase the interaction is so strong that configurations reflecting the periodicity of the angle cannot appear. Thus, at low temper- atures, the periodicity of the angle is irrelevant. This fact is expressed by the absence of single vortex, which means essentially no vortex. However, at finite temperatures, thermal fluctuation may still cause the Fig. 3. A vortex configuration of the two-dimensional XY model appearance of them in a form of pair of ‘±’ vortices [12] in a transient process of ordering. The open circles denote ‘ ’ ‘−’ which is localized in the space. This change from the the + vortices and the closed circles vortices. free single vortices to the bounded pair vortices is called “vortex association” at the Kosterlitz–Thouless P fi transition. In equilibrium con gurations of the XY then the condition m: around k skm ¼ 0 is auto- model, it is hard to identify vortices clearly. At high matically satisfied. Here, we have to be careful temperatures, configurations are too much disturbed to assign the suffix k and m. Thus, the partition to identify the vortices, and at low temperatures, function is expressed by the integer variable the probability for well-recognize vortex pair is very {hi} 20) low. In Fig. 3, we depict a typical configuration X1 Y ‘ ’ with vortices, where the ± vortices are shown by Z ¼ Ihi hj ðKÞ open and solid circles, respectively. It should be hi¼ 1 hiji fi zN=2 P noted that this is a transient con guration from a X1 1 2 1 ðhi hjÞ random configuration to an aligned one at a low ’ e 2K hiji ; ½19 2 K temperature. hi¼1 The model [12] is transformed to the so-called where z is the number of the nearest neighbor solid-on-solid (SOS) model by the dual transforma- sites. This partition function can be interpreted 21)–23) tion. Using the following identity relation as that a system of the integer variable {hi}: P X K cosð i jÞ 1 Z ¼ Tre hiji H ¼ Jðh h Þ2 ½20 Z Z SOS 2 i j 2 Y X1 2 d hiji isijð i jÞ isij þK cosð Þ ¼ d i e e ; at the temperature J/kT. Regarding hi as the 0 0 2 hiji sij¼ 1 height of solid, this model is called the solid-on- ½15 solid (SOS) model. It is used to study the and integrating over 3i, the partition function is given crystal growth where hi denotes the height of the by surface at the position i.24) Thus, we find that ! Y X1 Y the planar model and the SOS model are in the P dual relation.23) The SOS model is used to study Z ¼ s ;0 Isij ðKÞ; ½16 m: around k km hiji sij¼ 1 k phase transitions of the surface structures such as the roughening transition25) and facet tran- where sition.26) Z 2 Instead of using the planar model, if we use the d is þK cosð Þ 1 s2=2K 21) IsðKÞ¼ e ’ pffiffiffiffiffiffiffiffiffiffi e ½17 Villain model HV 0 2 2 K P X1 1 2 H Kð i j 2m Þ for large K. If we introduce the dual variables e V ¼ e2 hiji ; ½21 m¼ 1 s ¼ h h ; ½18 km k m we have exactly No. 7] Phase transition in spin systems with various types of fluctuations 647
P JF X1 1 2 ðhi hjÞ J Z ¼ e 2K hiji : ½22 AF
hi¼ 1 The model is further transformed by using an identity expression (Poisson sum): (a) (b) Z P P X1 1 1 2 2K ð j kÞ þ 2 ihj j Z ¼ d ie hjki j : ½23
hi¼ 1 1 If we perform the GaussP integration over ?,wefind J2 that only the case of j hj ¼ 0 is relevant, and there the partition function has the form P P X 2 a hi 2 K hihj lnðrijÞ Z ¼ Z0 e i i6¼j ; ½24 (c) (d) hi Fig. 4. Fully frustrated lattices. The bold lines denote antiferro- where Z0 and a are constants independent of {hi}. magnetic bonds and the thin lines ferromagnetic bonds. (a) This model is a neutral two-dimensional Coulomb Antiferromagnetic triangular lattice. (b) The Villain lattice 27) system of particles with the charges {hi}. (Fully frustrated square lattice. (c) The antiferromagnetic model on the triangular lattice with a next nearest neighbor interaction 3. Frustration I: Ising spin systems (dotted line). (d) The Villain model with next nearest neighbor interactions (dotted lines). In the models so far studied, the interaction tends to cause a perfect alignment of the spins. So, all the interactions work cooperatively, and how the function of this type of models decays in a power system is ordered is well defined. However, if there law32),33) are contradictions among the interactions, the order- 1=2 ing process becomes not trivial, and various interest- h i ji rij ; ½26 ing ordering phenomena occur. Although phase which means that the critical temperature is 0 in transitions in the frustrated or competing interac- these models. tions have been studied for a long time, the concept In the followings, we study phase transitions in of frustration was introduced by Toulouse28) in the this type of highly frustrated systems, for which some study of the spin glass. additional interaction is necessary to extract some In this section, we study the case of frustrated ordered configurations from the highly degenerate Ising spin systems in regular lattices.29) As a typical configurations. example, let us consider spins interacting antiferro- 3.1. Phase transitions induced by next-near- magnetically on a triangle lattice (Fig. 4(a)). On the est neighbor interactions in the fully-frustrated triangle, many degenerate ground states exist, that Ising model. Although no phase transition takes is, six states give the ground state for the anti- place in the fully frustrated Ising models, successive ferromagnetic case while two states (all up or down) phase transitions were observed in experiments 34) for the ferromagnetic case. If we consider a larger (CsCoCl3). In order to explain these phase lattice, number of the ground states in the anti- transitions, Mekata introduced next-nearest neighbor ferromagnetic case increases while that of the (nnn) interactions (Fig. 4(c)), and found successive ferromagnetic model remains two. It has been known phase transitions by a mean-field theory. In partic- by exact calculations30) that this antiferromagnetic ular, he found a phase in which a part of spins remain Ising model does not have a critical point at finite disorder due to the frustration and he called this temperatures, and the ground state has a macro- phase “partially disordered (PD) phase”. (Fig. 5(a)). scopic degeneracy W, and thus the system has At a lower temperature, the remaining spins order nonzero entropy at T =0: and form a kind of ferrimagnetic phase. Because of the new feature of the successive phase transitions, SðT ¼ 0Þ¼k ln W ’ 0:338Nk ; ½25 B B detailed investigations on this type of models have where N is the number of the lattice sites. Similar been done. properties have been obtained in the square version First, models in two dimensions were studied. of the fully frustrated model (Villain model31) The present author proposed an exactly solvable (Fig. 4(b))). It is known that the spin correlation model with a next nearest neighbor (nnn) interaction 648 S. MIYASHITA [Vol. 86, in the fully frustrated square lattice (Fig. 4(d)). It (1) was pointed out that the model without the nnn interaction (the original Villain model) is trans- (6) (2) formed into an exactly solvable symmetric eight vertex model,5),33) and it was shown that the spin correlation function in the ground state shows the power law decay [26]. Following the above mentioned (5) (3) Mekata’s idea, we expect that, by setting the nnn interaction, the sublattice order is enhanced and the (4) model exhibits a phase transition. The model with (a) (b) the nnn interaction only on one of the sublattices can still be transformed into the eight vertex model as Fig. 5. (a) The partially disordered (PD) state in the antiferro- well as the Villain model. It was shown that the magnetic Ising model with a next nearest neighbor interaction model with the nnn interaction has a finite critical (Mekata model). The circles denote the disordered sites. (b) Correspondence between the six-fold ground state of the Mekata temperature as a function the strength (say J2) of the model and the six-state clock model. nnn interaction.35) The critical temperature decreases with J2 and vanishes at J2 = 0. Moreover, it was shown that the critical exponent of the specificheat, Thus, the PD phase was regarded as the intermediate (Eq. [6]) of this system changes continuously with J2 temperature region with a large fluctuation. reflecting the peculiar property of the eight vertex But, finally it was discovered that the generalized 5) model. If the frustration is strong, i.e., for small J2, six-state clock model has a new type of ordered struc- the critical exponent , has a negative value with a ture. Although the antiferromagnets on the triangular large absolute value, which means a smooth change lattice (AFT) has the six-fold symmetry (Fig. 5(b)), of the specific heat although it is still singular. This the structure of energy levels for the states is not feature is natural and suggestive for phase transitions necessarily given by the standard six-state clock of frustrated models. But the feature of phase model [27], but it may have a generalized form transition is rather different from that of the original H ¼ " þ " þ " : ½28 Mekata problem. 1 ni;nj 1 2 ni;nj 2 3 ni;nj 3 More direct studies of the antiferromagnet on In the case C1 = C2 = C3 > 0, the model corresponds to the triangular lattice of the Ising spin (AFTI) with the six-state Potts model, which exhibits a first-order nnn interactions were performed. Reflecting the phase transition.9) symmetry of the ground state, the model was mapped It was discovered that, in the case "1 "2 ’ "3, to the six-state clock model by making use of the the model has a new type of intermediate-temper- relation depicted in Fig. 5(b).36)–39) In two dimen- ature phase in which two neighboring states mix sions, the six-state clock model, microscopically.43) In Fig. 6, we depict a snap shot of the intermediate phase. There we set different H ¼ J cosð Þ; ¼ n; n ¼ 0; ...5; ½27 i j i 3 boundary conditions in the left and right sides. At is known to have two KT-type phase transitions the left boundary, the states are restricted to the which are dual to each other, although the systems states 1 and 2 (left) and at the right the states 2 and 3 with n ¯ 4 have a unique self-dual transition. The (right). There are a mixed phase of the states 1 and 2 intermediate temperature phase is a massless phase in the left and that of 2 and 3 in the right. Between in which the correlation function decays in a power them, we find a sharp domain boundary which law.40) Thus, one may consider that the PD phase can indicates the stability of the mixed states. be understood in an analogy to this intermediate KT This state really possesses the property of the phase. PD phase proposed by Mekata,44) because, if we To study properties in three dimensions, the average two neighboring structures in Fig. 5, we have standard three dimensional six-state clock model [27] a disordered site in one of the three sites. Thus, was also studied. It turned out that the nature of finally it was proved that there exists an ordered fluctuation is different from that of the two dimen- state corresponding to the Mekata’s PD phase in sions. It was shown that the model has no inter- three dimensions. This type of phase with mixed mediate temperature phase though it shows a large states is not known so far, and gives a new type of intermediate crossover temperature region.41),42) ordered state. No. 7] Phase transition in spin systems with various types of fluctuations 649
s 1 1
J1 J0 σ1 σ2 C12
s2
T/J 0 0 2 4
2 tanh(J0/2kBT)=tanh (J1/kBT))
Fig. 7. Temperature dependence of the correlation function Fig. 6. Domain structure of mixed states of the generalized six- h 1 2i¼tanhðKeff ðTÞÞ of the decorated bond [29]. Inset shows < state clock model [28] with C1 = 0.1J and C2 = C3 = J. The six the bond structure, where the circles denote the system spins 1 states (n =1,³ 6) are plotted by red, yellow, green, blue, gray, and <2, and the squares the decoration spins s1 and s2. and black squares, respectively. The left part is a mixed phase of n = 1 and 2, and the right part is a mixed phase of n = 2 and 3. The lattice is a cubic lattice with L = 100 and T = 0.4J.An 2 ð J1Þ ). On the other hand, at low temperatures, the intersection of the lattice is depicted. effective coupling constant is approximately given by the sum of interactions of all the three paths, i.e., In three dimensions, it has been pointed out a K(T) ’ (2J1 − J0)/kBT. rich structure of the ordered states, so-called “devil’s Here, we consider a competing case, e.g., J0 =2J stair case” or “devil’s flower”, appears in the and J1 = 1.5J with a unit of energy J. Because the ferromagnetic Ising model with a nearest neighbor direct path is antiferromagnetic (J0 > 0), the effective interaction in one direction (ANNNI model).45) These interaction is antiferromagnetic at high temperatures observations indicate that various unknown rich K(T) < 0. At low temperatures, it is ferromagnetic structures of ordered states exist in higher dimensions because 2J1 > J0. In Fig. 7, temperature dependence for the future studies. of the correlation function C12 ¼h 1 2i¼ 3.2. Reentrant phase transition. Another tanh Keff ðTÞ is plotted. The effective coupling interesting property of the frustrated systems is the disappears at the temperature T0 at which 2 non-monotonic ordering process due to the peculiar tanhðJ0=2kBT0Þ¼tanh ðJ1=kBT0Þ. distribution of degeneracy. The non-monotonicity is We can also provide more complicated types of understood by the idea of the decorated bond.46),47) A reentrant phase transitions in exactly solvable two- typical example of the decorated bond is depicted in dimensional Ising models.48) This reentrant behavior Fig. 7. There, two spins, <1 and <2, which we call the can give a mechanism of temperature dependent system spin, are connected by a direct bond (J0) and configuration of ordered states in random systems, two paths with the spins si (i = 1 and 2) which we where interesting phenomena such as memory and call the decoration spin. The Hamiltonian of the rejuvenation take place.49) It is also known that this decorated bond is given by type of decoration causes a kind of screening effect on the spin state at each site and stabilize it, and thus H ¼ J J ð þ Þðs þ s Þ: ½29 decoratedbond 0 1 2 1 1 2 1 2 the spin dynamics becomes very slow.50) The effective interaction K(Teff) between the system spins is given by 4. Frustration II: Continuous spin systems
KðTÞ 1 2 H 4.1. XY model on the triangular lattice. So Z e eff Tr e decoratedbond=kBT ; 30 0 ¼ s1;s2 ½ far we studied the frustrated Ising model where the where Z0 is a factor independent of H/J HC3 (e) (d) (a) (b) (c) (d) (e) HC2 Fig. 9. Spin configurations of solutions of Eq. [35] for the antiferromagnetic anisotropic Heisenberg model in the mag- netic field H [34]: (a) Umbrella structure, for A =1, (c) ð 1; 1Þ¼ðacosðh=3Þ; 0Þ, ð 2; 2Þ¼ðacosðh=3Þ; 2 =3Þ, ð 3; 3Þ¼ ðacosðh=3Þ; 4 =3Þ. (b) distorted 120° structure (0 ¯ h ¯ 1), for A =1,ð 1; 1Þ¼ð ; 0Þð 2; 2Þ¼ðacosððh þ 1Þ=2Þ; 0Þ, ð 3; 3Þ¼ HC1 ðacosððh þ 1Þ=2Þ; Þ. (c) collinear up-up-down structure. (d) v- 2 shape structure (1 ¯ h ¯ 3), for A =1,ð 1; 1Þ¼ðacosðh þ 3Þ= (b) 4h; 0 , ; acos h2 3 =4h ; 0 , ; acos h2 T Þ ð 2 2Þ¼ð ð þ Þ Þ Þ ð 3 3Þ¼ð ð 0 3=2hÞ; Þ. (e) collinear ferromagnetic structure. Here h = H/3J. TC2 TC1 Fig. 10. A schematic phase diagram of the Ising-like Heisenberg modelpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the coordinate (T,H), where HC1 =3J, HC2 = axis), and the system shows a phase transition of the 2A 1 4A2þ4A 7 2 J, and HC1 =(6A +3)J. Ising universality with the order parameter of the Z2 symmetry. However, because of the frustration, the order parameter has a distorted 120° structure, and Mekata model for the successive phase transitions of nature of the phase transition is different from the frustrated Ising-like models in the triangular lattice. simple Ising case. Indeed successive phase transitions of this type were 63) There are several locally stable structures of the also found in experiments (VCl2, etc.). spins on a triangle in the field as depicted in Fig. 9. In the magnetic field, this model gives compli- The stable configurations in the ground state are cated phases.64) The phase diagram in the field is obtained by depicted in Fig. 10. This type of phase diagram has 8 been also observed in experiments.65) > cos 1ðsin 2 þ sin 3Þ > In the case of A = 1, all the configurations which > H > satisfy > ¼ A sin 1 cos 2 þ cos 3 ; > 3J > H <> cos ðsin þ sin Þ cos þ cos þ cos ¼ ½36 2 3 1 1 1 1 3 > H ½35 > ¼ A sin 2 cos 3 þ cos 1 ; fi > 3J are degenerate. At nite temperatures, however, the > ff 66) > cos ðsin þ sin Þ degeneracy is resolved by the entropy e ect. > 3 1 2 > Generally speaking, the collinear structure is en- :> H ¼ A sin 3 cos 1 þ cos 2 : tropically more favorable than the coplanar struc- 3J ture, and the coplanar structure is more favorable In the case of zero magnetic field, the system than the non-collinear structure (umbrella structure: shows successive phase transitions when we change Fig. 9(a)). The entropy effect is estimated by a the temperature. At a high temperature TC1,thez harmonic analysis of the model: component is ordered, and then at a low temperature H H 0 0 0 0 0 0 ð 1; 2; 3; 1; 2; 3Þ¼ ð 1; 2; 3; 1; 2; 3Þ TC2 the transverse component is ordered to form the 61) @H @H distorted 120° structure as shown in Fig. 10 at þ þ þ sin 0 þ txA^x; ½37 @ 1 @ 0 3 3 H = 0. The former phase transition belongs to the 1 sin 3 3 0 0 universality classes of the six-state clock model, and where i ¼ i þ i, i ¼ i þ i,(i = 1, 2, and 3), the latter to that of the XY model. Both of them are and of the Kosterlitz–Thouless type in two dimensions. In x ¼ð ; ; ; sin 0 ; sin 0 ; sin 0 Þ; ½38 three dimensions, the phase transitions are of normal 1 2 3 1 1 2 2 3 3 second order of the three dimensional XY universal- and A^ is a 6N ×6N matrix. The free energy F in the ity class.62) This mechanism of successive phase harmonic approximation for a given configuration 0 0 0 0 0 0 transition gives an alternate scenario to that of ð 1; 2; 3; 1; 2; 3Þ is given by 652 S. MIYASHITA [Vol. 86, −∆ 9 S/kB Umbrella−shape 2.4 Induced Ferro 2.2 H 5 V−shape − Y shape V−shape 2 Y−shape UUD 0510H =9 HC1=3 C2 Umbrella H/J 0 0 0.5 1 Fig. 11. The ground state entropy [40] for A = 1 for the umbrella, Y-shape and V-shape structures. The big circles show Temperature entropy for the collinear phases, i.e., the up-up-down and Fig. 12. A schematic phase diagram of a weakly XY-like ferromagnetic structures. Heisenberg model in the field. D = 0.01. ! Y fi 1 0 In the gure, we adopted a very small value of D = F= ¼ E þ N lnð Þþ ln ; ½39 0 2 i 0.01. If we use a larger value such as D = 0.1, the i Y-shape structure disappears, while the V shape ^ 68) where 6i is the i-th (non-zero) eigenvalue of A. The structure still remains. zero eigenvalues of A^ denote some continuous 4.3. Antiferromagnet on the Kagome lattice. degeneracies in the groundQ state, and they are Finally, let us refer to the case of the Kagome lattice. removed in the product 0. For example, the model This model has the so-called corner-sharing structure has U(1) symmetry, and thus the uniform rotation and effect of the frustration is more significant than around the z axis gives the zero eigenvalue, and also that in the edge-sharing lattices such as the the model of A = 1 has the degeneracy [36]. Moreover, triangular lattice. The Ising antiferromagnet in this it has been pointed out that the model with A > 1 has lattice has no phase transition and the correlation a non-trivial degeneracy that there is a ground state function decays exponentially even at T =0.69) 0 3 ? for all the values of 1. The zero modes for and give Because of the high degeneracy, even with the : and 2: to the product, respectively. We define the continuous spins, the XY and Heisenberg models do ground state entropy for the configuration by not have a phase transition, either.70) ! Y However, in the Ising-like Heisenberg model, the kB 0 fi S ¼ ln i : ½40 spins have to choose the same con guration from 2 i those given in Fig. 9 at all the triangles. For example, We plot −"S/kB for various configurations as a in a zero or weak field, they have the distorted 120° function of H in Fig. 11.67) structure. There are still two possible configurations Because of the entropy effect, at finite temper- at zero field as depicted in Figs. 13(a) and (b). The atures, phases of the Y-shape structure (Fig. 9(b)), system must choose one of them. This degree of and of the up-up-down structure (Fig. 9(c)), and freedom gives a two-fold degeneracy of the ground of the V-shape structure (Fig. 9(d)) appear.66) In state, and it causes a phase transition of the Ising systems with a weak XY anisotropy, the umbrella universality class at a finite temperature.71) Indeed structure is energetically favorable. However, the the structures of Figs. 13(a) and (b) have nonzero thermal fluctuation still causes the structures of magnetizations m0 ¼ ðA 1Þ=ðA þ 1Þ, respec- Figs. 9(b)–(d), and we have a very complicated phase tively. Therefore, the phase transition causes an diagram in the coordinate (T,H) as depicted in appearance of the spontaneous magnetization. Below Fig. 12 where we used a single-ion type anisotropy the critical temperature the two-fold degeneracy is model instead of the anisotropic coupling model [34] broken, but there still remains macroscopic degener- X X acy. The number of degeneracy is the same as that of H ¼ J S S þ D ðSzÞ2: ½41 i j i the Ising Kagome antiferromagnet in the magnetic ij i h i field.72) An example of the degenerate ground states No. 7] Phase transition in spin systems with various types of fluctuations 653 in the triangular lattice.73) These are examples of the “order by disorder”74) The relaxation process of this entropy-induced selection of configuration was also investigated.75) In three dimensions, the unit cell of frustration is (a) (b) a tetrahedron, and various interesting phenomena have been found in pyrochlore or spinel lattices, such as the spin ice phenomena.76) 5. Effect of randomness Spatial randomness of interactions also causes various peculiar features in the ordering processes. Generally the randomness smears out the critical phenomena.77)–80) In particular, if the specific heat diverges with the exponent , > 0 in the original pure model, the exponents is renormalized to ,X = −,/ (1 − ,)77) in the model with randomness. It was also pointed out that the Ising model on the square lattice with randomly distributed vertical coupling con- (c) stants keeping the horizontal coupling uniform, the fi Fig. 13. (a) Distorted chiral structure with a negative magnet- speci c heat has the essential singularity, i.e., the all 78) ization, (b) distorted chiral structure with a positive magnet- the derivatives of the free energy are continuous. ization, and (c) a snap shot of the ground state spin Whether a system with randomly distributed configuration of antiferromagnetic Ising-like Heisenberg model interaction can have a phase transition or not is a on the Kagome lattice. The dots denote the spins directing the z very interesting problem. Effects of the random field direction. The arrows denote the xy components of the spins. The dotted lines denote the weather-vane lines. on the phase transition were studied, where the reduction of lower critical dimensions for Ising models was discussed.81),82) is depicted in Fig. 13(c). There is no ferrimagnetic To study the phase transition for randomly structure, and the spontaneous magnetization ap- quenched interaction, we have to average the free pears uniformly over the lattice. Thus, we may regard energy, this phase transition as a ferromagnetic phase F ¼ k Tln Zfi fi ½42 transition, and we called this “exotic ferromagnetic quenchrandom B xedcon guration transition”. not the partition function Configurations of the dotted lines connecting the F ¼ k T ln Zfi fi : ½43 slanting spins characterize the degenerated ground annealrandom B xedcon guration state configurations, and we call the line “weather- For the former average, we need to calculate the free vane line”. In a weather-vane line, the slanting spins energy for each fixed random configuration, which is are parallel and each weather-vane loop contributes generally difficult, except for some special cases such 83) kB ln 2 to the entropy. Therefore, a configuration as the so-called Nishimori line. To avoid this with more weather-vane loops is more favorable. The difficulty, the so-called replica trick has been minimum length of the weather-vane loop is six. introduced.84),85) Thus, we expect that a state with maximum number Zn 1 of the weather-vane loop gives the thermodynamic fixedconfiguration Fquenchrandom ¼ lim : ½44 ground state, in which we have a ferrimagnetic long n!0 n range order of the z component of spins. Existence In particular, properties of spin glass have been of the long range order in the ground state due to studied in these decades extensively.82),84),85) The the entropy effect has been discussed also for the combination of the randomness and frustration Heisenberg model.70) causes a very complicated ordering process. Exper- Similar induction of long range order in the imentally, various interesting properties, such as ground state of fully frustrated model was also memory effect, rejuvenation phenomena, etc., have pointed out in the Ising antiferromagnet of S >1/2 been also pointed out.86) 654 S. MIYASHITA [Vol. 86, probability to find arbitrarily large clusters for any p, which causes the non-analytic effect on the free energy. While it has little effect on the equilibrium properties, it has significant effect on the dynamics. For example, the autocorrelation function of spin is affected by the very long relaxation time of the large clusters at the temperature below the critical temper- ature of the pure system (p = 0), and shows a slow relaxation than the simple exponential decay. This type of slow relaxation is called dynamical effect of the Griffiths singularity.95),96) 6. Quantum effect In this section, we discuss effects of the quantum Fig. 14. A diluted configuration with site occupation probability fluctuation on the phase transition. Generally, phase p = 0.5 which is below the critical probability of the percolation. transitions of magnetic systems take place as a macroscopic change of a classical order parameter If the system has a well-defined ground state, the such as the magnetization, and there the quantum system can be regarded as a generalized antiferro- effect is irrelevant at finite temperatures. This type of magnetic state. But, the spin glass shows several order parameter is called “Diagonal Long Range properties inherent to the randomly frustrated Order (DLRO)”. On the other hand, for the super- system, such as the divergence of the non-linear fluidity and superconductivity, the quantum effect is susceptibility.87),88) The analysis of the reprica trick essential. The latter type is called “Off-Diagonal Long was extended to the case of replica symmetry broken Range Order (ODLRO)”. (Parisi) solution which gives a new picture of ordered Usually, quantum fluctuation tends to destroy state of the random system.89) As an alternate the classical ordered state (DLRO). The ground state picture, an idea of extended antiferromagnetic state order–disorder transition in the transverse Ising with weak stiffness has been discussed.90) Moreover, a model97) X X scenario of chirality ordering for the spin glass has H ¼ J z z x ½45 been also proposed.91) Generally, the frustration i j i ij i among the interaction causes a distribution of h i strength of the correlation.92),93) We may define is the most typical example of the quantum phase domains in which spins strongly interact. The transition. When ! exceeds a critical value !C, the structure of the domains plays important role in the ground state becomes disordered, and it is called dynamics. Large clusters relax slowly, which causes “quantum disordered state”. The effect of quantum slow a relaxation of the autocorrelation function of fluctuation is taken into account by the Suzuki– spins in average. Although extensive studies for the Trotter expression of the partition function which nature of the spin glass have been done, here we will is a path-integral representation of the canonical not discuss on them. weight.98) This formula was proposed for the quan- In the diluted ferromagnet, the domains are well tum Monte Carlo method.99) The ground state defined. When dilution probability p exceeds the property of a d-dimensional quantum system is 94) critical percolation concentration pC, the lattice is expressed by the partition function separated into finite domains with probability one. In H Z ¼ lim Tre : ½46 Fig. 14, we depict a snap shot of a lattice of site !1 dilution with p = 0.5 which is above pC for the square This can be expressed by a path-integral or Suzuki– lattice with nearest neighbor interaction. We find Trotter formula in a (d +1)-dimensional space, P P fi z z x n nite clusters on it. In this case, we expect para- H nJ i j Z ¼ Tre ’ Trdþ1 e hiji en i i magnetic behavior. However, it has been pointed out X P P J z z x that randomly diluted ferromagnets have a non- 1 n hiji i j n i 2 2 ¼ hf i gje e i jf i gihf i gj analytic free energy below the critical temperature of f m¼ 1g i P P the non-diluted system. Although the probability for J z z x n hiji i j n i n large clusters is very small, there is a non-vanishing e e i jf i gi; ½47 No. 7] Phase transition in spin systems with various types of fluctuations 655 T=2.0J, Hx=0.8J T=0.02J, Hx=2.0J Imaginary time axis Imaginary time axis Real space Real space (a) (b) Fig. 15. Configurations of the quantum Monte Carlo simulation for the transverse Ising model. (a) Thermal fluctuation, and (b) Quantum fluctuation. where Trd+1 denotes the sum all over the spin states quantum disorder phase, which is called “quantum 100),101) includingP those inserted as a complete set of states Griffiths singularity”. m m fl f i jf i gihf i gj ðm ¼ 1; ...nÞg, and we call n In the above systems, the quantum uctuation “Suzuki–Trotter number”. If we regard the direction reduces the order, but in some cases the quantum of n as a new spatial direction, the model [45] is fluctuation induces an order. In a model of S = 1 with transformed into a (d + 1)-dimensional Ising model a tunneling effect between the state of S =±1 X X with the coupling constants: H z z z 2 X X ¼ J Si Sj þ D ðSi Þ J hiji i H ¼ i;k j;k K2 i;k i;kþ1 ½48 X n þ 2 2 hiji k þ ððSi Þ þðSi Þ Þ; ½50 where i 1 where Si denote the operations to change the K ¼ ln tanh : ½49 magnetization by one, we find the spontaneous 2 2 n magnetization appears in a region where no magnet- Thus, the critical property of a d-dimensional ization exists in the classical case (! = 0).102) quantum system is related to that of a (d + 1)- Recently, various peculiar ground states of dimensional classical system at finite temperature.98) quantum spin systems have been found in low In Fig. 15, we depict configurations in the (d + 1)- dimensions such as the Haldane state.103) In those dimensional system for an one dimensional transverse systems, spatial configuration of the singlet pairs Ising model. The left and right panels show typical plays an important role.104) A transition between configurations for the thermal fluctuation and the different types of ground states gives the quantum quantum fluctuation, respectively. The thermal phase transition.105),106) It is also known that, in fluctuation gives disorder in the real space while the quantum systems, inhomogeneity of the lattice shape quantum fluctuation gives disorder in the imaginary affects on the spin configuration in the ground state time space. significantly, e.g. the edge effect.107) Therefore, In the random systems such as the diluted randomness of lattice shapes such as the dilution system, the randomness exists in the real space but causes various non intuitive properties.108)–110) The not along the imaginary time axis. Thus, the clusters ground state phase transition has been studied in the (d + 1)-dimensional spaces has a rod type extensively,111) but we will not go in details. shape and they are not isotropic random clusters. It is also an interesting problem to study the Because of this fact, it has been known that the similarity of the quantum and thermal fluctuations. Griffiths singularity in random systems has serious As we see in the section 4.2, the fluctuation has effects on static properties in the ground state. important effects on the magnetization process of the Indeed, the magnetic susceptibility diverges in the antiferromagnetic XXZ model in the triangular diluted transverse Ising model in some regions of lattice. Classically, the umbrella structure is the 656 S. MIYASHITA [Vol. 86, fi lowest energy con guration for easy-plane (XY like) 3+ 3+ systems, and the magnetization process M(H)is Fe Fe linear until its saturated value at T = 0. However, existence of a 1/3-plataux was observed in the experiment on an easy-plane (XY-like) magnet 112) CsCuCl3. This observation implies an existence of some easy-axis (Ising like) anisotropy. Because the Low spin state High spin state temperature is low, we do not expect the thermal V(R) fluctuation, and the thermal-fluctuation induced metamagnetic structure which was explained in the section 4.2 is not applicable. However, Nikuni and Shiba pointed out that the quantum fluctuation takes the role to induce the 1/3 plateau by calculating the energy gain of the zero-temperature fluctuation in a spin wave theory.113) This problem was examined by direct numerical studies of the ground state wave- D function (PWFRG114) and DMRG115)).116) There, a metamagnetic process similar to the classical one was R revealed, and a phase diagram in the coordinate RL RH 116),117) (A,H) was obtained where A is the ratio of the Fig. 16. The High spin (HS) and low spin (LS) states, and a exchange energy of the xy component and that of the schematic energy structure as a function of the radius R of the z-component (see Eq. [34]). molecule. As to the ODLRO, their cooperative properties have been studied as the ordering of phase of the HS state (g+) is larger than that of the LS state (g−). wave function, because the ODLRO is regarded as We denote the HS and LS states by < = 1 and −1, the problem of phase coherence of the macroscopic respectively. The partition function of a molecule is wave function. Thus, phase transitions of the super- given by X conductivity were often analyzed by the XY mod- Z ¼ g e D ¼ g e D þ g e D: ½51 el.118) Recently, the nature of ODLRO has been 1 þ ¼ 1 studied more directly by using the quantum Monte Carlo method, and a transition between the Mott This can be rewritten as state and superfluidity,119) and also coexistence of the rffiffiffiffiffiffi X gþ ðD 1k T ln gÞ solid state and superfluidity (supersolid)120) have Z ¼ e 2 B ; ½52 1 g been clarified. ¼ 1 where g = g+/g−. This partition function can be 7. Phase transitions in the spin-crossover regarded as that of a model with a temperature type models dependent field 7.1. Spin-crossover transition. So far, we 1 studied orderings of the direction of spins on lattices. H ¼ D þ kBT ln g: ½53 2 However, the spin itself on each lattice site may also change as a function of parameters. One of typical The importance of the interaction among examples is the spin-crossover transition,121) where molecules for the SC transition was pointed out in the spin of an atom changes between the high spin the observation of the specific heat,122) and actually (HS) state and the low spin (LS) state depending on the transition often takes place as a discontinuous the ligand field as depicted in Fig. 16. When the transition. The variety of transitions, e.g., the ligand field is weak, the spins of electrons align due to smooth change and discontinuous change can be the Hund law and the total spin of the atom S is attributed to relations among parameters. If we large. On the other hand when the ligand field is include an interactions among spin states Hint, the strong the electrons occupy the low energy states and Hamiltonian of the system is given by S is small. X 1 Here, we consider cases in which the energy of H ¼ H ðf gÞ þ D k T lnðgÞ : ½54 int i 2 B i the LS state is low by 2D, while the degeneracy of the i No. 7] Phase transition in spin systems with various types of fluctuations 657 The simplest choice of the interactionP is the 2 H/TC (III) ferromagnetic Ising model123) H ¼ J . int i j (V) hiji For this model, we can obtain the phase diagram of (IV) the SC system by making use of that of the spinodal field ferromagnetic systems with the temperature depend- (I) ent magnetic field [53]. The free energy of the ferromagnetic Ising model (II) X X coexistence line disordered state 0 T/T Hint ¼ J i j H i: ½55 0 1 2 C hiji i in the mean-field theory is given by 1 Fðm; T; HÞ¼ zJm2 k T lnð2 coshð zJm þ HÞÞ; 2 B ½56 T3 T2 T1 2 LS state HS state where m ¼h ii. The spinodal point is given by metastable hysteresis 2 HS state @Fðm; T; HÞ @ Fðm; T; HÞ Spin−crossover states along the case (IV) ¼ 0 and ¼ 0; ½57 @m @m2 Fig. 17. Phase diagram of the ferromagnetic system in the and we have the spinodal field as coordinate (T,H) with the spinodal field [58]. The temperature sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 dependences of the SC models [54] for various cases (I)–(V) are 1 þ 1 1 plotted (see the text). Phases along the line (IV) are indicated 1 1 B zJ C fi HSP ¼ zJ 1 ln@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA: ½58 blow the gure, where T1 and T2 denote the temperatures of the zJ 2 1 upper limit and lower limit of the thermal hysteresis, respec- 1 1 zJ tively, and T3 denotes the temperature below which the Let the critical temperature of the system be TC = zJ, metastable HS state appears. and then the phase diagram in the coordinate (T,H) is schematically given by Fig. 17 with the lines If D > zJ and D < DC1, the line [53] crosses the indicating spinodal field. Within the spinodal fields spinodal lines as plotted by the dotted line (III). [58], there is a metastable state with the magnet- If D < zJ and D < DC1, beside the thermal ization opposite to the field H. In the present case, hysteresis between T1 and T2, the line [53] again the spinodal field at T =0iszJ. crosses the spinodal line [58] at a low temperature T3 It has turned out that there are various ordering (the line IV), because the slop of the spinodal line [58] patterns of the HS fraction124),125) is infinite at T = 0. Below this crossing point (T < T ), the HS state is metastable. This low f ðTÞ¼ðmðT;HÞþ1Þ=2: ½59 3 HS temperature metastable HS state may explain the In this phase diagram, temperature dependence of long life time of the LIESST (light-induced excited the model [54] is given by the straight lines (I–V) spin state trapping) state.124) Existence of such with the slope given by the relation [53]. For large metastable state has been confirmed experimentally D, the line of [53] crosses the line of H = 0 at a in a charge transfer material.126) higher temperature than TC (along the line I). In If D < zJ and D > DC1, although there is no this case the change of m(T,H) along this line is thermal hysteresis, the low temperature metastable smooth. That is, the change of fHS(T) for large D is HS state exists (along the line II). The critical value smooth. of D for the low temperature metastable state is On the other hand, for small D, the line crosses DC2 ¼ zJ: ½61 the line of H = 0 at a temperature below TC (the lines III–V). There, fHS(T) changes discontinuously at If the line [53] stays inside the metastable region H = 0, and fHS(T) shows a thermal hysteresis between (the line V), the HS state remains metastable at all the lines of the spinodal fields. The relation [53] the temperatures. The critical value of D for this case indicates that the thermal hysteresis takes place for is given by 1 1 D 2 Type I D/zJ Type III 1 Type IV Type II Type V 0 0 2 4 ln(g) D=12 g=20 D=5 g=5 D=6 g=20 1 1 1 fHS fHS fHS 0 0 0 0 5 10 15 0105 0 246 T T T (TypeI) (TypeII) (TypeIII) D=5.5 g=20 D=5 g=20 1 1 fHS fHS 0 0 0 2460 246 T T (TypeIV) (TypeV) Fig. 18. Phase diagram of the ordering patterns fHS(T) in the coordinate ðD=zJ; ln gÞ. Ordering patterns for various values of D: (Type I) smooth change for D > DC, (Type II) smooth change and a metastable branch at low temperatures for D > DC1 and D < DC2, (Type III) discontinuous change for DC1 > D > DC2, (Type IV) discontinuous change and a metastable branch at low temperatures for DC1 > D and D < DC2, and (Type V) HS is always stable or metastable for D < DC3. In Fig. 18, types of ordering processes corre- in the change of other parameters such as the sponding the lines (I–V), and the phase diagram in pressure and stiffness of the elastic constant. the coordinate ðln g; D=zJÞ are depicted.124),125) We This type of description can be applied to many find that the type IV appears for all the sequences in systems where the energy and the degeneracy of local the parameter space. We call this feature “generic degree of freedom compete, e.g. the charge transfer sequence of the temperature dependence of ordering (CT) materials,127) and we expect the present generic fHSðTÞ”. We found this type of sequence also appears sequence will be found in all the such materials. No. 7] Phase transition in spin systems with various types of fluctuations 659 (a) (b) Fig. 19. Deformation due to the size difference of molecules (a) A typical configuration near the critical point in the ferromagnetic Fig. 20. A typical configuration near the critical point. T = Ising model R = R . (b) The lattice deformation in the elastic H L 0.2 ’ TC, D = 0.6, k = 40, g = 20, RHS = 1.1RLS. model (RH > RL) with the same spin configuration. 7.2. Mean-field phase transition due to the ferromagnetic Ising model. It turned out that the elastic interaction. So far, we studied the spin- model exhibits critical behavior of the universality crossover phenomena in the analogy of ferromagnetic class of the mean-field model.131) model on a lattice. However, a significant feature of Realization of the mean-field universality class the SC system is the volume change between the HS has been discussed for the models in higher (D >4) and LS states. As we depicted in Fig. 19, the volume dimensions or models with a long range exchange change causes a distortion of the lattice. Therefore, in model.132) In contrast, the SC model consists of a this case, an elastic energy due to the deformation of normal short range elastic model in three dimensional the lattice must be taken into account.128) The elastic materials, and thus we expect this mean-field interaction favors non-deformed structure. In the universality class will be found in a wide range of previous subsection, we introduced a nearest neigh- real materials. bor ferromagnetic interaction Eq. [55] between the As a consequence of the mean-field criticality, we molecules to induce the cooperative behavior. How- found that the model does not show configurations ever, it is expected that even in systems without the with large domains even near the critical point131) as short range interaction, the elastic interaction can depicted in Fig. 20. That is, the model does not show induce a cooperative effect. Thus we studied a model divergence of the correlation length of the order with only an elastic interaction between the mole- parameter. The model exhibits a symmetry breaking cules, e.g., at the critical point keeping the uniformity of the X configuration. This fact indicates that the model will k1 2 V ¼ ½rij ðRi þ RjÞ not show the critical opalescence which is a symbol of 2 hi;ji the phase transition of the short range models. X k pffiffiffi At the end points of the hysteresis loop, the þ 2 ½r 2ðR þ R Þ 2; ½63 2 ij i j spinodal phenomenon takes place.133) We expect that i;j hh ii at the points the nucleation processes take place and where rij is the distance between the ith and jth sites, inhomogeneous configurations appear, which is an and Ri and Rj are the molecular radii, RHS and RLS essential feature of the relaxation from the metasta- for HS and LS, respectively. In the simulation we ble state in short-range interaction models. However, adopted the ratio RHS/RLS = 1.1 and the elastic in the present model, the configuration is kept constants k1/k2 =10 with k1 = 50. uniform. Moreover, the spinodal phenomenon occurs This scenario of phase transition due to the as a true critical change in the present elastic model, elastic interaction has been confirmed in both studies while it is a crossover in the short range model due to of molecular dynamics method129) and Monte Carlo the local nucleation processes. A similar threshold method.130) Because the elastic interaction causes singular behavior occurs in the photo-excitation an effective long range interaction among the spin process from the LS states to a photo-induced HS states, characteristics of the critical phenomena of states (LIESST), and the critical properties of the the model change from that of the short range switching have been studied.134) 660 S. MIYASHITA [Vol. 86, Acknowledgments The author would like to thank all the collabo- rators. 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(2006) Magnetic properties and metastable new aspects of dynamical processes. J. Phys. Conf. states in spin-crossover transition of Co-Fe Ser. 148, 012027. Prussian blue analogues. J. Phys. Soc. Jpn. 75, 114603. 126) Tokoro, H., Miyashita, S., Hashimoto, K. and (Received Feb. 4, 2010; accepted May 25, 2010) 666 S. MIYASHITA [Vol. 86, Profile Seiji Miyashita was born in 1954 and started research career in 1976 with the studies on the phase transition of the two-dimensional XY model under the guidance of Professor Masuo Suzuki at the graduate school of Science, after graduating from the Faculty of Science at the University of Tokyo. He received his doctorate (Doctor of Science) for the study on phase transitions without the long range order in 1981, and then became a Research Associate in the Department of Physics. In 1988, he became an associate Professor of Kyoto University. He was promoted to Professor at Osaka University in 1995. He moved to Department of Applied Physics of the University of Tokyo in 1999. He was awarded the Inoue Prize for Science in 2001 on theoretical studies on magnetic phase transition. In 2005, he moved to Department of Physics of the University of Tokyo where he is working now.