Applicability of effective field theory cluster approximations for investigation of geometrically frustrated magnetic systems: Antiferromagnetic model on kagome lattice

Physica A 514 (2019) 644–657

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Physica A journal homepage: www.elsevier.com/locate/physa

Applicability of effective field theory cluster approximations for investigation of geometrically frustrated magnetic systems: Antiferromagnetic model on kagome lattice E. Jurčišinová a,b , M. Jurčišin a,b,c ,

∗

a

Institute of Experimental Physics, SAS, Watsonova 47, 040 01 Košice, Slovakia Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Moscow Region, Russian Federation Department of Theoretical Physics and Astrophysics, Faculty of Science, P.J. Šafárik University, Park Angelinum 9, 040 01 Košice, Slovakia

b c

highlights • • • • •

Dependence of magnetization properties of the model on cluster size is studied. The formation of the magnetization plateaus for low temperatures is shown. The existence of the single-point ground states is also shown. Anomalies of the specific heat capacity are investigated. Dependence of the critical temperature of the ferromagnetic model on the size of cluster approximation is studied.

article

info

Article history: Received 21 March 2018 Received in revised form 4 June 2018 Available online xxxx Keywords: Large cluster effective field theory Geometric frustration Kagome lattice Ground states Specific heat capacity anomalies Critical temperatures

a b s t r a c t We investigate the antiferromagnetic spin-1/2 Ising model in the presence of the external magnetic field on the geometrically frustrated kagome lattice using the effective field theory cluster approximation up to the size of the cluster consisting of 12 connected sites which form typical basic geometrical structure of the kagome lattice. The magnetization properties, systems of the ground states, as well as the specific heat capacity behavior of the model are investigated depending on the cluster size. It is shown that properties of the model related to the frustration, such as the formation of discrete system of ground states, strongly depend on the used cluster approximation. It is shown that besides physically relevant ground states the model also exhibits the existence of ground states which are related to the used effective field theory technique and should disappear in the limit n → ∞ of the n-site cluster approximation with simultaneous formation of physically relevant ground states of the Ising model on the kagome lattice. The unphysical origin of these ground states is also demonstrated by the low temperature behavior of the specific heat capacity which exhibits even inverse Schottky peaks with negative values of the specific heat capacity which are reduced with increasing of the cluster size approximation towards physically acceptable behavior. For completeness, the dependence of the position of the critical temperature and the behavior of the spontaneous magnetization and specific heat capacity of the ferromagnetic model in the zero external magnetic field on the size of the cluster approximation is also briefly discussed. © 2018 Elsevier B.V. All rights reserved.

∗ Corresponding author at: Institute of Experimental Physics, SAS, Watsonova 47, 040 01 Košice, Slovakia. E-mail addresses: [email protected] (E. Jurčišinová), [email protected] (M. Jurčišin). https://doi.org/10.1016/j.physa.2018.09.147 0378-4371/© 2018 Elsevier B.V. All rights reserved.

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1. Introduction The phenomenon of frustration in magnetic systems related to the elementary geometrical properties of various regular lattices [1] represents one of the most phenomenologically interesting and important phenomenon widely studied experimentally as well as theoretically (see, e.g., Refs. [2–7] and references cited therein). Although traditionally it is supposed that typical properties of the frustrated magnetic systems, e.g., such as the formation of discrete system of ground states with the existence of the magnetization plateaus or the existence of various anomalies in the low temperature behavior of the specific heat capacity, are related to the quantum nature of the magnetism, nevertheless it is quite interesting that a lot of properties of frustrated systems can be investigated even by using well-defined classical lattice models of the statistical mechanics. Here, it is important to stress that from pure theoretical point of view, without doubt, the most significant are models which are exactly solvable because in the framework of such models many nontrivial physical properties of real magnetic systems can be understood on the fundamental level. In this respect, a notorious example of such a classical model is the antiferromagnetic spin-1/2 Ising model on the triangular lattice or kagome lattice which is exactly solvable on both these regular two-dimensional lattices but only in the zero external magnetic field [8,9]. At the same time, it is wellknown that the most interesting peculiarities in the behavior of frustrated systems are observed in the nonzero external field. However, even such simple classical model as the Ising model of uniaxial magnetic systems in the external magnetic field is exactly solvable only on one-dimensional lattices. Moreover, while exact solutions of the Ising model on regular two-dimensional lattices exist in zero external magnetic field (see, e.g., Refs. [10–12] and references cited therein) there is no exact solution of the Ising model in three dimensions at all. In this situation there are, in principle, two possible ways to investigate theoretically various magnetic and thermodynamic properties of two- and three-dimensional frustrated systems in nonzero magnetic field. Both of them represent certain approximations of a given statistical model on a regular lattice. First possibility is to try to investigate exactly a given model on an approximate lattice which take into account at least basic geometric properties of the corresponding regular lattice responsible, e.g., for the geometric frustration. On the other hand, the second possibility is to approximate a given statistical model in an appropriate way and then consider it on the regular lattice. A typical example of approximate models of the first kind are various statistical models on the well-defined so-called Husimi recursive trees and lattices [13–15], i.e., on the corresponding generalized Bethe-like trees and lattices [12], which are taken in a suitable form for taking into account important basic geometrical properties relevant for studied physical phenomenon. Such models can always be investigated numerically with arbitrary precision by using the recursive relation technique [16–30] but very often they can be solved exactly even in a closed analytical form [31–43]. Note that results obtained in the framework of the investigation of the Ising model in the presence of the external magnetic field on the kagome-like Husimi recursive lattice [33] or on the recursive lattice with pyrochlore structure [36] are in an excellent agreement with the corresponding results obtained by the Monte Carlo simulations [44,45]. Moreover, theoretical results obtained on the recursive lattices are also in a good agreement with thermodynamical properties of real frustrated materials. For example, the adiabatic cooling processes experimentally measured on the pyrochlore material Er2 Ti2 O7 [46] are in a very good agreement, at least qualitatively, with those obtained theoretically on the tetrahedron recursive lattice [42]. It means that classical spin models on appropriate recursive lattices represent a suitable and, at the same time, effective tool for theoretical investigation of frustrated systems in the nonzero external magnetic field. On the other hand, the mean field theory (see, e.g., Ref. [12]) and the effective field theory (EFT) cluster approximations based on the Callen–Suzuki identity [47,48] together with the differential operator technique [49,50] represent a typical examples of approximate models studied on regular two- and three-dimensional lattices [51]. Here, an interesting question arises immediately, namely, whether it is possible to describe, at least qualitatively, the basic magnetic and thermodynamic properties of various geometrically frustrated magnetic systems in nonzero external magnetic field in the same manner or similarly as, e.g., in the framework of the Husimi recursive lattice approximation. In this respect, in the present paper we shall investigate the antiferromagnetic spin-1/2 Ising model in the external magnetic field on the two-dimensional geometrically frustrated kagome lattice using the EFT cluster approximations up to the size of the cluster consisting of 12 connected sites which form typical basic geometrical structure of the kagome lattice. As we shall show, the EFT even in the framework of its simplest single-site cluster form describes qualitatively basic property of the studied frustrated model, namely, it predicts the formation of a discrete system of ground states of the model with defined values of magnetization in zero temperature limit. At the same time, besides the plateau-like ground states the model also exhibits the existence of the single-point ground states which are realized for exact values of the external magnetic field. This property is again in accordance with exact results obtained on the kagome-like Husimi recursive lattice [33]. Note that their formation can also be seen in the framework of the Monte Carlo simulations [44]. However, as we shall see, in addition to the well-defined ground states existence of which is independent of the used cluster approximation and which are in agreement with the Monte Carlo simulations as well as with the exact calculations on the corresponding recursive lattice the model also exhibits the existence of two artificial plateau ground states and one single-point ground state magnetization values of which tend to approach each other with increasing of the cluster approximation. At the same time, it can naturally be supposed that they will disappear in the n-site cluster approximation in the limit n → ∞ with simultaneous formation of the single plateau ground state with absolute value of the magnetization |m| = 1/3 in the interval 0 < |H /J | < 4 again in the full accordance with the Monte Carlo simulations [44] as well as with the exact results obtained on the kagome-like Husimi lattice [33].

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Fig. 1. The kagome lattice with explicit division into three different sublattices denoted as A, B, and C .

We shall also investigate the behavior of the specific heat capacity of the model and compare it to the corresponding behavior of the specific heat capacity in the model on the kagome-like Husimi recursive lattice. It will be shown that the specific heat capacity exhibits the Schottky type two-peak anomaly at low temperatures for the values of the external magnetic field from the left as well as right vicinity of the values for which the single-point ground states are formed. As we shall see, the temperature behavior of the specific heat capacity weakly depends on the cluster approximation for the magnetic fields for which the model exhibits formation of physically acceptable ground states and, in these cases, its behavior is similar as for the model on the kagome-like Husimi recursive lattice. On the other hand, however, the specific heat capacity shows strong dependence on the chosen cluster approximation for the magnetic fields for which ‘‘false’’ ground states are formed. Here, the behavior of the specific heat capacity can exhibit even unphysical ‘‘Schottky-type’’ behavior with negative values. However, again with increasing of the cluster approximation the low-temperature specific heat capacity behavior changes towards the specific heat capacity behavior in the framework of the model on the corresponding Husimi recursive lattice. Finally, we shall also briefly discuss the dependence of the position of the critical temperature of the ferromagnetic model in zero external magnetic field on the cluster approximation and obtained results will be compared to those obtained within the model on the kagome-like Husimi recursive lattice as well as to the exact value on the kagome lattice. Besides, we shall also investigate the dependence of the specific heat capacity with increasing of the cluster size approximation. It is shown that the specific heat capacity exhibits similar behavior for all cluster approximations with standard EFT jump at the critical point. 2. Model Let us consider the spin-1/2 Ising model in the presence of the external magnetic field on the regular two-dimensional kagome lattice (see Fig. 1) described by the Hamiltonian H = −J

∑ ⟨i j ⟩

si sj − H

∑

si

(1)

i

where the first sum runs over all nearest neighbor spin pairs and the second sum runs over all sites of the lattice, J represents the antiferromagnetic (J < 0) or ferromagnetic (J > 0) interaction parameter, H is the external magnetic field, and each variable si acquires one of two possible values ±1. As was already mentioned in the Introduction, our main aim is to investigate properties of the antiferromagnetic model by using the effective-field theory approximation based on the differential operator technique applied to spin identities [50,51].

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In the framework of the EFT with general n-site cluster approximation one divides the full Hamiltonian (1) into two parts H = H(n) + H′ ,

(2)

(n)

′

where H represents the part of the Hamiltonian related to the given n-site spin cluster and H is the remaining part of the whole Hamiltonian. In our case, the Hamiltonian H(n) can be written in the following general form H(n) = −J

∑

n ∑

si sj −

⟨i j ⟩

si (Hi + H ) ,

(3)

i=1

where the first sum runs over nearest neighbor spin pairs of the chosen n-site cluster and the second sum runs over all sites of the same cluster with Hi = J

zi ∑

sj ,

(4)

j=1

where the summation is performed over all remaining neighboring spin sites of si which do not belong into the cluster. Note also that a general EFT n-site cluster formulation of the spin-1/2 Ising model on an arbitrary lattice was performed in Ref. [52], therefore it is not necessary to repeat it here. In what follows, first of all, our aim is to investigate the influence of the cluster size in the EFT approximation on the thermodynamic properties of the antiferromagnetic system on the geometrically frustrated kagome lattice in the presence of the external magnetic field. Due to the geometric frustration of the studied magnetic system, to investigate it correctly, it is necessary to take into account the basic triangular symmetry of the kagome lattice, namely, it is necessary to divide the lattice into three independent sublattices denoted by A, B, and C as it is shown explicitly in Fig. 1. It means that, considering these three different sublattices, the total magnetization m of the system is given as follows m = (mA + mB + mC )/3,

(5)

where, in the framework of the n-site cluster EFT, the sublattice magnetizations mX , X = A, B, C are calculated as thermal (X ) average ⟨si ⟩ of the corresponding single spin

⟨ (X ) si

mX ≡ ⟨

⟩=

(X )

Trn si

(

exp −β H(n)

(

Trn exp −β H(n)

)

)⟩ ,

(6)

where β = 1/(kB T ), T is the temperature, kB is the Boltzmann constant, and Trn represents the trace over all lattice sites belonging into the considered n-site cluster of the kagome lattice. Now, using the differential operator technique [50,51] together with the well-known spin-1/2 van der Waerden identity (exp(asi ) = cosh a + si sinh a) and applying the simplest possible approximation to decouple the thermal average of spin products (⟨si1 si2 . . . sip ⟩ ≈ ⟨si1 ⟩⟨si2 ⟩ . . . ⟨sip ⟩) one finally comes to the system of three equations of the following general form mA = fA (mA , mB , mC ; . . .),

(7)

mB = fB (mA , mB , mC ; . . .),

(8)

mC = fC (mA , mB , mC ; . . .),

(9)

where fX , X = A, B, C are polynomial functions with respect to mX , explicit forms of which depend on the cluster size approximation, and dots represent all other model parameters on which functions fX depend, i.e., temperature, external magnetic field, and the nearest neighbor interaction parameter. For given values of the model parameters the system of Eqs. (7)–(9) gives all possible values of sublattice magnetizations mX , X = A, B, C . Direct numerical analysis of these system of equations for n-site cluster approximation up to n = 12 shows that in all studied cases, regardless of the model parameter values, the system of Eqs. (7)–(9) has only one solution which is symmetric, i.e., mA = mB = mC . It means that, in the present case of the antiferromagnetic system on the kagome lattice, all sublattices are completely equivalent and, as a consequence, the total magnetization per site is directly given by an arbitrary sublattice magnetization, i.e., m = mA = mB = mC . Here, it is important to bear in mind that the fact that in our case all sublattices are equivalent to each other does not mean that this conclusion is valid generally for all geometrically frustrated lattices. Rather it is a nontrivial exceptional fact related to the kagome lattice. It means that, when studying frustrated magnetic systems, it is always necessary to start the investigation taking into account the basic symmetry of the lattice responsible for the frustration [20,29]. However, in the case of the kagome lattice the situation seems to be very similar to the case of the model on the kagome-like Husimi recursive lattice where also only one solution exists which is symmetric [33]. Note also that the fact that the antiferromagnetic model on the kagome lattice exhibits only one solution regardless of the model parameter values as well as regardless of the cluster size approximation means that the model does not exhibit first order phase transitions.

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Fig. 2. The explicit form of all spin clusters on the kagome lattice investigated in the present study.

As it was already mentioned, in the present study we shall investigate the antiferromagnetic as well as ferromagnetic spin-1/2 Ising model on the kagome lattice using the EFT technique considered on various clusters up to the 12-site cluster approximation. The explicit form of all studied clusters is shown in Fig. 2. As it follows from this figure we shall consider 1, 2, 3, 5, 7, 9, and 12-site clusters taken in the form up to the elementary closed structure built up by six connected triangles which form basic typical structure on the kagome lattice. 3. Results and discussion 3.1. Magnetization and the system of ground states of the antiferromagnetic model As it follows from the Monte Carlo analysis [44] as well as, e.g., from the exact results obtained on the kagome-like Husimi recursive lattice [33], the antiferromagnetic spin-1/2 Ising model in the external magnetic field exhibits, in addition to the standard saturated ground state with |m| = 1, the formation of the plateau ground state with absolute value of magnetization |m| = 1/3 in the interval 0 < |H /J | < 4. However, as it follows from Fig. 3, where the dependence of the magnetization on the absolute value of the reduced external magnetic field for various values of the reduced temperature and for 1, 3, 7, and 12-site cluster approximations is present, the EFT approach also leads to the formation of nontrivial magnetization plateaus at low temperatures in the same interval 0 < |H /J | < 4 but now all considered cluster approximations exhibit the existence of two different plateaus in the interval 0 < |H /J | < 4, one of which is formed in the interval 0 < |H /J | < 2 and the second one is realized for 2 < |H /J | < 4. Moreover, while the intervals of the external magnetic field where these plateaus are formed do not depend on the size of the cluster approximation, the values of their magnetization strongly depend on the chosen approximation. At the same time, with increasing of the cluster approximation the difference between values of their magnetizations significantly decreases. This strong dependence of the magnetization properties of the formed plateaus on the chosen cluster approximation means that the formation of these two different plateaus can in fact be considered only as an artefact of the EFT approximation. Let us analyze the magnetization properties of the formed plateaus in more detail. In this respect, in Fig. 4 the low temperature behavior of the magnetization in the interval 0 < |H /J | < 4 is shown for kB T /|J | = 0.1 for all studied cluster approximations. Here, first of all, one can immediately see that the most significant shifts of the formed plateaus take place for cluster approximations which take into account additional structural information about the kagome lattice. For example, the corrections of the 2-site cluster approximation to the plateau magnetization values given by the single-site cluster approximation are negligible in comparison to the corresponding corrections obtained by the 3-site cluster approximation which takes into account basic triangular structure of the kagome lattice (see the corresponding curves for 1, 2, and 3−site

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Fig. 3. The dependence of the magnetization per site of the model on the absolute value of the external magnetic field for various values of the temperature and for the n-site cluster approximations with n = 1, 3, 7, and 12.

Fig. 4. The dependence of the magnetization of the model on the absolute value of the external magnetic field in the interval 0 < |H /J | < 4 for kB T /|J | = 0.1 and for all used cluster approximations.

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Fig. 5. The explicit form of full systems of the ground states of the model for the n-site cluster approximations with n = 1, 3, 7, and 12.

cluster approximations in Fig. 4). Similar significant changes in the plateau magnetization values are obtained when two connected triangles are taken into account, i.e, in the framework of the 5-site cluster approximation, and then when typical basic kagome lattice structure is taken into account which consists of six connected triangles in the framework of the 12-site cluster approximation (again see Fig. 4). The corresponding systems of the ground states of the model are formed in the limit T → 0. The full systems of the ground states of the model for 1, 3, 7, and 12−site cluster approximations are shown explicitly in Fig. 5. As it follows from this figure, besides the aforementioned plateau-like ground states the so-called single-point ground states are formed on the borders of the intervals of the external magnetic field for which the corresponding plateau ground states are realized (the filled circles in Fig. 5). The magnetization values of the single-point ground states, except of the one which is realized at H = 0, also depend on the chosen cluster approximation. At the same time, their positions with respect to the external magnetic field are fixed. The numerical values of the magnetization of all ground states of the model for all investigated cluster approximations are summarized and shown in Table 1. The table offers clear survey of the ground states magnetization dependence on the cluster approximation. Here, it is evident that the model exhibits two ground states which do not depend on the used cluster approximation, namely, the single-point ground state at zero external magnetic field with magnetization m = 0 and the saturated ground state with |m| = 1 which is realized for |H /J | > 4. Further, it is also evident that the very existence of the single-point ground state which is realized for |H /J | = 4 is not an artefact of the EFT approach since his magnetization only weakly depends on the used cluster approximation (see Table 1) and are very close to the corresponding value |m| = 0.6 obtained in the framework of the Husimi lattice approximation [33]. Weak dependence on the approximation is also seen for the second nontrivial single-point ground state which is realized for |H /J | = 2. Here, however, from the fact that the neighboring plateaus approach one to another with increasing of the cluster approximation (see Figs. 4 and 5 as well as Table 1) one can naturally come to the conclusion that in the limit n → ∞ of the n-site cluster approximation both plateau ground states together with the bordering single-point ground state join into the single plateau ground state with magnetization |m| = 1/3 in accordance with results obtained in the framework of the Monte Carlo approach [44] as well as in the framework of the exact analysis of the model on the kagome-like Husimi recursive lattice [33]. The fact that the existence of the single-point ground state at |H /J | = 2 together with two different neighboring plateaus seems to be only an artefact of the EFT approximation technique will be also demonstrated by the low temperature behavior of the specific heat capacity for external magnetic fields from the vicinity of |H /J | = 2. As we shall show in the next subsection the specific heat capacity can even exhibit unphysical behavior in this region.

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Table 1 The numerical values of the magnetization of all ground states of the model for all investigated n-site cluster approximations.

1-site 2-site 3-site 5-site 7-site 9-site 12-site

| HJ |= 0

0 < | HJ |< 2

| HJ |= 2

2 < | HJ |< 4

| HJ |= 4

| HJ |> 4

0 0 0 0 0 0 0

0.1444 0.1481 0.1609 0.1788 0.1926 0.2033 0.2312

0.2829 0.2862 0.2859 0.2847 0.2872 0.2880 0.2912

0.4490 0.4437 0.4160 0.3866 0.3789 0.3712 0.3548

0.5952 0.5983 0.5917 0.5907 0.5912 0.5913 0.5923

1 1 1 1 1 1 1

Fig. 6. The temperature behavior of the specific heat capacity for H = 0 and for various n-site cluster approximations with n = 2, 3, 5, and 12. The results are compared to the corresponding behavior of the specific heat capacity in the model on the kagome-like Husimi recursive lattice (red dashed curve).

3.2. Specific heat capacity of the antiferromagnetic model The specific heat capacity of the model can be investigated using the standard relation between the internal energy per site u = U /N, where U ≡ ⟨H⟩ and N is number of sites of the lattice, and the specific heat capacity in constant external magnetic field cH , namely, cH =

∂u . ∂T

(10)

It is well known that frustrated systems exhibit anomalous low temperature behavior of the specific heat capacity such as the appearance of an additional peak in the temperature dependence of the specific heat capacity for well defined regions of the external magnetic field which is generally known as the Schottky type anomaly. As was shown, e.g., in Ref. [43], the anomalous Schottky-type behavior of the specific heat capacity is directly related to the existence of highly macroscopically degenerated single-point ground states and, as a result, such kind of anomalous behavior is observed for magnetic fields from the vicinity of the values of the magnetic field for which these single-point ground states are formed. As we shall see, similar anomalous behavior of the specific heat capacity is also present in the studied model in the framework of the EFT approach. But, as was already discussed in the previous subsection, the magnetization behavior of the nontrivial plateau-like ground states together with the single-point ground state formed at H /|J | = 2 demonstrate a tendency to approach each other with improving approximation (see Fig. 4 and Table 1) and, as a consequence, it is natural to assume that they merge into one plateau ground state with |m| = 1/3 in the whole interval 0 < |H /J | < 4 in the n-site cluster EFT approximation when n → ∞ in accordance with other methods. Thus, the existence of these two different plateaus together with the existence of the corresponding separating single-point ground state represents only an artefact of the EFT approximation. The validity of this assumption is also supported by the unphysical temperature behavior of the specific heat capacity for the external magnetic field values from the interval 0 < |H /J | < 4 which is reduced with the increasing of the cluster approximation. First of all, let us analyze the behavior of the specific heat capacity for H /|J | = 0 where the single-point ground state with m = 0 is realized. The corresponding temperature behavior of the specific heat capacity for chosen various cluster approximations is shown in Fig. 6. Here, one can see that, except of the 2-site cluster approximation, the specific heat capacity behavior of all other cluster approximations, i.e., all cluster approximations which take into account at least one elementary

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Fig. 7. The temperature behavior of the specific heat capacity for |H /J | = 4 and for various n-site cluster approximations with n = 2, 3, 5, and 12. The results are compared to the corresponding behavior of the specific heat capacity in the model on the kagome-like Husimi recursive lattice (red dashed curve).

triangle of the kagome lattice, demonstrate very similar behavior. Moreover, starting from the 7-site cluster approximation the specific heat capacity curves become almost indistinguishable (this is the reason why the corresponding curves for n = 7 and 9 are omitted in Fig. 6). At the same time, as it also follows from Fig. 6, they are very similar with the corresponding specific heat capacity curve obtained in the framework of the kagome-like Husimi lattice approximation (see the red dash curve in Fig. 6). The fact that these two completely different approaches (the EFT large cluster approximations and the exactly solved model on the corresponding recursive lattice) give the same behavior of the specific heat capacity means that, on one hand, this single-point ground state is physically relevant and, on the other hand, moreover, one can also conclude that the temperature behavior of the entropy, the corresponding residual entropies, as well as the macroscopic degeneracy of this ground state in the framework of both models must also be very similar (although are not explicitly studied in this paper). Finally, let us also note that the 2-site cluster approximation seems to be too rough approximation for investigation of thermodynamical properties of the present frustrated antiferromagnetic model even in the zero external magnetic field (see Fig. 6). As it is shown in Fig. 7, almost the same conclusions are also valid for the second physically well-defined single-point ground state realized for |H /J | = 4. Here, we can see that all studied cluster approximations give very similar results for the specific heat capacity behavior. At the same time, again one can see very good agreement with the result obtained for the model on the kagome-like Husimi recursive lattice (the red dashed curve in Fig. 7). On the other hand, completely different situation is observed for |H /J | = 2 for which the corresponding single-point ground state is realized in the framework of the EFT cluster approximations (see Fig. 8). Although one can see rather strong stability of the magnetization properties of this single-point ground state (see Table 1) the temperature behavior of the specific heat capacity strongly depends on the approximation. It means that the entropy properties of this state will also significantly depend on the used cluster approximation. At the same time, with the increasing of the cluster approximation the specific heat capacity has tendency to approach the corresponding curve obtained in the framework of the Husimi lattice approximation (red dashed curve in Fig. 8). However, here it is necessary to realize that in the framework of the kagome-like Husimi lattice approximation for |H /J | = 2 the plateau ground state with |m| = 1/3 is realized. It means that the magnetization of this single-point ground state must also tend to |m| = 1/3 in the framework of the n-site cluster approximation in the limit n → ∞. Now, let us analyze the typical behavior of the specific heat capacity for the values of the external magnetic field from vicinity of the points for which the single-point ground states are realized. While directly at these points, i.e., at |H /J | = 0, 2, and 4, the standard one peak behavior of the specific heat capacity is observed (see Figs. 6–8) the Schottky-type anomaly behavior appears in the vicinity of these points for low temperatures. First of all, let us study the properties of the specific heat capacity in the left and right vicinity of |H /J | = 2. Here, as it follows from Fig. 9, where the dependence of the specific heat capacity on the temperature is demonstrated for |H /J | = 1.8 and 2.2, the Schottky-type two-peak structure in the low temperature behavior of the specific heat capacity appears. However, in both these cases, the behavior of the specific heat capacity strongly depends on the size of the cluster approximation and, moreover, for values of the magnetic field from the left vicinity of |H /J | = 2, i.e, for |H /J | < 2, the specific heat capacity shows even unphysical negative values (an inverse Schottky anomaly) for all cluster approximations with n > 2, i.e., for all cluster approximations which take into account elementary geometrical properties of the kagome lattice responsible for geometric frustration. On the other hand,

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Fig. 8. The temperature behavior of the specific heat capacity for |H /J | = 2 and for various n-site cluster approximations with n = 2, 3, 5, 7, 9, and 12. The results are compared to the corresponding behavior of the specific heat capacity in the model on the kagome-like Husimi recursive lattice (red dashed curve).

Fig. 9. The temperature behavior of the specific heat capacity for |H /J | = 1.8 (the left figure) and for |H /J | = 1.8 (the right figure) for various n-site cluster approximations. The results are compared to the corresponding behavior of the specific heat capacity in the model on the kagome-like Husimi recursive lattice (red dashed curves).

however, in both cases one can see that the Schottky-peaks (positive for |H /J | > 2 and negative for |H /J | < 2) are reduced with increasing of the cluster approximation and one can suppose that they will disappear completely in the limit n → ∞ of the n-site cluster approximation where only plateau ground state with |m| = 1/3 should exist in the whole interval 0 < |H /J | < 4 (see discussion in the previous subsection). At the same time, the standard first peaks in the specific heat capacity behavior show the same tendency with increasing of the cluster approximation as in the case |H /J | = 2 (see Fig. 8), i.e, they shift towards the corresponding curves for the specific heat capacity obtained for the model on the kagome-like Husimi recursive lattice (see the red dashed curves in Figs. 8 and 9) where the plateau ground state with |m| = 1/3 is formed in the limit T → 0. Similar situation with nonstandard or even unphysical behavior of the specific heat capacity in the framework of the EFT with various size cluster approximations can be observed for all magnetic fields inside the interval 0 < |H /J | < 4. This fact again demonstrates that the existence of two different plateau ground states together with the single-point ground state at |H /J | = 2 is only an artefact of the EFT approach. Finally, let us have a look at the properties of the specific heat capacity for the magnetic fields from the right near vicinity of |H /J | = 4 where, regardless of the cluster approximation, the saturated plateau-like ground state with |m| = 1 is formed in the limit T → 0 (see Figs. 3 and 5). In this respect, a typical temperature specific heat capacity behavior is shown in Fig. 10

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Fig. 10. The temperature behavior of the specific heat capacity for |H /J | = 4.2 and for various n-site cluster approximations with n = 3, 5, and 12. The results are compared to the corresponding behavior of the specific heat capacity in the model on the kagome-like Husimi recursive lattice (red dashed curve). Table 2 The critical temperatures of the ferromagnetic spin-1/2 Ising model on the kagome lattice in the n-site cluster EFT approximation for n = 1, 2, 3, 5, 7, 9, and 12 compared to the corresponding value on the kagome-like Husimi recursive lattice (HL) as well as to the exact value on the kagome lattice. kB Tc J

n=1

n=2

n=3

n=5

n=7

n=9

n = 12

HL

Exact

3.0898

2.9235

2.8169

2.7425

2.7083

2.6828

2.6127

2.4854

2.1432

for |H /J | = 4.2. As it follows from Fig. 10, here the specific heat capacity exhibits stable Schottky-type two-peak behavior which only weakly depends on the cluster approximation and is also in perfect accordance with the specific heat capacity behavior of the model on the kagome-like Husimi lattice (see the red dashed curve in Fig. 10). Moreover, the fact that two completely different approximations, one of which represents approximation of the model on regular kagome lattice and the second one is exactly solvable model on approximate kagome-like recursive lattice, gives almost the same results for the specific heat capacity for |H /J | > 4 means that one can suppose that the obtained behavior of the specific heat capacity is also close to the properties of the specific heat capacity of the exact solution of the model on the regular two-dimensional kagome lattice in this region of magnetic field values. 3.3. Critical temperatures and the specific heat capacity for ferromagnetic model For completeness of our investigation of the spin-1/2 Ising model on the kagome lattice, let us also briefly discuss the critical properties of the ferromagnetic model in zero external magnetic field, i.e., to investigate the dependence of the position of the critical temperature on the size of the cluster approximation for all clusters shown in Fig. 2. Using the standard EFT technique for determination of the critical temperature (see, e.g., Ref. [51]) the critical temperatures of the model for all cluster approximations can be expressed in the form of the corresponding polynomial equations for xc = exp[4J /(kB Tc )]. For example, for the 1, 2, and 3-site cluster approximations the critical temperatures are given by the following equations x3c − 3x2c − xc − 5 = 0,

(11)

9x8c − x7c − 51x6c − 197x5c − 333x4c − 595x3c − 497x2c − 295xc − 88 = 0,

(12)

13x8c − 24x7c + 8x6c − 392x5c − 166x4c − 1608x3c − 992x2c − 792xc − 143 = 0,

(13)

and

respectively. With increasing of the cluster approximation the polynomial equations for determination of the corresponding critical temperatures become huge, therefore we shall not present their explicit form here (For example, the corresponding equations in the framework of the 5 and 7-site cluster approximations are of the order 25 and 91, respectively).

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Fig. 11. The dependence of the critical temperature on the size of the cluster approximation. The results are compared to the corresponding value obtained on the kagome-like recursive lattice (the dashed line) as well as to the exact value on the kagome lattice (the solid line).

Fig. 12. The dependence of the temperature behavior of the spontaneous magnetization (the left figure) and the specific heat capacity (the right figure) of the spin-1/2 Ising model in zero external magnetic field for EFT n-site cluster approximation for various values of n up to n = 12 (see Fig. 2).

Note that the solution of Eq. (11) for the simplest single-site cluster approximation can be found in the exact analytical form which reads kB Tc 4 [ ]} . = { (14) √ )1/3 ( √ )1/3 ( 2 )2/3 ( J ln 1 + 3 9 − 69 + 9 + 69 Of course, this analytical result is completely the same as one obtained for the critical temperature of the Ising model on the regular square lattice in the framework of the single-site cluster approximation published recently in Ref. [53], because the single-site cluster EFT approximation take into account only the coordination number of the studied lattice which is the same for both lattices. The dependence of the critical temperature on the size of the cluster approximation is explicitly shown in Fig. 11 and the corresponding numerical values are summarized in Table 2 where they are also compared to the critical temperature values obtained in the framework of the recursive lattice approximation [28] as well as to the exact value (see, e.g., Ref. [12] and references cited therein). As it follows from Fig. 11 and Table 2, the value of the critical temperature decreases with the increasing of the size of the chosen cluster towards the exact value. However, it is quite interesting that even the 12-site cluster approximation gives less precise value than the model on the kagome-like Husimi recursive lattice. Finally, the temperature dependence of the spontaneous magnetization and the specific heat capacity for various cluster approximations is shown explicitly in Fig. 12. As it follows from this figure, the spontaneous magnetization as well as the

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specific heat capacity demonstrate standard EFT behavior with finite jump of the specific heat capacity directly at the critical temperature. 4. Summary and conclusion In this paper we have investigated in detail the magnetic and specific heat capacity properties of the spin-1/2 Ising model in the presence of the external magnetic field on the two-dimensional kagome lattice by using the EFT cluster approximation technique. The influence of the size of the cluster on the low temperature formation of magnetization plateaus in the geometrically frustrated antiferromagnetic model was analyzed in the framework of the studied model up to the 12-site cluster approximation which takes into account typical closed structure on the real kagome lattice with six site-connected triangles (see Fig. 2). It is shown that, regardless of the studied cluster approximation, the model exhibits the existence of two nontrivial plateaus which are formed in intervals 0 < |H /J | < 2 and 2 < |H /J | < 4 in the limit T → 0 (see Figs. 3– 5). However, while the positions of these plateaus do not depend on the used cluster approximation the magnetization properties of the formed plateaus strongly depend on the size of the cluster and their magnetization values approach each other with increasing of the cluster approximation. This behavior allows one to assume that the formation of two different plateaus is only an artefact of the EFT approximation and that both these plateaus will form one plateau in the whole interval 0 < |H /J | < 4 with magnetization |m| = 1/3 in the limit n → ∞ of the n-site cluster approximation in accordance with results obtained by various different techniques, e.g., by the Monte Carlo simulation [44] or in the framework of the exactly solvable model on the corresponding Husimi recursive lattice [33]. In the framework of the investigation of the system of ground states of the model it is also shown that the model exhibits the existence of three different single-point ground states at |H /J | = 0, 2, and 4. While two of them, namely, the single-point ground state at H = 0 with exact value of the magnetization m = 0 and the ground state at |H /J | = 4 which separates the saturated plateau-like ground state realized for |H /J | > 4, represent real single-point ground states which also exist in the framework of other approximation techniques the existence of the single-point ground state at |H /J | = 2 again seems to be only an artefact of the EFT cluster approximation which will disappear in the limit n → ∞ of the n-site cluster approximation. The unphysical nature of the existence of two plateau ground states in intervals 0 < |H /J | < 2 and 2 < |H /J | < 4 together with the existence of the single-point ground state at |H /J | = 2 in the EFT cluster approximations is also demonstrated in the temperature behavior of the specific heat capacity. It is shown that the temperature behavior of the specific heat capacity in the interval 0 < |H /J | < 4 strongly depends on the cluster approximation and the character of the behavior also strongly depends on the value of the magnetic field. Moreover, the specific heat capacity can even obtain negative values in the form of an inverse Schottky-type anomaly. However, the unphysical behavior of the specific heat capacity in the interval 0 < |H /J | < 4 is reduced with increasing of the cluster size approximation as it is explicitly demonstrated in Fig. 9. Further, it is shown that with increasing cluster approximation the specific heat capacity behavior in the interval 0 < |H /J | < 4 becomes more similar to the specific heat capacity behavior in the model on the kagome-like recursive lattice. On the other hand, the temperature behavior of the specific heat capacity for magnetic field values where relevant ground states are formed, i.e., for H = 0 and |H /J | ≥ 4, exhibits weak dependence on the cluster approximation and, at the same time, the behavior is in excellent agreement with the corresponding results obtained on the kagome-like Husimi recursive lattice. Thus, we can conclude that although the EFT cluster approximations feel the presence of the geometric frustration of the antiferromagnetic system on the kagome lattice, nevertheless for correct understanding and interpretation of obtained results it is necessary to study the system using at least a few cluster approximations especially those which take into account some symmetry of the lattice. Finally, for completeness, we have also investigated the dependence of the position of the critical temperature of the ferromagnetic spin-1/2 Ising model on the kagome lattice on the size of the EFT cluster approximation. It is shown that with increasing of the cluster size the position of the critical point shifts towards the exact one. Besides, dependence of the behavior of the spontaneous magnetization and the specific heat capacity on the cluster size is also briefly discussed. Acknowledgments M.J. gratefully acknowledge the hospitality of the TH division in CERN. The work was supported by the VEGA, Slovakia grant No. 2/0065/17 and by the realization of the project ITMS, Slovakia No. 26220120029, based on the supporting operational Research and Development Program financed from the European Regional Development Fund, Slovakia. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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