- Email: [email protected]
Four-state standard Potts model
Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Four-state standard Potts model A.I. Proshkin n, F.A. Kassan-Ogly Institute of Metal Physics, Ural Division, Russian Academy of Sciences, Ekaterinburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 12 June 2014 Available online 22 October 2014
We investigated magnetic and thermal properties of a one-dimensional 4-state Potts model by the method of Kramers–Wannier transfer-matrix. Not only magnetic field but also the interactions between nearest J and second J′ neighbors were taken into account. It was shown that in the case of a single crystal, this magnet could have up to four frustration fields at which magnetization has jumps and entropy tends to non-zero values at T → 0. We found the exact analytical expressions for these frustration fields. When an external field crosses over the frustration fields cardinal changes of magnetic structures take place. & 2014 Elsevier B.V. All rights reserved.
Keywords: Potts model Transfer-matrix Magnetization
1. Introduction
and the second one [3,4]
There is now a great interest devoted to the multi-k structures in monochalcogenides and monopnictides of lanthanides and actinides. Normile with co-authors [1] showed that the identification of USb neutron diffraction pattern in the 3-k model (Fig. 1a) gives better results than in the collinear model (Fig. 1b). This 3-k structure has no antiparallel directions of magnetic moments and is identical to a 4-state standard Potts model shown in Fig. 2. All the Potts models are a generalization of the Ising model that is widely used in the theory of magnetism. It is known that in the Ising model atomic magnetic moments (shortly “spins”) are assumed to be oriented only along two directions: either parallel or antiparallel to each other. In the Potts models, however, there are more than two possible spin directions. So far there are many articles concerned with Potts models. Wu [2] described the applications of the Potts models for solving problems of statistical physics and solid state physics, as well as various historical aspects of these models. In the standard q-state Potts model vectors can be oriented in the (q − 1)− dimensional space only in such a way that the angles between any two different orientations are the same. For such model there are two variants of determining the energy of interaction between the nearest neighbors. The first variant [2]
Jni, ni + 1
⎫ J⎧ = ⎨1 + (q − 1) e ni e ni + 1 ⎬ , q⎩ ⎭
Jni, ni + 1 = J0 δni, ni + 1 + J1 (1 − δni, ni + 1 ),
where eni are the unit vectors oriented along q symmetrical directions of a hypertetrahedron in the (q − 1)− dimensional space, ni = {1, 2 ,..., q} , δni, ni + 1 is the Kronecker symbol. In the first variant (1), it is assumed that the energy of interaction between parallel spins at the neighboring sites is a non-zero quantity J and that this energy is zero for a nonparallel orientation. In the second variant (2), the interaction energy is assumed to be a certain quantity J0 for a parallel orientation and another non-zero quantity J1 for a nonparallel orientation. For magnetism problems it is most convenient to introduce interaction in the form of a scalar product of spin vectors, namely
Jni, ni + 1 = Jσni σni + 1 cos θni, ni + 1,
Corresponding author. Tel.: þ 7 982 672 26 58. E-mail address: [email protected] (A.I. Proshkin).
http://dx.doi.org/10.1016/j.jmmm.2014.10.053 0304-8853/& 2014 Elsevier B.V. All rights reserved.
(3)
or, for the sake of brevity, as
Ji, i + 1 = J· (σ i, σ i + 1).
(4)
Now, we can write the Hamiltonian for any magnetic Potts model of a one-dimensional chain in the conventional form, including both next-nearest neighbors interaction J ′ and a magnetic field H
/ = − J ∑ (σ i , σ i + 1 ) i
(1)
− J ′ ∑ (σ i , σ i + 2 ) − i
n
(2)
∑ (σ i, H). i
(5)
For the sake of brevity, hereinafter, we will work in the system of units where Boltzmann constant and Bohr magneton are taken to be unity.
14
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
(see, for example, [10]) and having the dimensions 16 16
Wni, ni + 1 = 〈σ i σ i + 1|Wi, i + 1|σ i + 1σ i + 2 〉 ⎛J ⎞ J′ 1 = exp ⎜ (σ i, σ i + 1) + (σ i, σ i + 2 ) + (H , σ i ) ⎟. ⎝T ⎠ T T
(6)
The transfer-matrix does not actually depend upon the site indices and thus the partition function is equal to
Z N = Tr W N Fig. 1. Identification of the USb diffraction pattern in the 3-k structure (a) and in the collinear one (b).
N = λ1N + λ 2N + ⋯ + λ16 ⎫ ⎧ ⎛ λ 2 ⎞N ⎛ λ16 ⎞N ⎪ ⎪ = λ1N ⎨1 + ⎜ ⎟ + ⋯ + ⎜ ⎟ ⎬. ⎪ ⎝ λ1 ⎠ ⎝ λ1 ⎠ ⎪ ⎭ ⎩
(7)
The eigenvalues λi of the transfer-matrix (6) (λ1 ≡ λmax ) are determined from the secular equation
det |W − λE| = 0,
(8)
where E is the unit matrix of size 16. In the thermodynamic limit (i.e., at N → ∞) we have Z N = λ1N , and the quantities such as the free energy, magnetization and entropy can only be expressed in terms of the maximum eigenvalue λmax of the matrix (6)
F = − T ln λ max , M=−
S=−
Fig. 2. 4-State Potts model. Dashed lines are the directions of magnetic field.
It should be noted that it is impossible to draw the q-state standard Potts model in the real space for any q > 4 . Therefore for magnetic problems in magnets with cubic and other symmetries modified 6-, 8- and 12-state Potts models in the real three-dimensional space with spins oriented along cube edges, cube body diagonals and cube face diagonals, respectively, were introduced into the physics of magnetic phenomena. In the paper [5], authors obtained exact solutions for modified 6- and 8-state Potts models in the magnetic field. Using these solutions authors [6] have succeeded in developing the theory of simultaneous magnetic and structural phase transitions for magnetic crystals with easy axes [001] and [111]. Modified 6- and 8-state Potts models were used in works [7,8] for describing phase transitions in uranium arsenide and uranium monopnictides. In the paper [9] different Potts models including modified ones were investigated in the case of polycrystal.
2. Formulation of the problem Let us consider the 4-state standard Potts model in an applied magnetic field with interactions between the nearest and nextnearest neighbors on a one-dimensional monatomic equidistant chain described by Hamiltonian (5). Let us also introduce the Kramers-Wannier transfer-matrix defined by its matrix elements
∂F T ∂λ max = , ∂H λ max ∂H ∂F T ∂λ max = ln λ max + . ∂T λ max ∂T
(9)
(10)
(11)
3. 4-State Potts model without magnetic field As was mentioned in Section 1 and seen in Fig. 2 the 4-state Potts model is extraordinary one: it has no antiparallel directions of spins and it is absolutely frustrated in the absence of magnetic field with antiferromagnetic interactions between nearest (J < 0) and next-nearest (J ′ < 0) neighbors. The term frustrations, in the context of magnetic systems, have been introduced by Gerard Toulouse in 1977 [11]. Frustrations are a phenomenon when it is not possible to minimize all terms of the Hamiltonian (5) simultaneously. This leads to a strong degeneracy of a ground state of a system with non-zero entropy at T ¼0. In the 4-state Potts model if the interaction energy between nearest neighbors is antiferromagnetic ( J < 0) and between next-nearest neighbors is equal to zero ( J ′ ≡ 0) there are 4·3N − 1 possible configurations on 1D chain at T ¼0. All these configurations have the same energy and it is easy to show that the entropy of such system is equal to ln 3. In the case when J < 0 and J ′ < 0 there are 12·2N − 2 possible configurations of this model on 1D chain at T ¼0. Entropy of this configuration is equal to ln 2. To overcome frustrations in this antiferromagnetic system one has to take into consideration not only nearest and next-nearest neighbors but also third-nearest neighbors interactions. In this case there exist only 24 different configurations with the same energy and only in this case the entropy of such a system is equal to zero at T ¼0.
4. 4-State Potts model in a magnetic field In the absence of interaction between next-nearest neighbors it is possible to obtain analytical expressions for maximum eigenvalues of the matrix (6) of the 4-state Potts model for arbitrary
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
M
15
M
1
1.0
1.0
0.8
0.8
0.6
0.6
5 9 0.4
0.4
3 9
0.2
0.2
1 9 0.0 0.0
0.0 1
H 0.5
1.0
1.5
2.0
2.5
3.0
Fig. 3. Field dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2, T ¼ 0.023. The field is applied along the [111] axis.
direction of an external magnetic field. It can be done because the transfer-matrix (6) in this case has the dimensions 4 4 and it was done in the works [9,12]. If J ′ ≠ 0 then transfer-matrix generally has dimensions 16 16. In this work we have investigated only three directions of applied magnetic field defined by cubic symmetry of a crystal: [001], [110] and [111]. The axis [111] is the easy magnetization axis in the 4-state Potts model. The 4-state Potts model with nearest and next-nearest antiferromagnetic interactions has two different ground states. The first one occurs when the interaction between next-nearest neighbors is weak J ′ < J/2, and the second one when it is strong J ′ > J/2. When J ′ = J/2 the energies of these ground states are the same. 4.1. The [111] field direction Fig. 3 shows field dependence of magnetization in the 4-state Potts model with antiferromagnetic interactions when J ′ < J/2. It is seen that magnetization behavior is unusual: there are three plateaus of magnetization at 1/9, 3/9, 5/9 and the saturation magnetization is equal to 1. The fields at which magnetization have jumps that should be called frustration fields by the reasons mentioned in Section 3. Fig. 4 demonstrates magnetization behavior against temperature at different fields. Black thick lines are the magnetizations at frustration fields and gray dashed lines are the magnetizations at intermediate fields. It is seen that at low temperatures magnetization at intermediate fields tends to the values of magnetization plateaus, namely 1/9, 3/9, 5/9 and 1, but at frustration fields magnetization tends to different values. As one can see from Fig. 3 the first frustration field is equal to zero and this can be explained by the reasons mentioned in Section 3. We succeeded in deriving analytical expressions for other three frustration fields as functions of interaction energies
H2 = − 3J′,
H3 = − 2J + J′,
H4 = − 2J − 2J′.
2
MH3 =
4
5 4 ⎛ 2 ⎞⎟2 − 3 ⎜ 9 9 ⎝ 211 ⎠
⎡ · ⎢ 3 211 + 9 633 + ⎢ ⎣
MH4 =
3
⎤ 211 − 9 633 ⎥, ⎥ ⎦
(13)
5 2 ⎛ 2 ⎞2 + 3 ⎜ ⎟ 9 9 ⎝ 85 ⎠
⎡ ⎢ 3 85 + 9 85 + ⎢ ⎣
3
⎤ 85 − 9 85 ⎥. ⎥ ⎦
(14)
The peculiarity of the case considered is the non-zero values of the entropy at the plateaus and frustration fields at T¼ 0. As was mentioned in Section 3 the entropy in 4-state Potts model with both nearest and next-nearest interactions in the absence of field is equal to ln 2 because there are 12·2N − 2 different configurations having the same energy. With field increasing these structures of course are changing. For example, the first magnetization plateau M = 1/9 corresponds to the structure in which every 2 of 3 spins can be oriented in two different directions without changing the energy of the whole configuration. The entropy of such system is equal to
2 3
ln 2. Analyzing all these possible structures we obtained
six different values of entropy in the 4-state Potts model in the case J ′ < J/2 at T ¼0. Three of them at plateaus
0 < H < − 3J ′ ,
S=
− 3J ′ < H < − 2J + J ′ ,
2 ln 2, 3 S=
(12)
The magnetization value at the field H2 is merely 1/5. For other two fields the magnetization values are
3
Fig. 4. Temperature dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2. Black thick lines are the magnetizations at frustration fields and gray dashed lines are magnetizations at intermediate fields.
− 2J + J ′ < H < − 2J − 2J ′ ,
(15)
1 ln 2, 2 S=
1 ln 3. 3
(16)
(17)
16
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
S
0
1.4
S=
2 ln 2, 3
3 J − J
1.2
− 1.0
S=
J − 2J ′ < H < − 2J − 2J ′ , 2
⎡3 ⎢ 2 −1+ S H2 = ln ⎢ ⎢⎣
0.8
2 3
(22) 1 ln 6, 4 S=
(23)
1 ln 3, 3
(24)
⎤ 2 − 1 − ( 3 2 − 1)2 ⎥ ⎥. 2 32 −1 ⎥⎦ 3
(25)
The expression for the entropy in the third frustration field is too cumbersome but we have discovered that it can be easily found by the formula SH3 = ln xmax where xmax is the maximum real root of
0.6
the equation x 4 − 3x − 6 = 0. Entropy at field H4 = − 2J − 2J ′ in the case J ′ > J/2 is equal to that in (20).
0.4
0.2
4.2. The [110] field direction
0.0 0.0
T 0.1
0.2
0.3
0.4
Fig. 5. Temperature dependence of entropy in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2. Black thick lines are the entropies at frustration fields and the gray ones are the entropies at plateaus. The field is applied along the [111] axis.
Three others at frustration fields
S H2 = ln 2,
(18)
⎛ 3 S H3 = ln ⎜⎜ 3 − 2 ⎝
S H4
3 J, 2
⎡ ⎛ 1 = ln ⎢ ⎜⎜1 + ⎢3⎝ ⎣
633 + 18
3
3
⎞ 633 ⎟ , 18 ⎟⎠
3 − 2
83 9 85 − + 2 2
3
⎞⎤ 83 9 85 ⎟ ⎥ + . 2 2 ⎟⎠ ⎥⎦
J H3 = − − 2J ′ , 2
H4 = − 2J − 2J′.
is 2/3 and it is not equal to 1 because in the field [110] the ferromagnetic state cannot be reached (the impossibility to align all spins along the direction of magnetic field that does not coincide with the easy axis is the general peculiarity for monochalcogenides and monopnictides of lanthanides and actinides). Fig. 6 demonstrates three non-zero frustration fields at
H2 = − 2 6 J′, (19)
(20)
It is obvious that the entropy of ferromagnetic state (H > − 2J − 2J ′) is equal to zero at T ¼0. When T → ∞ the entropy of 4-state Potts model does not depend upon the magnetic field and approaches the asymptotic value ln 4 . Fig. 5 demonstrates the results of entropy calculation. Black thick lines are the entropies at frustration fields and the gray ones are the entropies at plateaus. Six different non-zero values of entropy at T → 0 are clearly visible. The lowest curve is the entropy of ferromagnetic state. The magnetization behavior in the case J ′ > J/2 is similar to the one shown in Fig. 3 with different values of the frustration fields (the first one is equal to zero), namely
3 H2 = − J , 2
Fig. 6 demonstrates the results of magnetization calculation in the 4-state Potts model with both nearest and next-nearest interactions in the case J ′ < J/2 and the field oriented along [110]. It is seen that there are three plateaus of magnetization, namely 6 /9, 6 /6, and 2 6 /9. The asymptotic value of magnetization
(21)
The expressions for magnetization values at these fields are too cumbersome, but as one can see from (21) the fourth frustration field in this case is equal to that in (14) and magnetization at these fields coincides for both J ′ < J/2 and J ′ > J/2 cases. As one can expect there will be non-zero entropy at these frustration fields and plateaus. Some of them coincide with entropies derived in (17)– (20), some of them do not
H3 = − 2
2 (2J − J′), 3
H4 = − 4
2 (J + J′). 3
(26)
The expressions for magnetizations at frustration fields are very cumbersome. But the values of entropy at different fields can M
2 3
0.8
0.6
2
6 9
6 6
0.4
6 9 0.2
H
0.0 0
1
2
3
4
5
6
Fig. 6. Field dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2, T ¼ 0.031. The field is applied along the [110] axis.
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
17
M
3 3
0.5
0.4
0.3
3
0.2
9 0.1
H 0.5
1.0
1.5
2.0
Fig. 7. Field dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2, T ¼ 0.031. The field is applied along the [001] axis.
be easily calculated by taking natural logarithm of maximum real root of the corresponding equation as it was shown in the end of the former Section 4.1. For example, to evaluate entropy in the field H2 = − 2 6 J ′ one should find the maximum root of the equation x 3 − x − 1 = 0 and to take the natural logarithm from it, the result can be written in the form
⎡ S = ln ⎢ 3 9 − ⎣
69 +
3
9+
⎤ 2 1 69 ⎥ − ln 3 − ln 2, ⎦ 3 3
H2 = −
6 J,
H3 = −
2 H4 = − 4 (J + J′), 3
(27)
1 ln 2, 4
S=
H3 < H < H4,
1 S = ln 2, 3
The asymptotic value of magnetization is strates the only one frustration field at
3 /9.
3 /3. Fig. 7 demon-
H2 = − 2 3 J′,
M=
·
(31)
1 + 3
2 3 87
3 28 − 3 87 − 3 28 + 3 87 , ⎞ 1⎛ ⎜1 + 3 28 − 3 87 + 3 28 + 3 87 ⎟ ⎠ 3⎝
(32)
and the entropy
(
S = ln 1 +
3
28 − 3 87 +
3
28 + 3 87
) − ln 3.
(33)
In the case J ′ > J/2 the only one non-zero frustration field is
(28)
The fourth frustration field coincides with that in (26). The equations for entropies are x 3 − x 2 + x − 2 = 0, 4 x − 2x − 2 = 0 and x 3 − x 2 − 2 = 0 for fields H2, H3 and H4, respectively
H2 < H < H3,
interactions in the case J ′ < J/2 and the field oriented along [001]. It is seen that there is only one magnetization plateau M =
at which the magnetization is equal to
which is exactly the value of entropy in this field. This answer can be analytically derived by applying formula (11) to the maximum eigenvalue of the Kramers–Wannier transfer-matrix (6). The entropies at other two frustration fields can be found from the equations x 3 − x − 2 = 0 and x 3 − x 2 − 2 = 0 for fields H3 and H4, 1 respectively. For the field H3 < H < H4 entropy is equal to 3 ln 2, but for other plateaus the entropy is equal to zero because the number of equal-energy configurations is not infinite. When J ′ > J/2 the frustration fields are
2 (J + 4J′), 3
Fig. 8. Experimental magnetization of a UThSb crystal in increasing the field received by [13].
3 J,
H2 = −
(34)
at which the magnetization is equal to
M=
1 + 3
2 3 78
(29) ·
(30)
3 27 − 3 78 − 3 27 + 3 78 , ⎛ ⎞ 1 3 ⎜ 27 − 3 78 + 3 27 + 3 78 ⎟ ⎠ 3⎝
(35)
In all the other field ranges entropy is equal to zero at T ¼0. 4.3. The [001] field direction
and the entropy
Fig. 7 demonstrates the results of magnetization calculation in the 4-state Potts model with both nearest and next-nearest
S = ln
(
3
27 − 3 78 +
3
27 + 3 78
) − ln 3.
(36)
18
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
5. Discussion
References
So far we have studied the 4-state Potts model in the magnetic field with both nearest and next-nearest antiferromagnetic interactions in an external magnetic field applied along [001], [110] and [111] directions. It was shown that in this model there exist up to four frustration fields (including a zero one) at which magnetization has jumps and entropy tends to non-zero values at T → 0. It should be noted that the model considered could be applied as a prototype for describing magnetic and thermal properties of real crystals from a wide class of monochalcogenides and monopnictides of lanthanides and actinides. For example, Fig. 8 demonstrates the experimental magnetization of a UThSb crystal [13]. As one can see that there are four fields at which magnetization has jumps. Fig. 8 can be compared with the Fig. 3. There is one distinction between these figures: experimental magnetization has no jump in zero field. We do believe that one should consider not only nearest and next-nearest but also third-nearest neighbors interactions. Only in this case the first magnetization plateau in Fig. 3 will be equal to zero. But despite this circumstance qualitatively the magnetization behavior is almost the same for both figures.
[1] P.S. Normile, W.G. Stirling, et al., Phys. Rev. B 66 (014405) (2002) 1–10. [2] F.Y. Wu, The Potts models, Rev. Mod. Phys. 54 (1982) 235–268. [3] R.B. Potts, Some generalized order–disorder transformations, Proc. Camb. Philos. Soc. 48 (1951) 106–109. [4] C. Domb, Configurational studies of the Potts models, J. Phys. A: Math. Gen. 7 (1974) 1335–1348. [5] F.A. Kassan-Ogly, Modified 6-state and 8-state Potts models in magnetic field, Phase Transit. 72 (2000) 223–237. [6] F.A. Kassan-Ogly, B.N. Filippov, Magnetic and structural transitions in crystals with a structure of a NaCl type, Phys. Met. Metallogr. 107 (4) (2009) 317–329. [7] F.A. Kassan-Ogly, B.N. Filippov, Phase transitions and magnetic scattering of neutrons in UAs, Phys. Met. Metallogr. 100 (2) (2005) 108–115. [8] F.A. Kassan-Ogly, B.N. Filippov, Magnetic and structural phase transitions in the monochalcogenides of uranium, Phys. Met. Metallogr. 105 (3) (2008) 211–218. [9] A.I. Proshkin, F.A. Kassan-Ogly, On the magnetization of polycrystalline magnets with strong anisotropy, Phys. Met. Metallogr. 115 (3) (2014) 211–225. [10] H.A. Kramers, G.H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Phys. Rev. 60 (1941) 252–262. [11] J. Vannimenus, G. Toulouse, Theory of the frustration effect: II. Ising spins on a square lattice, J. Phys. C: Solid State Phys. 10 (1977) L537–L542. [12] F.A. Kassan-Ogly, One-dimensional 3-state and 4-state standard Potts models in magnetic field, Phase Transit. 71 (2000) 39–55. [13] J. Rossat-Mignod, P. Burlet, S. Quezel, O. Vogt, H. Bartholin, in: Crystalline Electric Field Effects in f-Electron Magnetism, 1982. p. 501.
Acknowledgments This work was supported by Project no. 12-I-2-2020 of Ural Division RAS and No. 12-P-2-1041 of Presidium RAS.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Four-state standard Potts model A.I. Proshkin n, F.A. Kassan-Ogly Institute of Metal Physics, Ural Division, Russian Academy of Sciences, Ekaterinburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 12 June 2014 Available online 22 October 2014
We investigated magnetic and thermal properties of a one-dimensional 4-state Potts model by the method of Kramers–Wannier transfer-matrix. Not only magnetic field but also the interactions between nearest J and second J′ neighbors were taken into account. It was shown that in the case of a single crystal, this magnet could have up to four frustration fields at which magnetization has jumps and entropy tends to non-zero values at T → 0. We found the exact analytical expressions for these frustration fields. When an external field crosses over the frustration fields cardinal changes of magnetic structures take place. & 2014 Elsevier B.V. All rights reserved.
Keywords: Potts model Transfer-matrix Magnetization
1. Introduction
and the second one [3,4]
There is now a great interest devoted to the multi-k structures in monochalcogenides and monopnictides of lanthanides and actinides. Normile with co-authors [1] showed that the identification of USb neutron diffraction pattern in the 3-k model (Fig. 1a) gives better results than in the collinear model (Fig. 1b). This 3-k structure has no antiparallel directions of magnetic moments and is identical to a 4-state standard Potts model shown in Fig. 2. All the Potts models are a generalization of the Ising model that is widely used in the theory of magnetism. It is known that in the Ising model atomic magnetic moments (shortly “spins”) are assumed to be oriented only along two directions: either parallel or antiparallel to each other. In the Potts models, however, there are more than two possible spin directions. So far there are many articles concerned with Potts models. Wu [2] described the applications of the Potts models for solving problems of statistical physics and solid state physics, as well as various historical aspects of these models. In the standard q-state Potts model vectors can be oriented in the (q − 1)− dimensional space only in such a way that the angles between any two different orientations are the same. For such model there are two variants of determining the energy of interaction between the nearest neighbors. The first variant [2]
Jni, ni + 1
⎫ J⎧ = ⎨1 + (q − 1) e ni e ni + 1 ⎬ , q⎩ ⎭
Jni, ni + 1 = J0 δni, ni + 1 + J1 (1 − δni, ni + 1 ),
where eni are the unit vectors oriented along q symmetrical directions of a hypertetrahedron in the (q − 1)− dimensional space, ni = {1, 2 ,..., q} , δni, ni + 1 is the Kronecker symbol. In the first variant (1), it is assumed that the energy of interaction between parallel spins at the neighboring sites is a non-zero quantity J and that this energy is zero for a nonparallel orientation. In the second variant (2), the interaction energy is assumed to be a certain quantity J0 for a parallel orientation and another non-zero quantity J1 for a nonparallel orientation. For magnetism problems it is most convenient to introduce interaction in the form of a scalar product of spin vectors, namely
Jni, ni + 1 = Jσni σni + 1 cos θni, ni + 1,
Corresponding author. Tel.: þ 7 982 672 26 58. E-mail address: [email protected] (A.I. Proshkin).
http://dx.doi.org/10.1016/j.jmmm.2014.10.053 0304-8853/& 2014 Elsevier B.V. All rights reserved.
(3)
or, for the sake of brevity, as
Ji, i + 1 = J· (σ i, σ i + 1).
(4)
Now, we can write the Hamiltonian for any magnetic Potts model of a one-dimensional chain in the conventional form, including both next-nearest neighbors interaction J ′ and a magnetic field H
/ = − J ∑ (σ i , σ i + 1 ) i
(1)
− J ′ ∑ (σ i , σ i + 2 ) − i
n
(2)
∑ (σ i, H). i
(5)
For the sake of brevity, hereinafter, we will work in the system of units where Boltzmann constant and Bohr magneton are taken to be unity.
14
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
(see, for example, [10]) and having the dimensions 16 16
Wni, ni + 1 = 〈σ i σ i + 1|Wi, i + 1|σ i + 1σ i + 2 〉 ⎛J ⎞ J′ 1 = exp ⎜ (σ i, σ i + 1) + (σ i, σ i + 2 ) + (H , σ i ) ⎟. ⎝T ⎠ T T
(6)
The transfer-matrix does not actually depend upon the site indices and thus the partition function is equal to
Z N = Tr W N Fig. 1. Identification of the USb diffraction pattern in the 3-k structure (a) and in the collinear one (b).
N = λ1N + λ 2N + ⋯ + λ16 ⎫ ⎧ ⎛ λ 2 ⎞N ⎛ λ16 ⎞N ⎪ ⎪ = λ1N ⎨1 + ⎜ ⎟ + ⋯ + ⎜ ⎟ ⎬. ⎪ ⎝ λ1 ⎠ ⎝ λ1 ⎠ ⎪ ⎭ ⎩
(7)
The eigenvalues λi of the transfer-matrix (6) (λ1 ≡ λmax ) are determined from the secular equation
det |W − λE| = 0,
(8)
where E is the unit matrix of size 16. In the thermodynamic limit (i.e., at N → ∞) we have Z N = λ1N , and the quantities such as the free energy, magnetization and entropy can only be expressed in terms of the maximum eigenvalue λmax of the matrix (6)
F = − T ln λ max , M=−
S=−
Fig. 2. 4-State Potts model. Dashed lines are the directions of magnetic field.
It should be noted that it is impossible to draw the q-state standard Potts model in the real space for any q > 4 . Therefore for magnetic problems in magnets with cubic and other symmetries modified 6-, 8- and 12-state Potts models in the real three-dimensional space with spins oriented along cube edges, cube body diagonals and cube face diagonals, respectively, were introduced into the physics of magnetic phenomena. In the paper [5], authors obtained exact solutions for modified 6- and 8-state Potts models in the magnetic field. Using these solutions authors [6] have succeeded in developing the theory of simultaneous magnetic and structural phase transitions for magnetic crystals with easy axes [001] and [111]. Modified 6- and 8-state Potts models were used in works [7,8] for describing phase transitions in uranium arsenide and uranium monopnictides. In the paper [9] different Potts models including modified ones were investigated in the case of polycrystal.
2. Formulation of the problem Let us consider the 4-state standard Potts model in an applied magnetic field with interactions between the nearest and nextnearest neighbors on a one-dimensional monatomic equidistant chain described by Hamiltonian (5). Let us also introduce the Kramers-Wannier transfer-matrix defined by its matrix elements
∂F T ∂λ max = , ∂H λ max ∂H ∂F T ∂λ max = ln λ max + . ∂T λ max ∂T
(9)
(10)
(11)
3. 4-State Potts model without magnetic field As was mentioned in Section 1 and seen in Fig. 2 the 4-state Potts model is extraordinary one: it has no antiparallel directions of spins and it is absolutely frustrated in the absence of magnetic field with antiferromagnetic interactions between nearest (J < 0) and next-nearest (J ′ < 0) neighbors. The term frustrations, in the context of magnetic systems, have been introduced by Gerard Toulouse in 1977 [11]. Frustrations are a phenomenon when it is not possible to minimize all terms of the Hamiltonian (5) simultaneously. This leads to a strong degeneracy of a ground state of a system with non-zero entropy at T ¼0. In the 4-state Potts model if the interaction energy between nearest neighbors is antiferromagnetic ( J < 0) and between next-nearest neighbors is equal to zero ( J ′ ≡ 0) there are 4·3N − 1 possible configurations on 1D chain at T ¼0. All these configurations have the same energy and it is easy to show that the entropy of such system is equal to ln 3. In the case when J < 0 and J ′ < 0 there are 12·2N − 2 possible configurations of this model on 1D chain at T ¼0. Entropy of this configuration is equal to ln 2. To overcome frustrations in this antiferromagnetic system one has to take into consideration not only nearest and next-nearest neighbors but also third-nearest neighbors interactions. In this case there exist only 24 different configurations with the same energy and only in this case the entropy of such a system is equal to zero at T ¼0.
4. 4-State Potts model in a magnetic field In the absence of interaction between next-nearest neighbors it is possible to obtain analytical expressions for maximum eigenvalues of the matrix (6) of the 4-state Potts model for arbitrary
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
M
15
M
1
1.0
1.0
0.8
0.8
0.6
0.6
5 9 0.4
0.4
3 9
0.2
0.2
1 9 0.0 0.0
0.0 1
H 0.5
1.0
1.5
2.0
2.5
3.0
Fig. 3. Field dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2, T ¼ 0.023. The field is applied along the [111] axis.
direction of an external magnetic field. It can be done because the transfer-matrix (6) in this case has the dimensions 4 4 and it was done in the works [9,12]. If J ′ ≠ 0 then transfer-matrix generally has dimensions 16 16. In this work we have investigated only three directions of applied magnetic field defined by cubic symmetry of a crystal: [001], [110] and [111]. The axis [111] is the easy magnetization axis in the 4-state Potts model. The 4-state Potts model with nearest and next-nearest antiferromagnetic interactions has two different ground states. The first one occurs when the interaction between next-nearest neighbors is weak J ′ < J/2, and the second one when it is strong J ′ > J/2. When J ′ = J/2 the energies of these ground states are the same. 4.1. The [111] field direction Fig. 3 shows field dependence of magnetization in the 4-state Potts model with antiferromagnetic interactions when J ′ < J/2. It is seen that magnetization behavior is unusual: there are three plateaus of magnetization at 1/9, 3/9, 5/9 and the saturation magnetization is equal to 1. The fields at which magnetization have jumps that should be called frustration fields by the reasons mentioned in Section 3. Fig. 4 demonstrates magnetization behavior against temperature at different fields. Black thick lines are the magnetizations at frustration fields and gray dashed lines are the magnetizations at intermediate fields. It is seen that at low temperatures magnetization at intermediate fields tends to the values of magnetization plateaus, namely 1/9, 3/9, 5/9 and 1, but at frustration fields magnetization tends to different values. As one can see from Fig. 3 the first frustration field is equal to zero and this can be explained by the reasons mentioned in Section 3. We succeeded in deriving analytical expressions for other three frustration fields as functions of interaction energies
H2 = − 3J′,
H3 = − 2J + J′,
H4 = − 2J − 2J′.
2
MH3 =
4
5 4 ⎛ 2 ⎞⎟2 − 3 ⎜ 9 9 ⎝ 211 ⎠
⎡ · ⎢ 3 211 + 9 633 + ⎢ ⎣
MH4 =
3
⎤ 211 − 9 633 ⎥, ⎥ ⎦
(13)
5 2 ⎛ 2 ⎞2 + 3 ⎜ ⎟ 9 9 ⎝ 85 ⎠
⎡ ⎢ 3 85 + 9 85 + ⎢ ⎣
3
⎤ 85 − 9 85 ⎥. ⎥ ⎦
(14)
The peculiarity of the case considered is the non-zero values of the entropy at the plateaus and frustration fields at T¼ 0. As was mentioned in Section 3 the entropy in 4-state Potts model with both nearest and next-nearest interactions in the absence of field is equal to ln 2 because there are 12·2N − 2 different configurations having the same energy. With field increasing these structures of course are changing. For example, the first magnetization plateau M = 1/9 corresponds to the structure in which every 2 of 3 spins can be oriented in two different directions without changing the energy of the whole configuration. The entropy of such system is equal to
2 3
ln 2. Analyzing all these possible structures we obtained
six different values of entropy in the 4-state Potts model in the case J ′ < J/2 at T ¼0. Three of them at plateaus
0 < H < − 3J ′ ,
S=
− 3J ′ < H < − 2J + J ′ ,
2 ln 2, 3 S=
(12)
The magnetization value at the field H2 is merely 1/5. For other two fields the magnetization values are
3
Fig. 4. Temperature dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2. Black thick lines are the magnetizations at frustration fields and gray dashed lines are magnetizations at intermediate fields.
− 2J + J ′ < H < − 2J − 2J ′ ,
(15)
1 ln 2, 2 S=
1 ln 3. 3
(16)
(17)
16
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
S
0
1.4
S=
2 ln 2, 3
3 J − J
1.2
− 1.0
S=
J − 2J ′ < H < − 2J − 2J ′ , 2
⎡3 ⎢ 2 −1+ S H2 = ln ⎢ ⎢⎣
0.8
2 3
(22) 1 ln 6, 4 S=
(23)
1 ln 3, 3
(24)
⎤ 2 − 1 − ( 3 2 − 1)2 ⎥ ⎥. 2 32 −1 ⎥⎦ 3
(25)
The expression for the entropy in the third frustration field is too cumbersome but we have discovered that it can be easily found by the formula SH3 = ln xmax where xmax is the maximum real root of
0.6
the equation x 4 − 3x − 6 = 0. Entropy at field H4 = − 2J − 2J ′ in the case J ′ > J/2 is equal to that in (20).
0.4
0.2
4.2. The [110] field direction
0.0 0.0
T 0.1
0.2
0.3
0.4
Fig. 5. Temperature dependence of entropy in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2. Black thick lines are the entropies at frustration fields and the gray ones are the entropies at plateaus. The field is applied along the [111] axis.
Three others at frustration fields
S H2 = ln 2,
(18)
⎛ 3 S H3 = ln ⎜⎜ 3 − 2 ⎝
S H4
3 J, 2
⎡ ⎛ 1 = ln ⎢ ⎜⎜1 + ⎢3⎝ ⎣
633 + 18
3
3
⎞ 633 ⎟ , 18 ⎟⎠
3 − 2
83 9 85 − + 2 2
3
⎞⎤ 83 9 85 ⎟ ⎥ + . 2 2 ⎟⎠ ⎥⎦
J H3 = − − 2J ′ , 2
H4 = − 2J − 2J′.
is 2/3 and it is not equal to 1 because in the field [110] the ferromagnetic state cannot be reached (the impossibility to align all spins along the direction of magnetic field that does not coincide with the easy axis is the general peculiarity for monochalcogenides and monopnictides of lanthanides and actinides). Fig. 6 demonstrates three non-zero frustration fields at
H2 = − 2 6 J′, (19)
(20)
It is obvious that the entropy of ferromagnetic state (H > − 2J − 2J ′) is equal to zero at T ¼0. When T → ∞ the entropy of 4-state Potts model does not depend upon the magnetic field and approaches the asymptotic value ln 4 . Fig. 5 demonstrates the results of entropy calculation. Black thick lines are the entropies at frustration fields and the gray ones are the entropies at plateaus. Six different non-zero values of entropy at T → 0 are clearly visible. The lowest curve is the entropy of ferromagnetic state. The magnetization behavior in the case J ′ > J/2 is similar to the one shown in Fig. 3 with different values of the frustration fields (the first one is equal to zero), namely
3 H2 = − J , 2
Fig. 6 demonstrates the results of magnetization calculation in the 4-state Potts model with both nearest and next-nearest interactions in the case J ′ < J/2 and the field oriented along [110]. It is seen that there are three plateaus of magnetization, namely 6 /9, 6 /6, and 2 6 /9. The asymptotic value of magnetization
(21)
The expressions for magnetization values at these fields are too cumbersome, but as one can see from (21) the fourth frustration field in this case is equal to that in (14) and magnetization at these fields coincides for both J ′ < J/2 and J ′ > J/2 cases. As one can expect there will be non-zero entropy at these frustration fields and plateaus. Some of them coincide with entropies derived in (17)– (20), some of them do not
H3 = − 2
2 (2J − J′), 3
H4 = − 4
2 (J + J′). 3
(26)
The expressions for magnetizations at frustration fields are very cumbersome. But the values of entropy at different fields can M
2 3
0.8
0.6
2
6 9
6 6
0.4
6 9 0.2
H
0.0 0
1
2
3
4
5
6
Fig. 6. Field dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2, T ¼ 0.031. The field is applied along the [110] axis.
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
17
M
3 3
0.5
0.4
0.3
3
0.2
9 0.1
H 0.5
1.0
1.5
2.0
Fig. 7. Field dependence of magnetization in the 4-state Potts model with nearest and next-nearest antiferromagnetic interactions, J ′ < J/2, T ¼ 0.031. The field is applied along the [001] axis.
be easily calculated by taking natural logarithm of maximum real root of the corresponding equation as it was shown in the end of the former Section 4.1. For example, to evaluate entropy in the field H2 = − 2 6 J ′ one should find the maximum root of the equation x 3 − x − 1 = 0 and to take the natural logarithm from it, the result can be written in the form
⎡ S = ln ⎢ 3 9 − ⎣
69 +
3
9+
⎤ 2 1 69 ⎥ − ln 3 − ln 2, ⎦ 3 3
H2 = −
6 J,
H3 = −
2 H4 = − 4 (J + J′), 3
(27)
1 ln 2, 4
S=
H3 < H < H4,
1 S = ln 2, 3
The asymptotic value of magnetization is strates the only one frustration field at
3 /9.
3 /3. Fig. 7 demon-
H2 = − 2 3 J′,
M=
·
(31)
1 + 3
2 3 87
3 28 − 3 87 − 3 28 + 3 87 , ⎞ 1⎛ ⎜1 + 3 28 − 3 87 + 3 28 + 3 87 ⎟ ⎠ 3⎝
(32)
and the entropy
(
S = ln 1 +
3
28 − 3 87 +
3
28 + 3 87
) − ln 3.
(33)
In the case J ′ > J/2 the only one non-zero frustration field is
(28)
The fourth frustration field coincides with that in (26). The equations for entropies are x 3 − x 2 + x − 2 = 0, 4 x − 2x − 2 = 0 and x 3 − x 2 − 2 = 0 for fields H2, H3 and H4, respectively
H2 < H < H3,
interactions in the case J ′ < J/2 and the field oriented along [001]. It is seen that there is only one magnetization plateau M =
at which the magnetization is equal to
which is exactly the value of entropy in this field. This answer can be analytically derived by applying formula (11) to the maximum eigenvalue of the Kramers–Wannier transfer-matrix (6). The entropies at other two frustration fields can be found from the equations x 3 − x − 2 = 0 and x 3 − x 2 − 2 = 0 for fields H3 and H4, 1 respectively. For the field H3 < H < H4 entropy is equal to 3 ln 2, but for other plateaus the entropy is equal to zero because the number of equal-energy configurations is not infinite. When J ′ > J/2 the frustration fields are
2 (J + 4J′), 3
Fig. 8. Experimental magnetization of a UThSb crystal in increasing the field received by [13].
3 J,
H2 = −
(34)
at which the magnetization is equal to
M=
1 + 3
2 3 78
(29) ·
(30)
3 27 − 3 78 − 3 27 + 3 78 , ⎛ ⎞ 1 3 ⎜ 27 − 3 78 + 3 27 + 3 78 ⎟ ⎠ 3⎝
(35)
In all the other field ranges entropy is equal to zero at T ¼0. 4.3. The [001] field direction
and the entropy
Fig. 7 demonstrates the results of magnetization calculation in the 4-state Potts model with both nearest and next-nearest
S = ln
(
3
27 − 3 78 +
3
27 + 3 78
) − ln 3.
(36)
18
A.I. Proshkin, F.A. Kassan-Ogly / Journal of Magnetism and Magnetic Materials 383 (2015) 13–18
5. Discussion
References
So far we have studied the 4-state Potts model in the magnetic field with both nearest and next-nearest antiferromagnetic interactions in an external magnetic field applied along [001], [110] and [111] directions. It was shown that in this model there exist up to four frustration fields (including a zero one) at which magnetization has jumps and entropy tends to non-zero values at T → 0. It should be noted that the model considered could be applied as a prototype for describing magnetic and thermal properties of real crystals from a wide class of monochalcogenides and monopnictides of lanthanides and actinides. For example, Fig. 8 demonstrates the experimental magnetization of a UThSb crystal [13]. As one can see that there are four fields at which magnetization has jumps. Fig. 8 can be compared with the Fig. 3. There is one distinction between these figures: experimental magnetization has no jump in zero field. We do believe that one should consider not only nearest and next-nearest but also third-nearest neighbors interactions. Only in this case the first magnetization plateau in Fig. 3 will be equal to zero. But despite this circumstance qualitatively the magnetization behavior is almost the same for both figures.
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Acknowledgments This work was supported by Project no. 12-I-2-2020 of Ural Division RAS and No. 12-P-2-1041 of Presidium RAS.