- Email: [email protected]
Breaking of degeneracy in classical Heisenberg antiferromagnets
PHYSICA
Physica B 194-196 (1994) 237-238 North-Holland
Breaking of Degeneracy in Classical Heisenberg Antiferromagnets M.T. Heinil~ and A.S. Oja Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland According to arguments given by Henley, thermal effects select collinear spin structures when the classical ground state exhibits continuous degeneracy in the mean-field theory and the spin-spin interactions are isotropic. We generalize this result and derive an approximate free energy describing ground-state selection for anisotropic spin-spin interactions using thermal perturbation theory. Reasonable agreement is found with Monte Carlo simulations for Type I ~cc antiferromagnets.
PACS numbers: 75.25.+z 75.50.Ee
Let us consider the low temperature behavior of a classical spin system with the Hamiltonian j>i k where t~kij is an interaction matrix which, in our applications, includes isotropic exchange coupling and the dipolar interaction. Introducing the local frame (~'/, ~'y/,g'/) of the site i we expand the spin vectors about the T = 0 configuration at Oi = 0, defining complex quantities ~i:
S,
=
where Go is the free energy associated with H0 and
up to a normalization factor. The extension of the integration over the whole complex plane of ai introduces the unphysical states lail ~ > 2 s of the conventional spin-wave theories. Owing to this effect, our analysis applies for low-temperature region only. One obtains
S [ ~ cosO, + s i n 0 , ( ~ cos ¢, + ~j sin ¢i)] G ~ E o - T S o - T N In T -
+(s - 1~,t212)'i:(,~;~'~. + ~,~ )
(2)
Here c~i ---- S l l 2 e i ¢ ' ( 1 - cosOi) 1/2 = ui + ivi and g'~/ = 2-1/2(g'~/ -4- i~*~). This is the classical analog of the tIolstein-Primakoff transformation. The usefulness of Eq. (2) relies on the fact sin Oid¢idOi o¢ duidvi. Retaining only the parts bilinear in oq and c~ we rewrite Eq. (1) as H=H0+Ha with Ho
=
T x ( B~oc / S)2
x E
(5)
+
j>i
i
where S0 is a constant. This result, is an estimate of the lowest-order spin-wave free energy. When the classical ground-state at T = 0 has infinite degeneracy, Eq. (5) can be used to find the configuration favored by thermal effects. When the interactions are isotropic, A q is proportional to the unit matrix. Since
~ije+~io~ j
i~.~_ " e~_~12 + i~.~_ " (,j- i~ = 71 + 7 -1~ ( s~' s-j )
So + B I°c Z
c~*cq
(3)
and H1 = S j>i
+~'.~,,.~,~:'o,j +c.c.},
(41
where B I°c is the magnitude of the site independent local field. The free energy can be expanded with respect to Hi: G ~ Go-
(H~)/(2T)
,
the stable spin configuration is the one which minimizes - ~ ( S , . Sv) ~, where S~s are the sublattice spin vectors. This ground-state selection term was proposed by Henley [1] for antiferromagnets with isotropic spin-spin interactions [1, 2]. The term shows that the collinear structures are favored by thermal fluctuations.
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0698-G -
,
238 An interesting application of the theory is an isotropic Type I f e ¢ antiferromagnet in an external field. The evolution of the spin structures given by minimizing - Z ( S u "Sv)2 was considered in Ref. [3]. We have also studied the (T, B) phase diagram of this system by means of a Monte Carlo calculation [4]. The results of the simulations are in reasonable agreement with the present theory. Our main application of Eq. (5) in the presence of an anisotropy is the Type I f e e system with an exchange and dipolar interaction. This model is relevant in the case of nuclear magnetic ordering in copper and silver at nanokelvin temperatures [5]. In this case there is a continuous degeneracy with respect to two parameters in the MF ground state. It is possible to compare the thermally favored structures given by Eq. (5) with those found in the Monte Carlo simulations [6, 7]. In Ref. [6] a model with relatively large dipolar anisotropy was studied. Four antiferromagnetic structures (AF1-4) were found in order of increasing magnetic field directed along a crystalline axis, B tt [001]. Analysis based on Eq. (5) predicts the evolution scheme AF1 --~ AF2 --~ a two-sublattice structure, thus failing for AF3 and AF4. Both AF2 and AF3 are four-sublattice structures but their detailed forms are slightly different. The ordering vector for the two-sublattice structure predicted by Eq. (5) at high fields is transversal [(1, O, O) or (0, 1, 0)1 whereas it is longitudinal for AF4 [(0, 0, 1)]. The ordered states of a model with a relatively weak dipolar interaction was studied in Ref. [7] for /~ [[ [011]. A transition between a lowfield two-sublattice structure [T(0, a/2, a/2)] and a high-field four-sublattice state (P~v) was found in agreement with an analysis based on Eq. (5). However, according to this equation, another twosublattice state should appear in the lowest fields. Substituting ai ~ ai, a~' ---* a~ in Eqs. (3 - 4) where a~, ai are the bosonic creation and annihilation operators, respectively, one gets the bilinear Hamiltonian of the Holstein-Primakoff approach. In analogy with our treatment of thermal effects, one can estimate the quantum correction to the classical ground state energy at T = 0 by treating H1 within the second order perturbation
theory [8, 9]. This gives [3] 1 S2
tfEo =
- -2 - B- ~°c
Z
gi+ A ~ gj+ 2
.
(6)
j>i
In the isotropic case the main difference between this result and Eq. (5) is that the correction of Eq. (6) vanishes for the fully polarized state in B > B¢ whereas the thermal correction remains non-zero. Eq. (5) and the result Eq. (6), studied in detail in Ref. [3], turned out to favor the same spin configurations among the investigated Type I states. Therefore, the spin structures selected by thermal and quantum effects appear to be identical. ACKNOWLEDGMENTS We would like to thank O.V. Lounasmaa for valuable comments on the manuscript. This work has been supported by the Academy of Finland. REFERENCES
1. C.L. Henley, J. Appl. Phys. 61, 3962 (1987). 2. C.L. Henley, Phys. Rev. Lett. 62, 2056 (1989). 3. M.T. Heinil~ and A.S. Oja, submitted to Phys. Rev. B. 4. M.T. Heinil~ and A.S. Oja, in these proceedings. 5. P.J. Hakonen, O.V. Lounasmaa, and A.S. Oja, J. Magn. & Magn. Mater. 100, 394 (1991). A.S. Oja, Physica B 169,306 (1991). 6. S.J. Frisken and D.J. Miller, Phys. Rev. Lett. 61, 1017 (1988). 7. H.E. ViertiS, Physica Scripta T33, 168 (1990). 8. P.-A. Lindg£rd, Phys. Rev. Lett. 61, 629 (1988). 9. M.W. Long, J. Phys.: Condens. Matter (UK) 1, 2857 (1989).
Physica B 194-196 (1994) 237-238 North-Holland
Breaking of Degeneracy in Classical Heisenberg Antiferromagnets M.T. Heinil~ and A.S. Oja Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland According to arguments given by Henley, thermal effects select collinear spin structures when the classical ground state exhibits continuous degeneracy in the mean-field theory and the spin-spin interactions are isotropic. We generalize this result and derive an approximate free energy describing ground-state selection for anisotropic spin-spin interactions using thermal perturbation theory. Reasonable agreement is found with Monte Carlo simulations for Type I ~cc antiferromagnets.
PACS numbers: 75.25.+z 75.50.Ee
Let us consider the low temperature behavior of a classical spin system with the Hamiltonian j>i k where t~kij is an interaction matrix which, in our applications, includes isotropic exchange coupling and the dipolar interaction. Introducing the local frame (~'/, ~'y/,g'/) of the site i we expand the spin vectors about the T = 0 configuration at Oi = 0, defining complex quantities ~i:
S,
=
where Go is the free energy associated with H0 and
up to a normalization factor. The extension of the integration over the whole complex plane of ai introduces the unphysical states lail ~ > 2 s of the conventional spin-wave theories. Owing to this effect, our analysis applies for low-temperature region only. One obtains
S [ ~ cosO, + s i n 0 , ( ~ cos ¢, + ~j sin ¢i)] G ~ E o - T S o - T N In T -
+(s - 1~,t212)'i:(,~;~'~. + ~,~ )
(2)
Here c~i ---- S l l 2 e i ¢ ' ( 1 - cosOi) 1/2 = ui + ivi and g'~/ = 2-1/2(g'~/ -4- i~*~). This is the classical analog of the tIolstein-Primakoff transformation. The usefulness of Eq. (2) relies on the fact sin Oid¢idOi o¢ duidvi. Retaining only the parts bilinear in oq and c~ we rewrite Eq. (1) as H=H0+Ha with Ho
=
T x ( B~oc / S)2
x E
(5)
+
j>i
i
where S0 is a constant. This result, is an estimate of the lowest-order spin-wave free energy. When the classical ground-state at T = 0 has infinite degeneracy, Eq. (5) can be used to find the configuration favored by thermal effects. When the interactions are isotropic, A q is proportional to the unit matrix. Since
~ije+~io~ j
i~.~_ " e~_~12 + i~.~_ " (,j- i~ = 71 + 7 -1~ ( s~' s-j )
So + B I°c Z
c~*cq
(3)
and H1 = S j>i
+~'.~,,.~,~:'o,j +c.c.},
(41
where B I°c is the magnitude of the site independent local field. The free energy can be expanded with respect to Hi: G ~ Go-
(H~)/(2T)
,
the stable spin configuration is the one which minimizes - ~ ( S , . Sv) ~, where S~s are the sublattice spin vectors. This ground-state selection term was proposed by Henley [1] for antiferromagnets with isotropic spin-spin interactions [1, 2]. The term shows that the collinear structures are favored by thermal fluctuations.
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0698-G -
,
238 An interesting application of the theory is an isotropic Type I f e ¢ antiferromagnet in an external field. The evolution of the spin structures given by minimizing - Z ( S u "Sv)2 was considered in Ref. [3]. We have also studied the (T, B) phase diagram of this system by means of a Monte Carlo calculation [4]. The results of the simulations are in reasonable agreement with the present theory. Our main application of Eq. (5) in the presence of an anisotropy is the Type I f e e system with an exchange and dipolar interaction. This model is relevant in the case of nuclear magnetic ordering in copper and silver at nanokelvin temperatures [5]. In this case there is a continuous degeneracy with respect to two parameters in the MF ground state. It is possible to compare the thermally favored structures given by Eq. (5) with those found in the Monte Carlo simulations [6, 7]. In Ref. [6] a model with relatively large dipolar anisotropy was studied. Four antiferromagnetic structures (AF1-4) were found in order of increasing magnetic field directed along a crystalline axis, B tt [001]. Analysis based on Eq. (5) predicts the evolution scheme AF1 --~ AF2 --~ a two-sublattice structure, thus failing for AF3 and AF4. Both AF2 and AF3 are four-sublattice structures but their detailed forms are slightly different. The ordering vector for the two-sublattice structure predicted by Eq. (5) at high fields is transversal [(1, O, O) or (0, 1, 0)1 whereas it is longitudinal for AF4 [(0, 0, 1)]. The ordered states of a model with a relatively weak dipolar interaction was studied in Ref. [7] for /~ [[ [011]. A transition between a lowfield two-sublattice structure [T(0, a/2, a/2)] and a high-field four-sublattice state (P~v) was found in agreement with an analysis based on Eq. (5). However, according to this equation, another twosublattice state should appear in the lowest fields. Substituting ai ~ ai, a~' ---* a~ in Eqs. (3 - 4) where a~, ai are the bosonic creation and annihilation operators, respectively, one gets the bilinear Hamiltonian of the Holstein-Primakoff approach. In analogy with our treatment of thermal effects, one can estimate the quantum correction to the classical ground state energy at T = 0 by treating H1 within the second order perturbation
theory [8, 9]. This gives [3] 1 S2
tfEo =
- -2 - B- ~°c
Z
gi+ A ~ gj+ 2
.
(6)
j>i
In the isotropic case the main difference between this result and Eq. (5) is that the correction of Eq. (6) vanishes for the fully polarized state in B > B¢ whereas the thermal correction remains non-zero. Eq. (5) and the result Eq. (6), studied in detail in Ref. [3], turned out to favor the same spin configurations among the investigated Type I states. Therefore, the spin structures selected by thermal and quantum effects appear to be identical. ACKNOWLEDGMENTS We would like to thank O.V. Lounasmaa for valuable comments on the manuscript. This work has been supported by the Academy of Finland. REFERENCES
1. C.L. Henley, J. Appl. Phys. 61, 3962 (1987). 2. C.L. Henley, Phys. Rev. Lett. 62, 2056 (1989). 3. M.T. Heinil~ and A.S. Oja, submitted to Phys. Rev. B. 4. M.T. Heinil~ and A.S. Oja, in these proceedings. 5. P.J. Hakonen, O.V. Lounasmaa, and A.S. Oja, J. Magn. & Magn. Mater. 100, 394 (1991). A.S. Oja, Physica B 169,306 (1991). 6. S.J. Frisken and D.J. Miller, Phys. Rev. Lett. 61, 1017 (1988). 7. H.E. ViertiS, Physica Scripta T33, 168 (1990). 8. P.-A. Lindg£rd, Phys. Rev. Lett. 61, 629 (1988). 9. M.W. Long, J. Phys.: Condens. Matter (UK) 1, 2857 (1989).