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On the ground state of Ising models with arbitrary spin quantum number
10 June 2002
Physics Letters A 298 (2002) 236–237 www.elsevier.com/locate/pla
On the ground state of Ising models with arbitrary spin quantum number Terufumi Yokota Department of Materials Science, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan Received 20 September 2001; accepted 6 December 2001 Communicated by A.R. Bishop
Abstract The ground state of a class of Ising models with site dependent arbitrary spin quantum number is shown to be restricted to ±SiMAX state where SiMAX is the spin quantum number at the site i. 2002 Elsevier Science B.V. All rights reserved. PACS: 75.10.Hk Keywords: Ising models; Ground state; Spin quantum number
Ising models with arbitrary spin quantum number S have been studied for a long time. Blume, Emery and Griffiths proposed a spin-1 Ising model with biquadratic exchange interaction and crystal field [1]. It has been extensively studied in connection with the theory of tricritical points. Also a general S spin glass model with crystal fields has been investigated [2]. Recently the ground-state phase diagrams for a generalized axial-next-nearest-neighbor Ising chain with alternating arbitrary spin quantum number were determined [3]. The authors of the paper postulated that every spin is its saturate state at the ground states. That is the spin state at site i Si is restricted to |Si | = SiMAX where SiMAX is the spin quantum number at the site. The postulation has been verified repeatedly to determine ground state phase diagrams for an axialnext-nearest-neighbor Ising model [4], an axial-third-
E-mail address: [email protected] (T. Yokota).
nearest-neighbor Ising model [5] and the mixed spin model mentioned above [6]. Here the speculation is shown to be true for more general models including further neighbor interactions, random bilinear exchange interactions and random external fields at arbitrary dimension. We study about the ground state of models represented by the following Hamiltonian: Jij Si Sj − hi Si , H=− (1) i,j
i
where Si is the Ising spin variable at site i and the spin quantum number SiMAX is assumed to be site dependent. Therefore Si = −SiMAX , −SiMAX + 1, . . . , SiMAX − 1, SiMAX at the site. The bilinear exchange coupling Jij may be random or non-random with arbitrary range of interactions. The summation in the first term is over all pairs with non-zero interactions. Also the external field hi can be random or non-random. This Hamiltonian includes the mixed
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 4 2 3 - 1
T. Yokota / Physics Letters A 298 (2002) 236–237
spin model [3], the axial-next-nearest-neighbor Ising model [4] and the axial-third-nearest-neighbor Ising model [5] as special cases. To show that the ground states of the models represented by the Hamiltonian (1) are restricted to ±SiMAX state, we first assume that there would be a site A at which SA = ±SiMAX is a ground state. The bar on the spin variable represents the ground-state value. The ground-state energy E0 may be written as E0 = E0 + EA ,
(2)
where EA is the energy that includes the site A and E0 is the residual part of the ground-state energy. EA is given by EA = − JiA Si SA − hA SA i
= SA − JiA Si − hA .
237
a similar way, when SA 0, SA = −SAMAX will give the ground state except for the special cases mentioned above. The above argument shows that each Si is +SiMAX or −SiMAX for the ground state of the models represented by the Hamiltonian (1) except for the special cases. It should be noted that the argument would be invalid if there were even powers of a spin variable in Hamiltonian. This includes the cases with biquadratic exchange interactions and/or crystal fields. In conclusion, we have proved that the ground state of a class of Ising models with arbitrary spin quantum number is restricted to the saturate state Si = ±SiMAX . This includes the proof of the speculation in Ref. [3]. The case of spin quantum number 1/2 is enough to investigate the ground state of the class of Ising models.
(3)
i
When SA 0, the quantity inside the bracket should be equal to or less than zero. Otherwise the energy EA would be lower reversing the sign of SA and keeping all other values of spins unchanged, which does not affect the value of E0 . The zero value inside the bracket may occur in fully frustrated cases like phase boundaries and also in random systems with accidental degeneracy. Except for these special cases, SA = +SAMAX will give lower value of EA , which contradicts the assumption of SA = ±SAMAX . In
References [1] M. Blume, V.J. Emery, R.B. Griffiths, Phys. Rev. A 4 (1971) 1071. [2] S.K. Ghatak, D. Sherrington, J. Phys. C. 10 (1977) 3149. [3] J.-J. Kim, S. Mori, I. Harada, Phys. Lett. A 202 (1995) 68. [4] T. Idogaki, K. Oda, Y. Muraoka, J.W. Tucker, J. Magn. Magn. Mater. 171 (1997) 83. [5] Y. Muraoka, M. Kanemaru, T. Idogaki, J. Magn. Magn. Mater. 177-181 (1998) 773. [6] Y. Muraoka, K. Oda, T. Idogaki, Physica B 284-288 (2000) 1553.
Physics Letters A 298 (2002) 236–237 www.elsevier.com/locate/pla
On the ground state of Ising models with arbitrary spin quantum number Terufumi Yokota Department of Materials Science, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan Received 20 September 2001; accepted 6 December 2001 Communicated by A.R. Bishop
Abstract The ground state of a class of Ising models with site dependent arbitrary spin quantum number is shown to be restricted to ±SiMAX state where SiMAX is the spin quantum number at the site i. 2002 Elsevier Science B.V. All rights reserved. PACS: 75.10.Hk Keywords: Ising models; Ground state; Spin quantum number
Ising models with arbitrary spin quantum number S have been studied for a long time. Blume, Emery and Griffiths proposed a spin-1 Ising model with biquadratic exchange interaction and crystal field [1]. It has been extensively studied in connection with the theory of tricritical points. Also a general S spin glass model with crystal fields has been investigated [2]. Recently the ground-state phase diagrams for a generalized axial-next-nearest-neighbor Ising chain with alternating arbitrary spin quantum number were determined [3]. The authors of the paper postulated that every spin is its saturate state at the ground states. That is the spin state at site i Si is restricted to |Si | = SiMAX where SiMAX is the spin quantum number at the site. The postulation has been verified repeatedly to determine ground state phase diagrams for an axialnext-nearest-neighbor Ising model [4], an axial-third-
E-mail address: [email protected] (T. Yokota).
nearest-neighbor Ising model [5] and the mixed spin model mentioned above [6]. Here the speculation is shown to be true for more general models including further neighbor interactions, random bilinear exchange interactions and random external fields at arbitrary dimension. We study about the ground state of models represented by the following Hamiltonian: Jij Si Sj − hi Si , H=− (1) i,j
i
where Si is the Ising spin variable at site i and the spin quantum number SiMAX is assumed to be site dependent. Therefore Si = −SiMAX , −SiMAX + 1, . . . , SiMAX − 1, SiMAX at the site. The bilinear exchange coupling Jij may be random or non-random with arbitrary range of interactions. The summation in the first term is over all pairs with non-zero interactions. Also the external field hi can be random or non-random. This Hamiltonian includes the mixed
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 4 2 3 - 1
T. Yokota / Physics Letters A 298 (2002) 236–237
spin model [3], the axial-next-nearest-neighbor Ising model [4] and the axial-third-nearest-neighbor Ising model [5] as special cases. To show that the ground states of the models represented by the Hamiltonian (1) are restricted to ±SiMAX state, we first assume that there would be a site A at which SA = ±SiMAX is a ground state. The bar on the spin variable represents the ground-state value. The ground-state energy E0 may be written as E0 = E0 + EA ,
(2)
where EA is the energy that includes the site A and E0 is the residual part of the ground-state energy. EA is given by EA = − JiA Si SA − hA SA i
= SA − JiA Si − hA .
237
a similar way, when SA 0, SA = −SAMAX will give the ground state except for the special cases mentioned above. The above argument shows that each Si is +SiMAX or −SiMAX for the ground state of the models represented by the Hamiltonian (1) except for the special cases. It should be noted that the argument would be invalid if there were even powers of a spin variable in Hamiltonian. This includes the cases with biquadratic exchange interactions and/or crystal fields. In conclusion, we have proved that the ground state of a class of Ising models with arbitrary spin quantum number is restricted to the saturate state Si = ±SiMAX . This includes the proof of the speculation in Ref. [3]. The case of spin quantum number 1/2 is enough to investigate the ground state of the class of Ising models.
(3)
i
When SA 0, the quantity inside the bracket should be equal to or less than zero. Otherwise the energy EA would be lower reversing the sign of SA and keeping all other values of spins unchanged, which does not affect the value of E0 . The zero value inside the bracket may occur in fully frustrated cases like phase boundaries and also in random systems with accidental degeneracy. Except for these special cases, SA = +SAMAX will give lower value of EA , which contradicts the assumption of SA = ±SAMAX . In
References [1] M. Blume, V.J. Emery, R.B. Griffiths, Phys. Rev. A 4 (1971) 1071. [2] S.K. Ghatak, D. Sherrington, J. Phys. C. 10 (1977) 3149. [3] J.-J. Kim, S. Mori, I. Harada, Phys. Lett. A 202 (1995) 68. [4] T. Idogaki, K. Oda, Y. Muraoka, J.W. Tucker, J. Magn. Magn. Mater. 171 (1997) 83. [5] Y. Muraoka, M. Kanemaru, T. Idogaki, J. Magn. Magn. Mater. 177-181 (1998) 773. [6] Y. Muraoka, K. Oda, T. Idogaki, Physica B 284-288 (2000) 1553.