The magnetization plateaus of the ferro and anti-ferro spin-1 classical models with S 2 z term

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1 Te magnetization plateaus of te ferro and anti-ferro spin-1 classical models wit S 2 z term arxiv: v1 [cond-mat.stat-mec] 18 Apr 2013 S.M de Souza 1 and M.T. Tomaz 2 1 epartamento de Ciências Exatas, Universidade Federal de Lavras, Caixa Postal 3037, CEP , Lavras-MG, Brazil. 2 Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n o, CEP , Niterói-R, Brazil. November 6, 2018 Abstract We study in detail te exact termodynamics of te one-dimensional standard and staggered spin-1 Ising models wit a single-ion anisotropy term in te presence of a longitudinal magnetic field at low temperatures. Te results are valid for te ferromagnetic and anti-ferromagnetic AF) models and for positive and negative values of te crystal field for T > 0. Altoug te excited states of te ferro and anti-ferro models are igly degenerate, we sow tat te temperature required for reacing te first excited state in te classical spin-1 ferro model gives a scale of temperature tat permits fitting te z-component of te magnetization only by te contribution of two ground states of te model. Tis approximation is not true for te equivalent AF function due to te fact tat te AF model is gapless along te lines separating te pases in its pase diagram at T = 0. We relate te number of plateaus in te magnetization of eac model to teir respective pase diagrams at T = 0. Te specific eat per site of te AF model distinguises, at low temperature, te transitions A E and G E as te external magnetic field is varied. Te exact Helmoltz free energy of te classical spin-1 model is mapped onto te equivalent function of te ionic limit of te 1 extended Hubbard model by proper transformations. Keywords: Quantum statistical mecanics, Ising model, spin-1, single-ion anisotropy term, staggered, termodynamics, optical device. PACS numbers: d, Hk, m Corresponding autor: mtt@if.uff.br 1

2 1 Introduction Exactly solvable models elp us aving insigts on more complex systems. In one space dimension 1), te termodynamics of te spin- 1 Ising model in te presence of a 2 longitudinal magnetic field as been solved in te wole interval of temperature using te density matrix approac[1, 2, 3]. To do so, it is essential te commutative nature of all operators in its Hamiltonian. Tis model is usually called a classical system. Cold atoms ave made possible simulations of spin models. Recently Simon et al.[4] simulated a one-dimensional spin-1/2 model in te presence of a magnetic field wit longitudinal and transverse components by using a Mott insulator of spinless bosons in a tilted optical lattice. Te device was used to study te pases of tis classical model at low temperatures. Tese optical devices ave permitted te experimental study of properties of cain models, including te spin models. Te spin-1 Ising model wit single-ion anisotropy term, te Blume[5]-Capel[6] model, in te presence of a longitudinal magnetic field, as also a classical nature. Tis fact permits us to apply te transfer matrix metod[1, 2, 3] to solve it. Wit tis approac, Aydiner and Akyüz[7] obtained te numerical solution of te 1 spin-1 anti-ferromagnetic AF) Ising model wit a single-ion anisotropy term in te presence of a longitudinal magnetic field. Tey studied te magnetic and termal beavior of te model at very low temperatures. In 2003 Cen et al.[8] applied te classical Monte CarloCMC) tecnique to te numerical calculation of te pase diagram at T = 0 of te spin-1 anti-ferromagnetic Ising model in te presence of a single-ion anisotropy term for positive crystal field. Te CMC was applied also to te study of magnetization plateaus of tis model at low temperature. In 2005 Mancini[9] used te mapping between te 1 extended Hubbard model in te ionic limit i.e., t U, V, in wic t is te opping excange, U is te on-site Coulomb interactions and V is te inter-site Coulomb interaction) and te spin-1 Ising model wit a single-ion anisotropy term in te presence of a longitudinal magnetic field to write te exact Helmoltz free energy HFE) of te latter[9] as a set of coupled equations. Te ierarcy of te equations of motion is closed, and tey can be solved numerically. In Ref.[10] tey ave extended te approac to include a biquadratic nearest-neigbour interaction term, numerically solving te coupled equations and presenting te beavior of some termodynamic functions at low temperatures. We derived in Ref.[11] te exact expression of te HFE of te 1 spin-1 Ising model wit te S z ) 2 term in te absence of a magnetic field. Tis HFE as a simple analytic expression valid for te ferromagnetic and AF models in te wole range of temperature. Te absence of a magnetic field in tis solution prevents accessing some information about te system, e.g.: information about te magnetization and te susceptibility of te model. All te previously mentioned papers on te one-dimensional spin-1 Ising model seem to be unaware of te nice work by Krinsky and Furman[12] in 1976, wic presents te calculation of te exact HFE for te 1 ferromagnetic spin-1 Ising model wit a single-ion anisotropy term, a biquadratic nearest-neigbour interaction term and a nonsymmetric term, in te presence of a longitudinal magnetic field. More recently Litaiff et al. also applied te transfer-matrix tecnique to calculate te exact expression of te HFE of tis model. Te autors did not use tis termodynamic function to explore its low 2

3 temperature beavior. Altoug te 1 spin-s Ising model wit te single-ion anisotropy term in te presence of a magnetic field is called a classical model, it as a quantum nature tat manifests itself as plateaus in te z-component of te magnetization at very low temperature. Te existence of tose plateaus as been demonstrated in a number of 1 models[14] and experimentally measured in some materials[15, 16, 17] described by 1 spin-models. Magnetization plateaus in te te AF spin-1 Ising model wit positive crystal field in te presence of a longitudinal magnetic field ave been obtained numerically in Refs.[7, 8, 10]. Te autors of Ref.[12] mentioned te discontinuous cange of te magnetization of te ferromagnetic model, but teir focus was on te exact renormalization group of te model and te discussion of its critical points. Te study of te simple one-dimensional spin-1 Ising model elps to understand te origin of plateaus in te magnetization function. In te present communication we present te exact analytic expressions of te HFE s of te standard and te staggered versions of te 1 spin-1 Ising model wit te S z i) 2 term in te presence of a longitudinal magnetic field, valid for T > 0 in section 2. Our solutions apply equally well to te ferromagnetic and to te AF models of bot versions, extending te validity of te solution derived in Ref.[12] for te ferromagnetic model. We study te z-component of te magnetization and te entropy per site at very low temperatures for te ferro and AF versions of te classical spin-1 model in sections 3 and 4, respectively. We relate te number of plateaus in te magnetization to te number of pases in te ground state of te model at T = 0. Finally in section 5 we present our conclusions. More detailed calculations are sown in Appendices A and B. 2 Te Hamiltonians of te classical spin-1 models in te presence of a longitudinal magnetic field Te Hamiltonian of te one-dimensional of te classical spin-1 Ising model wit a singleion anisotropy term, te Blume-Capel model, in te presence of a longitudinal magnetic field is[5, 6] H = = N i=1 N i=1 [ S z i S z i+1 S z i + S z i )2] H i,i+1, 1) were S z i is te z- component of te spin-1 operator wit norm: S 2 = 2. is te excange strengt and it can ave eiter a negative value ferromagnetic model) or a positive value AF model). Te crystal field can assume positive, negative or null values. Te model satisfies spacial periodic boundary condition in a cain wit N sites. Te external magnetic field is oriented along te easy-axis z. In tis paper we use natural units, e = m = = 1. Comparing our Hamiltonian1) tote Hamiltonian25)in Ref.[13] 3

4 we ave =, = µ B H and =, in wic, H and are te parameters of te Hamiltonian of te single spin-1 Ising model in Ref.[13]. Mancini[9] sowed tat by te substitution S z i = n i 1 i were 1 i is te identity operator at te i-t site, in wic n i = n i, +n i,, n i,σ c i,σ c i,σ wit σ {, }, c i,σ is te fermionic creation operator of an electron at te i-t site wit te spin component σ and c i,σ is te corresponding destruction operator, te Hamiltonian 1) is mapped onto te ionic limit of te Hamiltonian of te 1 extended Hubbard model in te presence of a cemical potential, tat is were[18, 19] H = H Hub +N + + ) 1, 2) H Hub = N i=1 Un i, n i, +2Vn i n i+1 µn i ). 3) Te last equality is valid for U = 2, V = and µ = As a consequence of relation 2) we ave, for tese values of parameters, tat W Hub U,V,µ;β) = W 1,, ;β) + + ), were W Hub W 1 ) is te HFE of te extended Hubbard model in te ionic limit te HFE of te spin-1 Ising model 1)). One is reminded tat β = 1, were k is te Boltzmann s constant and T is te absolute temperature in kelvin. kt Te Hamiltonian 1) in te presence of a staggered longitudinal magnetic field is H stag = 2M i=1 [ S z i S z i+1 1) i S z i + S z i )2]. 4) Te Hamiltonian continues to satisfy space periodic condition, but now we ave an even number of sites in te cain: N = 2M. Applying te metod presented in Ref.[20] we calculated in Ref.[21] te β-expansion of te HFE of te one-dimensional spin-s Ising model, wit single-ion anisotropy term, in te presence of a longitudinal magnetic field, W S,, ;β), up to order β 17, in te termodynamic limit. Te β-expansion of te HFE of te staggered Hamiltonian 4), wit = 0, W stag S,,0;β), was derived in Ref.[22] up to te same order in β also for M. Unfortunately tose expansions do not permit us to study te termodynamics of te standard and te staggered 1 spin-1 Ising model close to T = 0. In our web page 1 we provide te data files wit te quantum arbitrary spin-s) and te classical HFE s of te normalized Hamiltonians 1) and 4) up to order β 17 for bot versions standard and staggered) of te model. In te present work we apply te transfer matrix metod[3] togeter wit te β- expansion of te function W 1,, ;β) obtained from te results of Ref.[21] wit S = 1 to calculate te exact HFE of te Hamiltonian 1) in te termodynamic limit N 1 Our web page: ttp:// 4

5 ), valid for T > 0. Te expression of W stag 1,, ;β) is obtained by using a well known result in te literature[22], namely, W stag S,, ;β) = W S,, ;β), wit S = 1,1, 3,. Tis equality is valid for T [0, ). 2 2 In order to calculate te exact function W 1,, ;β), valid for T > 0 and in te limit of N, using te transfer matrix metod, we rewrite Hamiltonian 1) as a symmetric operator in te i-t and i+1)-t sites, tat is, H S) i,i+1 = Sz i Sz i+1 Sz i Sz i+1 +Sz i )2 +S z i+1 )2. 5) Bycomparingeqs.1)and5), weverifytatteβ-expansionoftehfeoftismodel, in terms of te parameters, and in Hamiltonian 5) may be obtained by applying te following cange of variables: 2 and 2, in te expansion of Ref.[21] wit S = 1. In appendix A we calculate te tree roots of te tird degree equation derived from te transfer matrix metod for te classical spin-1 model wit a single-ion anisotropy term in te presence of a longitudinal magnetic field. Te root wit te largest modulus gives te expression of te HFE of te model 1)/5) see eq. A.4)). In order to verify wic root, eqs.a.8a) - A.8c), corresponds to te eigenvalue λ 1 of matrix U, assumed to be te root wit te largest modulus, we calculate te β-expansion of te functions 1 β ln[s i], i = 1,2,3, and compare eac one wit te expansion of te HFE of Ref.[21] wit S = 1 aving made te cange of variables = 2 and = 2 in te β-expansion of Ref.[21]). By direct comparison of te expansions, we obtain for finite value of β, up to order β 10, tat te root s 1, eq.a.8a), is te eigenvalue λ 1, tat as te largest modulus among te eigenvalues of matrix U. Our analysis is valid for finite value of β, wic excludes te value T = 0. Oursolutions 1 isequaltoteeigenvalueoftecubicequationwittelargestmodulus calculated inref.[12] forteferromagnetic model < 0). Teway we writete solutions A.9a) - A.9c) avoids te necessity of defining a cut in te complex plane for teir calculation. Our results extended tose derived in Ref.[12] to include te HFE of te AF spin-1 Ising model in te presence of a longitudinal magnetic field. Te exact HFE of te standard one-dimensional spin-1 Ising model wit te single-ion anisotropy term in te presence of a longitudinal magnetic field is W 1,,;β) = 1 [2 β ln Q cos θ 3 )+ P ], 6) 3 in wic P, θ and Q are given respectively by eqs.a.7a), A.9a), A.9b). Te HFE 6) is an even function of : W 1,,;β) = W 1,,;β). One is reminded tat W stag 1,, ;β) = W 1,, ;β) and W Hub U,V,µ;β) = W 1,, ;β) + + ), wit U = 2, V = and µ = From te simple expression 6) one can calculate te termodynamic functions of te standard and te staggered models, as well of te extended Hubbard model in te ionic 5

6 limit in te absence of an external magnetic field, for any finite value of β, tat does not include te temperature T = 0, tat is, T 0, ). It is a well known fact tat at T = 0, = 0 and = 0, two eigenvalues of matrix U are degenerated. Our results are valid in te limit of T 0. ue to te presence of te longitudinal magnetic field in te HFE 6) we can derive from it te z-component of te magnetization and te magnetic susceptibility per site for te ferromagnetic < 0) and AF > 0) versions of te standard Hamiltonian 1)/ 5) as well as te staggered magnetization and te staggered magnetic susceptibility[22] of te Hamiltonian 4) for any value of te excange strengt. For = 0, te expression 6) of te HFE W 1,0,;β) coincides wit te exact result presented in Ref.[11]. Our exact result permits us to study te plateaus of te standard and te staggered z component of te magnetization per site of te ferro and te AF models of te standard and te staggered versions, respectively, of te one-dimensional spin-1 Ising model, wit te single-ion anisotropy term, in te presence of a longitudinal magnetic field for temperatures close to T 0. Te autors of Ref.[12] mentioned te discontinuity of te magnetization and te magnetic quadrupolar moment of te ferromagnetic model at T = 0, not going any furter ontis point. Aydiner and Akyüz [7] and Cen et al [8] studied numerically tese plateaus in te AF version of Hamiltonian 1)/5). More recently, Mancini and Mancini[10] solved also numerically) te self consistent equations tat yield te exact HFE of te AF S = 1 Ising model wit a biquadratic nearest-neigbour term in te presence of a longitudinal magnetic field. In te rest of tis paper we let = 1 for te ferromagnetic model and = 1 for te AF model. Te values of te parameters, and T are given in units of, tat is:, and T, respectively. We restrict our presentation mainly to te beavior of two termodynamic functions associated to te Hamiltonians 1)/5) and 4) tat elps us to understand te presence of plateaus in te classical spin-1 model: te z- component of te magnetization[24], M z,,;β) ) M z = 1 W 1 2, of te standard model 1)/5), and te staggered z-component ) of te magnetization[22], M stag z,,;β) M stag z = 1 2 [ Sz 2 S2 ] z = 1 W stag 1, of te staggered Hamiltonian 2 ) 4); and te entropy per site, S,, ; β) S = β 2 W, were W = W β 1 W stag 1 ) for te standard staggered) model. Tose are studied at very low temperatures. Te HFE s W 1 and W stag 1 are even functions of, terefore S,,;β) and te magnetization functions M z,,;β) and M stag z,,;β)) are even and odd functions of, respectively. For tis reason we restrict ourselves to te case of 0. In Fig.1 we present te pase diagram of te ferro = 1) and AF versions of Hamiltonian1)/5)atT = 0. Initscaptionwedescribetemeaning oftepases ineac diagram. It is interesting to notice tat te excange coupling term in te ferromagnetic Hamiltonian 1)/5) favors te states wit z component of te spin s z i = 1 and parallel neigboring spins. Te same term in te AF version of tis Hamiltonian also favors te states wit s z i = 1 but for anti-parallel Néel state) neigboring spins. In te ferro and AFmodelsteZeemantermfavorsstateswits z i = 1alignedwitteexternalmagnetic field, wereas te single-ion anisotropy term sows two distinct beaviors: for < 0 6

7 states wit s z i = 1 are favored, independently of teir relative alignment; and for > 0 states wit s z i = 0 are favored, te spin being perpendicular to te external magnetic field applied. For < 0 all te terms in Hamiltonian 1)/5) force te neigboring spins to align to eac oter and consequently to te external magnetic field at T = 0. Te ground state is a collective stable state under small temperature fluctuations, a point wic will be made clear after we discuss te necessary temperature to excite te first excited state of te ferromagnetic cain. For > 0 te effect of te crystal field competes wit te excange strengt and te external magnetic. In te near future we will verify tat in tis case we need a smaller energy to break te alignment between neigboring spins and excite te first excited energy level of te ferromagnetic cain. For 0 and only for 1 te Fig.1a as two distinct pases. Te line between 2 te pases is: = 1. In te region A of Fig.1a, at T = 0, te single-ion anisotropy 2 term gives te main contribution to te ground state and te spins are perpendicular to te longitudinal magnetic field. Our pase diagram Fig.1a coincides wit Fig.2 of Ref.[12] wit K = 0. Fig.1b presents te pase diagram of te standard AF model = 1) at T = 0. For < 0 and 0 tere are two distinct pases tat correspond to te competition between te excange coupling term and te Zeeman term. One of tem is te pase G tat is te Néel state wen 0 < 1; te excange term gives te largest contribution to te ground state energy. For > 0 and 0 te tree terms in te AF Hamiltonian 1)/5) favor distinct states at T = 0. Teir competition is responsible for a ricer pase diagram at T = 0 for te AF model in tis region of parameters. On te oter and, tis competition among te terms of Hamiltonian 1)/5) makes te ground state of te one-dimensional system less stable under small fluctuations of te termal energy, as we will verify in te discussion of te first excited state of te AF cain. Te diagram of tis model as two tricritical points: P = 1 and = 0) and Q = 1 2 and = 1 2 ) in Fig.1b. For 0 < < 1 and 0, te AF model as tree distinct pases at T = 0, 2 namely: te G Néel) pase, te E pase in wic alf of te spins in te cain are perpendicular to te external magnetic field), and te B pase in wic all te spins are aligned wit te magnetic field). We also ave tree pases for 1, but in tis region 2 of te pase diagram at T = 0 te Néel state is not one of te possible ground states of te system; tere is an A pase instead, in wic all spins are perpendicular to te longitudinal magnetic field, besides te pases B and E. Te pase diagram Fig.1b agrees wit Fig.2b of Ref.[8] and sows qualitative agreement wit Fig.2 of Ref.[7]. Te transitions between te ground states of te standard AF model appen for te 7

8 following values of magnetic field 0 and te intervals of : G B : < 0 and = 1, G E : 0 < < 1 2 and A E : E B : > 1 2 and = 1, =, > 0 and = 1+. 7a) 7b) 7c) 7d) Te pase diagram of te staggered ferromagnetic = 1) Hamiltonian 4) at T = 0 is sown in Fig.1b wereas te diagram of te staggered AF = 1) model at T = 0 is depicted in Fig.1a. To te best of our knowledge, tere is no detailed discussion in te current literature regarding te presence of plateaus in te z-component of te magnetization in 1 ferromagnetic spin models at very low temperatures, altoug its discontinuity is mentioned in Ref.[12]. From te exact expressions of te z-component of te magnetizations of te ferro and AF spin-1 classical models, valid in te termodynamic limit, we verify tat tey ave no discontinuity for T > 0. Rater, tere is a range of values of te magnetic field for wic a continuous transition from one magnetization plateau to anoter takes place. Our result 6) is equally valid for te ferro and AF models of te standard and staggered spin-1 Ising model wit Si) z 2 term in te presence of a longitudinal magnetic field T > 0. Our result 6) of te HFE of te ferro = 1) and te AF = 1) Hamiltonians 1)/5) are valid for T > 0. A natural way to describe tose models at finite temperature is troug teir respective density matrix operator. From now on, we restrict our discussions to te condition 0. 3 Te ferromagnetic classical spin-1 model Let us consider te ferromagnetic model 1)/5) wit = 1. Te ground states of te cain at T = 0 in te pases A, B and C in Fig.1a are Ψ 0 A = N, 8a) Ψ 0 B = N, 8b) Ψ 0 C = N. 8c) were Si s z i = s s i, s = 0,±1 and i = 1,2,,N. Teir respective ground state energy are named E0 A, EB 0 and EC 0. Te ground state 8c) is presented since at = 0 its energy degenerates into te energy of te state Ψ 0 A. Te values of te ground state energies in units of are 8

9 E0 A E0 B E0 C = 0, 9a) = N ), 9b) = N ), 9c) were N is te number of sites in te cain. Te density matrix operator of te ferromagnetic Hamiltonian 1)/5) is ρ 1, ), ;β = e βea 0 Ψ 0 A A Ψ 0 + e βeb 0 Ψ 0 B B Ψ 0 + Z 1 Z 1 + e βec 0 N Ψ 0 C C Ψ 0 + e βe1 Φ l) 1 Φ l) 1, 10) Z 1 Z 1 were Z 1 is te partition function associated to te ferromagnetic Hamiltonian 1)/5), Z 1 = Tr[e βh ], wit β = 1. E kt 1 is te first excited state of te cain and its degeneracy is equal to N for all tree ferromagnetic pases. Te first excited state is obtained from te ground state vectors 8a)-8c) by flipping one of its spin-1 in te cain. In te termodynamic limit, te first excited state is igly degenerate as well te iger excited states[25]. In principle, te contribution of all excited states of te cain sould be taken into account for non null temperatures. We ave no matematically sound argument to affirm tat, at low temperatures, te expansion 10) sould be cut after te contribution of te first excited state. Te exact expressions of te termodynamic functions can be derived from result 6) of te HFE 6) of te HFE, and tey sow plateaus in te z-component of te magnetization of tis model at low temperature. Te comparison between te exact result and a proposed approximation expression of M z sould be able to say ow good is te latter. It is common sense tat te te least value of temperature for wic te contribution oftefirstexcitedstatemust betakenintoaccount fortebeavior oftetermodynamic functions is determined by te factor e E 1 E 0 kt. But in te present model te excited states ave degeneracy at least of order N, so all of tem sould contribute to te functions at non zero temperature. In te following discussion we determine in eac pase of te diagram 1a te value of temperature T max were l=1 e E 1 E 0 ktmax e , 11) were E 0 is te ground state energy of te pase and E 1 is its first excited state. Ten kt max E 1 E 0. 12) 15 We want to verify if te temperature T max at eac ferromagnetic pase is suc tat, for temperatures lower tan T max, te dependence of te z-component of te magnetization on te external magnetic field as a step-like form. 9

10 We ave tree distinct first excited states of te ferro classical spin-1 model 1)/5). Tey depend on te values of te parameters and. Tere are two distinct first excited states in pase B see Fig.1a): 1.1) for > 0 and 1 Φ l) 1 B = l 1 N, l = 1,2,,N, 13) and E B 1 EB 0 = 4 1+ ). 14) Replacing te result 14) in eq.12), we obtain kt max ). 15) Tat gives te lowest temperature for wic te first excited state is expected to contribute to te beavior of te termodynamic functions. Eq.15) sows tat T max is independent of te value of. In tis region of parameters of te Hamiltonian 1)/5), te action of te crystal field term is more relevant tan te excange coupling term and te Zeeman term. 1.2) for and > 0 > 1 2 and 1 2 Φ l) 1 B = l 1 N, l = 1,2,,N, 16) and E B 1 EB 0 = 2 1+ ). 17) For tis set of parameters and kt max 2 15, eq.12) gives 1+ ). 18) Te results 15) and 18) belong to te same pase B and teir difference wit respect to te first excited state come from te fact tat tey describe te beavior of te cain of spins-1 in distinct ranges of te parameters of te Hamiltonian: in case 1.1) case 1.2)) te effect of is more less) important tan te excange coupling and te term Zeeman effect togeter. 2) te first excited state in pase A see Fig.1a): 1 2 and Φ l) 1 A = l 0 N, l = 1,2,,N, 19) and E A 1 E A 0 = 2 + ). 20) 10

11 Replacing te variation of energy 20) in eq.12), we ave kt max ). 21) In te present case te value of T max depends on te value of te crystal field and T max is a decreasing function of. Te latter implies tat te ground state vector 8a) is less stable tan te vector 8b) and 8c) under an increasing external magnetic field. In pase A of Fig.1a we ave a competition between te crystal field term and te oter two terms in te ferromagnetic Hamiltonian 1)/5). In Fig.2 we plot te curves ktmax as a function of for tree values of, covering all te regions of 0 described in te discussion of te first excited states of te ferromagnetic classical spin-1 model 1)/5). For = 2.5 we ave pase A at T = 0 and tere is a competition between te action of te single-ion anisotropy term and te two oter terms in te Hamiltonian 1)/5). For 0 2 we ave a negative slope in te curve ktmax. Tis negative slope means tat te first excited state can be reaced witout increasing te temperature. In Figs.3a and 3b we plot te exact curves solid lines) of M z wit = 2 and 2.5 at kt = and , respectively. In Fig.3a te z-component of te magnetization of te ferromagnetic model begins at null value wit = 0 and reaces its saturated value M z = 1 for variations of te magnetic field so tat Wit = 2.5 te function M z in Fig.3b as a continuous transition from M z = 0 to M z = 1 for Te maximum value of te entropy per site for tese two cases is equal to at = 0 and = 2. Tese are te values of te external magnetic field tat appens te ferromagnetic pase transitions at T = 0 in diagram 1a. In Fig.3cwe compare tegraps ofte z-component ofte magnetizationof temodel wit kt = 2.5 at two different temperatures: = dotted line) and 0.11 solid line). Te latter value of temperature is almost twice te value of ktmax wit = 0. It is clear from Fig.3b and 3c te presence of te two plateaus at M z = 0 and M z = 1 at tose low temperatures. Tese plateaus of M z satisfy te Osikawa, Yamanaka and Affleck OYA) condition[26]. We remind tat tis condition, wen applied to te periodic Hamiltonian 1)/5) wit spin-1, imposes tat p1 M z ) be an integer, were p is te spatial period of te ground state. Tis is a necessary condition for te occurrence of a plateau in te magnetization curve of te one-dimensional spin system[8, 26]. Te same condition is satisfied in Fig.3a. In Fig.3b, te transition between te two plateaus appens at = 2, te value of te magnetic field per unit of in wic tis ferromagnetic model suffers a transition between te pases A and B at = 2.5. Note tat doubling te temperature per unit of does not cange muc te step-like form of te plateaus in te curve of M z. We could say tat in te standard ferromagnetic model te plateaus are stiff. From Figs.3 we verify tat outside te transition region were te value of M z suffers a finite transition, te values of tis termodynamic function correspond to teir respective value of te ground state in te pase diagram at T = 0. We can use a penomenological approac to fit M z in te wole interval of at T < T max. We assume 11

12 tat te contributions to M z in te transition region of at T < T max come only from te ground states in te density matrix operator 10). We ave two distinct situations to discuss. Situation 1): > 0 and 1. In tis approximation we obtain for 2 > 0: M z,,;β) = e βeb 0 e βe C 0 e βeb 0 +e βec 0 = E B 0 <EC 0 tan2βn). 22) Since we consider to be non-zero and positive we ave E B 0 < E0 C. We do not ave, a priori, a matematical sound argument to affirm tat te approximation 22) gives te rigt picture of te z-component of te magnetization in te ferromagnetic model at low temperatures. On te last term on te r..s. of result 22) we ave a product of two limit processes in te region of 0: N termodynamic limit) and 0. Te result 22) depends on te assumption tat lim N = l;β), 23) N 0 in wic te function l; β) is assumed to be finite. Te simplest penomenological function l; β) is a linear function were we take a as a constant. l;β) = a, 24) Situation 2: 1 and 1. Assuming tat for T 2 2 < T max only te ground states of te ferromagnetic pases A and B contribute to M z, we derive te approximate expression of tis termodynamic function M z valid for 1 at low temperature, 2 were M z,,;β) e βnε 1+e βnε, 25a) [ ε )]. 25b) Again in te exponential functions on te r..s. ) of eq.25a) we need to calculate ) te product of two limits in te region 1 + : N and 1 +. In 2 2 analogy to eqs. 23) and 24) we write lim N ) 0 N [ )] = l;β). 26) 12

13 In tis case we also take te simplest linear function for te penomenological function l;β), [ l;β) a )], 27) were a is a constant. In Table 1 we present te values of te parameter a for various values of at distinct low temperatures. Some of te values of a were used in te approximate curves of M z in Figs.3a and 3b. 4 Te antiferromagnetic classical spin-1 model Fig.1b sows te six distinct pases at T = 0 for te AF spin-1 Ising model wit a singleion anisotropy term in te presence of a longitudinal external magnetic field. Wit null crystal field = 0), te model as only tree different pases at T = 0. Te tree terms in te AF = +1) Hamiltonian 1)/5) compete: te excange coupling term favors neigbor spins to align anti-parallel. Te most stable configurations of te AF ground state appens wen te effect of two terms on eac spin in te cain are in te same direction. We present in appendix B te ground state vectors and energies of te six pases of te AF Hamiltonian 1)/5) at T = 0 and teir respective energy difference between te first excited state and its ground state. We also present in tat appendix te relation between ktmax and te parameters and suc tat te condition 11) is valid for te classical AF spin-1 model. In Figs.4 we plot te curves of ktmax for various values of tat spans all AF pases at T = 0 in te pase diagram Fig.1b. Te fundamental difference between Figs. 2 and 4 of te ferromagnetic and AF models, respectively, is tat te temperature required to excite te first excited ferromagnetic state never vanises. In Fig.4a wit < 0 we ave T max = 0 at = 1. Te line = 1 wit < 0 separates te pases B and G in diagram 1b. Te result T max = 0 along tis line means tat te classical AF spin-1 model 1)/5) is gapless along te separation of tese two pases. We also ave T max = 0 in Fig.4b wit = 1 at = 2 and 4. Tese values of are on te lines tat separate te pases E G and B E respectively. For te interval 0 1 te AF model 1)/5) is gapless along te lines = 1 and = 1+. At 2 = 2.5 and = 3.5 wit = 2.5 we also ave T max = 0 in Fig.4b. Tese values of are on te lines tat separate te pases A E and B E respectively. Te AF model 1)/5) is gapless for 1 along te lines = and = 1+. We partially summarize 2 te contents of te curves in Figs.4 by saying tat te classical AF spin-1 Hamiltonian 1)/5) is gapless along te lines tat separate te pases in te diagram 1b at T = 0. Te presence of plateaus in te termodynamic functions M z of te AF Hamiltonian 1) as been reported previously by some autors[7, 8, 10] for positive values of [8]. In te following we discuss te beavior of te termodynamic functions M z and S of te AF version of Hamiltonian 1), tat is = 1, by varying te value of te crystal field 13

14 per unit of,, to span part of te pase diagram of te AF model 1) at T = 0 see Fig.1b). Here we draw a more detailed comparison between our pase diagram in Fig.1b wit te pase diagram of Ref.[8], in wic only te case > 0 is considered. Before we continue te comparison, we sould notice tat Cen = 2 and Cen = 2, were and are our parameters in te Hamiltonian 5). Teir Fig.2b agrees wit our diagram 1b for > 0, except for te fact tat in teir pase diagram te pase M z = m = 0 does not distinguis te pases A and G see te caption of our Fig.1b) tat correspond to different ground states. In our pase diagram of te AF model we point out te presence of two tricritical points, tat is, te points: P = 1 and = 0) and Q = 1 and 2 = 1). 2 For < 0 and > 0 we ave at T = 0 te pases B and G in diagram 1b for te standard AF model. In tis region of parameters of te Hamiltonian we ave a competition between te states favored by te excange coupling term and tose favored by te Zeeman term. Figs.5 sow te termodynamic functions M z and S versus wit = 2 longdased lines). Te z-component of te magnetization for < 0 as two plateaus, M z = 0 and M z = 1, at very low temperatures. Tey satisfy te OYA condition[26]. In tis region of, te transition between tese two plateaus appens at = 1, tat is te value in wic te transition between te pases B and G at T = 0 also occurs, for < 0. Since along any vertical line in tis region of only two pases are crossed at T = 0 see Fig.1b), te AF spin-1 Ising model 1) as only two plateaus at very low temperatures for < 0. Fig.5a sows te curve M z at T = 0.266, tat is te same temperature used in plotting tis function for te ferro model in Fig.3a wit = 2. Te curve of M z at T = still as little resemblance to a step-like function. Tis beavior differs from tat of te magnetization of te standard ferromagnetic model see Fig.3a). By comparing te beavior of te plateaus in te magnetization in te standard ferro and te AF models at T = 0.266, we can say tat te plateaus of te latter smear out even at low temperatures. Tis appens because altoug te neigbour spins are aligned anti-aligned) in pase B G) te excange coupling term and te Zeeman term togeter wit te crystal field compete in promoting opposing effects on tese spins. Fig.5b presents te entropy per site wit = 2 at T = long-dased line). From tis plot we verify tat we cannot approximate te function M z by taking into account only te contribution of te two ground states of te pases B and G as was done in te ferromagnetic version of te model. At T = 0.266, te interval of te magnetic field for wic te transition between te plateaus M z = 0 and M z = 1 occurs is suc tat 1. For temperatures of tree orders of magnitude lower, te function M z as a function of as a step-like form and Next we consider te AF model 1) wit = 2.5. In Fig.5a we plot M z versus at T = solid line). We verify tat at tis temperature tree plateaus still occur; namely, at M z {0, 1,1}, for wic te OYA condition is satisfied. We ave tree 2 plateaus in tis case because te vertical line in te diagram 1b, localized at = 2.5, crosses tree pases B, E and G). Te values = 2.5 and 3.5 were te transitions between te plateaus of te z-component of te magnetization occur is te same as tat 14

15 of te transition between te AF pases A E and E B, respectively, of te standard model 1) at T = 0. Also in tis case te curve of M z looses its step-like form at T = Te widt of te transition between te plateaus of M z is 0.5. Tis widt reduces to 10 3 for temperatures tree orders of magnitude lower tan To understand wy te AF function M z looses its step-like form at T = and wit = 2 and 2.5 respectively, we plot in Fig.5b te function S as a function of for tese two set of parameters. Te function S is non-null around = 1 wit = 2 long-dased line) and around te values = 2.5 and 3.5 solid line). Tese points in te pase diagram 1b are on te transitions line between te pases A E and E B respectively, of te standard model 1)/5) wit = 2.5 at T = 0. Altoug te AF models are presented at very low temperature, te excited states of te AF model already contribute to te termodynamic function M z around te values of tat correspond to te lines tat separate te AF pases in Fig.1b. Te curve M z wit = 1 and 2.5 are very similar at low temperatures. For 3 = 0.25, te transition from te plateaus M z = 0 to M z = 1 corresponds, in te 2 pase diagram 1b, to te transition G E, wereas for = 2.5 see Fig.5a) te same transition corresponds to te pase crossover A E. Te magnetization functions are almost insensible if te initial configuration of te spins is eiter a Néel state or a state wit all te spins perpendicular to te longitudinal magnetic field. Comparing Figs.6a and6b, weseetattespecificeatpersite, C,,;β) C = β 2 2 [βw 1 ] )distinguises β 2 te transitions A E and G E at very low temperatures. In Fig.6a te function C is symmetric around = From Fig. 5a we could say tat te plateaus of M z in te AF model 1)/5) are soft, in te sense tat tey are already smeared out at T = 0.26 and wit = 2 and 2.5 respectively. Since W stag 1,, ;β) = W 1,, ;β), we obtain and S stag,,;β) = S,,;β) 28) M stag z,,;β) = M z,,;β). 29) Te plateaus in te z-component of te magnetization of te standard ferromagnetic AF) model 1) appear in te z-component of te staggered magnetization of te staggered AF ferromagnetic) model 5). Tis fact implies tat te plateaus in te staggered magnetization of Hamiltonian 4) satisfy te OYA condition, but for tis termodynamic function te AF model as only two plateaus for > 0. Te beavior of te curves of S stag versus of te staggered ferro and AF models is identical to te curves of te entropy per site of te standard AF and ferromagnetic models 1)/5) respectively. It is very simple to understand tis situation once te pase diagram at T = 0 of te staggered ferromagnetic and AF models are depicted by Figs.1b and 1a respectively. 15

16 5 Conclusions Te density matrix metod[3] and te β-expansion of te HFE[20] ave been applied to te obtainment of te exact expressions of te HFE s of te one-dimensional standard[21] and te staggered[22] spin-1 Ising models wit S z ) 2 term in te presence of a longitudinal magnetic field. Te analytic expressions of tese functions ave been written in terms of exponentials of te parameters of te Hamiltonians 4) and 1)/5). Our results are valid for teir respective ferromagnetic < 0) and AF > 0) models in te interval T 0, ), extending te results of Ref.[12] to te AF models. Tese results do not coincide wit te results of Ref.[13]. We present te pase diagram of te standard and te staggered spin-models 1)/5) and 4), respectively, at T = 0. Our result also gives te exact HFE of te one-dimensional extended Hubbard model in te ionic limit but in te absence of an external magnetic field. We ave studied te beavior of te z-component of te magnetization M z and te entropy S, bot per site, of te ferromagnetic and AF models of Hamiltonian 1)/5) at low temperatures and teir respective staggered versions. We ave presented te two plateaus of M z at low temperatures of te standard ferromagnetic model for > 1 and sow tat te value in wic te transition between 2 plateaus at low temperatures occurs is te same as tat of diagram 1a. For te AF model 1)/2) we sow tat te number of plateaus of te z-component of te magnetization at very low temperatures depends on te number of pases of te model at T = 0 for a given value of. Our results for te AF model wit > 0 agree wit previous results in te literature[7, 10, 8]. Te ferromagnetic classical spin-1 model 1)/5) as as igly degenerate excited states, but we sowed tat for temperatures T < T max te function M z can be approximated by te contribution of only two ground states. Tis fact is not true for te AF model because it is gapless for values of and along te line separating te AF pases in te diagram 1b at T = 0. Te AF specific eat at very low temperatures distinguises te pase transitions E G at = 0.75) and E B at = 1.25) at = 0.25 and 2.5, respectively, in Figs.6. By comparing te plots of te M z as a function of at T = and to = 2 and 2.5 respectively, we verify tat te plateaus of tis function of te standard ferro and AF models can be called stiff and soft, respectively. All te plateaus in te z-component of te magnetization of te ferro and AF models of te standard and staggered version satisfy te OYA condition. As a final comment we sould say tat te presence of plateaus in te function M z in te classical spin-1 Ising model wit single-ion anisotropy term in te presence of a longitudinal magnetic field comes from te stability at low temperature of te ground state vector under te action of increasing te norm of te external magnetic field and te temperature. S.M.S. Fellowsip CNPq, Brazil, Proc.No.: /2009-9) tanks CNPq and FAPEMIG for te partial financial support. M.T.T. also tanks FAPEMIG. M.T.T. 16

17 tanks. Florêncio r and.f. Stilck for enligtening discussions. A Calculation of te roots of te tird degree equation of te spin-1 classical model. Krinsky and Furman applied te transfer matrix metod[1, 2, 3] to calculate te HFE of te classical spin-1 model in Ref.[12]. Te roots coming from te cubic equations ave to be real, but tey write tem in terms of complex quantities. Wen we andle tose roots numerically tese complex quantities may not disappear. In tis appendix we recalculate teir results for Hamiltonian 1)/5), sowing explicitly tat te roots of te cubic equation are real. Following Ref.[3] we obtain tat te partition function Z 1,,;β) of te spin-1 Hamiltonian 1)/5) is equal to Z 1,,;β) = Tr[U N ], A.1) were N is te number of sites in te periodic cain and te matrix U for te symmetric Hamiltonian 5) is U,,;β) = e β+2+2) e β+) e β +2) e β+) 1 e β +). A.2) e β +2) e β +) e β 2+2) Te matrix U,,;β) is ermitian for any value of,, and β. Its tree eigenvalues λ i, i = 1,2 and 3, are real. Te matrices U see eq.a.2)) and T in Ref.[12]) differ by a rearrangement of lines, only. In terms of te eigenvalues of U, te partition function A.1) becomes: Z 1,,;β) = λ 1 ) N [1+ λ2 λ 1 ) N + λ3 λ 1 ) N ], A.3) in wic λ 1 is assumed to be te eigenvalue of matrix U wit te largest modulus. In te termodynamic limit N ), te partition function A.1) of te model is Z 1,,;β) = λ 1 ) N, A.4) for non-degenerate eigenvalues of U. Te HFE of te model is W 1,,;β) = 1 β ln[λ 1,,;β)]. A.5) Te eigenvalues λ i, i = 1,2,3, are roots of te cubic equation 17

18 λ 3 +Pλ 2 +Qλ+R = 0, A.6) were were P = 1+2e β+2) cos2β) = tr[u] A.7a) ) β Q = 4e 2β e β 2 cos2β)sin +2e 4β sin2β), A.7b) 2 )] 2 β R = 8e [sin 4β sinβ). A.7c) 2 Te roots of te cubic equation A.6) are well known[23] ) θ s 1 = 2 Q cos + P 3 3, A.8a) θ s 2 = 2 Q cos 3 + 2π ) + P 3 3, A.8b) θ s 3 = 2 Q cos 3 2π ) + P 3 3, A.8c) cosθ) = R Q) 3 A.9a) wit and Q = 3Q+P2 9 A.9b) R = 9QP +27R+2P3. A.9c) 54 Ourprevious result s 1 doesnotagreewitteexpression ofλ max ofref.[13]; webelieve tat tere are some misprints in teir eqs.18)-24). 18

19 B Te ground states of te classical AF spin-1 and te calculation of T max For te AF version of te Hamiltonian 1)/5) we assume tat te cain as a even number of sites. Let N = 2M, in wic M is a positive integer. In te termodynamic limit N ) we ave M. Te ground state vectors at eac pase in te diagram 1b at T = 0 are Ψ 0 A = M, B.1a) Ψ 0 B = M, B.1b) Ψ 0 C = V, B.1c) Ψ 0 E = M 1 1 2M, B.1d) Ψ 0 F = M 1 1 2M, B.1e) Ψ 0 G = M 1 1 2M. B.1f) One is reminded tat Si z s i = s s i, s = 0,±1 and i = 1,2, 2M. We are interested in te discussion of te termodynamic function M z at low temperatures. Tis function as an even dependence on ; ten we restrict our discussion to 0. In te following we do not mention te quantum beavior of te pases C and F at low temperature. Te value of te ground state energy of te pases A, B, E and G are, respectively, E A 0 E0 B E0 E E0 G = 0, B.2a) = N ), B.2b) = N ), B.2c) = N 1+ 2 ). B.2d) We ave a muc more complex distribution of first excited states E 1 in te AF spin-1 Ising model 1)/5) tan in te ferromagnetic version of te model. Let us present te difference between te first excited and te ground states of eac pase A, B, E and G in diagram 1b. 1) Pase A: 1 and. We ave 2 E 1 E A 0 = 2 + ), B.3) wic is substituted in eq.12) to give kt max ). B.4) 19

20 2) Pase B: we ave two distinct first excited states 2.1) 0 and 1 0 and 1+. Ten E 1 E B 0 = 2 1+ ) B.5) and eq.12) gives kt max ). B.6) 2.2) 0 and 1 1. Ten E 1 E B 0 = 4 1+ ), B.7) and eq.12) becomes kt max ). B.8) 3) Pase E: in tis pase we ave tree distinct first excited states. 3.1) and 1 1. We ave E 1 E E 0 = ), B.9) wit eq.12) giving kt max ), B.10) 3.2) 0 1 and and We ave E 1 E E 0 = ), B.11) wic gives kt max ). B.12) 3.3) 1 1 and and 1 +. We ave

21 E 1 E E 0 = 2 ), B.13) tat substituted in eq.12) gives kt max 2 15 ). B.14) 4) Pase G: in tis pase we ave two distinct first excited states. 4.1) 1 and and 1+. We ave E 1 E G 0 = 4 1 ). B.15) From eq.12) we obtain kt max ). B.16) 4.2) 1 0 and and and We ave 2 E 1 E G 0 = 2 1 ), B.17) wit eq.12) given References kt max ). B.18) [1] H.A. Kramers and G.H. Wannier, Pys. Rev ) 252. [2] H.A. Kramers and G.H. Wannier, Pys. Rev ) 263. [3] R.. Baxter, Exactly Solved Models in Statistical Mecanics, Academic Press 1989), section 2.1. [4]. Simon et al., Nature 472, 04/21/2011, p. 307, and references terein. [5] M. Blume, Pys. Rev )

22 [6] H.W. Capel, Pysica Utrect) ) 966; ) 295. [7] E. Aydiner and C. Akyüz, Cin. Pys. Lett ) [8] X.Y. Cen et al,. of Mag. and Mag. Mat ) 258. [9] F. Mancini, Europys. Lett ) 485. [10] F. Mancini and F.P. Mancini, Cond. Matt. Pys ) 543. [11] W.A. Moura-Melo et al., Pys. A ) 393. Please notice a misprint in eq.28) of tis reference: te correct coefficient of te term p + +p +r ), inside te square root, is 8 3. [12] S. Krinsky and. Furman, Pys. Rev. B ) [13] F. Litaiff, R. de Sousa and N.S. Branco, Solid State Communication ) 494. [14] C.Ekiz, H. Yaraneri,. of Mag. and Mag. Mat ) 49, and references terein. [15] Y. Narumi et al., Pys. B ) 509. [16] Y. Narume et al.,. of Mag. and Mag. Mat ) 685. [17] T. Goto et al.. Pys. B ) 43. [18]. Luz, R.R. dos Santos, Pys. Rev. B ) [19] M. Moutino, E.V. Corra Silva and M.T. Tomaz, Pys. A ) 477. [20] O. Rojas, S. M. de Souza, and M. T. Tomaz,. Mat. Pys ) [21] M.T. Tomaz and O. Rojas, Condensed Matter Pysics ) 13706: 110 [dx.doi.org/ /cmp , ttp:// [22] E.V. Corrêa Silva, S.M de Souza and M.T. Tomaz, Pys. A ) [23] M.R. Spiegel, Matematical Handbook of Formula and Tables, Scaum s Outline Series, Singapore 1990), page 32. [24] Te factor 1 2) in te relation between Mz and W 1 comes from te symmetric form of Hamiltonian 5) in te sites i and i+1. [25] F.C. Alcaraz, S.R. Salinas and W.F. Wreszinski, Pys. Rev. Lett ) 930. [26] M. Osikawa, M. Yamanaka and I. Affleck, Pys. Rev. Lett )

23 Figure 1: Te pase diagrams of te ferromagnetic = 1) and te anti-ferromagnetic = 1) models 1)/2) at T = 0. a) Te ferromagnetic pases represent te following configurations of te z-components of te neigbour spins: A 0,0), B 1,1) and C 1, 1). b) Te AF pases include te configurations A, B and C tat are found in te ferromagnetic diagram plus te pases: E 1,0), F 0, 1) and G 1,1). Tis diagram as also two tricritical points: P = 1 and = 0) and Q = 1 and 2 = 1 ). Te pase diagram of te staggered ferromagnetic and te AF models of te 2 Hamiltonian 4), at T = 0, are given by te figures b) and a) respectively. Figure2: Tecurvesof ktmax for te tree distinct first excited states of te ferromagnetic model. Te dotted line as = 2, te solid line as = 1 and te piecewise solid 2 line as = 2.5. Te vertical dased line gives te value of te magnetic field were te first excited state in te ferromagnetic pase A canges. 23

24 Figure 3: Te exact and approximate expressions of M z of te ferromagnetic spin-1 model 1)/5) versus. In a) we ave = 2 at kt = Te solid dotted) line is its exact approximate) expression. In b) we ave = 2.5 at kt = Te solid dotted) line is its exact approximate) expression. In c) we compare te exact expressions of M Z wit = 2.5 at temperatures kt = 0.11 solid line) and dotted line). Figure 4: Te functions ktmax for te AF classical spin-1 model. In a) we ave te curves wit = 2 dased line) and = 1 solid line), and in b) = 1 dased 2 3 line) and = 2.5 solid line) 24

25 Figure 5: Te termodynamic functions M z and S of te AF spin-1 model 1)/5) are plotted versus wit = 2 and 2.5. a) M z as a function of wit = 2 at T = long-dased line) and wit T = 2.5 at = solid line). Te entropy T per site, S, is plotted in b) for te same set of parameters: = 2 at = long-dased line) and T = 2.5 at = solid line). Figure 6: Comparison of te beavior of te function C under te transitions G E and A E at low temperature. In a) we ave te transition G E wit = In b) is plotted te specific eat per site of te transition A E wit = 2.5. For bot curves we ave kt =

26 kt A Table 1: Te values of te parameter a in eqs.24) and 27) for several values of and different values of temperature. is te interval of variation of te magnetic field in wic te function M z varies continuously between M z = 0 and M z = 1. 26