The high temperature expansion of the classical $XYZ$ chain

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by  E. V. Correa Silva

The high temperature expansion of the classical XYZ chain
E.V. Corrˆea Silva,1 Onofre Rojas,2 James E.F. Skea,3 S.M. de Souza,4 and M.T. Thomaz5,∗
1Departamento de Matem´atica e Computac¸˜ao, Faculdade de Tecnologia,
Universidade do Estado do Rio de Janeiro. Estrada Resende-Riachuelo,
6 s/no, Morada da Colina, CEP 27523-000, Resende-RJ, Brazil
0 2Departamento de Ciˆencias Exatas, Universidade Federal de Lavras,
0 Caixa Postal 3037, CEP 37200-000, Lavras-MG, Brazil
2 3Instituto de F´ısica, Universidade do Estado do Rio de Janeiro,
R. S˜ao Francisco Xavier no 524, Bloco B, CEP- 20559-900, Rio de Janeiro -RJ, Brazil
n
4Departamento de Ciˆencias Exatas, Universidade Federal de Lavras,
a
J Caixa Postal 37, CEP 37200-000, Lavras-MG, Brazil
5Instituto de F´ısica, Universidade Federal Fluminense,
6
Av. Gal. Milton Tavares de Souza s/no, CEP 24210-340, Niter´oi-RJ, Brazil
2
(Dated: February 6, 2008)

]
Wepresenttheβ-expansionoftheHelmholtzfreeenergyoftheclassicalXYZmodel,withasingle-
h
ionanisotropytermandinthepresenceofanexternalmagneticfield,uptoorderβ12. Wecompare
c
e ourresultstothenumericalsolutionofJoyce’s[Phys. Rev. Lett. 19,581(1967)]expressionforthe
m thermodynamics of the XXZ classical model, with neither single-ion anisotropy term nor external
magneticfield. Thiscomparisonshowsthatthederivedanalyticalexpansionisvalidforintermediate
-
t temperatures such as kT/Jx ≈ 0.5. We show that the specific heat and magnetic susceptibility of
a
the spin-2 antiferromagnetic chain can be approximated by their respective classical results, up to
t
s kT/J ≈0.8,withinanerrorof2.5%. Intheabsenceofanexternalmagneticfield,theferromagnetic
t. andantiferromagnetic chainshavethesame classical Helmholtzfree energy. Weshow howthistwo
a typesof media react tothe presence of an externalmagnetic field.
m
-
d 1. INTRODUCTION where S is the spin value (S = 1/2,1,3/2,2,···,∞).
n The calculation of this thermodynamical function of the
o
quantum model was done by using the cummulant se-
c The classical limit of quantum spin chains has at-
riesfortheone-dimensionalmodelspresentedinRef. [6].
[
tractedattentionsince this classofmodelswasproposed This approach can also be applied to classical periodic

1 to describe the magnetic interactions of one-dimensional
chain models with nearest-neighbor interactions. To the
v models. In 1964 Fisher[1] calculated the analytical ex-
bestofourknowledge,analyticalexpressionsoftheHFE
5 pressionoftheHelmholtzfreeenergy(HFE)oftheXXX
forintermediateandhightemperatureregionsoftheclas-
1
chaininthepresenceofanexternalmagneticfield. Some
6 sical XYZ model are currently unknown.
yearslater,Joyce[2]includedinthatmodelananisotropy
1
term of interaction between the z components of neigh- In Ref. [5], we showed that certain thermodynamical
0
6 bouringspins,andobtaineditsexactHFEforavanishing functions like the magnetization and magnetic suscepti-
0 external magnetic field. Since then the thermodynamics bility per site of the quantum XYZ chain can be well
/ of the classical XXX chain, in the presence of in-plane approximated,inthe intermediateandhightemperature
t
a ortransversemagneticfieldhasbeenstudiedatverylow regions, by their classical results for S ≥ 3/2. The ab-
m temperatures(limitoflongwavelength)andthemapping senceofresultsforthequantumXYZ chainforS >1/2,
- ofthis classicalspinchainintotheSine-Gordonmodel[3] makestheknowledgeoftheclassicalmodelveryinterest-
d has permitted one to obtain analytical expressions for ing in the region of kT & Jx, where Jx is the coupling
n
sometermodynamicalfunctionsinthisregionoftemper- constantbetween the closestspins along the x direction.
o
c ature. Ontheotherhand,theclassicalXXZchaininthe Even for other thermodynamical functions, like the spe-
: presence of an external magnetic field in the z direction cific heat per site, the classical models give results with
v
has also been studied numerically[4]. the correct order of magnitude for those regions of tem-
i
X perature. Nowadaysthe possibilityof designingdifferent
Recently we obtained the high temperature expansion
materialswithsuitable propertiesturnthe analyticalex-
r
(HTE) ofthe HFE ofthe quantumspin-S XYZ chain[5],
a pressions of the classical thermodynamical functions to
with a single-ion anisotropy term and in the presence of
be very helpful.
anexternalmagneticfield,uptoorderβ5. (Herewehave
β = 1 , where k is the Boltzmann’s constant, T is the InRef. [5]wecalculatedtheβ-expansionoftheHFEof
kT
temperature in Kelvin.) The HFE of the classicalmodel the quantum and classical XYZ model, with single-ion
is obtained from Ref. [5] by taking the limit S → ∞, anisotropy term and in the presence of an external mag-
neticfield,uptoorderβ5. Whenapplyingthemethodof
Ref. [6]directlytotheclassicalmodelwehavetheadvan-
tage of calculating traces of c-numbers,thus diminishing
∗Correspondingauthor: [email protected]ff.br the number of terms to be evaluated. Here is the first
2
time thatthis method is being applieddirectly to aclas- 3. THE METHOD OF CUMMULANT SERIES
sical periodic chain with first-neighbour interactions. APPLIED TO CLASSICAL CHAINS
Insection2,wepresenttheHamiltonianthatdescribes
the classical XYZ chain with unitary spin (|~si| = 1) at In Ref. [7] we presented a survey of the cummulant
thei-thsite. Inthismodelwetakeintoaccountthepres- series method and its applicationto one-dimensionalpe-
enceofasingle-ionanisotropytermandanexternalmag- riodic quantum chain models with first-neighbour inter-
netic field along the z direction. In section 3, we present actions. Here, we discuss the application of this method
the results of Ref.[6] when applied directly to the clas- to any classical one-dimensional chain model subject to
sical XYZ chain; a rule is obtained that allows, in the periodic boundary conditions, spatial translation invari-
present case, optimization of the algebraic calculations ance, and nearest-neighbor interactions. Following Ref.
of the HFE. In section 4, we compare Joyce’s solution[2] [6], we obtain the analytical expressions for the HTE of
withourresultsfortheparticularcaseofaclassicalXXZ the HFE in the thermodynamical limit of such models,
chain (Jx = Jy) without the single-ion anisotropy term which can be written as the β-expansion
(D =0) andin the absence of anexternalmagnetic field
(h=0),forintermediateandhightemperatures. Joyce’s
1
solutionofthe classicalXXZ chainallowsone to obtain W(β)=− [ln(tri(1i))+ln(1+ξ(β))], (4)
the behaviour of the specific heat per site at low tem- β
peratures, and then apply Pad´e’s method to extend the
wheretr (1 )equalsthedimensionoftheclassicalHilbert
validity of the HTE to lower temperatures. In section 5 i i
space of the i-th site,
we study some thermodynamical functions of the classi-
cal XYZ chain in the presence of an external magnetic
field. Finally, in section 6 we present a summary of our ∞ 1 ∂n
results. ξ(β)= (ϕ(λ)n+1) (5)
(n+1)! ∂λn (cid:12)
InappendixA,wepresentsomeusefulintegralresults. nX=0 (cid:12)λ=1
(cid:12)
As an example of our HTE of the HFE of the classical (cid:12)
XYZ chain, we present in appendix B its expansion up and the auxiliary function ϕ(β) is given by
to order β6 in the absence of an external magnetic field.
Its complete expansion,up to order β12, is quite lengthy ∞ ∞ (−β)n
and can be obtained under request to the autors. ϕ(λ)= H(n). (6)
λm 1,m
X X
m=1 n=m
2. THE CLASSICAL XYZ CHAIN WITH The functions H(n) correspond to the “connected”
UNITARY SPIN strings with n ope1r,amtors H (in the present case, see
i,i+1
Eq. (1)) so that m of them are distinct, that is,
The classical version of the hamiltonian of the
anisotropic XYZ chain with unitary spin-S, as shown
in Eq.(2) of Ref. [5], is n m Hni
H(n) = ′′ i,i+1 . (7)
1,m n !
N N {Xni} (cid:10)Yi=1 i (cid:11)
H= H = {J sxsx +J sysy +J szsz
i,i+1 x i i+1 y i i+1 z i i+1 The normalized trace is defined as
X X
i=1 i=1
−hsz+D(sz)2}, (1)
i i
hH ...H i≡
where i1,i1+1 im,im+1
ssxizi == csions((θθii));cos(φi), syi =sin(θi) sin(φi), (2) tri1,t..r.,ii1m(1+i11()H..i1.,ti1r+im1+.1.(.1Himim+,1im)+1), (8)
θi and φi are the polar and azimuthal angles, respec- wheretheindicesi1,...,im canbeequalordistinct. The
tively, of the classical spin ~si; N is the number of sites notationtri1,...,im+1 stands forthe traceoverthe degrees
in the periodic chain; h is the external magnetic field offreedomofm+1distinctsitesintheset: {i1,...,im+
raalomnegtetrh.eTzh-aexciosnasntadnDts Jisx,thJey sainndgleJ-ziognivaentishoetrsotrpeyngptah- 1ov}e,rwtihtehdmegr≤eesn.ofIfnreeaddodmitioonf,thteriikk-rtehprseitseenatnsdt1hiektirsaictes
of first-neighbour interactions between the components identity operator.
ofthe spins. From(1)and(2), wedefine the functionH, Eq.(7) differs from its quantum version on showing a
so that normalized trace, rather then a g-trace: in the classical
case, all H are commuting functions. This property
i,i+1
greatlysimplifies allcalculations (see Appendix A). The
N
H= H(θi,φi,θi+1,φi+1). (3) notation n ′′ stands for the restriction m′′ ni = n and
Xi=1 {Pni} iP=1
3
n 6= 0 for i = 1,2,..,m. The index m satisfies the con- than the expansion up to β5. However, a closer look at
i
dition 16m6n. Equations (4) to (7) are valid for any the interval kT/J = [0.4,1] (see the detail box) shows
classicalone-dimensionalchainmodelsubjecttoperiodic that the β12-curve coincides with the exact solution up
boundary conditions, spatial translation invariance, and to kT/J = 0.53, whereas the β5-curve coincides with it
nearest-neighbors interactions. In order to reach higher only up to kT/J =0.9. In Fig. 1b, we take ∆=±1 and
orders in β in the HTE of the HFE, it is important to weseethattheexpansionuptoβ5 coincideswiththenu-
optimize the calculation of the normalized traces in Eq. mericalsolutionofJoyce’sexpressionupto kT/J =0.56
(7). Although the classical model has less terms to be whereas the series up to β12 goes up to kT/J = 0.39,
calculatedthanitsquantumversion,westillhavealarge whichcancertainlybeconsideredanintermediateregion
number of integrals to be done for each order in β. For of temperature.
Hamiltonian (1) it is straightforward to verify that, at In Fig. 2 we compare the specific heat per site,
each site, using the results (A.4) and (A.5) of appendix C (β)=−β2 ∂2 (βW (β)), obtainedfromJoyce’s
A, that we only have a non-vanishing normalized trace class ∂β2 class
solution of the classical XXZ chain (with h = 0 and
h(sx)l(sy)m(sz)pi, if and only if l, m and p are all even.
D = 0) and the HTE up to β6 (see Ref. [5]) and up to
β13. We also include (Figs. 2c and 2d) the respective
relative percental difference of these expansions to the
4. THE β-EXPANSION OF THE HFE OF THE
exact solution. In Fig. 2a we have J =1 and ∆=±0.5,
CLASSICAL XYZ CHAIN
whereasinFig. 2bwe haveJ =0.5and∆=±2. Again,
inbothplotsweverifythattheβ13 expansionextendsto
¿From the results of Ref. [5], we can obtain the HTE lower temperature the validity of the HTE of the classi-
of the HFE of the classical XYZ model, up to order cal model in Refs. [5, 8], as shown in the detail box of
β5, by taking the limit of S → ∞ in its Eq.(3). By Fig. 2c. It is simple to derive from Joyce’s solution the
using the results of section 3, we obtain here the HTE correlation function between the x, y and z components
of the HFE, Wclass, up to order β12, for a non-vanishing of the classical spins of first nearest neighbours. Due to
externalmagnetic field h and in the presence of a single- the symmetry in the x and y directions in Hamiltonian
ionanisotropyterminthe Hamiltonian(1). The derived (1), for J = J , we have hsxsx i = hsysy i. In Figs.
x y i i+1 i i+1
expansion is very lengthy and it can be obtained from 3 we plot the correlation function hsxsx i= ∂Wclass for
the authors by request. Although this series has a large i i+1 ∂Jx
J =1 and ∆=0.5 (Fig. 3a) and for J =0.5 and ∆=2
number of terms, it can be easily handled by any CAS
(Fig. 3b), derived from the exact result and from our β-
system (we used Maple). In appendix B we present the
expansion (up to order β12) for the HFE of the classical
expansion of the HFE of the classical XYZ chain, with
model. FromFig. 3b we see that the β12-curvegives the
unitary spin (see Hamiltonian (1), in the absence of an
correct maximum of the function hsxsx i for J = 0.5
external magnetic field (h = 0)), up to order β6. The i i+1
and ∆=2.
resulting expansion (B.1) agrees with the series of this
thermodynamical function obtained from eq.(3) of Ref. In Figs. 4 we plot the correlation function hsziszi+1i =
[5], in the limit S →∞. ∂Wclass fortheexactresultoftheclassicalmodelandour
∂Jz
Inthis section,wehaveJ =J =J andJ =J∆,for β-expansion,forthe samesetof constantsJ and∆as in
x y z
the constants in Hamiltonian (1) for the particular case Figs. 3. FromFig. 4aweseethatthederivedHTEupto
of the classical XXZ model. β12 gives the correctmaximum of this thermodynamical
function.
One way to extrapolate the results of our high tem-
4.1. Comparison of the HTE of the classical XXZ perature series to higher orders in β is through the
chain with Joyce’s solution Pad´e approximants (PA), which allows us to combine
thermodynamical information from both high and low
Next, we perform some comparisons with known re- temperatures. Among the several approaches to Pad´e
sults, to verify the interval of temperature where our approximants[9], here we employ the two-point Pad´e
HTEs still hold. Joyce[2] obtained the exact solution of approximant[10] to extend our β-expansion of the spe-
theclassicalXXZ chain,withoutasingle-ionanisotropy cific heat per site to low temperatures.
term (D = 0) and in the absence of an external mag- ¿From the numerical analysis of Joyce’s solution[2] of
netic field (h = 0). In order to check our HTE of the the specific heat per site at very low temperatures, for
classicalmodelandtoverifyhowitextends ourprevious J 6= 1 and ∆ 6= 1, we realize that it has a polynomial
results[5], we compare in Figs. 1a and 1b the Joyce’s behaviourinT. FortherangeoftemperatureT ∈[0,0.1],
numerical solution for the HFE with our expansions up it can be chosen to be of fourth order in T, C(T) ≈
to orders β5 and β12. In Figs. 1c and 1 d we present 1+a T +a T2+a T3+a T4, where the coefficients a
1 2 3 4 i
the respective percental difference between each of those can be adjusted appropriately by linear regression. In
expansions and the exact Joyce’s solution. In Fig. 1a this case, we have 5 known terms in the region of low
we plot the HFE for ∆ = ±0.5. At first glance of this temperature and 12 known terms in the region of high
graph suggests that the expansion up to β12 is poorer temperature, resulting in a Pad´e approximant with 17
4
terms. S =2antiferromagneticchain[14]toitsequivalentclassi-
Fig.5a compares results for the specific heat per site, cal function obtained from our β-expansion, up to order
namelythebestPAsandJoyce’sexactsolution,forJ =1 β12. In the same token, Fig. 7b shows their percental
and ∆=0.5. For this constants, we have: a =0.51519, difference, whichis smaller than2.5%up to kT/J ∼0.8.
1
a2 = 1.42213, a3 = −2.10687 and a4 = 62.1676. Below In Ref. [8] we verified that the higher the spin, the
the graphin 5a, we show the percentaldeviation of each closerquantum andclassicalthermodynamicalfunctions
PA with respect to the numerical result of the exact so- get. The curves, for a given thermodynamical function
lution. It turns out that P11,6 is the best approximation of the quantum models, do not cross for different values
oftheexactresult,withinadifferenceoflessthan2%for of spin. Although we do not have a numerical study of
all values of temperature. the thermodynamics of the antiferromagneticchain with
Fig. 5b shows a similar comparison, but for the pa- S =5/2,wecanaffirmthatitsspecificheatandmagnetic
rametersvalues J =0.5and∆=2. Forthoseconstants, susceptibility per site, for kT/J up to 0.8, differs from
we obtain numerically: a1 = 0.529029, a2 = 1.63073, the classical result in less than 2.5%. This also applies
a3 = −8.64101 and a4 = 86.746. In Fig. 5c, we present to the TMMC[15], since it anisotropy in the z directions
the percental deviation of each PA to the numerical re- is small, namely ∆=0.016.
sult of the exact solution. We see that P is the best
10,7
approximation to the exact result. In the whole interval
oftemperature,itspercentaldifferencetotheexactsolu-
5. THE THERMODYNAMICAL FUNCTIONS
tionis lessthan10%. AlthoughFigs. 5aand5brefer to
OF THE CLASSICAL XYZ CHAIN
the same thermodynamical function, each case demands
specific PAs for the best fitting.
¿From expression(B.1) of Appendix B, we verify that
the HFE of the classical XYZ chain in the absence of
4.2. The classical model as an approximation to anexternalmagnetic field(h=0),is anevenfunctionof
the quantum model. the constants Jx, Jy and Jz, so that this function is the
same for classical ferromagnetic and antiferromagnetic
materials with the same single-ion anisotropy D-term.
The classical XYZ chain is a good approxima-
As a consequence of this property, we have that some
tion to the quantum chain, for all values of S
thermodynamical functions have well defined parity un-
(S = 1/2,1,3/2,2,···), in the high temperature region
der the transformation (J ,J ,J ) → (−J ,−J ,−J ):
(J β ≪ 1). In Ref. [8] we showed that some thermo- x y z x y z
x
the specific heat, h(sz)2i, the entropy and the mean en-
dynamical functions of the quantum XXZ chain can be i
ergyareevenfunctions,whereasthefirst-neighborcorre-
well approximated by their classical version for not so
lation functions, hsxsx i, hsysy i and hszsz i are odd
high temperatures. This region of temperature, where i i+1 i i+1 i i+1
functions. None of the quantum versions of the above
theclassicalandquantummodelsareequivalent,depends
mentioned functions have defined parity for h=0 [5].
on the termodynamical function and on the spin. The
higherthespinofthequantummodel,themoreinvolved Figs. 8 show how each type of medium responds to
its numerical solution gets, due to the growing number the presence of an external magnetic field. We use the
degreesoffreedom. However,asfarastheXYX modelis fact that the mean energy per site, hEi = ∂(βWclass), is
∂β
concerned, for S ≥2, the quantum HTE of a given ther- the same for both ferromagnetic and antiferromagnetic
modynamicalquantity canthus be wellapproximatedto media at h = 0. Fig. 8a shows the curve of the mean
its classical HTE. energy as a function of J β, for J /J = 1/3, J /J =
x y x z x
Some materials are well described by XXZ mod- ±2/3 and D/J =−0.6, at h=0.
x
els with higher values of spin. For example, the Fig. 8b shows, for the ferromagnetic (J /J = −2/3)
z x
(C10H8N2)MnCl3, described by a S = 2 model[11]; and and antiferromagnetic (Jz/Jx = 2/3) cases, the differ-
the(CH3)4NMnCl3,alsoknownasTMMC,describedby enceofthe meanenergyper sitewithandwithoutexter-
aS =5/2model[12,13]. InRef. [14],Yamamotocarried nal magnetic field (cf. Fig. 8b, ∆E = hEi| −hEi| ),
h h=0
out Monte Carlo calculations of the thermodynamics of plottedasafunctionofh/J ,forJ β =0.5andJ β =1.
x x x
the S =2 XXZ chain with 96 sites, with ∆=1, D =0 Fig. 8cquantifiesthisdifference,showingthecorrespond-
and h = 0. He obtained the temperature dependence ingpercentaldifferences(∆E(%)= hEi|h−hEi|h=0×100%).
of the specific heat per site and the magnetic suscepti- hEi|h=0
The behavior of the mean square of the z-component
bility per site for any temperature. Fig. 6a compares
of each spin in the chain at h = 0 as a function of J β,
the numerical results of Yamamoto [14] for the specific x
h(sz)2i = ∂Wclass, can be seen in Fig. 9a. The reac-
heat per site, for the S = 2 antiferromagnetic case, to i ∂D
tion of the ferromagnetic and antiferromagnetic media
our β-expansionof the classical specific heat per site, up
to the presence of an external magnetic field, h(sz)2i−
to order β13. Fig. 6b shows the relative percental error i
h(sz)2i| , canbe seeninFig. 9b,asafunctionofh/J
between this two curves, which is less than 2.5% up to i h=0 x
at J β =0.5 and J β =1. For both graphs,we take the
kT/J =0.75. In Fig. 7a we compare Yamamoto’s quan- x x
same set of values for J /J , J /J and D/J as that
tum magnetic susceptibility per site χ=−∂∂2hW2 ofthe of Fig. 8 for the ferromyagnxeticz anxd antiferroxmagnetic
(cid:16) (cid:17)
5
media. Fig. 9c is similar to Fig. 9b, showing the corre- Thislow-temperatureinformationcanbe combinedwith
spondingpercentaldifferencewithrespecttoh(sz)2i| . the high-temperatureinformationbythe two-pointPad´e
i h=0
In Fig. 10 we compare the magnetization per site, approximants (PA)[11]. The best PA gives a very good
Mclass = −∂W∂chlass, at Jxβ = 0.8, for the ferromag- description of the classical specific heat per site in the
netic case (with J /J = 1/3, J /J = −2/3, D/J = whole interval of temperature.
y x z x x
−0.6)andtheantiferromagneticcase(withJy/Jx =1/3, In the high temperature region (Jxβ ≪ 1), the quan-
Jz/Jx = 2/3, D/Jx = −0.6). For these same cases, in tum spin-S XYZ chain (S = 1/2,1,3/2,2,···) is well
Fig. 11 we compare the classic magnetic susceptibility described by the classical model. Certainly, the best ap-
per site at h/Jx =0.35. plication of the HFE of the classical XYZ chain is to
Since we are working with unitary classical spins supportthe study the thermodynamicalproperties of its
(|~s | = 1), we define hcosθ i ≡ h~s · ~s i, where θ quantum models for finite spin-S at lower temperatures.
i i i i+1 i
is the angle between the i-th and (i + 1)-th spins in For S ≥ 2 we have very few numerical analysis of the
the chain. In Fig. 12a we plot the function hcosθ i ≡ thermodynamicalfunctionsofthesequantummodelsdue
i
∂ + ∂ + ∂ W fortheantiferromagneticcase tothe largenumber ofdegreesoffreedomto be handled.
(cid:16)∂Jx ∂Jy ∂Jz(cid:17) class By using the Monte Carlonumericalcalculationdoneby
(withJ /J =1/3,J /J =2/3,D/J =−0.6),whereas
y x z x x Yamamoto[14], foraspin-2antiferromagneticchainwith
in Fig. 12b we plot hcosθ i for the ferromagnetic case
i 96 sites, we showed that its specific heat and magnetic
(with J /J = 1/3, J /J = −2/3, D/J = −0.6), both
y x z x x susceptibility per site can be approximated by their re-
ath/J =0.35. FromFig. 12bweseethatatJ β =0.9,
x x spective classical results, up to kT/J ≈ 0.8, within a
x
on the average,neighboringspins are orthogonalto each
precisionof2.5%. Thisgivesushopethatthethermody-
other. For these cases, we also plot in Fig. 13 the corre-
namics ofthe TMCC (S =5/2)canbe approximatedby
lation function hszsz i.
i i+1 theHFEpresentedinthispaper,uptothistemperature,
Within the range of the independent variable (h/J )
x within a precision higher than 2.5%.
shown in Fig. 10, the HTE of the magnetization M is
Finally, from the HTE for the HFE of the classical
almost equal to its exact solution. This range has been
XYZ chain,whichcanbefoundinEq.(B.1)ofAppendix
determined so that the HTEs of leading orders β11 and
B,weverifiedthatintheabsenceofanexternalmagnetic
β12 differsby0.1%therein. Asimilarthoughtguidedthe
field the quantum ferromagnetic and antiferromagnetic
determination of the ranges of J β in Figs. 11 and 13,
x chains have the same classical limit. As a consequence
regarding the magnetic susceptibility χ and correlation
ofthisessentiallyclassicalresult,somethermodynamical
function hszsz i, respectively.
i i+1 functions like: mean energy, h(sz)2)i, entropy and the
i
correlationfunctions between first-neighborspin compo-
nents have defined parity, at h = 0, under the parame-
6. CONCLUSIONS ter transformation (J ,J ,J )→(−J ,−J ,−J ). This
x y z x y z
fact permits studying the reaction of each type of chain
The method developed in Ref. [6] can be equally ap- (ferromagneticandantiferromagnetic)to the presenceof
plied to both quantum and classical chains with first- an external magnetic field.
neighbor interactions, spatial periodic boundary condi-
tions and translational invariance. In Ref. [5] we cal-
culated the high temperature expansion (HTE) of the
Helmholtz free energy (HFE), up to order β5, of the Acknowledgments
quantumspin-S XYZ chain,withasingle-ionanisotropy
term and in the presence of an external magnetic field.
The authors are in debt to CNPq for partial finan-
From this result, we obtained the HTE of the classical
cial support. S.M. de S. and O.R. thank FAPEMIG and
version of the model by taking the limit S →∞. In the
M.T.T. thanks FAPERJ for partial financial support.
presentpaper,weapplythemethodofRef. [6]directlyto
theclassicalXYZ (alsowithasingle-ionanisotropyterm
and in the presence of an external magnetic field) thus
simplifying enormously algebraic calculations. By this
APPENDIX A: USEFUL INTEGRALS
way,we are able to calculate the β-expansionof its HFE
uptoorderβ12. Eachcoefficientofβnintheexpansionis
exact (n = −1,0,1,2,···,12). Having a higher order in Thenormalizedtracesofproductsofoperatorsareex-
β allowedextendingtheknowledgeofthethermodynam- pressedintermsofsurfaceintegralsoverunitaryspheres,
ics of the classicalXYZ chainup to kT/J ∼0.5,which each of which represents the state space of a chain site.
x
mightbeconsideredasanintermediateregionoftemper- For example,
ature. Joyce’sexactresult[2]oftheclassicalXXZchain,
with no single-ionanisotropyterm and no externalmag-
neticfield(D =0,h=0),permitsknowingthebehaviour
hH i≡ dΩ˜ dΩ˜ H(θ ,φ ,θ ,φ ), (A.1)
of the specific heat per site at very low temperatures. 1,2 Z 1Z 2 1 1 2 2
S1 S2
6
where the normalized solid angles dΩ˜ are defined[16] as or
i
sinθ
dΩ˜ ≡ i dθ dφ , (A.2)
i i i
4π
2π
and the function H has been defined in Eq.(3). We have I(m,n) ≡ dφ (sinφ)m(cosφ)n
φ Z
θ ∈[0,π] and φ ∈[0,2π]. Another example is 0
i i
1 Γ(m+1)Γ(n+1)
= (1+(−1)m)(1+(−1)n) 2 2 ,
2 Γ(m+n+2)
2
hH21,2H2,3i ≡ Z dΩ˜1Z dΩ˜2Z dΩ˜3 H(θ1,φ1,θ2,φ2)2× (A.5)
S1 S2 S3
× H(θ ,φ ,θ ,φ ). (A.3)
2 2 3 3
The generalization of these formulas is straightforward.
where n,m=0,1,2,... and Γ is the gamma function.
Because of the structure of the Hamiltonian (1) and the
expression of spin operators (2), the integrals related to
thei-thsitewhichcontributetoH(n) (seeEq. (7)),turn
1,m
out to be either
I(m,n) ≡ πdθ (sinθ)m(cosθ)n APPENDIX B: THE β-EXPANSION OF THE
θ Z HELMHOLTZ FREE ENERGY IN THE ABSENCE
0 OF A MAGNETIC FIELD
1 Γ(m+1)Γ(n+1)
= (1+(−1)m)(1+(−1)n) 2 2
4 Γ(m+n+2)
2
1 (1+(−1)n)Γ(n+3)Γ(m+1) The high temperature expansion of the HFE of the
+ (1−(−1)m) 2 2 , classical XYZ (S → ∞) in the absence of an external
2 n+1 Γ(m+n+2)
2 magneticfield (Hamiltonian(1)with h=0),up to order
(A.4) β6, is
7
ln(2S) D J 2 J 2 J 2 2D2
Wh=0 = − + +(− z − y − x − )β
class β 3 18 18 18 45
2J 2D 2J 2D 8D3 4J 2D 4D4 32J 2D2
+ (− y − x + + z )β2+( − z
135 135 2835 135 14175 4725
2J 2D2 7J 4 7J 4 J 2J 2 2J 2D2 J 2J 2 7J 4
y x z x z x x y y
+ − − + + + −
4725 2700 2700 225 4725 225 2700
J 2J 2 16J 2D3 2J 4D 32D5 4J 4D 2J 4D
+ y z )β3+( y − y − + z − x
225 42525 2835 467775 2835 2835
16J 2D3 8J 2J 2D 4J 2J 2D 16J 2D3 4J 2J 2D
+ x + x y − y z + z − x z )β4
42525 2835 2835 42525 2835
736D6 107J 6 107J 6 107J 6 844J 2J 2J 2
x y z z x y
+ (− − − − −
1915538625 2679075 2679075 2679075 893025
106J 4J 2 106J 4J 2 106J 4J 2 106J 2J 4
z x z y x y x y
+ + + +
893025 893025 893025 893025
106J 2J 4 106J 2J 4 8768J 2D4 106D2J 4 106D2J 4
z y z x z x y
+ + + + +
893025 893025 49116375 1488375 1488375
16J 4D2 2008D4J 2 2008D4J 2 232J 2D2J 2
z y x z x
− − − −
297675 49116375 49116375 1488375
232J 2D2J 2 584D2J 2J 2 2944D7 736D5J 2
− z y + x y )β5+( − x
1488375 1488375 1915538625 70945875
25024J 2D5 736D5J 2 8912J 4D3 52DJ 6 52DJ 6
z y z y x
− − − + +
638512875 70945875 49116375 1913625 1913625
904D3J 4 904D3J 4 104J 6D 6976J 2D3J 2
x y z z x
+ + − +
49116375 49116375 1913625 49116375
6976J 2D3J 2 4736D3J 2J 2 52J 4DJ 2 52J 2DJ 4
z y x y z y z x
+ − + −
49116375 49116375 637875 637875
52J 2DJ 4 52J 4DJ 2
− z y + z x )β6+O(β7). (B.1)
637875 637875
In the absence of an external magnetic field (h = 0), and J of the Hamiltonian (1). We have confirmed this
z
this is an even function of the coupling constants J , J property up to order β12.
x y
[1] M.E. Fisher, Am. J. of Phys. 32, 343 (1964). [10] G. A. Baker Jr., G. S. Rushbrooke, H.E. Gilbert, Phys.
[2] G. S.Joyce, Phys.Rev. Lett. 19, 581 (1967). Rev. 135, A1272 (1964).
[3] H.J. Mikeska, J. Phys.: Solid St. Phys. 13, 2913 (1980), [11] G. E. Granroth, M. W. Meisel, M. Chaparala, Th.
and references therein. Jolicœur,B.H.WardandD.R.Taham,Phys.Rev.Lett.
[4] T. Delica, and H. Leschke,Physica A 168, 768 (1990). 77, 1616 (1996).
[5] O.Rojas, S.M.deSouza,E.V.CorrˆeaSilva,andM. T. [12] R.J.Birgeneau, R.Dingle,M.T.Hutchings,G.Shirane
Thomaz, Phys.Rev.B 72, 172414 (2005). and S.L. Holt, Phys. Rev.Lett. 26, 718 (1971).
[6] O. Rojas, S. M. de Souza and M. T. Thomaz, J. Math. [13] M. T. Hutchings, G. Shirane, R. J. Birgeneau and S. L.
Phys.43, 1390 (2002). Holt, Phys.Rev.B 5, 1999 (1972).
[7] O. Rojas, E. V. Corrˆea Silva, W.A. Moura-Melo, S. M. [14] S. Yamamoto, Phys. Rev. B 53, 3364 (1996-II); Phys.
de Souza, and M. T. Thomaz, Phys. Rev. B 67, 115128 Lett. A 213, 102 (1996).
(2003). [15] H.J.Jensen,O.G.Mouritsen,H.C.Fogedby,P.Hedeg˚ard
[8] O. Rojas, S. M. de Souza, E. V. Corrˆea Silva and M. T. and A.Svane,Phys. Rev.B 32, 3240 (1985).
Thomaz, Eur. Jour. of Phys. B47, 165 (2005). [16] M. Takahashi. Thermodynamics of One-dimensional
[9] A.Bu¨ller,U.L¨ow,G.S.Uhrig,Phys.Rev.B64,024428, Solvable Models. Cambridge, Cambridge University
(2001); B. Bernu and G. Misguich, Phys. Rev. B63, Press, 1999.
134409 (2001).
8
∆ = ±0.5 ∆ = ±1
0 0
(a) (b)
-0.2 -0.2
)
J
/
T-0.4 -0.4
k
(
s
as-0.6 -0.6
cl
W
-0.8 -0.8
3-01 3-01
25 0.01 20 (d)
(c)
20 0.005 10
)
%15 0 0
(
W10 -0.005 -10
∆ -0.01
5 0.4 0.6 0.8 1 -20
0 -30
-5 -40
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
kT/J kT/J
FIG. 1: (a) For ∆ = ±0.5, comparison of the numerical solution of Joyce’s expression (solid line) and our HTEs up to order
β5 (dashed line) and order β12 (dashed-dotted line), for theHFEof theclassical XXZ chain (W ), as a function of kT/J.
class
In (c), the relative percental differences (∆W(%)), with respect to Joyce’s exact solution, of the β5 (dashed line) and the β12
(dashed-dotted line) expansions, also as functions of kT/J. In (b) and (d) a similar comparison is performed, for ∆=±1.
9
J = 0.5 and ∆ = ±2
J = 1 and ∆ = ±0.5
2
1.2 exact 1.6 (b)
β6
T) 1 β13 T)1.2
C(kclass00..68 (a) C(kclass0.8
0.4
0.4
0.2
0 100
0.4
24
5 (d)
) )
% 0.2 %
(16 ( 0
C C
0
∆ ∆
8 (c) -5
0.4 0.6 0.8 1
0 -10
0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
kT kT
FIG.2: (a)SpecificheatpersiteC oftheclassical XXZ chain,asafunctionofkT,T beingtheabsolutetemperature,for
class
J =1 and ∆=±0.5. The upperpaneshows Joyce’s solution (solid line), and our expansions up to β6 (dotted-dashed line)[5]
and up to β13 (dashed line). The lower pane (c) shows the percental differences (∆C(%)), with respect to Joyce’s solution,
of the β6 (dotted-dashed line) and β13 (dashed line) expansions, also as functions of kT. Panes (b) and (d) show a similar
comparison, for J =0.5 and ∆=±2.
10
J = 1 and ∆ = 0.5 J = 0.5 and ∆ = 2
1 0.2
numeric
(b)
β12
0.8 0.16
(a)
0.6 0.12
i i
1 1
+ +
xsi xsi
xsi xsi
h h
0.4 0.08
0.2 0.04
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
kT kT
FIG. 3: The correlation function hsxsx i between the classical spin x-components of first neighbouring sites. The solid line
i i+1
standsfor theexact function,whereas thedashed linestandsfor theHTEuptoorderβ12. In(a)we haveJ =1and∆=0.5;
in (b) we haveJ =0.5 and ∆=2.
J = 1 and ∆ = 0.5 J = 0.5 and ∆ = 2
0.1 1
numeric
β12 (b)
0.08 0.8
0.06 0.6
i i
1 1
+ +
zsi zsi
zsi zsi
h h
0.04 0.4
0.02 0.2
(a)
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
kT kT
FIG. 4: The correlation function hszsz i between the classical spin z-components of first neighbouring sites. The solid line
i i+1
standsfor theexact function,whereas thedashed linestandsfor theHTEuptoorderβ12. In(a)we haveJ =1and∆=0.5;
in (b) we haveJ =0.5 and ∆=2.