Thermodynamic properties of ferromagnetic mixed-spin chain systems

Noboru Fukushima,1, ∗ Andreas Honecker,1 Stefan Wessel,2 and Wolfram Brenig1

1
Institut für Theoretische Physik, Technische Universität Braunschweig, D-38106 Braunschweig, Germany
2
Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland
Using a combination of high-temperature series expansion, exact diagonalization and quantum
Monte Carlo, we perform a complementary analysis of the thermodynamic properties of quasi-one-
arXiv:cond-mat/0307761v2 [cond-mat.str-el] 16 Jan 2004

dimensional mixed-spin systems with alternating magnetic moments. In addition to explicit series
expansions for small spin quantum numbers, we present an expansion that allows a direct evaluation
of the series coefficients as a function of spin quantum numbers. Due to the presence of excitations
of both acoustic and optical nature, the specific heat of a mixed-spin chain displays a double-peak-
like structure, which is more pronounced for ferromagnetic than for antiferromagnetic intra-chain
exchange. We link these results to an analytically solvable half-classical limit. Finally, we extend
our series expansion to incorporate the single-ion anisotropies relevant for the molecular mixed-spin
ferromagnetic chain material MnNi(NO2 )4 (ethylenediamine)2 , with alternating spins of magnitude
5/2 and 1. Including a weak inter-chain coupling, we show that the observed susceptibility allows
for an excellent fit, and the extraction of microscopic exchange parameters.

PACS numbers: 75.10.Pq, 75.50.Gg, 75.40.Cx

I. INTRODUCTION ries expansion (HTSE), exact diagonalization (ED), and

quantum Monte Carlo (QMC). In particular, our HTSE
Promoted by the synthesis of various one-dimensional will be derived for arbitrary alternating spins S and s.
(1D) bimetallic molecular magnets, the physics of This will allow not only for a direct comparison with ex-
quantum spin chains with mixed magnetic moments perimental data, but also for a study of a gradual tran-
is of great interest. Typically, quasi-1D mixed- sit to the ‘half-classical’ limit S → ∞ which is exactly
spin (MS) compounds display antiferromagnetic (AFM) solvable14,15,16 . Finally, while our ED and QMC data will
intra-chain exchange1,2,3 , which has stimulated theoret- be obtained on systems of the smallest possible mixed-
ical investigations of AFM MS models using a vari- spin magnitude, i.e. S = 1 and s = 1/2, this is a case
ety of techniques such as spin-wave theory4,5,6,7,8,9,10 , also included in the HTSE. In addition, for the sake of
variational methods11 , the density-matrix renormaliza- completeness and to compare with existing literature, we
tion group (DMRG)4,5,9 , and quantum Monte Carlo will also include results for the AFM case.
calculations6,7,8,9,10 . Interestingly, since a unit cell of the In this paper, we focus on a chain with two kinds
MS chain comprises of two different magnetic moments, of spin species, i.e. S and s, arranged alternatingly
the spectrum will allow for excitation of ‘acoustic’ as well and coupled by a nearest-neighbor Heisenberg exchange.
as ‘optical’ nature7,9 . While not identified unambigu- Namely, the Hamiltonian reads
ously in present day experiments, the character of these N
excitations should appear in thermodynamic and other Hint = −J
X
(Si · si + si · Si+1 ) . (1)
observable properties as two independent energy scales.12
i=1
In addition, and apart from the preceding, what re-
mains less well studied are MS chains with ferromagnetic The subscript i = 1 . . . N refers to the unit cells, and we
(FM) intra-chain exchange, which arise in materials of always use periodic boundary conditions.
recent interest such as MnNi(NO2 )4 (en)2 with en = In section II, our HTSE approach is detailed. In sec-
ethylenediamine13 . This compound is regarded as a tion III, we turn to a comparison with QMC and ED
quasi-1D MS material with spins S = 5/2 and s = 1 and the results of the ‘half-classical’ limit. In section IV,
at the Mn and Ni ions, respectively. The susceptibil- we discuss the result of fitting the HTSE to susceptibil-
ity displays an easy-axis anisotropy. At temperatures ity data obtained for MnNi(NO2 )4 (en)2 . Conclusions are
below TN = 2.45K, a weak AFM inter-chain coupling presented in section V.
induces AFM ordering. If this antiferromagnetic order
is suppressed by a magnetic field of approximately 1.6T,
the low-temperature specific heat shows a maximum at II. HIGH-TEMPERATURE SERIES
T = 6K and a shoulder at T = 1.5K. These features EXPANSION
could possibly reflect the two aforementioned character-
istic energy scales, however for the case of an FM, rather The HTSE is an expansion in powers of βJ, where
than an AFM MS chain. β is the inverse temperature. Here we use the linked-
Motivated by this, it is the purpose of this paper to cluster expansion of Ref. 17. In this method, the series
perform a complementary analysis of the thermodynamic coefficients for the thermodynamic limit are obtained ex-
properties of FM MS chains, using high-temperature se- actly from those calculated on finite-size clusters. In gen-
 2

eral, this includes the subtraction of contributions from the initial state is represented by |0 . . . 0). Spin opera-
a large number of so-called subclusters. In one dimen- tors act on these states as follows. Suppose | ± n) (n¿0)
sion, however, significant simplifications occur due to represents the state at site i in the notation (5). Then,
cancellation18,19 . This is true also for the MS chain sys-
tems. That is, in the absence of a magnetic field, the free sz | ± n) = (m ± n)| ± n), (6)
energy F in the thermodynamic limit is represented by ±
s | ± n) = | ± n ± 1), (7)
F/N = Fℓ (S, s) + Fℓ (s, S) and
−Fℓ−1 (S, s) − Fℓ−1 (s, S) + O[(βJ)2ℓ ], (2)
s∓ | ± n) = s∓ s± | ± n ∓ 1)
where Fℓ (S, s) is the free energy of the ℓ-site open-chain
= | ± n ∓ 1) ×
system described by
{s(s + 1) − (m + n ∓ 1)(m + n)} . (8)
(ℓ−1)/2
X
Hℓ = −J (Si · si + si · Si+1 ) (ℓ : odd) Note that the norm of |n) is not unity, namely,
i=1
n
= Hℓ−1 − JS ℓ · s ℓ (ℓ : even), (3)
Y
2 2 (±n| ± n) = {s(s + 1) − (m + n′ ∓ 1)(m + n′ )} .
n′ =1
and Fℓ (s, S) is obtained by exchanging S and s in (9)
Fℓ (S, s). Then, a calculation of Tr[(Hℓ )n ] is needed on a Besides the methods (i) and (ii), the contribution to
finite system, i.e., the specific heat from the largest cluster is calculated
X separately. Namely, contributions from ℓ-site chain to
TrHℓn = hm1 . . . mℓ |Hℓn |m1 . . . mℓ i, (4) O[(βJ)2ℓ−2 ] and O[(βJ)2ℓ−1 ] have a simple form; with
{mi } notation x ≡ s(s + 1) and X ≡ S(S + 1), for ℓ = 2l it is
proportional to 2l xl X l and for ℓ = 2l + 1 to xl X l+1 +
where mi represents the magnetic quantum numbers at
xl+1 X l . The prefactors of these terms can be determined
site i. We apply Hℓ order by order on the ket |m1 . . . mℓ i.
by comparing with those of s = S = 1/2 for any ℓ in
This operation yields linear combinations of kets with
Ref. 19. The methods (i) and (ii) are used only for the
coefficients which are functions of {mi }. To evaluate
rest of the contribution.
TrHℓ2n , products of kets of type Hℓn |m1 . . . mℓ i are needed
We have computed the specific heat for the model (1)
at most, while in the case of TrHℓ2n+1 , one can use that with s = 1/2 and S = 1, up to 29th order using the
Hℓ2n+1 = Hℓn Hℓn+1 . In order to evaluate this trace, we method (i). Furthermore, for arbitrary s and S, the series
use two different algorithms. has been calculated up to 11th order using the method
Method (i) is based on a direct matrix multiplication (ii) and standard symbolic packages21 . By setting S =
for fixed S and s. A linear combination of kets with s = 1/2, our model is reduced to the homogeneous spin-
coefficients is regarded as a sparse vector. It is stored as a half Heisenberg chain, and our series agrees with that in
compressed array of non-zero elements and another array the literature18 . Furthermore, we have also checked that
of their pointers to the kets. These pointers are stored in the S → ∞ limit of the series agrees with the Taylor
the ascending order so that one can find a needed element series of the exact solution14 which will be shown in the
using binary search in the array. All the operations are next section.
performed using integers, and thus there is no loss of
precision.
Method (ii) is designed for arbitrary spins, which is III. LARGE- AND SMALL-SPIN LIMITS
based on an analytic approach to the matrix elements
in Eq. (4). It has an advantage for large spins because
In this section, we provide evidence for the presence
the method (i) will fail for very large spins due to time
of a double-peak-like structure in the specific heat of the
and/or memory constraints. PsAfter symbolic operations, FM MS chain. We begin by recalling the elementary
the summation of the form m=−s mn can be calculated
excitations in an MS chain. The dispersion relation of
analytically for arbitrary n. For example, when n = 2,
the one-magnon excitations in the FM case reads
the sum is equal to 13 s (s + 1)(2s + 1). As a result, the
series coefficients are obtained as an expression valid for  p 
arbitrary S and s. Naively it may seem that operators of ω(k) = J S + s ± S 2 + s2 + 2Ss cos(k) , (10)
type s± will causes square roots in the matrix elements
of the Hamiltonian. However, such square roots are ab- and is shown in Fig. 1 for the extreme quantum case,
sent in the final result. In fact, the calculation can be s = 1/2 and S = 1. Similar to the AFM case7,9 , the
carried out disregarding square roots as explained below. spectrum consists of both an acoustic and an optical
Introducing the simplified notation, branch reflecting the presence of two different spins in
a unit cell. These branches indicate two energy scales in
| ± n1 . . . ± nℓ ) ≡ (s±
1)
n1
. . . (s± nℓ
ℓ ) |m1 . . . mℓ i. (5) the thermodynamics of the MS chains. The appearance
 3

3 the FM and the AFM case. Namely, the AFM specific

heat will be obtained using the substitutions e → −e
SsSsSs SsSsSs and βJ → −βJ from the FM case. Then, if the Padé
2 approximant in e is applied to

w HkL −3  −1  −1

 e e
S 3 ln(2S+1)(2s+1) 1 − FM 1 + AFM ,
1 e0 e0
the low-temperature behavior and the high-temperature
SsSsSs entropy are correctly reproduced. For s = 1/2 and S = 1,
0 we use the AFM ground-state energy eAFM 0 = −1.45408J
0 0.2 0.4 0.6 0.8 1 from Ref. 4. The extrapolation is found to be rather in-
kp sensitive to errors in eAFM
0 . For example, an error 10−4 J
AFM
in e0 affects the AFM specific heat only by 4×10−4 at
T = 0.1J, and even less at higher temperatures. In the
FIG. 1: The spin-wave dispersion relation for the FM MS following we denote by P [m/n] the rational approximant
chain with s = 1/2 and S = 1. The spatial distance of two function in e resulting from a polynomial of order m over
spins of the same species in neighboring unit cells is taken to a polynomial of order n.
be unity. In Fig. 2, a comparison is shown between different Padé
approximants for the low-temperature specific heat for
the s = 1/2, S = 1 case. We also include results from
of two one-magnon branches can be qualitatively under-
stochastic series expansion QMC simulations24 for both
stood as follows: At k = π, the excitations correspond to
the FM and the AFM case for chains of 100 sites. No
an alternating tilting of either the shorter or the longer
deviations were found to the QMC data for 50 sites, and
spins as indicated by the arrows in Fig. 1. The magni-
thus we regard these results to represent the thermody-
tude of neighboring spins determines the energies of these
namic limit. For the AFM case, the DMRG results from
modes, leading to a splitting of the branches for S 6= s.
Ref. 9 are shown in Fig. 2a as well. In contrast to the
As k is reduced away from k = π, the magnon eigen-
AFM case, the HTSE result for the low-temperature spe-
states gradually loose these alternating tilting form, and
cific heat for the FM case shows large oscillation upon
near k = 0 become similar in nature to those of uniform
increasing the order of the series. Comparison with the
chains. The low-temperature specific heat as computed
QMC data shows that the exact solution is located within
using linear spin-wave theory is expected to be exact22
the range of these oscillations. We take the arithmetic
to leading order in T . Since ω(k) is proportional to k 2
average of P [12/11], P [13/12], P [14/13] and P [15/14] as
at low energies, the specific heat is thus proportional to
the final HTSE result, which in the temperature range
T 1/2 for T ≪ J.
0.03J < T < 0.45J has an error of order 5%.
We can establish a connection between the high-
The specific heat in a larger temperature regime is
temperature series for the specific heat and this low-
shown in Fig. 3. We find overall good agreement be-
temperature scaling using a suitable Padé analysis. In
tween the QMC and the HTSE data, both for the AFM
extrapolating the high-temperature series for the specific
and the FM case. In contrast to the homogeneous FM
heat, information from the ground-state energy, the low-
spin-1/2 chain, the FM MS chain displays at least two
energy excitations, and the high-temperature entropy can
distinct structures in C(T ), namely a peak at T ∼ 0.54J
be used as follows23 : The expansion variable is changed
and a shoulder at T ∼ 0.25J. The HTSE result indicates
from β to the internal energy per unit cell, e = hHint i/N .
the presence of an additional weak shoulder at T ∼ 0.1J,
Here, e = 0 for β = 0. The power series of e(β) is thus
which is however difficult to check for using QMC due to
inverted to obtain β(e). Let S(e) denote the entropy per
increasing statistical errors at that low temperatures.
unit cell of the MS chain. From β = dS/de, one obtains
The features in the intermediate temperature regime
Z e become more pronounced with increasing order of the se-
S = ST =∞ + β(e′ )de′ , (11) ries expansion, probably because the higher-order poly-
0
nomials can reproduce the involved rapid changes more
where ST =∞ = ln(2S + 1) + ln(2s + 1). The low- accurately.
temperature behavior of the specific heat, C ∝ T 1/2 , We have included in Fig. 3 ED results for small finite
translates into S ∝ (e−e0 )1/3 at e ∼ e0 for the new series, chains for both the AFM and the FM case. The full spec-
where e0 is the ground state energy per site. In Ref. 23, trum has been calculated for finite chains with up to 7
only the FM e0 is used to extrapolate the specific heat of unit-cells (14 sites). The ED result for 14 sites agrees
a HTSE for a FM model. Here, we also employ the AFM with the QMC and the HTSE down to temperatures of
e0 , since this additional constraint from the other sign of T ∼ 0.5J for the FM case, and T ∼ 0.3J for the AFM
J does not drastically change the final result, but makes case. Fig. 3 clearly shows that finite-size effects in the
the extrapolation less sensitive to the used Padé approx- specific heat are significantly larger in the FM than in
imant. We thus obtain the same extrapolation for both the AFM case. In the AFM case, the differences between
 4

0.35 (a) 0.7 (a)

0.3 0.6
0.25 0.5
CR 0.2 P[15/14] CR 0.4 8 sites
10 sites
0.15 P[14/13]
P[12/11] 0.3 12 sites
14 sites
0.1 P[11/10] 0.2 HTSE
QMC
0.05 DMRG 0.1 QMC

0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 1.2 1.4
TJ TJ

0.4 0.4
(b) (b)
0.35
0.3
CR 0.3 P[15/14]
CR
0.2 8 sites
P[14/13] 10 sites
0.25 P[12/11] 12 sites
P[11/10] 0.1 14 sites
HTSE
QMC
0.2 QMC
0
0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4
TJ TJ

FIG. 2: Low-temperature behavior of the specific heat per FIG. 3: Specific heat per unit cell of the AFM (a) and FM
unit cell of the AFM (a) and FM (b) mixed-spin chain with (b) mixed-spin chain with s = 1/2, S = 1 as obtained from
s = 1/2, S = 1 using several different Padé approximants of HTSE, QMC and full diagonalizations for small finite chains.
the HTSE and from QMC. In (a), DMRG results from Ref. 9
are also shown. Here, R is the gas constant.
mode in the FM case is larger than in the AFM7,9 case,
the 12- and 14-site data are already rather small, with a so that the splitting of the structures in C(T ) is more
weak shoulder at low temperature signaling the second evident in the former case. Apart from (i) and (ii), the
energy scale in C (T ). For the FM case the ED data elementary excitation spectra of the FM and the AFM
show a strong finite-size shift of the specific heat max- case resemble each other, pointing towards additional ef-
imum, which gradually develops into the shoulder near fects from magnon interactions.
T ∼ 0.25J, which is found both from the HTSE and After analyzing the extreme quantum case with s =
QMC. 1/2 and S = 1, we now discuss the specific heat of MS
This suggests that long-range collective modes domi- chains for larger values of S, keeping s = 1/2 fixed. In
nate the thermodynamics up to higher temperatures for particular, we want to connect to the exactly solvable14
the FM than for the AFM case. To shed more light onto ‘half-classical’ limit, S → ∞.
this difference, let us compare the elementary excitation Fig. 4 shows the results from the 11th order HTSE
spectra for the FM and the AFM case: (i) At low tem- for s = 1/2 and various values of S. The temperature
peratures, the specific heat reflects the dispersion of the p
is normalized to |S|J ≡ J S(S + 1) in order to render
acoustic branch of excitations. In the long-wavelength the ’half-classical’ limit finite. In the limit S → ∞ the
limit, the dispersion in linear spin-wave theory reads specific heat of the chain with FM couplings is identi-
Ss cal to that of the AFM chain, with a distinct peak at
ω(k) ∼ k2 , T ∼ 0.5 |S|J. At T = 0 the specific heat of the ‘half-
2|S ± s|
classical’ model is finite. In the quantum case, however,
where the plus (minus) sign represents the FM (AFM4,6 ) the specific heat vanishes as T → 0. Using the variable S
case. Therefore, the slope of the acoustic branch in HTSE, we can link both limiting situations as follows: In
the AFM case is larger than in the FM case, making the limit S → ∞, the series coefficients of (βSJ)n with
low-temperature shoulders more pronounced for FM MS odd n converge to zero. Let us compare the (2n)-th and
chains. (ii) The gap between the acoustic and the optical (2n+1)-th order terms at finite S. The ratio of the latter
 5

to the former is of O(t/S), where t ≡ T /(SJ). Hence, if 1

the limit S → ∞ is taken with t fixed, the latter becomes ‡
S= S=1
negligible compared to the former as S → ∞. However, S=3
if S is finite, the two terms are comparable for temper- CR
atures below t ∼ S −1 . Therefore, even if S is large, the
specific heat deviates from the ’half-classical’ behavior
below t ∼ S −1 , and approaches zero as t → 0. As a
0.5
AFM
result, the specific heat will display a double-peak-like
structure for large S as well. Summarizing these results
from the HTSE, there are two different ways of taking FM
the large-S and T → 0 limit,
0
lim lim C/R = 1, (12) 0 1 2
T →0 S→∞ T (|S|J)
lim lim C/R = 0. (13)
S→∞ T →0

As seen from Fig. 4 the difference between the FM FIG. 4: The HTSE results for the dependence on S of
and the AFM cases are most pronounced in the extreme the specific heat per unit cell with s = 1/2 for antiferro-
quantum limit s = 1/2 and S = 1, and in the low- magnetic
p (AFM) and ferromagnetic (FM) coupling, where
temperature regime. The large- and small-spin limits ex- |S| = S(S + 1). Here, S = ∞ is from Ref. 14. Details
hibit similar structures, suggesting that the specific heat are in the text.
of MS chain systems in general show a double-peak-like
or peak-shoulder structure, both for AFM and FM intra- Here, we derive the power series of χ in βD as well as
chain exchange, and for any combination of spins. In the βJ up to O(β 7 ). When D = 0 and g = G, the series
FM case this structure is more pronounced if |S − s| is coefficients coincide with those in the literature25,26 as-
large, reflecting the size of the gap between the acoustic suming a misprint27 in Ref. 25. Since the contribution
and the optical mode, similar to the AFM case.12 Hmag is used to evaluate the susceptibility we will only
We note that the specific heat in the high-temperature consider the case of small Zeeman energies |h| = h → 0
limit of the AFM case is larger than in the FM case be- in the following. The orientation of the magnetic field
cause of quantum effects. This is evident from the first h will be chosen to be either hkz or hkx. Most theo-
two terms of the series for the specific heat, which we retical studies of MS systems are limited to the case of
present in Appendix A. These two terms stem from two- D =P 0 and g = G, in order to make use of total spin-z,
site correlations only and dominate the high-temperature i.e. z z
i (Si + si ), conservation. However, for a proper
behavior. Since the total entropy difference between zero comparison to experimental data, D 6= 0 and g 6= G has
and infinite temperature is the same in both cases, the to be accepted, leading to
FM and AFM specific-heat curves therefore have to in-
tersect at low temperatures. [Hint + Hani , Hmag ] 6= 0. (17)
We emphasize, that our HTSE is carried out taking into
IV. FITTING TO EXPERIMENTAL DATA account this non-commutativity.
Since 3D AFM ordering of MnNi(NO2 )4 (en)2 below
We now turn to a comparison to the susceptibility data TN = 2.45K signals the presence of a non-negligible inter-
observed on MnNi(NO2 )4 (en)2 . In this compound, the chain exchange, we will enhance our 1D analysis to incor-
symmetry around Ni ions is nearly cubic with however a porate this coupling on a phenomenological basis. That
fairly large anisotropy at the Mn site to be expected13 . is to say, fits to the experimental results will be performed
Hence, we take into account a single-site anisotropy only using an RPA expression
on one of the spins, i.e. S. The g-factors of the spins S χ1D
and s are represented by G and g, respectively. There- χ≃ , (18)
1 − J⊥ χ1D
fore, the total Hamiltonian reads
where χ1D is the susceptibility of the pure 1D system
H = Hint + Hani + Hmag , (14) obtained by HTSE and extrapolated by a simple Padé
approximation (PA). Here, J⊥ effectively models the av-
where erage inter-chain exchange. Figure 5 shows the results of
N our fits of χ to experimental data of the susceptibility28
2
X
Hani = D (Siz ) , (15) with a magnetic field oriented both, perpendicular and
i=1 parallel to the c-axis. Apart from s = 1 and S = 5/2
N we have used g = 2.24 and G = 2 as listed in Ref. 13.
Best fits are obtained for J = 2.8 K, J⊥ = −0.036 K and
X
Hmag = −h · (G Si + g si ). (16)
i=1 D = −0.36 K. As estimated from the PA of the HTSE,
 6

chain exchange. A symbolic high-temperature series has

4 experiment been derived for general values of the spin quantum num-
fitting by HTSE bers, and including single ion anisotropies. Using this
c@emumolD

3 series expansion, we were able to extract the microscopic

h||c model parameters for the quasi-one-dimensional FM
mixed-spin chain compound MnNi(NO2 )4 (en)2 . Com-
2 h||a paring our results to the analytically solvable limit S →
∞, we found that not only the AFM but also the FM
1 TN case displays a double-peak-like structure in the specific
heat which is due to the presence of both ‘optical’ and
‘acoustic’ excitations. In fact, we find that for FM intra-
0 chain coupling this structure is more pronounced than
0 5 10 15 20
for AFM exchange. The low-temperature specific heat of
T@KD MnNi(NO2 )4 (en)2 shows a weak shoulder around 1.5K if
magnetic order is suppressed by a finite magnetic field. It
is thus suggestive to associate this shoulder with the pres-
FIG. 5: Fitting to susceptibility of MnNi(NO2 )4 (en)2 by ence of ‘acoustic’ excitations in this compound. However,
Eq.(18). Here, (µ2B NA /kB )χ is plotted, where µB , NA and a quantitatively accurate description of the relevant tem-
kB are Bohr magneton, Avogadro’s number and Boltzmann
perature range for the FM mixed-spin chain with s = 1
constant, respectively.
and S = 5/2 in a magnetic field requires further numer-
ical studies and is left for future investigations. Finally,
in the range plotted, the error involved in the theoretical comparing the AFM with the FM case in the extreme
curve for χ1D is within the width of the line. As is ob- quantum limit, i.e. s = 1/2, S = 1, we found finite-
viously from the figure our theory allows for an excellent size effects to be more pronounced in the FM than in
fit to the experimental data down to T ∼4K. Only the the AFM case. This suggests long-range collective exci-
high-temperature data, for 10K ≤ T ≤ 25K have been tations to be more relevant for the low-temperature ther-
utilized to set the parameters J, D, and J⊥ . Keeping modynamics of FM mixed-spin chains.
these fixed, the splitting between hkc and h ⊥ c of the
experimental data tends to become larger than that of
the theory for T < 4K. Acknowledgments
In Ref. 13, values of J ∼ 1.9K and D ∼ −0.45K have
been reported for MnNi(NO2 )4 (en)2 . These have been The authors are grateful to R. Feyerherm and S. Süllow
obtained by fitting a directional average of the suscep- for fruitful discussions about the experimental situation
tibility with respect to the magnetic field to the ‘half- on MnNi(NO2 )4 (en)2 and the provision of data. This
classical’ limit,16 i.e. S → ∞, applicable for D = 0. work was supported in part by the Deutsche Forschungs-
Moreover the effects of interchain coupling have been ne- gemeinschaft through grant SU 229/6-1. S.W. acknowl-
glected. To compare with this result we may use J⊥ ≡ 0 edges support from the Swiss National Science Foun-
in Eq. (18). In fair agreement with Ref. 13, we find dation. Parts of the numerical calculations were per-
J ∼ 2.4K and D ∼ −0.36K in this case, with a qual- formed on the Asgard cluster at the ETH Zürich and on
ity of the fit however, which is inferior to that shown on the cfgauss at the computing center of the TU Braun-
Fig. 5. schweig.

V. SUMMARY APPENDIX A: SERIES DATA

We have studied the specific heat and uniform suscep- Using X ≡ S(S + 1) and x ≡ s(s + 1), the specific heat
tibility of mixed-spin chain systems using a combination series without the single-site anisotropy is given by
of high-temperature series expansion, exact diagonaliza-
C = CD + CND (x, X) + CND (X, x) + O (βJ)12 ,
 
tion, and quantum Monte Carlo techniques. In particu- (A1)
lar, we have contrasted the cases of FM and AFM intra-
 7

2 x X (βJ)2 x X (βJ)3 2 x2 X 2 2 x2 X 2
   
2xX −x X
CD = − + − (βJ)4 + + (βJ)5
3 3 15 15 18 27
643 x2 X 2 4 x3 X 3 4043 x2 X 2 2 x3 X 3
   
8xX −23 x X
+ + + (βJ)6 + − − (βJ)7
315 7560 189 1800 32400 135
1162 x2 X 2 622 x3 X 3 2 x4 X 4
 
19 x X
+ + − − (βJ)8
2700 10125 18225 675
1489333 x2 X 2 115043 x3 X 3 4 x4 X 4
 
−2231 x X
+ − + + (βJ)9
529200 15876000 2976750 1575
15768563 x2 X 2 5996723 x3 X 3 36427 x4 X 4 4 x5 X 5
 
15901 x X
+ + − + + (βJ)10
5821200 209563200 943034400 3742200 10395
822853 x2 X 2 111661717 x3 X 3 1448899 x4 X 4 2 x5 X 5
 
−72557 x X
+ − − − − (βJ)11 , (A2)
38102400 13395375 3857868000 128595600 5103

−8 x X 2 (βJ)4 4 x X 2 (βJ)5 −179 x X 2 2 x X3 10 x2 X 3

 
CND (x, X) = + + + + (βJ)6
45 27 1890 63 189
157 x X 2 x X3 82 x2 X 3
 
+ − − (βJ)7
2700 25 2025
−124 x X 2 124 x X 3 6091 x2 X 3 16 x X 4 1064 x2 X 4 8 x3 X 4
 
+ + − − − − (βJ)8
3375 3375 364500 3375 91125 675
3229 x X 2 1019 x X 3 23281 x2 X 3 8 x X4 128 x2 X 4 244 x3 X 4
 
+ − + + + + (βJ)9
132300 33075 441000 945 10125 23625
−17137 x X 2 24727 x X 3 86018257 x2 X 3 11489 x X 4 6287 x2 X 4

+ + − − +
997920 970200 1257379200 1091475 11642400
3 4 5 2 5 3 5 4 5

25411 x X 4xX 316 x X 218 x X 2x X
+ + + + + (βJ)10
2619540 6237 155925 66825 891
732629 x X 2 757 x X 3 11884871 x2 X 3 10279 x X 4 5676361 x2 X 4

+ − + + −
57153600 35280 163296000 893025 367416000
3 4 5 2 5 3 5
32 x4 X 5

3918727 x X 1382 x X 2672 x X 30181 x X
− − − − − (βJ)11 . (A3)
214326000 893025 893025 8037225 14175

We have computed the series of the susceptibility with the request. Since the non-commutativity, Eq. (17), is neglected
single-site anisotropy up to up to O(β 7 ), which is too lengthy in Ref. 16, the S → ∞ limit of the series below with D = 0
to be fully listed in this paper. Hence, we show here only the and g 6= G is different from the function given in Ref. 16.
first four terms of the series, and the rest will be provided on

(
g2 x G2 X 4 X2 2 x2 X
      
2 4gGJ xX 2 −X 3 − (x X)
χzz = β + +β −DG + +β g2 J 2 +
3 3 9 15 45 27 27
" #  )
− J2 x X


16 x X 2 2 x X2 4 X2 8 X3
     
−4 x X − (x X) X
+g G −DJ + + G2 J 2 + + D2 − +
27 45 135 27 27 42 105 945
(  
x2 X 16 x2 X 8 x X2 64 x2 X 2 4 x X2
     
xX 2xX xX
+β 4 g 2 J 3 − − D J2 − − + + g G D J2 −
108 81 675 675 2025 2025 405 135
2 2 2 2 2 3
   
xX 16 x X 16 x X 8x X 2xX 16 x X 32 x X
+J 3 − − + + D2 J − +
90 405 405 405 63 315 2835
2 2 3
 )
97 X 2 32 X 3 16 X 4
     
xX xX xX 22 x X 16 x X −X
+G2 J 3 − − D J2 − + − D3 + − −
108 81 54 405 405 90 4725 4725 14175
+..., (A4)
 8
(
g2 x G2 X 2 X2 2 x2 X
      
4gGJ xX X − (x X)
χxx = β + + β2 − D G2 − + β3 g2 J 2 +
3 3 9 30 45 27 27
" #  )
− J2 x X


8 x X2 2 x X2 X2 4 X3
     
2xX 2 2 − (x X) 2 X
+g G −DJ − +G J + +D − −
27 45 135 27 27 210 315 945
(  
2 2 2 2 2
  
xX x X − (x X) 8x X 4xX 32 x X
+β 4 g 2 J 3 − − D J2 + + −
108 81 675 675 2025 2025
2 x X2 16 x2 X 16 x X 2 8 x2 X 2
    
xX xX
+g G D J 2 − + J3 − − +
270 405 90 405 405 405
4 x X2 16 x X 3 x X2 11 x X 2 8 x X3
      
2 2xX 2 3 xX 2 − (x X)
+D J − − +G J − −DJ + −
315 945 2835 108 81 108 405 405
)
2 3 4
 
−X X 2X 8X
−D3 − + + + .... (A5)
2520 1350 1575 14175

∗
Electronic address: n.fukushima@tu-bs.de K123 (1975).
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6 21
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22
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11
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