Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems
We present the formulation of the Replica Ornstein-Zernike equations for a model of positionally frozen disordered Heisenberg spin system. The results are obtained for various models, one in which the particle positions correspond to a frozen hard sphere fluid, another system in which the con- f...
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| Цитувати: | Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems / E. Lomba, C. Martin // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 551-560. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1207432017-06-13T03:03:52Z Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems Lomba, E. Martin, C. We present the formulation of the Replica Ornstein-Zernike equations for a model of positionally frozen disordered Heisenberg spin system. The results are obtained for various models, one in which the particle positions correspond to a frozen hard sphere fluid, another system in which the con- figurations are generated by a random insertion of hard spheres, a system of randomly distributed spins, and finally a system corresponding to a soft sphere fluid quenched at high and low temperatures. We will see that the orientational structure of the spin system is fairly well reproduced by the integral equation which, however, does not correctly account for the critical behaviour. Ми формулюємо реплічні рівняння Орнштейна-Церніке для моделі з позиційно замороженою невпорядкованою гайзенбергівською спіновою системою. Результати отримані для різних моделей, в одній з яких положення частинки відповідає замороженому твердосферному плину. Інша система, в якій конфігурація генерується випадковим вміщенням твердих сфер, також система з випадково розподіленими спінами і нарешті система, яка відповідає плину м’яких кульок, розглянуті при високих і низьких температурах. Ми можемо бачити, що орієнтаційна структура спінової системи досить добре відтворюється методом інтегральних рівнянь, які, проте, не дають коректного опису критичної поведінки. 2003 Article Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems / E. Lomba, C. Martin // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 551-560. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 75.50.Lk, 61.20.Ja, 64.60.Fr DOI:10.5488/CMP.6.3.551 http://dspace.nbuv.gov.ua/handle/123456789/120743 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We present the formulation of the Replica Ornstein-Zernike equations for
a model of positionally frozen disordered Heisenberg spin system. The results
are obtained for various models, one in which the particle positions
correspond to a frozen hard sphere fluid, another system in which the con-
figurations are generated by a random insertion of hard spheres, a system
of randomly distributed spins, and finally a system corresponding to a soft
sphere fluid quenched at high and low temperatures. We will see that the
orientational structure of the spin system is fairly well reproduced by the
integral equation which, however, does not correctly account for the critical
behaviour. |
| format |
Article |
| author |
Lomba, E. Martin, C. |
| spellingShingle |
Lomba, E. Martin, C. Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems Condensed Matter Physics |
| author_facet |
Lomba, E. Martin, C. |
| author_sort |
Lomba, E. |
| title |
Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems |
| title_short |
Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems |
| title_full |
Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems |
| title_fullStr |
Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems |
| title_full_unstemmed |
Replica Ornstein-Zernike equations for positionally frozen Heisenberg systems |
| title_sort |
replica ornstein-zernike equations for positionally frozen heisenberg systems |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120743 |
| citation_txt |
Replica Ornstein-Zernike equations for
positionally frozen Heisenberg systems / E. Lomba, C. Martin // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 551-560. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT lombae replicaornsteinzernikeequationsforpositionallyfrozenheisenbergsystems AT martinc replicaornsteinzernikeequationsforpositionallyfrozenheisenbergsystems |
| first_indexed |
2025-07-08T18:30:15Z |
| last_indexed |
2025-07-08T18:30:15Z |
| _version_ |
1837104539357937664 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 3(35), pp. 551–560
Replica Ornstein-Zernike equations for
positionally frozen Heisenberg systems
E.Lomba, C.Martı́n
Instituto de Quı́mica Fı́sica Rocasolano (CSIC),
Serrano 119, E-28006 Madrid, Spain
Received March 28, 2003
We present the formulation of the Replica Ornstein-Zernike equations for
a model of positionally frozen disordered Heisenberg spin system. The re-
sults are obtained for various models, one in which the particle positions
correspond to a frozen hard sphere fluid, another system in which the con-
figurations are generated by a random insertion of hard spheres, a system
of randomly distributed spins, and finally a system corresponding to a soft
sphere fluid quenched at high and low temperatures. We will see that the
orientational structure of the spin system is fairly well reproduced by the
integral equation which, however, does not correctly account for the critical
behaviour.
Key words: Heisenberg system, integral equations, ferromagnetic
transition, spinglass
PACS: 75.50.Lk, 61.20.Ja, 64.60.Fr
1. Introduction
The study of positionally frozen dipolar fluids has been the focus of various works
in recent years [1–4], using both simulation techniques [1,3], mean field theory [2]
and quite recently the Replica Ornstein-Zernike (ROZ) integral equation theory
[4]. In related works [5,6], the authors focused on the ferromagnetic transition of a
positionally frozen Heisenberg spin system, which is amenable to be treated more
accurately using simulation techniques adapted for near critical conditions.
In this paper we will explore the capabilities of the ROZ integral equation treat-
ment to describe the ferromagnetic transition in positionally frozen Heisenberg sys-
tems. For this purpose we will solve the ROZ equations in the Hypernetted Chain
(HNC) approximation for the models simulated in references [5] and [6], namely,
Heisenberg spin systems in which the spin positions are frozen according to the
configurations of hard sphere (HS) fluids (model A) [5], a random distribution of
spins (model B) [5], and soft spheres (SS) quenched at high and low temperatures
(models C and D, respectively) [6]. Additionally, we have considered a system in
c© E.Lomba, C.Martı́n 551
E.Lomba, C.Martı́n
which the spatial distribution is generated by random insertion of non-overlapping
hard spheres in a given volume until the desired density is attained (model E). The
critical behavior of this system studied by means of Monte Carlo (MC) simulation
and finite size scaling analysis is presented in this paper.
The spin-spin interaction in all the models considered is defined by
U(r12, ω1, ω2) = −J(r12)(s1 · s2), (1)
where exchange coupling is given by
J(r) = ε
σ
r
exp[(σ − r)/σ],
si being the unit vector that describes the orientation of the spin i, and ε is a
positive constant, favoring ferromagnetic alignment. With this, the reduced temper-
ature will be defined by T ∗ = kBT/ε. For practical purposes J(r) is truncated at
R = 2.5σ. In the case of the SS model J(r) is shifted at the truncation radius, so that
limr→R− J(r) = limr→R+ J(r) = 0. As mentioned before, the spatial distribution of
model A is simply generated by taking independent HS configurations (in which
the particle diameters are σ0 = σ) at the desired density, which in this case will be
ρσ3 = 0.6. The random positions of model B are generated by simple insertion of
hard spheres of diameter σ0 = 0.1σ in a given volume until the density ρσ3 = 0.6
is reached. These configurations are practically random and the small hard cores
prevent the divergence of the interactions at zero separation. Models C and D are
generated by placing the spins onto the positions of a frozen SS fluid which interacts
via a potential of the form
ψsoft(r) = 4ε
[
(
σ
r
)12
−
(
σ
r
)6
]
+ ε, r < 21/6σ
and ψsoft(r12) = 0 otherwise. Now, in this case the spatial distribution of the spins
will be defined by quenching the positions of the SS fluid at a quench temperature
T ∗
0 = kT0/ε (with T ∗
0 = 2.1 for model C and T ∗
0 = 100 for model D). As a matter
of fact, in the case of Heisenberg interactions, the spatial distribution is hardly
sensitive to the spin-spin correlations for temperatures as low as the Curie point.
One might envisage then the situation depicted here as the effect of quenching the
particle positions on the spin fluid itself, and the same can be said for the frozen HS
Heisenberg system. Finally, model E is constructed exactly as the random system
B, but using a particle diameter σ0 = σ. The distribution thus generated has some
similarities with the HS model but is somewhat more random.
The rest of the paper is organized as follows. In the next section we introduce
the ROZ equations for the positionally frozen Heisenberg system. In section 3 we
present our most significant results and conclusions.
2. A summary of the theory
The application of the replica trick [7] to a system like ours, in which the orienta-
tional degrees are allowed to equilibrate and the particle positions are frozen, implies
552
ROZ equations for Heisenberg systems
taking the s→ 0 limit of a fully equilibrated system composed of soft spheres with
embedded s replicas of the spins. The Hamiltonian of such a system reads
Hrep =
β0
β
∑
i>j
ψ0(rij) +
s
∑
α=1
∑
i>j
J(rij)(s
α
i · sα
j ), (2)
where β = 1/kBT and β0 = 1/kBT0, T0 and T being the temperature at which the
particle positions have been frozen, and the equilibrium temperature of the spins,
respectively. In equation (2), ψ0(r) is either the hard sphere potential, ψsoft(r), or
a given radial potential capable of generating the spatial distributions of models B
or E. As prescribed by the replica procedure, only replicas of the same family, α,
interact.
The Ornstein-Zernike equation in Fourier space for the replicated system reads
h̃rep(12) = c̃ rep(12) + ρ
∫
c̃ rep(13)h̃rep(32)d{ω3}, (3)
where d{ω3} = dω1
3 . . .dω
s
3 denotes the integration over the orientations of the s
replicas of the spin in particle 3, ρ is the number density, and h̃rep and c̃ rep are the
Fourier transforms of the total and direct correlation functions, respectively. The
replicated functions can be expanded in Legendre polynomials as
f rep(12) =
∑
αβ
∑
l
fαβ
l (r)Pl(cos θαβ
12 ), (4)
with fαβ
0 (r) = f0(r)/s
2, and in general
fαβ
l (r) =
2l + 1
2
∫
d cos θαβ
12 f
rep(12)Pl(cos θαβ
12 ). (5)
Inserting (4) in equation (3) and taking the limit s→ 0, the ROZ equations read
h̃0 = c̃0 + ρc̃0h̃0 , (6)
h̃f
l = c̃fl +
ρ
2l + 1
[
c̃fl h̃
f
l − c̃bl h̃
b
l
]
, (7)
h̃b
l = c̃bl +
ρ
2l + 1
[
c̃fl h̃
b
l + c̃bl h̃
f
l − 2c̃fl h̃
f
l
]
, (8)
where l 6= 0, and
f b
l (r) = lim
s→0
fαβ
l (r), α 6= β,
f f
l (r) = lim
s→0
fαα
l (r).
As usual, the connected functions are defined by f c = f f − f b, by which the last
two equations transform into
h̃f
l = c̃fl +
ρ
2l + 1
[c̃cl h̃
f
l + c̃bl h̃
c
l ], (9)
h̃c
l = c̃cc +
ρ
2l + 1
c̃cl h̃
c
c. (10)
553
E.Lomba, C.Martı́n
These equations are just a particular case of those derived by Klapp and Patey [4]
for the dipolar case.
As to the closure relation, here we will use the HNC approximation, which reads
crep(12) = hrep(12) − log[grep(12)] − βurep(12), (11)
where according to (2)
βurep(12) = β0ψ0(r12) + β
∑
α
J(r12)(s
α
1 · sα
2 ). (12)
The s → 0 limit of (11) can be obtained if one uses Fries and Patey’s form of the
HNC closure [8], which expanded in Legendre’s polynomials leads to
c0(r) = −β0u0(r) −
∫
∞
r
dr′h0(r
′)
∂X0(r
′)
∂r′
, (13)
cfl (r) = −βul(r) − (2l + 1)
∑
λλ′
(
λ λ′ l
0 0 0
)2
∫
∞
r
dr′hf
λ(r
′)
∂Xb
λ′(r′)
∂r′
, (14)
cbl (r) = −(2l + 1)
∑
λλ′
(
λ λ′ l
0 0 0
)2
∫
∞
r
dr′hb
λ(r
′)
∂Xb
λ′(r′)
∂r′
(15)
after the limit s→ 0 is taken. Above we have used X(12) = h(12)− c(12)−βu(12),
and the quantities in brackets are the 3 − j Wigner symbols, u0(r) = ψ0(r) and
u1(r) = J(r) (with ul = 0 for l > 1.) From equations (15) and (6)–(8) it turns out
that hb
l = cbl = 0 when l 6= 0 (and hf
0 = hb
0 = h0 by definition). With this, the
equations can be recast in the form
h0(r) = exp[−β0u0(r) + s0(r)] − 1, (16)
hf
l (r) =
2l + 1
2
∫
d cos θPl(cos θ)(h0(r) + 1)
× exp
[
∑
l′=1
(
−βul′(r) + sf
l′(r)
)
Pl′(cos θ)
]
, (17)
where s = h− c, and equation (17) holds for l > 0. Equation (6) can be solved cou-
pled with (16) and this gives the spatial distribution of the particles at the quenching
inverse temperature β0. Equations (17) and (7) will describe the orientational struc-
ture of the spins at the equilibrium inverse temperature β.
3. Results
The ROZ-HNC equations have been solved by a standard mixing iterates proce-
dure, with the r-space discretized in 8192 points and a grid size of 0.01σ. The spatial
distribution of the spins can be visualized in figure 1, where we have plotted the pair
distribution function, g0(r) = h0(r) + 1 for models A–E. The only features worth
mentioning are, on the one hand, the slight difference between the g0(r) of mod-
els A and E, despite the fact that the hard sphere configurations are generated by
554
ROZ equations for Heisenberg systems
0
0.5
1
1.5
2
2.5
3
g 0(r
)
Model A
Model E
Model B
0 1 2 3
r/σ
0
0.5
1
1.5
g 0(r
)
model C
model D
Figure 1. Radial distribution function of the positionally frozen Heisenberg sys-
tems studied in this work.
completely different procedures (this simply illustrates how the hard core condition
completely dominates the structure of the fluid even at relatively low densities). On
the other hand, the comparison between the high and low temperature SS quench
shows that the high temperature quench allows for a closer approach distance be-
tween the particles, which will have consequences on the critical temperature.
The results of the ROZ equations for the structure of the positionally frozen
Heisenberg system are illustrated in figure 2 where the most significant angular co-
efficient of the fluid-fluid correlation function, hf
1(r), is depicted. This coefficient is
proportional to the ensemble average 〈s1 ·s2〉(r) which measures the relative orienta-
tion of the two spins separated by a distance r. The results presented correspond to
the SS model D, in which the spatial correlations of the quenched system differ most
notably from those of the corresponding fully equilibrated (i.e. spin fluid) system.
In contrast, the correlations for the HS quenched systems are completely dominated
by the HS repulsion (which is temperature independent) and thus the structures of
quenched and equilibrated systems are extremely similar. We observe that the ROZ
equation rather accurately reproduces the structure for the highest temperature. As
555
E.Lomba, C.Martı́n
0.00
0.25
0.50
h 1f (r
)
0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
r/σ
0.00
0.25
0.50
h 1f (r
)
MC
ROZ-HNCT* = 3.2
T* = 4.0
Figure 2. Leading angular coefficient of the pair distribution function expansion
for the positionally frozen Heisenberg system constituted by spins embedded in
frozen soft spheres (high temperature quench, model D).
the temperature is lowered the theoretical correlation becomes somewhat more long
ranged that its MC counterpart. We will see that this is consistent with a systematic
overestimation of the critical temperature on the part of the ROZ results.
The critical parameters of model E have been calculated following the prescrip-
tion of references [5] and [6] (essentially similar to the procedure used in the spin fluid
cases studied in reference [10] and [9]). In figure 3 we present the evolution with tem-
perature of Binder’s cumulant [11], U4, and the percolation fraction, φ, (defined as
the fraction of configurations for which, at least one of the Swendsen-Wang clusters
percolates through the periodic system) for various system sizes. Binder’s cumulant
is defined by
U4 = 1 −
〈m4〉
3〈m2〉2
, (18)
where the magnetization per particle is
m =
1
N
∣
∣
∣
N
∑
j
si
∣
∣
∣,
556
ROZ equations for Heisenberg systems
0.50
0.55
0.60
0.65
U
4
3.00 3.05 3.10 3.15 3.20
T*
0.20
0.30
0.40
0.50
0.60
φ
256
500
1372
4000
Figure 3. Evolution of Binder’s cumulant, U4, and percolation fraction,φ, with
temperature and system size for the positionally frozen Heisenberg system with
the spatial distribution described by model E (randomly inserted non-overlapping
hard spheres of diameter σ0 = σ).
N being the number of particles in the sample. From the analysis of the intersections
of the U4 and φ curves one obtains the critical parameters for this model which are
presented in table 1 together with the parameters for the rest of the models taken
from references [5] and [6], as well as the results for the hard sphere Heisenberg
spin fluid from reference [9], and for the soft sphere spin fluid of reference [10]. The
values of β/ν are obtained by a fit of the critical magnetization, mc, which is known
to scale with the system size as mc ∝ L−β/ν , and γ/ν is obtained from a fit of the
magnetic susceptibility maxima near the critical point, which also scale with system
size as χm ∝ Lγ/ν . In table 1 we also include the values of T ∗
ns, the temperature at
which the ROZ integral equation breaks down with diverging zero-field magnetic
susceptibility,
χ/ρβµ2 =
1
3
[
1 +
ρ
3
h̃f
1(0)
]
=
1
3
[
1 −
ρ
3
c̃f1(0)
]
−1
, (19)
where µ is the coupling parameter between the spins and the external field. The first
striking feature in table 1 is the fact that despite the equilibrium HNC approxima-
tion predicts the transition temperatures within less than a 6% deviation from the
557
E.Lomba, C.Martı́n
Table 1. Critical parameters of positionally frozen Heisenberg systems obtained
from FSS analysis of the MC simulation, and non-solution temperature of the
ROZ-HNC (or HNC for the spin fluid systems) equations, T ∗
ns.
Model T ∗
0 σ0/σ T ∗
c U4 γ/ν β/ν T ∗
ns
A [5] – 1.0 3.145(3) 0.609(3) 1.87(2) 0.53(1) 3.517
B [5] – 0.1 3.947(2) 0.616(4) 1.86(1) 0.55(2) 6.520
E – 1.0 3.120(3) 0.609(2) 1.86(2) 0.57(1) 3.463
HS spin fluid [9] – 1.00 3.150(5) 0.608(1) 1.85(1) 0.56(2) 3.186
C [6] 2.1 – 2.050(6) 0.617(2) 1.92(2) 0.52(4) 2.420
D [6] 100.0 – 2.196(3) 0.618(2) 1.93(2) 0.51(4) 2.826
SS spin fluid [10] – – 2.054(1) 0.619(2) 1.90(3) 0.54(2) 2.187
simulation in the spin fluid cases, the theoretical results for the quenched systems
are considerably away from the simulated estimates. As a general trend the ROZ
overestimates the critical temperatures, but the results are particularly off in the
case of the random model B. Moreover, comparing model A with the HS spin fluid,
and model C with the corresponding SS spin fluid, one observes that the ROZ pre-
dicts an increase in the critical temperature when quenching the particle positions
for these systems with positional correlations. This is in agreement with the ROZ
results reported in reference [4] for dipolar systems, but it is in clear disagreement
with the simulation results that unequivocally indicate that freezing the particle po-
sitions does not affect the critical behaviour, at least in the case of separable angular
interactions. Aside from this, what seems to be correct in the ROZ predictions is
the qualitative change in the critical temperature when comparing different posi-
tionally frozen systems. Thus, we observe that the ROZ critical temperatures are
ordered as T ∗
ns(C) < T ∗
ns(D) < T ∗
ns(E) < T ∗
ns(A) < T ∗
ns(B). This is the same order
observed in the simulation results. Moreover, when comparing models E and A, one
observes that T ∗
ns(E) < T ∗
ns(A). The only difference between these two models lies
in the fact that model A presents fluid-like positional correlations, whereas model E
is somewhat more random (hardly noticeable in the distribution functions, cf. fig-
ure 1). This agrees with the results reported by Klapp and Patey for the dipolar fluid
[4] who based on their ROZ calculations concluded that the presence of positional
correlations tends to increase the critical temperature. Interestingly, the simulation
results also support this view. As a general observation, it can be mentioned that
the marked increase in T ∗
c when going from model C to D and, very especially, from
models A or E to model B, simply reflects the fact that the core size in models D
and B is smaller (cf. figure 1), by which particles can get in closer contact, thus
enhancing angular correlations and subsequently rising the critical temperature.
In summary, we have presented the results of the ROZ-HNC integral equation
for a variety of positionally frozen Heisenberg systems in order to assess its ability
to describe the critical behaviour. Although the theory yields reasonably accurate
results for the structure, it presents a severe deficiency: it systematically predicts a
558
ROZ equations for Heisenberg systems
substantial increase in the critical temperature due to the freezing of the particle
positions, in contrast to the evidence provided by the simulation. As a consequence,
all the theoretical critical temperatures are overestimated by far. On the other hand,
when considering just the positionally frozen systems, the ROZ theory predicts the
correct dependence of the critical temperature on the topology. Finally, both theory
and simulation seem to support the view that positional correlations tend to raise
the critical temperature.
4. Acknowledgement
Dr. N.G.Almarza is gratefully acknowledged for his assistance in the simula-
tion calculations. The authors acknowledge support from the Dirección General de
Investigación Cient́ıfica y Técnica under grant No. BFM2001-1017-C03-01.
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E.Lomba, C.Martı́n
Реплічні рівняння Орнштейна-Церніке для
позиційно заморожених гайзенбергівських систем
Е.Ломба, С.Мартін
Інститут фізичної хімії
Серрано 119, Е-28006 Мадрід, Іспанія
Отримано 28 березня 2003 р.
Ми формулюємо реплічні рівняння Орнштейна-Церніке для моделі
з позиційно замороженою невпорядкованою гайзенбергівською спі-
новою системою. Результати отримані для різних моделей, в одній з
яких положення частинки відповідає замороженому твердосферно-
му плину. Інша система, в якій конфігурація генерується випадковим
вміщенням твердих сфер, також система з випадково розподілени-
ми спінами і нарешті система, яка відповідає плину м’яких кульок,
розглянуті при високих і низьких температурах. Ми можемо бачити,
що орієнтаційна структура спінової системи досить добре відтворю-
ється методом інтегральних рівнянь, які, проте, не дають коректного
опису критичної поведінки.
Ключові слова: гайзенбергівська система, інтегральні рівняння,
феромагнетний перехід
PACS: 75.50.Lk, 61.20.Ja, 64.60.Fr
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