Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling
The magnetic properties of the two-dimensional S = 1/2 (quantum) antiferromagnetic Heisenberg model on a honeycomb lattice with and without interlayer coupling are studied by means of a continuous Euclidean time Quantum-Monte-Carlo algorithm. The internal energy, the magnetic susceptibility and th...
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Інститут фізики конденсованих систем НАН України
2009
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| Цитувати: | Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling / U. Löw // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 497-506. — Бібліогр.: 30 назв. — англ. |
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Löw, U. 2017-06-11T14:38:36Z 2017-06-11T14:38:36Z 2009 Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling / U. Löw // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 497-506. — Бібліогр.: 30 назв. — англ. 1607-324X PACS: 71.70.Ch, 61.72.Yx, 64.60.Cn DOI:10.5488/CMP.12.3.497 https://nasplib.isofts.kiev.ua/handle/123456789/120290 The magnetic properties of the two-dimensional S = 1/2 (quantum) antiferromagnetic Heisenberg model on a honeycomb lattice with and without interlayer coupling are studied by means of a continuous Euclidean time Quantum-Monte-Carlo algorithm. The internal energy, the magnetic susceptibility and the staggered magnetization are determined in the full temperature range. For the two-dimensional system the groundstate energy/bond is found to be Ehc 0 = -0:36303(13), and the zero temperature staggered magnetization mst = 0:2681(8). For coupled planes of honeycomb systems a phase transition from an ordered phase to a disordered phase is found at T/J = 0:695(10). Магнiтнi властивостi двовимiрної S = 1/2 (квантової) антиферомагнiтної моделi Гайзенберга на шестикутнiй ґратцi з мiжшаровою взаємодiєю i без мiжшарової взаємодiї дослiджено методом квантового Монте Карло алгоритму з неперервним евклiдовим часом. Внутрiшня енергiя, магнiтна сприйнятливiсть i стрибаюча (пiдґраткова) намагнiченiсть визначенi у повному температурному iнтервалi. Для двовимiрної системи енергiя основного стану на зв’язок рiвна Ehc 0 = -0:36303(13), а стрибаюча намагнiченiсть при нульовiй температурi є mst = 0:2681(8). Для взаємодiючих площин шестикутних систем фазовий перехiд мiж впорядкованою i невпорядкованою фазою вiдбувається при T/J = 0:695(10). It is a pleasure to acknowledge helpful discussions with B. L uthi, W. Weber and S. Bienas. Part of the data were produced on the parallel cluster of the RWTH Aachen Rechenzentrum and on the Jump in J ulich. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling Властивостi двовимiрної спiн-1/2 моделi Гайзенберга на шестикутнiй ґратцi з мiжшаровою взаємодiєю Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling |
| spellingShingle |
Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling Löw, U. |
| title_short |
Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling |
| title_full |
Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling |
| title_fullStr |
Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling |
| title_full_unstemmed |
Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling |
| title_sort |
properties of the two-dimensional spin-1/2 heisenberg model on a honeycomb lattice with interlayer coupling |
| author |
Löw, U. |
| author_facet |
Löw, U. |
| publishDate |
2009 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Властивостi двовимiрної спiн-1/2 моделi Гайзенберга на шестикутнiй ґратцi з мiжшаровою взаємодiєю |
| description |
The magnetic properties of the two-dimensional S = 1/2 (quantum) antiferromagnetic Heisenberg model on
a honeycomb lattice with and without interlayer coupling are studied by means of a continuous Euclidean
time Quantum-Monte-Carlo algorithm. The internal energy, the magnetic susceptibility and the staggered
magnetization are determined in the full temperature range. For the two-dimensional system the groundstate
energy/bond is found to be Ehc 0 = -0:36303(13), and the zero temperature staggered magnetization
mst = 0:2681(8). For coupled planes of honeycomb systems a phase transition from an ordered phase to a
disordered phase is found at T/J = 0:695(10).
Магнiтнi властивостi двовимiрної S = 1/2 (квантової) антиферомагнiтної моделi Гайзенберга на шестикутнiй ґратцi з мiжшаровою взаємодiєю i без мiжшарової взаємодiї дослiджено методом квантового Монте Карло алгоритму з неперервним евклiдовим часом. Внутрiшня енергiя, магнiтна сприйнятливiсть i стрибаюча (пiдґраткова) намагнiченiсть визначенi у повному температурному iнтервалi. Для двовимiрної системи енергiя основного стану на зв’язок рiвна Ehc 0 = -0:36303(13), а стрибаюча намагнiченiсть при нульовiй температурi є mst = 0:2681(8). Для взаємодiючих площин шестикутних систем фазовий перехiд мiж впорядкованою i невпорядкованою фазою вiдбувається при T/J = 0:695(10).
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120290 |
| citation_txt |
Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling / U. Löw // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 497-506. — Бібліогр.: 30 назв. — англ. |
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| first_indexed |
2025-11-26T22:50:51Z |
| last_indexed |
2025-11-26T22:50:51Z |
| _version_ |
1850779029673607168 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 3, pp. 497–506
Properties of the two-dimensional spin-1
2
Heisenberg
model on a honeycomb lattice with interlayer coupling
U. Löw
Theoretische Physik II, Technische Universität Dortmund, 44227 Dortmund, Germany
Received June 17, 2009, in final form July 17, 2009
The magnetic properties of the two-dimensional S = 1
2
(quantum) antiferromagnetic Heisenberg model on
a honeycomb lattice with and without interlayer coupling are studied by means of a continuous Euclidean
time Quantum-Monte-Carlo algorithm. The internal energy, the magnetic susceptibility and the staggered
magnetization are determined in the full temperature range. For the two-dimensional system the ground-
state energy/bond is found to be Ehc
0
= −0.36303(13), and the zero temperature staggered magnetization
mst = 0.2681(8). For coupled planes of honeycomb systems a phase transition from an ordered phase to a
disordered phase is found at T/J = 0.695(10).
Key words: three-dimensional quantum systems, phase transitions, Quantum-Monte-Carlo
PACS: 71.70.Ch, 61.72.Yx, 64.60.Cn
Introduction
Besides spin models on the two-dimensional square lattice and on the cubic lattice other lattice
topologies are interesting both due to their principal importance as quantum systems and because
of their phenomenological relevance.
In particular, the honeycomb lattice which is a two-dimensional bipartite lattice with mini-
mal coordination number has attracted continuous attention over the years. Early studies of its
finite temperature properties by means of high temperature expansions date back to 1973 [1,2].
A Quantum-Monte-Carlo study was performed by Reger, Riera and Young [3] in 1989 using the
discrete world line algorithm. Bishop and Rosenfeld [4] used a coupled cluster method to evaluate
the ground-state properties of the XXZ-model on the honeycomb lattice and compared with the
series expansions by Weihong, Oitmaa, and Hamer[5,6]. An elaborate study of the two-dimensional
honeycomb lattice in the presence of site-percolation can be found in [7].
Also, only recently InCu2/3V1/3O3 has been discussed as a possible candidate for a spin- 1
2
substance [8,9] on a honeycomb lattice. The advent of this and possible other upcoming new
substances as well as the availability of high precision Monte-Carlo methods [10,11] prompted
us to make a renewed study of spin systems on the honeycomb lattice. We study both the two-
dimensional honeycomb system focusing on issues which have not been treated in the literature
and on honeycomb planes coupled in the third dimension. To our knowledge this type of three-
dimensional system has not been studied thus far by quantum Monte-Carlo.
For our investigation the continuous time loop algorithm [11] was employed, which allows us
to study spin systems with unprecedented precision even at very low temperatures. The short
autocorrelation times of the method are achieved by nonlocal loop updates and no extrapolation
in Trotter number is needed, since the algorithm works with a continuous Trotter time variable.
Beard and Wiese [11] originally suggested this algorithm for the two-dimensional square lattices,
but it can also be used for three-dimensional systems and to some extent also to other lattice
topologies. Simulation of a honeycomb structure is achieved by “viewing” the honeycomb lattice
as a double dimerized model with Hamiltonian
H = Jx
∑
x, y, z
(
1 + (−1)x+yδ
)
~Sx,y,z
~Sx+1,y,z + Jy
∑
x,y,z
~Sx,y,z
~Sx,y+1,z + Jz
∑
x,y,z
~Sx,y,z
~Sx,y,z+1 (1)
c© U. Löw 497
U. Löw
and δ = 1, Jx = J
2 and Jy = J . I. e. the Hamiltonian equation (1) has different dimerizations in
x-direction for y even and y odd, where J is the exchange coupling of the honeycomb system. The
Hamiltonian equation (1) for one layer is depicted in figure 1. The coupling in z-direction between
the layers is Jz = J . Note that the cubic lattice in terms of equation (1) is readily given by δ = 0
and Jx = Jy = Jz = J .
Figure 1. Honeycomb lattice mapped onto a two-dimensional dimerized lattice.
It is well known that including such a dimerization-structure does not pose a problem to
the Monte Carlo-algorithm. Using dimerizations δ = ±1 to calculate ladder properties is a well
established technique. Also the data shown here for the one- and two-dimensional systems indicate
that breaking higher dimensional structures down to lower dimensions by putting bonds to zero
works perfectly with this algorithm.
The paper is organized as follows. In section 1 the ground-state energy and the internal energy
of the honeycomb lattices are studied as well as the presence of a phase transition in the three-
dimensional system is demonstrated. Section 2 is devoted to the susceptibility and section 3 to the
staggered magnetization.
1. The internal energy
The ground-state energy of the one-dimensional Heisenberg model is exactly known since the
seminal works of H. Bethe [12,13] and L.Hulthén [14]. However, there is no counterpart of exact
results for higher-dimensional versions of the model. Therefore, over the past years a substantial
effort has been made both with approximate analytical and numerical techniques to obtain a value
of the ground-state energy of the two- and three-dimensional Heisenberg models.
At low temperatures for the one-dimensional model, Bethe ansatz [18] and conformal theory
[19] predict a quadratic dependence of the internal energy U(T ) on temperature. Whereas for
two-(three)-dimensional models, U(T ) depends in the leading order spin wave theory on the third
(fourth) power of temperature [20,21]. For models with linear spin wave dispersion, this behavior
does not depend on the details of the coupling, but only on the dimension of the system.
To compare with the above predictions and to determine the coefficients of the leading power
laws, low temperature data for the internal energy were calculated both for a honeycomb lattice,
and a honeycomb system with three-dimensional coupling Jz = J . For reasons of reference the
data for a chain, a square and a cubic lattice are also presented.
The coefficients for the one-dimensional system follow from the analysis of [18] and [19]. For
the square lattice they were determined by Takahashi [22] using modified spin-wave theory and by
Avoras and Auerbach using Schwinger boson mean-field theory [23]. Our fits to the one-dimensional
case are in very good agreement with these predictions. Also for a 32×32 square lattice a fit to the
data in the temperature range 0.005 6 T/J 6 1.1 reproduces the results of Takahashi [22] with
good accuracy. For the honeycomb lattice we find (lower inset in 2) for a 32× 64 lattice and a fit
498
Properties of the two-dimensional spin- 1
2
Heisenberg model on a honeycomb lattice with interlayer coupling
range (T/J 6 0.8)
Uhex/J = −0.363 + 0.799(T/J)3 (2)
which corroborates the universal results of Oguchi and Kubo, [21,20] of a T 3 behavior, however
the validity is confined to smaller temperatures as compared to the square lattice.
For the honeycomb lattice with interlayer-coupling we find a best fit with a somewhat higher
exponent of the temperature,
U3dim
hc /J = −0.3124 + 0.325 (T/J)4.8 (3)
hinting at higher order contributions in spin wave theory. Forcing a T 4 fit gives
U3dim
hc /J = −0.3127 + 0.186 (T/J)4. (4)
For a graphical depiction of the results see inset of figure 2.
Figure 2. Monte-Carlo results for the internal energy U/J per bond as a function of temperature,
for a 400 site chain (circles), a 32 × 32 square lattice (squares), a honeycomb lattice of size
32 × 64 (diamonds), a honeycomb lattice with three-dimensional coupling (triangles), and a
cubic system of size 163 (crosses). Upper inset: Fit (equation (3)) to low temperature behavior
of the three-dimensional system(red), and fit (equation (4)) using a T 4 law (green). Lower inset:
Low temperature behavior of the internal energy for the honeycomb lattice equation (2).
Next, the internal energy and the specific heat of the honeycomb lattice with three-dimensional
coupling in an intermediate temperature range are discussed. Analogous to the Heisenberg model on
a three-dimensional cubic lattice, one expects that the three-dimensionally coupled honeycomb lat-
tices undergo a second-order phase transition at a finite temperature Tc. Since rigorously speaking
phase transitions can be seen only in infinite size systems, in numeric calculations involving systems
of limited size one is in general restricted to the study of size dependent quantities, e.g. the increase
of thermodynamic quantities with growing system size or scaling laws, to show the presence of the
singularity connected to the phase transition.
Here we estimate the critical temperature of the coupled honeycomb planes by analyzing the
specific heat. As can be seen in figure 3 the height of the peak of the specific heat grows with
the system size in a temperature range, where one does not expect finite size effects. (Note that
for not too low temperatures and T < 0.6J and also for T > J the curves of the internal energy
U(T ) for different system sizes fall together.) Thus, the unconventional finite size behavior can
be interpreted as a manifestation of a phase transition. Also, the location of the maximum of the
specific heat is shifted to lower temperatures with growing system size. From this behavior we
499
U. Löw
estimate (see inset in figure 3) that the transition is located at Tc/J = 0.695(10). The temperature
Tc corresponds approximately to the steep decay of the staggered magnetization analyzed in the
section 3, indicating that the transition is between an ordered phase at low temperatures and a
magnetically disordered high-temperature phase with vanishing staggered magnetization.
Figure 3. Monte-Carlo results for the internal energy U(T ) per bond of coupled honeycomb
planes as a function of temperature, for systems of different size. Upper inset: Specific heat of
the same systems. Lower inset: Location of the peak as a function of 1/N with N = 6, 8, 10, 12, 16.
Figure 4. Linear extrapolation of the internal energy U(T = 0.005J) per bond as a function of
1
N3 forN × N -lattice with N = 16 . . . 36.
As expected, the high temperature behavior of the three curves approaches the same value
E∞ = 0. Since we considered the energy per bond, the fact that the internal energy for the
honeycomb lattice lies between the results for the square lattice and the chain is an effect of the
quantum fluctuations independent of coordination number.
To obtain the value of the ground-state energy of the honeycomb lattice for the infinite size
system the energies U(T = 0.005J) (see figure 4) are extrapolated for system sizes N = 16 . . . 36.
Taking into account the previous result of equation (2) one can expect that the error due to the
finite temperature should be ≈ (0.005)3 which is smaller than the statistical error.
500
Properties of the two-dimensional spin- 1
2
Heisenberg model on a honeycomb lattice with interlayer coupling
As in the case of the square lattice the ground-state energy scales in leading order with 1
N3 .
For a N ×N lattice with N = 16 . . . 36 a linear fit gives Ehc
0 = −0.363035(13) for the ground-state
energy. This agrees with the value Ehc
0 = −0.3630(3) of [3]. From the nonlinear σ model description
of the two-dimensional Heisenberg model [24–26] it is known that for the square lattice, the next
order of the finite size behavior is given by a 1/N 4 correction. However, a significant deviation
from the 1/N3 law cannot be extracted from the data.
As a technical point we remark that for this analysis a rectangular lattice with N × 2N sites
has been chosen, which amounts to an equal number of unit cells of the honeycomb lattice in
x and y-direction. We found, however, that extrapolated quantities do not depend on the shape
of the system. This is demonstrated in some detail in case of the staggered magnetization, a
quantity which is particularly sensitive to the size of the system, because it is closely related to
the correlations.
2. Susceptibility
In this section the uniform susceptibility is discussed. Figure 5 displays an overview of the
results for the chain, the square, the honeycomb lattice and the three-dimensional systems. The
susceptibility of the honeycomb lattice reaches its maximum at Tmax ≈ 0.72J . Both low and high
temperature series expansions fail to reach this value and earlier Monte-Carlo simulations did not
concentrate on this point. Note that data for the two-dimensional honeycomb-model were already
used in [8] to confirm that the magnetically active ions of InCu2/3V1/3O3 form a honeycomb-
structure.
Figure 5. Susceptibility of the Heisenberg model on a honeycomb lattice and a honeycomb
lattice with interlayer coupling Jz = J compared to the susceptibility of a square lattice, a one-
dimensional chain with and a cubic system. The dot dashed line shows the high-temperature-
series expansion of [1,2] for the honeycomb lattice.
For a comparison with experimental data the most prominent feature of a susceptibility curve
is its maximum, which most often serves as a first estimate of the exchange coupling. It is, there-
fore, certainly of phenomenological interest, that Tmax for the honeycomb lattice lies close to
the maximum of the susceptibility of the chain at Tmax ≈ 0.64J [27], but it is well distinguish-
able from the square lattice whose maximum is at Tmax ≈ 0.94J . As for the honeycomb lattice
with three-dimensional coupling the maximum is located at Tmax ≈ 1.03J . For the cubic system
Tmax ≈ 1.22(2)J . Thus, inter-layer couplings shift the maximum of the susceptibility to higher
temperatures. Also the slow ascent of the curves is typical of three-dimensional systems.
501
U. Löw
Next we study the low temperature behavior of the susceptibility of the two-dimensional hon-
eycomb lattice, which has been discussed extensively by spin wave theory and series expansions
[5,6]. In contrast to the susceptibility of the one-dimensional Heisenberg model, which approaches
its zero temperature value Jχ(T = 0) = 0.101322 [28] with infinite slope for two-dimensional
lattices spin-wave theory predicts a linear behavior of χ(T ) at low temperatures. We clearly see
this linear behavior in our system (see figure 6). Note that the exponential drop of χ(T ) is a finite
size effect, originating from the energy gap of the finite systems. But in the temperature range
T/J > 0.04, where the results fall both for the 32 × 32 and the 24 × 24 size system, the curves
definitely represent the thermodynamic limit. A linear fit to the data for 0.04 6 T/J 6 0.15 gives
Jχ(T = 0) = 0.05188(8), which agrees well with the result of series expansion [6], whereas spin-
wave theory in the first order gives Jχ(T = 0) = 0.111 and thus lies out of the scale of figure 6.
Second order spin-wave theory is ≈ 40% too small.
Figure 6. The susceptibility of the honeycomb lattice at low temperatures for three system sizes.
The solid line is a linear fit to the Monte-Carlo data of the 32 × 32 lattice in the range between
T = 0.04J and T = 0.15J . The star marks the series expansion for χ(T = 0) from [6], the
triangle the result of second order spin-wave [5].
The failure of spin-wave theory for the honeycomb lattice can be intuitively understood by the
fact that quantum fluctuations are larger for the honeycomb than for the square lattice.
3. The staggered magnetization
It is rigorously known [29] that for the two-dimensional model the thermal expectation value
of the staggered magnetization operator
~Mst =
1
N2
∑
〈x,y〉
(−1)(x+y)~Sx,y (5)
vanishes at finite temperatures. But also the zero-temperature magnetization 〈0| ~Mst|0〉, where |0〉
is the ground-state of the finite system, does vanish, since Mst is an operator with momentum
~q = (π, π) whose matrix elements for all non-degenerate states with fixed momentum are zero. It
is thus not obvious how to conclude on the existence of a non-vanishing staggered magnetization
from finite size systems, working at low, but nonzero temperatures.
502
Properties of the two-dimensional spin- 1
2
Heisenberg model on a honeycomb lattice with interlayer coupling
Instead, it is common use to calculate at the lowest possible temperatures the expectation value
of the square of ~Mst
(mst)
2 = 〈0| ~M2
st|0〉 (6)
which is proportional to the static structure factor at ~q = (π, π). Calculations of this type have
a long history [15] for the two-dimensional Heisenberg model, yielding limN→∞ mst = 0.3070(3)
[16,30] for the square lattice.
Figure 7. The staggered magnetization (mst)
2 for square systems of size N × N (squares) and
rectangular system of size N/2×N (diamonds) and the correlations function at maximal distance
C(N
2
, N
2
) (circles) of the honeycomb lattice at T/J = 0.01 as a function of 1/N . The insets show
the temperature dependence of (mst)
2 (left figure) and C(N/2, N/2) (right figure) for N = 16
(triangles), N = 24 (diamonds) and N = 32 (squares).
The finite size effects of the staggered magnetization are very large as compared to, e.g. finite
size effects of the ground-state energy. However (mst)
2 shows an almost perfect linear behavior in
1/N (see figure 7) with some small correction proportional to 1/N 2, which allows to extrapolate
with good accuracy.
We thus obtain for the two-dimensional system
lim
N→∞
mst = 0.2681(8) (7)
as result for the infinite size system. Figure 7 also shows data points for rectangular systems, which
show a slightly different slope and extrapolate to mst = 0.2671(10).
Another way to proceed is to calculate the correlation at maximal distance
C (N/2, N/2) := 〈0|~S(0, 0), ~S (N/2, N/2)|0〉
= 3〈0|Sz(0, 0), Sz (N/2, N/2)|0〉 (8)
on the periodic lattice [3] and extrapolate it to N → ∞. From this fit one concludes that
lim
N→∞
√
C = 0.2683(6) (9)
which is within the error equal to mst.
The results are compatible with the previous mst = 0.22(3) of [3] and in excellent agreement
with [7] who find mst = 0.2677(6) by means of the stochastic series expansion Monte-Carlo. Again
the error due to finite temperature effects is smaller than the statistical error (see insets of figure 7).
503
U. Löw
Figure 8. Staggered magnetization of the honeycomb lattice with interplane coupling Jz = J
for different system sizes. Lower inset: Staggered magnetization for T = 0.9J (circles), T = J
(squares), T = 1.1J (triangles) as a function of N−
3
2 for N = 6 . . . 16. The dotted lines are linear
fits. Upper inset: Staggered magnetization for T = 0.2J (circles), T = 0.4J (squares), T = 0.6J
(diamonds) as a function of 1
N
for N = 6 . . . 16. The dotted lines are three parameter fits of the
form aN b + c.
Thus, the honeycomb lattice displays a reduced magnetic order as compared to the square
lattice. However, the Néel-type structure is not destroyed by quantum fluctuations.
The staggered magnetization mst for a honeycomb system with three-dimensional coupling is
shown in figure 8. In this case one observes a sudden decrease of mst with temperature, which is
a sign of the phase transition observed in section 1. For an increasing system size, the slope of
mst increases. To corroborate that the peak found in the specific heat corresponds to a transition
between an ordered phase at T < Tc and a disordered phase with vanishing staggered magnetization
at T > Tc we extrapolated the data for mst for system size N × 2N × N , as a function of 1
N . For
T > Tc we find a very convincing N− 3
2 law extrapolating to mst = 0 for N → ∞. The lower inset
in figure 8 shows a linear fit of mst for T/J = 0.9, 1.0, 1.1. The linear decrease to zero as a function
of N− 3
2 is almost perfect.
The finite size dependence for T < Tc is shown in the upper inset, where the data for T/J =
0.2, 0.4, 0.6 and a fit to the data with fit function aN b + c is presented. For all three temperatures,
mst approaches a finite value. Since, however, the steepest drop in mst depends on the system size,
this type of extrapolations can only be performed not too close to the critical temperature.
The above analysis was performed to demonstrate the existence of the two phases. With vani-
shing and non-vanishing staggered magnetization. It cannot be used as a reliable method to deter-
mine a critical exponent. We wish to postpone an analysis of critical exponents to a future project
beyond the scope of this work, which includes an examination of spin-spin-correlation functions. A
study of this type will hopefully give further insight in the interesting physics of three-dimensional
quantum spin systems. Also we hope, that analysing systems with variable inter-plane couplings
and anisotropies will be of experimental interest for S = 1
2 systems with honeycomb lattice struc-
ture, like InCu2/3V1/3O3.
Acknowledgements
It is a pleasure to acknowledge helpful discussions with B. Lüthi, W. Weber and S. Bienas.
Part of the data were produced on the parallel cluster of the RWTH Aachen Rechenzentrum and
on the Jump in Jülich.
504
Properties of the two-dimensional spin- 1
2
Heisenberg model on a honeycomb lattice with interlayer coupling
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505
U. Löw
Властивостi двовимiрної спiн-1/2 моделi Гайзенберга на
шестикутнiй ґратцi з мiжшаровою взаємодiєю
У. Льов
Теоретична фiзика II, Технiчний унiверситет Дортмунда, 44227 Дортмунд, Нiмеччина
Отримано 17 червня 2009 р., в остаточному виглядi – 17 липня 2009 р.
Магнiтнi властивостi двовимiрної S = 1/2 (квантової) антиферомагнiтної моделi Гайзенберга на
шестикутнiй ґратцi з мiжшаровою взаємодiєю i без мiжшарової взаємодiї дослiджено методом
квантового Монте Карло алгоритму з неперервним евклiдовим часом. Внутрiшня енергiя, магнiт-
на сприйнятливiсть i стрибаюча (пiдґраткова) намагнiченiсть визначенi у повному температурному
iнтервалi. Для двовимiрної системи енергiя основного стану на зв’язок рiвна Ehc
0
= −0.36303(13),
а стрибаюча намагнiченiсть при нульовiй температурi є mst = 0.2681(8). Для взаємодiючих площин
шестикутних систем фазовий перехiд мiж впорядкованою i невпорядкованою фазою вiдбувається
при T/J = 0.695(10).
Ключовi слова: тривимiрнi квантовi системи, фазовi переходи, квантове Монте Карло
PACS: 71.70.Ch, 61.72.Yx, 64.60.Cn
506
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