Green Function theory vs. Quantum Monte Carlo Calculation for thin magnetic films
S. Henning,∗ F. Körmann, J. Kienert, and W. Nolting
Lehrstuhl Festkörpertheorie, Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany
S. Schwieger
Technische Universität Ilmenau, Theoretische Physik I, Postfach 10 05 65, 98684 Ilmenau, Germany
(Dated: October 31, 2018)
In this work we compare numerically exact Quantum Monte Carlo (QMC) calculations and Green
function theory (GFT) calculations of thin ferromagnetic films including second order anisotropies.
Thereby we concentrate on easy plane systems, i.e. systems for which the anisotropy favors a
magnetization parallel to the film plane. We discuss these systems in perpendicular external field, i.e.
B parallel to the film normal. GFT results are in good agreement with QMC for high enough fields
and temperatures. Below a critical field or a critical temperature no collinear stable magnetization
exists in GFT. On the other hand QMC gives finite magnetization even below those critical values.
This indicates that there occurs a transition from non-collinear to collinear configurations with
increasing field or temperature. For slightly tilted external fields a rotation of magnetization from
out-of-plane to in-plane orientation is found with decreasing temperature.
PACS numbers: 75.10.Jm, 75.40.Mg, 75.70.Ak, 75.30.Gw
I. INTRODUCTION
The fast development of technological applications
based on magnetic systems in the last years, e.g.
magnetic data storage devices, causes a high interest
in thin magnetic films. One precondition for the tech-
nological development is the investigation of magnetic
anisotropies and spin reorientation transitions connected
therewith. Those reorientation transitions can occur
from out-of-plane to in-plane or vice versa for increasing
film thickness d1, temperature T 2,3,4,5,6,7,8, or external
field B0.
Quantum Monte Carlo (QMC) calculations give the
possibility to compare numerically exact results with
analytical approximations. In Ref. 9 the authors inves-
tigated a ferromagnetic monolayer including positive
second order anisotropy (easy axis perpendicular to the
film plane). They discuss the temperature dependence
of the magnetization 〈Sz〉(T ) as well as field induced
reorientation transitions from out-of-plane to in-plane
and compare the QMC results with Green function
theory (GFT). They found good agreement in the case
of applied external field in the easy direction (here
z-axis). However, their GFT fails for external field
applied in arbitrary direction, especially in the hard
direction (within the film plane). As shown in Ref. 10
for getting closer to the QMC results for magnetic field
induced reorientation from out-of-plane to in-plane a
more careful treatment of the local anisotropy terms is
needed. In Refs. 10,11,12,13 a decoupling scheme was
presented which yields excellent agreement with QMC
results for out-of-plane systems.
The availability of theories such as GFT and their
check against state-of-the-art numerical algorithms
is highly desirable because of the size limitations of
systems where QMC can be performed. On the other
hand the extension of GFT from a monolayer (where it
can be compared to QMC as in the the present work)
to multilayer systems is a straightforward task without
further approximations11.
Up to now, to our knowledge, there is no comparison
between QMC and approximative theories for easy-plane
systems and it is not obvious that the theory presented
in Refs. 10,11,12,13 can reproduce the QMC results for
in-plane systems as accurately as for the out-of-plane
case. In contrast to the easy-axis case where a certain
direction is preferred by the single ion anisotropy in
easy-plane systems the full xy-plane is favored and no
particular direction is distinguished within the plane. A
magnetic field applied perpendicular to the plane does
not destroy the xy-symmetry.
For systems exhibiting this kind of symmetry it was
shown in a classical treatment that for external fields
smaller than a critical field 0 ≤ B < Bcrit (B || z)
stable vortices, i.e. a non-collinear arrangement of
spins, can exist15,16,17,18,19. These vortices can undergo
a Berezinskii-Kosterlitz-Thouless (BKT) transition14.
Depending on the strength of the anisotropy K2 there
might be vortices with or without a finite z-component
of magnetization15. In the small anisotropy case (which
is considered in this work, |K2| < 0.1J) there is a
finite out-of-plane component and for zero field the
two possible directions of magnetization (±z) are en-
ergetically degenerate. For increasing magnetic field in
z-direction the vortices antiparallel to the field become
more and more unstable (heavy vortices). However the
so called light vortices (parallel to the field) are stable
up to a critical field Bz = Bcrit and contribute a finite
z-component to the net-magnetization of the considered
system19.
The vortices in connection with a finite z-component
of the net-magnetization emerge because of two reasons:
first the competition between the anisotropy (favoring
a orientation of the magnetization within the xy-plane)
http://arxiv.org/abs/0704.1552v1
and the external field (favoring a perpendicular magne-
tization), and second: the xy-symmetry of the system,
which does not allow for a rotated homogeneous phase.
In this paper we investigate both aspects, i.e. the
field vs. anisotropy competition as well as the symmetry
properties in detail for a quantum mechanical system.
We will compare the results of QMC and GFT calcula-
tions.
As explained in more detail below, the QMC al-
gorithm used here allows only for an external field
applied in z-direction. Thus the xy-symmetry can not
be broken and no comparison between xy-symmetric
and asymmetric systems is possible. We will use GFT
to clarify the influence of this symmetry breaking on
the homogeneous phase. On the other hand, the GFT
used here is by ansatz limited to the homogeneous
phase. Therefore it can not describe a non-collinear
(e.g, vortex-) magnetic phase, which is expected for
B || z and small field strengths. The breakdown of
magnetization in GFT as well as an exposed maximum
in the magnetization in QMC at certain critical values
of the external field or temperature gives, however, a
clear fingerprint of non-collinear configurations, at least
if there is no meta-stable homogeneous phase. Below
these critical values there will be a finite z-component
in QMC and a vanishing magnetization in GFT.
For parameters, where both theories are applicable,
QMC serves as a test for the approximations needed in
In this work we find indications for non-collinear
spin configurations below a critical field or temperature
for B || z by comparing results of QMC and GFT as
explained in the last clause. Above the critical field
we obtain good agreement between QMC and GFT
results. Breaking the xy-symmetry by adding a small
x-component to the external field yields a stable collinear
solution in GFT. The z-component of the magnetization
in this case is in good agreement with the QMC results
calculated with untilted field. Thus we can conclude
that except for the restriction to collinear magnetic
states GFT describes the competition between external
field and anisotropy quite well.
The paper is organized as follows: First we ex-
plain the basics of the GFT and the QMC calculations.
Then we apply both approaches to easy-plane systems
in external magnetic fields and report the results of our
calculations.
II. THEORY
A. Green Function Theory
In the following we present our theoretical approach us-
ing Green function theory. The focus of this work lies on
the translational invariant system of a two-dimensional
monolayer. Therefore the following Hamiltonian is used:
H = −
JijSiSj −B
Si −K2
(Sz,i)
2. (1)
The first term describes the Heisenberg coupling Jij be-
tween spins Si and Sj located at sites i and j. The second
term contains an external magnetic field B in arbitrary
direction (the Landé factor gJ and the Bohr magneton
µB are absorbed in B). The third term represents second
order lattice anisotropy due to spin-orbit coupling. Sz,i
is the z-component of Si (the z-axis of the coordinate
system is oriented perpendicular to the film-plane). The
lattice anisotropy favours in-plane (K2 < 0) or out-of-
plane (K2 > 0) orientation. Our Hamiltonian is similar
to that used in Refs. 10,11,13,22,23 for the investigation
of the magnetic anisotropy and the field induced reori-
entation transition. To simplify calculations we consider
nearest neighbor coupling only
Jij =
J (i), (j) n.n.
0 otherwise.
The main idea of the special treatment presented in
Refs. 10,11,12,13 is that, before any decoupling is applied,
the coordinate system Σ is rotated to a new system Σ′
where the new z′-axis is parallel to the magnetization im-
plying a collinear alignment of all spins within the layer.
Then a combination of Random Phase approximation
(RPA)24 for the nonlocal terms in Eq. (1) (Heisenberg
exchange interaction term) and Anderson-Callen approx-
imation (AC)25 for the local lattice anisotropy term is
applied in the rotated system. After application of the
approximation one gets an effective anisotropy
Keff (T ) = 2K2
S(S + 1)− 〈S2z′〉
〈Sz′〉 (3)
where 〈Sz′〉 is the norm of the magnetization and S is the
spin quantum number, that we have chosen to be S = 1
in all our calculations.
As shown in comparison with an exact treatment of the
local anisotropy term in Ref. 26 this approximation still
holds up to anisotropy strengths K2 ∼ 1/2J . Therefore
we restrict ourselves in the following to small anisotropies
(K2 ≤ 0.1J) as found in most real materials
33. For a
magnetic field applied in the xz-plane (B = (Bx, 0, Bz))
our theory gives a condition for the polar angle θ of the
magnetization:
sin θBz − cos θBx +Keff sin θ cos θ = 0 (4)
The uniform magnon energies (q = 0) which dominate
the physical behavior of the magnetic system can easily
be extracted from the theory12,13:
E2q=0 =
cos θBz + sin θBx +Keff(cos
2 θ − sin2 θ)
cos θBz + sin θBx +Keff cos
This result coincides with the spin-wave result13 if one
replaces 〈Sz′〉 by the spin quantum number S and Keff
by the bare anisotropy constant K2 in Eq. (5). For an
easy-plane system (Keff < 0) with external field B in
z-direction the polar angle θ of the magnetization34 is
given by:
cos θ =
−B/Keff(T ) for B < |Keff (T )|
1 otherwise
By inserting Eq.(6) into Eq.(5) one immediately gets:
Keff<0
q=0 (B) =
0 B < |Keff (T )|
B +Keff (T ) otherwise.
For gapless magnon energies Eq=0 = 0 the magnon oc-
cupation number φ diverges (φ → ∞) in film systems
with ferromagnetic coupling J > 0 and the magnetiza-
tion becomes zero 〈Sz′〉 = 0 in the collinear phase. This
can be seen by following an argument of Bloch20 already
given in 1930. Since the spin wave dispersion is E ≈ q2
in the vicinity of q = 0 the spin-wave density of states
N(E) is independent of E for a two-dimensional system
for E close to zero. The excitation of spin-waves at finite
temperature leads to a variation of the magnetization of
the order:
∆m(T ) ∼
N(E)dE
exp(E/kBT )− 1)
∼ kBT
exp(x)− 1
. (8)
Since the integral in Eq. (8) diverges for T 6= 0 and exited
spin-waves lead to a reduction of the magnetization one
can conclude that the magnetization should be zero at
finite temperature. However for an infinitesimally small
contribution of the external field parallel to the plane,
i.e. Bx 6= 0, a finite gap in the excitation spectrum
at q = 0 opens. This can be seen in Fig.1 where the
uniform magnon modes Eq=0(B) are shown for different
orientations θB, where θB is the polar angle of the ex-
ternal field. The integral (8) is now finite and a stable
finite magnetization in the collinear phase having a well
defined orientation in the xz-plane is possible.
Let us now come back to the case where the applied
field is aligned in z-direction. It can be seen from Eq.
(7) that for external field B (B || z) larger than a critical
field B > Bcrit given by:
Bcrit
= |Keff (T,B)| (9)
a stable collinear solution exists. Since Keff (T ) is a de-
creasing function of temperature T a transition from non-
collinear to collinear phase with increasing temperature
is possible. In Fig. 2 we show the normalized critical
field (9) Bcrit/K2 as a function of temperature T . For
a constant magnetic field B (B || z) at a temperature
T1 with B < Keff (T1, B) no stable collinear phase ex-
ist. Then by increasing the temperature up to T2 the ef-
fective anisotropy Keff is sufficiently reduced such that
0.01 0.02 0.03 0.04
B / J
=0.2°
FIG. 1: The energies of the uniform magnon mode Eq=0(B)
for different polar angles θB of the external field. Eq=0 is
zero below B/J ≈ 0.03 for θB = 0
◦. The prefactors gJµB
and kB are absorbed in B and T respectively. The latter are
given in units of the nearest neighbor Heisenberg coupling J .
Parameters: S = 1, K/J = −0.03 and T/J = 10−4.
FIG. 2: The normalized critical field Bcrit/K2 as a function
of temperature. Parameters: S = 1.
B > Keff (T2, B), and the collinear phase becomes sta-
ble. Before we come to the results let us briefly sketch
the main aspects of the QMC.
B. QMC
In the last section we gave a short description of the
theory used to treat a system described by a Hamilto-
nian of form (1). This theory applies to the thermody-
namic limit (films of infinte size) but contains certain
approximations. Additionally the GFT is restricted to
ordered phases with a collinear alignment of all spins.
Therefore it would be very useful to have exact results at
hand to crosscheck the predictions of GFT. A Quantum
Monte Carlo method, particularly well suited for spin
systems, is the stochastic series expansion (SSE) with
directed loop update. We will sketch this method here
only briefly as detailed descriptions can be already found
elsewhere28,29,30.
Our starting point is the series expansion of the parti-
tion function
Z = Tre−βH =
〈α|(−H)n|α〉 (10)
whereH denotes the Hamiltonian, {|α〉} are basis vectors
of a proper Hilbert space and β is the inverse tempera-
ture. The Hamiltonian is then rewritten in terms of bond
Hamiltonians:
H = −J
Hb (11)
where Hb can be further decomposed into a diagonal and
an off-diagonal part:
HD,b = C + S
i(b)S
j(b) + bb[S
i(b) + S
j(b)] (12)
+k2b[(S
i(b))
2 + (Szj(b))
HO,b =
] (13)
Here we have renormalized the anisotropy constant k2b
and the magnetic field bb in such a way that (11) coincides
with (1). i(b) and j(b) denotes the lattice sites connected
by the bond b and the additional constant C in HD,b will
be chosen such that all matrix elements of this term be-
come positive, a condition necessary to interpret them as
probabilities. Note that for a finite system at finite tem-
perature the power series of the partition function can
be truncated at a finite cutoff length Λ without intro-
ducing any systematic error in practical computations29.
Therefore reinserting (11) into (10) and rewriting the re-
sult yields:
βn(Λ− n)!
〈α|SCΛ |α〉. (14)
Here SCΛ denotes a product of operators (operator
string) consisting of n non-unity operators and (Λ − n)
unity operators H0 = Id which were inserted to get op-
erator strings of equal length Λ.
In fact it is impossible to evaluate all operator strings in
(14). The SSE-QMC replaces such an evaluation there-
fore by importance sampling over the strings according to
their relative weight. Hence an efficient scheme for gen-
erating new operator strings is needed. In the directed
loop version of the SSE this is done by dividing the up-
date into two parts. In a first step a diagonal update is
performed by traversing the operator string and replac-
ing some unity operators by diagonal bond operators and
vice versa (the probabilities for both substitutions have
to fulfill the detailed balance criterion). Then the loop
update follows in which new non-diagonal bond opera-
tors can appear in the operator string. For details of the
update procedure we refer the interested reader to the
0 0.5 1 1.5 2
GFT (RPA+AC)
QMC (N=16)
QMC (N=32)
QMC (N=64)
FIG. 3: Magnetization vs. temperature for an out-of-plane
easy-axis system (K2 > 0). Straight line: GFT (RPA+AC)
result; symbols: QMC results for different system sizes N2.
Parameters: S = 1, B/J = 0.01 (B || z) and K2/J = 0.01.
according literature28,29,30.
A full implementation of the SSE with directed loop
update which we have used for all QMC calculations
in this work can be found in the ALPS project30,31.
Since the SSE-QMC used by us is implemented in z-
representation (spin quantization axis along z-axis) in-
plane correlation functions e.g. the in-plane magnetiza-
tion are not accessible. Further B || z is the only possible
field direction in the used QMC implementation because
a traverse field (in-plane field component) would lead to
non-closing loops (see Ref. 9).
III. RESULTS
As mentioned in Sec. II A the results for the in-plane
systems are very sensitive to the effective anisotropy
Keff (T ). This sensitivity of the anisotropy is less pro-
nounced for out-of-plane systems (K2 > 0) since the ap-
plied field B (B || z) and the intrinsic easy axis are par-
allel. In order to test our decoupling scheme (RPA+AC)
we first compare GFT and QMC for an out-of-plane
system.35
In Fig. 3 the magnetization 〈Sz〉 as a function of tem-
perature T is shown. The straight line belongs to the
GFT whereas the symbols show the result of the QMC
for different system sizes. Let us first comment on finite
size effects in the QMC results.
It can be seen in Fig. 3 that the QMC results converge
for increasing system size N2 (for N ×N square lattice).
Indeed forN ≥ 32 the QMC results are unbiased by finite
size effects and resulting magnetization curves are almost
equal for increasing N ≥ 32. Note that we have omitted
error bars in the figures showing QMC results because
the relative errors are of the order 10−4.
We now compare the GFT with the QMC results
(N = 64). For low temperatures (T/J ≤ 0.5) we ob-
tain excellent quantitative agreement. This is plausible
0 0.05 0.1 0.15 0.2
QMC (16)
QMC (32)
QMC (64)
QMC (128)
FIG. 4: z-component of magnetization as a function of ex-
ternal magnetic field for fixed temperature T/J = 0.4. In
contrary to the GFT the magnetization obtained by QMC
remains finite for all fields. The QMC results are converged
for N ≥ 64. Parameters: S = 1, K2/J = −0.06 and θB = 0
0 0.05 0.1 0.15 0.2
GFT (θ
 = 0°)
GFT (θ
 = 0.5°) 
QMC (128)
FIG. 5: z-component of magnetization vs. external field for
T/J = 0.4 with slightly tilted field (θB = 0.5
◦)in the GFT
result (solid line). The dotted line shows GFT result for (θB =
0◦). Other parameters as in Fig. 4.
because in this region the GFT result coincides with the
result of the spin-wave theory which is known to be re-
liable (exact for T = 0) for low temperatures. For the
intermediate region T/J = 0.5..1 the RPA slightly un-
derestimates the magnetization which was also found in
Ref. 9. The opposite is the case in the region near the
extrapolated Curie temperature TC
36, where the magne-
tization is overestimated. The reason is the presence of
longitudinal fluctuations, which play an important role
in this region and it is well known that the RPA fails to
treat them properly.
We consider now the case of in-plane systems (K2 < 0)
and applied field in the hard direction (B || z). As al-
ready mentioned there is no ’collinear’ magnetization in
the GFT for Bz < |Keff (T )|. In Fig. 4 the z-component
of the magnetization is shown as a function of the ex-
ternal field B for a constant temperature T/J = 0.4.
0 0.05 0.1 0.15
T/J = 0.4
T/J = 0.6
T/J = 0.9
T/J = 1.2
T/J = 1.4 
FIG. 6: z-component of magnetization vs. external field for
different temperatures T/J and fixed system size N2 (N =
128). Solid lines: GFT (θB = 0.5
◦), dashed lines: GFT (θB =
0◦) other parameters as in Fig. 4.
As in Fig. 3 we see that the QMC results for N ≥ 64
are almost converged and the finite size of the calculated
system in QMC should not influence the results anymore.
The dotted line marks a critical field Bcrit. For magnetic
fields larger than the critical one B > Bcrit we obtain
good agreement between QMC and GFT results. Below
the critical field B < Bcrit GFT does not yield a stable
homogeneous magnetization. However the QMC results
show that there is a finite z-component of the magneti-
zation in the considered system for 0 ≤ B ≤ Bcrit.
In order to compare QMC with GFT results we have
tilted the magnetic field by θB = 0.5
◦ which corresponds
to Bx < 10
−2Bz in the GFT. As explained before any
symmetry breaking field Bx 6= 0 leads to a stable homo-
geneous magnetization with well-defined orientation in
the xz-plane. However such a small contribution of the
external field within the plane should hardly influence the
z-component of the magnetization. This is confirmed by
Fig. 5 where we show QMC results (N = 128, θB = 0
as well as the corresponding GFT results with θB = 0
and θB = 0.5
◦. As expected for |B| > Bcrit the two so-
lutions in the GFT are nearly the same and agree well
with QMC. Below the critical field only the solution with
the slightly tilted field yields a stable homogeneous mag-
netization and its z-component compares well with the
QMC result in the untilted case.
The above results can be interpreted within a semi-
classical picture of non-collinear vortex configurations
which are stable below a critical field Bcrit in z-direction
and contribute a finite z-component to the magnetiza-
tion in case of an applied field.19 Despite the lack of di-
rect, quantitative access to such states (or correspond-
ing physical in-plane observables) within the QMC al-
gorithm they are included in principle and one can ob-
serve their consequences, namely a finite z-component of
the magnetization below the critical GFT field. On the
other hand GFT can only describe homogeneous collinear
0 0.5 1 1.5 2
GFT (θ
=0.5°)
GFT (AC)
GFT (MF)
QMC (16)
QMC (64)
QMC (128)
QMC (256)
FIG. 7: The z-component of magnetization as function of
temperature for a fixed external field. Below a critical tem-
perature Tcrit there is a breakdown of magnetization in GFT
where is no in QMC. Parameters: B/J = 0.03, S = 1,
K2/J = −0.06.
configurations of spins therefore showing a breakdown of
magnetization. However by applying a small field in x-
direction the xy-symmetry is broken and the spins ro-
tate in the field direction (the vortices vanish) and the
collinear phase is retrieved. Our results corroborate this
interpretation based on the classical picture. Let us em-
phasize that both, GFT for slightly tilted field and QMC
for B || z, describe the competition between the external
field (which favors magnetization parallel to z) and the
anisotropy favoring in-plane magnetization. Comparing
the z-components of the magnetization for both cases,
one can conclude that the ratio of the competing forces
are comparable for QMC and GFT. This indicates that
this competition is correctly taken into account in GFT.
In Fig. 6 the same field dependence of the z-component
of the magnetization is shown for different temperatures.
We have plotted the result for the tilted field in case of
GFT, the point of breakdown in the untilted case is in-
dicated by the dotted line. It can be seen that for higher
temperatures no breakdown of collinear magnetization
occurs, meaning that the condition for the critical field
(B ≤ |Keff (T,B)|) is never fulfilled in this case. The dis-
crepancies at intermediate temperatures (T = 0.9..1.2)
are due to the RPA decoupling in the GFT as was dis-
cussed already.
In Figs. 7, 8 and 9 the z-component of the magnetiza-
tion is plotted as a function of temperature obtained by
GFT (straight line RPA+AC) as well as QMC (symbols)
for different system sizes and a constant applied magnetic
field.
Let us first discuss the qualitative behavior of the mag-
netization as a function of temperature which is found in
all three figures. For high T (T ≫ Tcrit) the magneti-
zation is reduced by thermal fluctuations (where the tail
of the curve above T/J ≈ 1.5 is due to the applied ex-
ternal field). In the vicinity of Tcrit, T − Tcrit → 0
competition between two effects sets in and has a pro-
0 0.5 1 1.5 2
GFT (θ
GFT (θ
=0.5°)
GFT (two layers)
QMC (128)
FIG. 8: Same situation as in Fig. 7 for K2/J = −0.04 (other
parameters as in Fig.7). The result for a two layer film treated
by GFT is plotted also (dashed-dotted line).
nounced influence on the magnetization. On the one
side the effective anisotropy acts against the external
field (Beff = B − |Keff (T )|, (B || z)). The effective
anisotropy Keff (T ) is reduced with increasing temper-
ature T and thus the effective field Beff increases with
T . This effect tends to enhance the magnetization with
T . On the other side thermal fluctuations suppress the
magnetization with increasing T . The flattening of the
magnetization curve near Tcrit is a result of this com-
petition. For low temperatures T < Tcrit the effective
anisotropy in the GFT cannot be overcome by the exter-
nal field (B < |Keff (T )|, (B ||z)). Therefore the collinear
magnetization in our approximation vanishes due to the
mentioned gapless excitations, in contrast to QMC which
yields again a finite magnetization because non-collinear
states are taken into account as discussed above. The
reduction of the z-component of magnetization in QMC
below Tcrit can be pictured classically as the spins being
in a non-collinear phase with an angle θ with respect to
the z-axis. Since in general anisotropy effects (which fa-
vor in-plane magnetization) increase when temperature is
lowered the z-component of the magnetization decreases.
Now we discuss the three figures in detail. In Fig. 7
we have plotted QMC results for different system size
showing again that these are well converged for N ≥ 64.
Thus we conclude that the striking difference between
GFT and QMC is not a mere finite size effect. The
breakdown of magnetization in GFT occurs at a critical
temperature Tcrit/J = 0.5 whereas no such breakdown
exists in QMC. However the exposed maximum of the
magnetization in QMC lies near the breakdown point.
The differences between QMC and GFT in the tempera-
ture range T/J ≈ 0.3 . . . 1.3 are due to the decoupling of
the exchange and anisotropy term in GFT as also seen
in Fig. 3. It is worth mentioning that the value of the
z-component of the magnetization is nearly the same at
the breakdown point in GFT and the maximum in the
QMC. Thus we have the result that although GFT can-
0 0.5 1 1.5 2
GFT (<S
>;  θ
=0.5°)
GFT (<S
>;  θ
=0.5°)
QMC (θ
FIG. 9: Same situation as in Fig. 7 for K2/J = −0.01, B/J =
0.005 and slightly tilted field (θB = 0.5
◦) for the GFT results.
not describe the non-collinear phase by ansatz its break-
down coincides rather well with the onset of this phase,
which we attribute to the maximum of the QMC curve.
Fig.8 shows the same situation for a different anisotropy
constant K2 = −0.04. The critical temperature is lower
than in Fig.7 since the ratio Bz/K2 becomes larger. The
tilted field case is also shown for the GFT results. Again
the qualitative agreement of the z-component of magne-
tization with QMC is good. To confirm this point we
have plotted the temperature dependence for an other
set of parameters in Fig. 9. There is as good qualitative
agreement of the two approaches. Additionally one gets
a finite component in x-direction in GFT which is also
plotted in the figure. The two effects of the external field
vs. anisotropy competition are nicely to be seen: a non-
collinear state for B || z (z-component only in QMC but
not in GFT) and rotation of magnetization for slightly
tilted external field (seen only in GFT). The ratio of the
competing forces agree well again in both treatments.
In Fig. 7 we have plotted the results of a different
decoupling scheme of the anisotropy terms (namely a
mean field decoupling, dashed line in Fig. 7). Although
the overall characteristic resembles the RPA+AC result
(breakdown of magnetization) the mean field results dif-
fer extremely from the QMC for a large range of tempera-
ture and underestimates the magnetization. This demon-
strates the reliability of the Anderson-Callen treatment of
the local anisotropy terms presented in Refs. 10,11,12,13.
The extension of the GFT method to multi-layer films
is straightforward.11 We have also included results for a
two-layer film in Fig. 8 for the same parameters as in
the monolayer case. One finds that for a double layer
magnetism is stabilized, which can be attributed to the
increased coordination number and thus higher exchange
energy. Just like for a monolayer, one observes a break-
down of collinear magnetization at some critical temper-
ature. This is due to the fact that the same reasoning
regarding the vanishing excitation gap also applies for
multilayer (slab) systems32. The effective anisotropy per
layer is essentially the same as for a single layer, thus the
critical 〈Sz〉-value (magnetization at critical field Bcrit) is
practically the same. The critical temperature is higher
than that of a monolayer due to the increased magnetic
stiffness of the double layer.
IV. SUMMARY AND CONCLUSIONS
Using GFT and QMC calculations we studied easy-
plane systems as well as easy-axis systems with an exter-
nal field applied perpendicularly to the film. The GFT
treatment of the Hamiltonian Eq. (1) consists of a RPA-
decoupling for the nonlocal terms and an AC-decoupling
for the local terms performed in a rotated frame, where
the new z′-axis is parallel to the magnetization. For the
QMC calculations we have used the stochastic series ex-
pansion (SSE) with directed loop updates, which is well
suited for spin-systems.
We have calculated the magnetization as a function of
the external field as well as temperature. We found a
critical field and critical temperature respectively below
which is no magnetization in GFT whereas there is one in
QMC. By tilting the field slightly in GFT so that it has
a small component in x-direction we get a stable magne-
tization even below the critical field or temperature. The
z-component of the magnetization in this case coincides
well with the z-component obtained by QMC for the un-
tilted field confirming that GFT and QMC agree well in
the description of the external field vs. anisotropy com-
petition. However, this comparison can be only some-
what indirect, since QMC has access to the non-collinear
(B || z) state only, while GFT is limited to collinear fer-
romagnetic states (rotated homogeneous magnetization)
found for slightly tilted external fields.
For parameters that are accessible by both QMC and
GFT (B || z; B > Bcrit(T )) QMC and GFT are in good
agreement. Thus one can conclude that the GFT is ap-
plicable to the homogeneous phases of systems described
by Eq. (1) and can be used also for system configurations
not accessible by QMC due to too large system size as
e.g. multilayer systems.
It would be an interesting task for a succeeding work
to extend the GFT in order to get a deeper insight into
the non-collinear configurations also.
APPENDIX A: MAGNETIZATION ANGLE
Here we will discuss the second mathematical solution
which occurs besides Eq. 6. For an external field in
the z-direction the angle dependent part of the free en-
ergy including second order anisotropy can be expanded
as1,27:
F = −MzBz cos θ − K̃2 cos
whereMz is the z-component of the magnetization and
K̃2 is the first nonvanishing coefficient in an expansion of
the free energy for a system with second order anisotropy.
For the equilibrium angle one gets:
∂F (θ)
= MzBz sin θ + 2K̃2 cos θ sin θ
= 0. (A1)
Therefore one gets two solutions for in-plane systems
(K̃2 < 0). For sin θ 6= 0 one gets immediately the so-
lution of Eq. 6 if 2K̃2/Mz ≡ Keff holds. This is the
stable solution. The trivial (second) solution sin θ = 0 is
unstable for Bz < |Keff | because
∂2F (θ)
|sin θ=0 =
< 0 for Bz < |Keff |
> 0 otherwise
holds. For a detailed discussion of stability conditions in
film systems we refer to Refs. 1,27.
∗ Electronic address: henning@physik.hu-berlin.de
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and K. Baberschke, Phys. Rev. B 55, 3708 (1997).
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electronic correlations in local-moment systems, edited by
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(2000).
33 Besides some rare earth materials where the anisotropy can
be of the order of J .
34 For B < |Keff | there is another mathematical solution
(sin θ = 0) which however is unstable (see appendix A).
35 Note that a similar result has already been published in
Ref. 9.
36 Strictly speaking there is no phase transition because of
the applied magnetic field as can be seen from the large
tail of the magnetization curve. However one can extract
a TC from the curves by extrapolating to the zero field
case and additionally to an infinte system size in the QMC
calculations.
mailto:henning@physik.hu-berlin.de
http://arxiv.org/pdf/cond-mat/0607675
http://alps.comp-phys.org/

  In this work we compare numerically exact Quantum Monte Carlo (QMC)
calculations and Green function theory (GFT) calculations of thin ferromagnetic
films including second order anisotropies. Thereby we concentrate on easy plane
systems, i.e. systems for which the anisotropy favors a magnetization parallel
to the film plane. We discuss these systems in perpendicular external field,
i.e. B parallel to the film normal. GFT results are in good agreement with QMC
for high enough fields and temperatures. Below a critical field or a critical
temperature no collinear stable magnetization exists in GFT. On the other hand
QMC gives finite magnetization even below those critical values. This indicates
that there occurs a transition from non-collinear to collinear configurations
with increasing field or temperature. For slightly tilted external fields a
rotation of magnetization from out-of-plane to in-plane orientation is found
with decreasing temperature.

Green Function theory vs. Quantum Monte Carlo Calculation for thin magnetic films
S. Henning,∗ F. Körmann, J. Kienert, and W. Nolting
Lehrstuhl Festkörpertheorie, Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany
S. Schwieger
Technische Universität Ilmenau, Theoretische Physik I, Postfach 10 05 65, 98684 Ilmenau, Germany
(Dated: October 31, 2018)
In this work we compare numerically exact Quantum Monte Carlo (QMC) calculations and Green
function theory (GFT) calculations of thin ferromagnetic films including second order anisotropies.
Thereby we concentrate on easy plane systems, i.e. systems for which the anisotropy favors a
magnetization parallel to the film plane. We discuss these systems in perpendicular external field, i.e.
B parallel to the film normal. GFT results are in good agreement with QMC for high enough fields
and temperatures. Below a critical field or a critical temperature no collinear stable magnetization
exists in GFT. On the other hand QMC gives finite magnetization even below those critical values.
This indicates that there occurs a transition from non-collinear to collinear configurations with
increasing field or temperature. For slightly tilted external fields a rotation of magnetization from
out-of-plane to in-plane orientation is found with decreasing temperature.
PACS numbers: 75.10.Jm, 75.40.Mg, 75.70.Ak, 75.30.Gw
I. INTRODUCTION
The fast development of technological applications
based on magnetic systems in the last years, e.g.
magnetic data storage devices, causes a high interest
in thin magnetic films. One precondition for the tech-
nological development is the investigation of magnetic
anisotropies and spin reorientation transitions connected
therewith. Those reorientation transitions can occur
from out-of-plane to in-plane or vice versa for increasing
film thickness d1, temperature T 2,3,4,5,6,7,8, or external
field B0.
Quantum Monte Carlo (QMC) calculations give the
possibility to compare numerically exact results with
analytical approximations. In Ref. 9 the authors inves-
tigated a ferromagnetic monolayer including positive
second order anisotropy (easy axis perpendicular to the
film plane). They discuss the temperature dependence
of the magnetization 〈Sz〉(T ) as well as field induced
reorientation transitions from out-of-plane to in-plane
and compare the QMC results with Green function
theory (GFT). They found good agreement in the case
of applied external field in the easy direction (here
z-axis). However, their GFT fails for external field
applied in arbitrary direction, especially in the hard
direction (within the film plane). As shown in Ref. 10
for getting closer to the QMC results for magnetic field
induced reorientation from out-of-plane to in-plane a
more careful treatment of the local anisotropy terms is
needed. In Refs. 10,11,12,13 a decoupling scheme was
presented which yields excellent agreement with QMC
results for out-of-plane systems.
The availability of theories such as GFT and their
check against state-of-the-art numerical algorithms
is highly desirable because of the size limitations of
systems where QMC can be performed. On the other
hand the extension of GFT from a monolayer (where it
can be compared to QMC as in the the present work)
to multilayer systems is a straightforward task without
further approximations11.
Up to now, to our knowledge, there is no comparison
between QMC and approximative theories for easy-plane
systems and it is not obvious that the theory presented
in Refs. 10,11,12,13 can reproduce the QMC results for
in-plane systems as accurately as for the out-of-plane
case. In contrast to the easy-axis case where a certain
direction is preferred by the single ion anisotropy in
easy-plane systems the full xy-plane is favored and no
particular direction is distinguished within the plane. A
magnetic field applied perpendicular to the plane does
not destroy the xy-symmetry.
For systems exhibiting this kind of symmetry it was
shown in a classical treatment that for external fields
smaller than a critical field 0 ≤ B < Bcrit (B || z)
stable vortices, i.e. a non-collinear arrangement of
spins, can exist15,16,17,18,19. These vortices can undergo
a Berezinskii-Kosterlitz-Thouless (BKT) transition14.
Depending on the strength of the anisotropy K2 there
might be vortices with or without a finite z-component
of magnetization15. In the small anisotropy case (which
is considered in this work, |K2| < 0.1J) there is a
finite out-of-plane component and for zero field the
two possible directions of magnetization (±z) are en-
ergetically degenerate. For increasing magnetic field in
z-direction the vortices antiparallel to the field become
more and more unstable (heavy vortices). However the
so called light vortices (parallel to the field) are stable
up to a critical field Bz = Bcrit and contribute a finite
z-component to the net-magnetization of the considered
system19.
The vortices in connection with a finite z-component
of the net-magnetization emerge because of two reasons:
first the competition between the anisotropy (favoring
a orientation of the magnetization within the xy-plane)
http://arxiv.org/abs/0704.1552v1
and the external field (favoring a perpendicular magne-
tization), and second: the xy-symmetry of the system,
which does not allow for a rotated homogeneous phase.
In this paper we investigate both aspects, i.e. the
field vs. anisotropy competition as well as the symmetry
properties in detail for a quantum mechanical system.
We will compare the results of QMC and GFT calcula-
tions.
As explained in more detail below, the QMC al-
gorithm used here allows only for an external field
applied in z-direction. Thus the xy-symmetry can not
be broken and no comparison between xy-symmetric
and asymmetric systems is possible. We will use GFT
to clarify the influence of this symmetry breaking on
the homogeneous phase. On the other hand, the GFT
used here is by ansatz limited to the homogeneous
phase. Therefore it can not describe a non-collinear
(e.g, vortex-) magnetic phase, which is expected for
B || z and small field strengths. The breakdown of
magnetization in GFT as well as an exposed maximum
in the magnetization in QMC at certain critical values
of the external field or temperature gives, however, a
clear fingerprint of non-collinear configurations, at least
if there is no meta-stable homogeneous phase. Below
these critical values there will be a finite z-component
in QMC and a vanishing magnetization in GFT.
For parameters, where both theories are applicable,
QMC serves as a test for the approximations needed in
In this work we find indications for non-collinear
spin configurations below a critical field or temperature
for B || z by comparing results of QMC and GFT as
explained in the last clause. Above the critical field
we obtain good agreement between QMC and GFT
results. Breaking the xy-symmetry by adding a small
x-component to the external field yields a stable collinear
solution in GFT. The z-component of the magnetization
in this case is in good agreement with the QMC results
calculated with untilted field. Thus we can conclude
that except for the restriction to collinear magnetic
states GFT describes the competition between external
field and anisotropy quite well.
The paper is organized as follows: First we ex-
plain the basics of the GFT and the QMC calculations.
Then we apply both approaches to easy-plane systems
in external magnetic fields and report the results of our
calculations.
II. THEORY
A. Green Function Theory
In the following we present our theoretical approach us-
ing Green function theory. The focus of this work lies on
the translational invariant system of a two-dimensional
monolayer. Therefore the following Hamiltonian is used:
H = −
JijSiSj −B
Si −K2
(Sz,i)
2. (1)
The first term describes the Heisenberg coupling Jij be-
tween spins Si and Sj located at sites i and j. The second
term contains an external magnetic field B in arbitrary
direction (the Landé factor gJ and the Bohr magneton
µB are absorbed in B). The third term represents second
order lattice anisotropy due to spin-orbit coupling. Sz,i
is the z-component of Si (the z-axis of the coordinate
system is oriented perpendicular to the film-plane). The
lattice anisotropy favours in-plane (K2 < 0) or out-of-
plane (K2 > 0) orientation. Our Hamiltonian is similar
to that used in Refs. 10,11,13,22,23 for the investigation
of the magnetic anisotropy and the field induced reori-
entation transition. To simplify calculations we consider
nearest neighbor coupling only
Jij =
J (i), (j) n.n.
0 otherwise.
The main idea of the special treatment presented in
Refs. 10,11,12,13 is that, before any decoupling is applied,
the coordinate system Σ is rotated to a new system Σ′
where the new z′-axis is parallel to the magnetization im-
plying a collinear alignment of all spins within the layer.
Then a combination of Random Phase approximation
(RPA)24 for the nonlocal terms in Eq. (1) (Heisenberg
exchange interaction term) and Anderson-Callen approx-
imation (AC)25 for the local lattice anisotropy term is
applied in the rotated system. After application of the
approximation one gets an effective anisotropy
Keff (T ) = 2K2
S(S + 1)− 〈S2z′〉
〈Sz′〉 (3)
where 〈Sz′〉 is the norm of the magnetization and S is the
spin quantum number, that we have chosen to be S = 1
in all our calculations.
As shown in comparison with an exact treatment of the
local anisotropy term in Ref. 26 this approximation still
holds up to anisotropy strengths K2 ∼ 1/2J . Therefore
we restrict ourselves in the following to small anisotropies
(K2 ≤ 0.1J) as found in most real materials
33. For a
magnetic field applied in the xz-plane (B = (Bx, 0, Bz))
our theory gives a condition for the polar angle θ of the
magnetization:
sin θBz − cos θBx +Keff sin θ cos θ = 0 (4)
The uniform magnon energies (q = 0) which dominate
the physical behavior of the magnetic system can easily
be extracted from the theory12,13:
E2q=0 =
cos θBz + sin θBx +Keff(cos
2 θ − sin2 θ)
cos θBz + sin θBx +Keff cos
This result coincides with the spin-wave result13 if one
replaces 〈Sz′〉 by the spin quantum number S and Keff
by the bare anisotropy constant K2 in Eq. (5). For an
easy-plane system (Keff < 0) with external field B in
z-direction the polar angle θ of the magnetization34 is
given by:
cos θ =
−B/Keff(T ) for B < |Keff (T )|
1 otherwise
By inserting Eq.(6) into Eq.(5) one immediately gets:
Keff<0
q=0 (B) =
0 B < |Keff (T )|
B +Keff (T ) otherwise.
For gapless magnon energies Eq=0 = 0 the magnon oc-
cupation number φ diverges (φ → ∞) in film systems
with ferromagnetic coupling J > 0 and the magnetiza-
tion becomes zero 〈Sz′〉 = 0 in the collinear phase. This
can be seen by following an argument of Bloch20 already
given in 1930. Since the spin wave dispersion is E ≈ q2
in the vicinity of q = 0 the spin-wave density of states
N(E) is independent of E for a two-dimensional system
for E close to zero. The excitation of spin-waves at finite
temperature leads to a variation of the magnetization of
the order:
∆m(T ) ∼
N(E)dE
exp(E/kBT )− 1)
∼ kBT
exp(x)− 1
. (8)
Since the integral in Eq. (8) diverges for T 6= 0 and exited
spin-waves lead to a reduction of the magnetization one
can conclude that the magnetization should be zero at
finite temperature. However for an infinitesimally small
contribution of the external field parallel to the plane,
i.e. Bx 6= 0, a finite gap in the excitation spectrum
at q = 0 opens. This can be seen in Fig.1 where the
uniform magnon modes Eq=0(B) are shown for different
orientations θB, where θB is the polar angle of the ex-
ternal field. The integral (8) is now finite and a stable
finite magnetization in the collinear phase having a well
defined orientation in the xz-plane is possible.
Let us now come back to the case where the applied
field is aligned in z-direction. It can be seen from Eq.
(7) that for external field B (B || z) larger than a critical
field B > Bcrit given by:
Bcrit
= |Keff (T,B)| (9)
a stable collinear solution exists. Since Keff (T ) is a de-
creasing function of temperature T a transition from non-
collinear to collinear phase with increasing temperature
is possible. In Fig. 2 we show the normalized critical
field (9) Bcrit/K2 as a function of temperature T . For
a constant magnetic field B (B || z) at a temperature
T1 with B < Keff (T1, B) no stable collinear phase ex-
ist. Then by increasing the temperature up to T2 the ef-
fective anisotropy Keff is sufficiently reduced such that
0.01 0.02 0.03 0.04
B / J
=0.2°
FIG. 1: The energies of the uniform magnon mode Eq=0(B)
for different polar angles θB of the external field. Eq=0 is
zero below B/J ≈ 0.03 for θB = 0
◦. The prefactors gJµB
and kB are absorbed in B and T respectively. The latter are
given in units of the nearest neighbor Heisenberg coupling J .
Parameters: S = 1, K/J = −0.03 and T/J = 10−4.
FIG. 2: The normalized critical field Bcrit/K2 as a function
of temperature. Parameters: S = 1.
B > Keff (T2, B), and the collinear phase becomes sta-
ble. Before we come to the results let us briefly sketch
the main aspects of the QMC.
B. QMC
In the last section we gave a short description of the
theory used to treat a system described by a Hamilto-
nian of form (1). This theory applies to the thermody-
namic limit (films of infinte size) but contains certain
approximations. Additionally the GFT is restricted to
ordered phases with a collinear alignment of all spins.
Therefore it would be very useful to have exact results at
hand to crosscheck the predictions of GFT. A Quantum
Monte Carlo method, particularly well suited for spin
systems, is the stochastic series expansion (SSE) with
directed loop update. We will sketch this method here
only briefly as detailed descriptions can be already found
elsewhere28,29,30.
Our starting point is the series expansion of the parti-
tion function
Z = Tre−βH =
〈α|(−H)n|α〉 (10)
whereH denotes the Hamiltonian, {|α〉} are basis vectors
of a proper Hilbert space and β is the inverse tempera-
ture. The Hamiltonian is then rewritten in terms of bond
Hamiltonians:
H = −J
Hb (11)
where Hb can be further decomposed into a diagonal and
an off-diagonal part:
HD,b = C + S
i(b)S
j(b) + bb[S
i(b) + S
j(b)] (12)
+k2b[(S
i(b))
2 + (Szj(b))
HO,b =
] (13)
Here we have renormalized the anisotropy constant k2b
and the magnetic field bb in such a way that (11) coincides
with (1). i(b) and j(b) denotes the lattice sites connected
by the bond b and the additional constant C in HD,b will
be chosen such that all matrix elements of this term be-
come positive, a condition necessary to interpret them as
probabilities. Note that for a finite system at finite tem-
perature the power series of the partition function can
be truncated at a finite cutoff length Λ without intro-
ducing any systematic error in practical computations29.
Therefore reinserting (11) into (10) and rewriting the re-
sult yields:
βn(Λ− n)!
〈α|SCΛ |α〉. (14)
Here SCΛ denotes a product of operators (operator
string) consisting of n non-unity operators and (Λ − n)
unity operators H0 = Id which were inserted to get op-
erator strings of equal length Λ.
In fact it is impossible to evaluate all operator strings in
(14). The SSE-QMC replaces such an evaluation there-
fore by importance sampling over the strings according to
their relative weight. Hence an efficient scheme for gen-
erating new operator strings is needed. In the directed
loop version of the SSE this is done by dividing the up-
date into two parts. In a first step a diagonal update is
performed by traversing the operator string and replac-
ing some unity operators by diagonal bond operators and
vice versa (the probabilities for both substitutions have
to fulfill the detailed balance criterion). Then the loop
update follows in which new non-diagonal bond opera-
tors can appear in the operator string. For details of the
update procedure we refer the interested reader to the
0 0.5 1 1.5 2
GFT (RPA+AC)
QMC (N=16)
QMC (N=32)
QMC (N=64)
FIG. 3: Magnetization vs. temperature for an out-of-plane
easy-axis system (K2 > 0). Straight line: GFT (RPA+AC)
result; symbols: QMC results for different system sizes N2.
Parameters: S = 1, B/J = 0.01 (B || z) and K2/J = 0.01.
according literature28,29,30.
A full implementation of the SSE with directed loop
update which we have used for all QMC calculations
in this work can be found in the ALPS project30,31.
Since the SSE-QMC used by us is implemented in z-
representation (spin quantization axis along z-axis) in-
plane correlation functions e.g. the in-plane magnetiza-
tion are not accessible. Further B || z is the only possible
field direction in the used QMC implementation because
a traverse field (in-plane field component) would lead to
non-closing loops (see Ref. 9).
III. RESULTS
As mentioned in Sec. II A the results for the in-plane
systems are very sensitive to the effective anisotropy
Keff (T ). This sensitivity of the anisotropy is less pro-
nounced for out-of-plane systems (K2 > 0) since the ap-
plied field B (B || z) and the intrinsic easy axis are par-
allel. In order to test our decoupling scheme (RPA+AC)
we first compare GFT and QMC for an out-of-plane
system.35
In Fig. 3 the magnetization 〈Sz〉 as a function of tem-
perature T is shown. The straight line belongs to the
GFT whereas the symbols show the result of the QMC
for different system sizes. Let us first comment on finite
size effects in the QMC results.
It can be seen in Fig. 3 that the QMC results converge
for increasing system size N2 (for N ×N square lattice).
Indeed forN ≥ 32 the QMC results are unbiased by finite
size effects and resulting magnetization curves are almost
equal for increasing N ≥ 32. Note that we have omitted
error bars in the figures showing QMC results because
the relative errors are of the order 10−4.
We now compare the GFT with the QMC results
(N = 64). For low temperatures (T/J ≤ 0.5) we ob-
tain excellent quantitative agreement. This is plausible
0 0.05 0.1 0.15 0.2
QMC (16)
QMC (32)
QMC (64)
QMC (128)
FIG. 4: z-component of magnetization as a function of ex-
ternal magnetic field for fixed temperature T/J = 0.4. In
contrary to the GFT the magnetization obtained by QMC
remains finite for all fields. The QMC results are converged
for N ≥ 64. Parameters: S = 1, K2/J = −0.06 and θB = 0
0 0.05 0.1 0.15 0.2
GFT (θ
 = 0°)
GFT (θ
 = 0.5°) 
QMC (128)
FIG. 5: z-component of magnetization vs. external field for
T/J = 0.4 with slightly tilted field (θB = 0.5
◦)in the GFT
result (solid line). The dotted line shows GFT result for (θB =
0◦). Other parameters as in Fig. 4.
because in this region the GFT result coincides with the
result of the spin-wave theory which is known to be re-
liable (exact for T = 0) for low temperatures. For the
intermediate region T/J = 0.5..1 the RPA slightly un-
derestimates the magnetization which was also found in
Ref. 9. The opposite is the case in the region near the
extrapolated Curie temperature TC
36, where the magne-
tization is overestimated. The reason is the presence of
longitudinal fluctuations, which play an important role
in this region and it is well known that the RPA fails to
treat them properly.
We consider now the case of in-plane systems (K2 < 0)
and applied field in the hard direction (B || z). As al-
ready mentioned there is no ’collinear’ magnetization in
the GFT for Bz < |Keff (T )|. In Fig. 4 the z-component
of the magnetization is shown as a function of the ex-
ternal field B for a constant temperature T/J = 0.4.
0 0.05 0.1 0.15
T/J = 0.4
T/J = 0.6
T/J = 0.9
T/J = 1.2
T/J = 1.4 
FIG. 6: z-component of magnetization vs. external field for
different temperatures T/J and fixed system size N2 (N =
128). Solid lines: GFT (θB = 0.5
◦), dashed lines: GFT (θB =
0◦) other parameters as in Fig. 4.
As in Fig. 3 we see that the QMC results for N ≥ 64
are almost converged and the finite size of the calculated
system in QMC should not influence the results anymore.
The dotted line marks a critical field Bcrit. For magnetic
fields larger than the critical one B > Bcrit we obtain
good agreement between QMC and GFT results. Below
the critical field B < Bcrit GFT does not yield a stable
homogeneous magnetization. However the QMC results
show that there is a finite z-component of the magneti-
zation in the considered system for 0 ≤ B ≤ Bcrit.
In order to compare QMC with GFT results we have
tilted the magnetic field by θB = 0.5
◦ which corresponds
to Bx < 10
−2Bz in the GFT. As explained before any
symmetry breaking field Bx 6= 0 leads to a stable homo-
geneous magnetization with well-defined orientation in
the xz-plane. However such a small contribution of the
external field within the plane should hardly influence the
z-component of the magnetization. This is confirmed by
Fig. 5 where we show QMC results (N = 128, θB = 0
as well as the corresponding GFT results with θB = 0
and θB = 0.5
◦. As expected for |B| > Bcrit the two so-
lutions in the GFT are nearly the same and agree well
with QMC. Below the critical field only the solution with
the slightly tilted field yields a stable homogeneous mag-
netization and its z-component compares well with the
QMC result in the untilted case.
The above results can be interpreted within a semi-
classical picture of non-collinear vortex configurations
which are stable below a critical field Bcrit in z-direction
and contribute a finite z-component to the magnetiza-
tion in case of an applied field.19 Despite the lack of di-
rect, quantitative access to such states (or correspond-
ing physical in-plane observables) within the QMC al-
gorithm they are included in principle and one can ob-
serve their consequences, namely a finite z-component of
the magnetization below the critical GFT field. On the
other hand GFT can only describe homogeneous collinear
0 0.5 1 1.5 2
GFT (θ
=0.5°)
GFT (AC)
GFT (MF)
QMC (16)
QMC (64)
QMC (128)
QMC (256)
FIG. 7: The z-component of magnetization as function of
temperature for a fixed external field. Below a critical tem-
perature Tcrit there is a breakdown of magnetization in GFT
where is no in QMC. Parameters: B/J = 0.03, S = 1,
K2/J = −0.06.
configurations of spins therefore showing a breakdown of
magnetization. However by applying a small field in x-
direction the xy-symmetry is broken and the spins ro-
tate in the field direction (the vortices vanish) and the
collinear phase is retrieved. Our results corroborate this
interpretation based on the classical picture. Let us em-
phasize that both, GFT for slightly tilted field and QMC
for B || z, describe the competition between the external
field (which favors magnetization parallel to z) and the
anisotropy favoring in-plane magnetization. Comparing
the z-components of the magnetization for both cases,
one can conclude that the ratio of the competing forces
are comparable for QMC and GFT. This indicates that
this competition is correctly taken into account in GFT.
In Fig. 6 the same field dependence of the z-component
of the magnetization is shown for different temperatures.
We have plotted the result for the tilted field in case of
GFT, the point of breakdown in the untilted case is in-
dicated by the dotted line. It can be seen that for higher
temperatures no breakdown of collinear magnetization
occurs, meaning that the condition for the critical field
(B ≤ |Keff (T,B)|) is never fulfilled in this case. The dis-
crepancies at intermediate temperatures (T = 0.9..1.2)
are due to the RPA decoupling in the GFT as was dis-
cussed already.
In Figs. 7, 8 and 9 the z-component of the magnetiza-
tion is plotted as a function of temperature obtained by
GFT (straight line RPA+AC) as well as QMC (symbols)
for different system sizes and a constant applied magnetic
field.
Let us first discuss the qualitative behavior of the mag-
netization as a function of temperature which is found in
all three figures. For high T (T ≫ Tcrit) the magneti-
zation is reduced by thermal fluctuations (where the tail
of the curve above T/J ≈ 1.5 is due to the applied ex-
ternal field). In the vicinity of Tcrit, T − Tcrit → 0
competition between two effects sets in and has a pro-
0 0.5 1 1.5 2
GFT (θ
GFT (θ
=0.5°)
GFT (two layers)
QMC (128)
FIG. 8: Same situation as in Fig. 7 for K2/J = −0.04 (other
parameters as in Fig.7). The result for a two layer film treated
by GFT is plotted also (dashed-dotted line).
nounced influence on the magnetization. On the one
side the effective anisotropy acts against the external
field (Beff = B − |Keff (T )|, (B || z)). The effective
anisotropy Keff (T ) is reduced with increasing temper-
ature T and thus the effective field Beff increases with
T . This effect tends to enhance the magnetization with
T . On the other side thermal fluctuations suppress the
magnetization with increasing T . The flattening of the
magnetization curve near Tcrit is a result of this com-
petition. For low temperatures T < Tcrit the effective
anisotropy in the GFT cannot be overcome by the exter-
nal field (B < |Keff (T )|, (B ||z)). Therefore the collinear
magnetization in our approximation vanishes due to the
mentioned gapless excitations, in contrast to QMC which
yields again a finite magnetization because non-collinear
states are taken into account as discussed above. The
reduction of the z-component of magnetization in QMC
below Tcrit can be pictured classically as the spins being
in a non-collinear phase with an angle θ with respect to
the z-axis. Since in general anisotropy effects (which fa-
vor in-plane magnetization) increase when temperature is
lowered the z-component of the magnetization decreases.
Now we discuss the three figures in detail. In Fig. 7
we have plotted QMC results for different system size
showing again that these are well converged for N ≥ 64.
Thus we conclude that the striking difference between
GFT and QMC is not a mere finite size effect. The
breakdown of magnetization in GFT occurs at a critical
temperature Tcrit/J = 0.5 whereas no such breakdown
exists in QMC. However the exposed maximum of the
magnetization in QMC lies near the breakdown point.
The differences between QMC and GFT in the tempera-
ture range T/J ≈ 0.3 . . . 1.3 are due to the decoupling of
the exchange and anisotropy term in GFT as also seen
in Fig. 3. It is worth mentioning that the value of the
z-component of the magnetization is nearly the same at
the breakdown point in GFT and the maximum in the
QMC. Thus we have the result that although GFT can-
0 0.5 1 1.5 2
GFT (<S
>;  θ
=0.5°)
GFT (<S
>;  θ
=0.5°)
QMC (θ
FIG. 9: Same situation as in Fig. 7 for K2/J = −0.01, B/J =
0.005 and slightly tilted field (θB = 0.5
◦) for the GFT results.
not describe the non-collinear phase by ansatz its break-
down coincides rather well with the onset of this phase,
which we attribute to the maximum of the QMC curve.
Fig.8 shows the same situation for a different anisotropy
constant K2 = −0.04. The critical temperature is lower
than in Fig.7 since the ratio Bz/K2 becomes larger. The
tilted field case is also shown for the GFT results. Again
the qualitative agreement of the z-component of magne-
tization with QMC is good. To confirm this point we
have plotted the temperature dependence for an other
set of parameters in Fig. 9. There is as good qualitative
agreement of the two approaches. Additionally one gets
a finite component in x-direction in GFT which is also
plotted in the figure. The two effects of the external field
vs. anisotropy competition are nicely to be seen: a non-
collinear state for B || z (z-component only in QMC but
not in GFT) and rotation of magnetization for slightly
tilted external field (seen only in GFT). The ratio of the
competing forces agree well again in both treatments.
In Fig. 7 we have plotted the results of a different
decoupling scheme of the anisotropy terms (namely a
mean field decoupling, dashed line in Fig. 7). Although
the overall characteristic resembles the RPA+AC result
(breakdown of magnetization) the mean field results dif-
fer extremely from the QMC for a large range of tempera-
ture and underestimates the magnetization. This demon-
strates the reliability of the Anderson-Callen treatment of
the local anisotropy terms presented in Refs. 10,11,12,13.
The extension of the GFT method to multi-layer films
is straightforward.11 We have also included results for a
two-layer film in Fig. 8 for the same parameters as in
the monolayer case. One finds that for a double layer
magnetism is stabilized, which can be attributed to the
increased coordination number and thus higher exchange
energy. Just like for a monolayer, one observes a break-
down of collinear magnetization at some critical temper-
ature. This is due to the fact that the same reasoning
regarding the vanishing excitation gap also applies for
multilayer (slab) systems32. The effective anisotropy per
layer is essentially the same as for a single layer, thus the
critical 〈Sz〉-value (magnetization at critical field Bcrit) is
practically the same. The critical temperature is higher
than that of a monolayer due to the increased magnetic
stiffness of the double layer.
IV. SUMMARY AND CONCLUSIONS
Using GFT and QMC calculations we studied easy-
plane systems as well as easy-axis systems with an exter-
nal field applied perpendicularly to the film. The GFT
treatment of the Hamiltonian Eq. (1) consists of a RPA-
decoupling for the nonlocal terms and an AC-decoupling
for the local terms performed in a rotated frame, where
the new z′-axis is parallel to the magnetization. For the
QMC calculations we have used the stochastic series ex-
pansion (SSE) with directed loop updates, which is well
suited for spin-systems.
We have calculated the magnetization as a function of
the external field as well as temperature. We found a
critical field and critical temperature respectively below
which is no magnetization in GFT whereas there is one in
QMC. By tilting the field slightly in GFT so that it has
a small component in x-direction we get a stable magne-
tization even below the critical field or temperature. The
z-component of the magnetization in this case coincides
well with the z-component obtained by QMC for the un-
tilted field confirming that GFT and QMC agree well in
the description of the external field vs. anisotropy com-
petition. However, this comparison can be only some-
what indirect, since QMC has access to the non-collinear
(B || z) state only, while GFT is limited to collinear fer-
romagnetic states (rotated homogeneous magnetization)
found for slightly tilted external fields.
For parameters that are accessible by both QMC and
GFT (B || z; B > Bcrit(T )) QMC and GFT are in good
agreement. Thus one can conclude that the GFT is ap-
plicable to the homogeneous phases of systems described
by Eq. (1) and can be used also for system configurations
not accessible by QMC due to too large system size as
e.g. multilayer systems.
It would be an interesting task for a succeeding work
to extend the GFT in order to get a deeper insight into
the non-collinear configurations also.
APPENDIX A: MAGNETIZATION ANGLE
Here we will discuss the second mathematical solution
which occurs besides Eq. 6. For an external field in
the z-direction the angle dependent part of the free en-
ergy including second order anisotropy can be expanded
as1,27:
F = −MzBz cos θ − K̃2 cos
whereMz is the z-component of the magnetization and
K̃2 is the first nonvanishing coefficient in an expansion of
the free energy for a system with second order anisotropy.
For the equilibrium angle one gets:
∂F (θ)
= MzBz sin θ + 2K̃2 cos θ sin θ
= 0. (A1)
Therefore one gets two solutions for in-plane systems
(K̃2 < 0). For sin θ 6= 0 one gets immediately the so-
lution of Eq. 6 if 2K̃2/Mz ≡ Keff holds. This is the
stable solution. The trivial (second) solution sin θ = 0 is
unstable for Bz < |Keff | because
∂2F (θ)
|sin θ=0 =
< 0 for Bz < |Keff |
> 0 otherwise
holds. For a detailed discussion of stability conditions in
film systems we refer to Refs. 1,27.
∗ Electronic address: henning@physik.hu-berlin.de
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33 Besides some rare earth materials where the anisotropy can
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34 For B < |Keff | there is another mathematical solution
(sin θ = 0) which however is unstable (see appendix A).
35 Note that a similar result has already been published in
Ref. 9.
36 Strictly speaking there is no phase transition because of
the applied magnetic field as can be seen from the large
tail of the magnetization curve. However one can extract
a TC from the curves by extrapolating to the zero field
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calculations.
mailto:henning@physik.hu-berlin.de
http://arxiv.org/pdf/cond-mat/0607675
http://alps.comp-phys.org/
