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this is tyte...they give u all the info u need...include where u can find the closest arcade near u... 8)

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btw the ps2 version coming out March 27th 2003...stay tuned 8)




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[quote:be278f552e][quote:be278f552e][quote:be278f552e][quote:be278f552e]November 21, 2002
Abstract
Simple arguments are given, related to the apparent universality with
which Bloch's famous T3=2-law, and generalizations thereof, are not
only found in d=3-dimensional ferromagnetic systems, but astonish-
ingly also in lower dimensions. It is argued that
_ one should not simply apply the usual isotropic dispersion re-
lation !(~k) = D _ ~k2 known to almost everyone but only valid
for circular precession of the spins (i.e. where only the exchange
interaction is taken into account), but instead one should con-
sider also the other interactions and use the less-known relation
!(~k) = q!a(~k) _ !b(~k) for elliptical precession,
_ one might consider the apparent universality of Bloch's T3=2-law
as some kind of 'apparent quantum universality' in a certain
_nite-temperature crossover region from a quantum phase tran-
sition at T=0,
_ one should use certain simple crossover-scaling arguments for bet-
ter understanding of the phenomena, instead of the usual more
complicated derivations by direct integration.
This is also exempli_ed for more general cases in three appendices on
the signi_cance of a crossover from a quantum phase transition.
PACS numbers: 75. Magnetic properties;
05.50 Fh Phase Transitions: General Studies;
Keywords: Bloch's law; Quantum Phase Transitions; Universality
Introduction: This letter grew out of discussions following a recent presentation
of certain experimental results in our institute, which showed
1
the T3=2 Bloch law for Ms(T) for a nanostructured planar system, [1];
and although I originally thought that the considerations presented below
were too simple for publication, some of the participants of those
discussions suggested that I should write them down. So here I do so,
just hoping to broaden and intensify more personal discussions in this
way.
Bloch's law: According to this law (which is derived in any textbook
on solid-state magnetism and is one of the most prominent results of
theoretical physics) the temperature dependence of the magnetization
Ms(T) of a three-dimensional ferromagnet is simply given by
Ms(T) _ Ms(0) _ const: _ Z
BZ
d3k
1
exp __(~k) _ 1
; (1)
because each excited magnon reduces the magnetic moment of a ferromagnetic
sample by 2 Bohr magnetons. Here T is the Kelvin temperature,
_ = (kBT)_1 with the Boltzmann constant kB, and the fraction
1
exp(__(~k)_1
represents the thermal expectation value of the number of
magnetic excitations ('magnons') with excitation energy _(~k), where
the wave-vector ~k has its usual meaning. The integration is over the
Brillouin zone BZ of the crystal.
For the excitation energy _(~k) of a magnon in a Heisenberg ferromagnet
(and also in itinerant ferromagnets when the spin-orbit interaction,
and also the Stoner excitations, are neglected against the
collective magnon-like spin excitations) one simply has (for simplicity
we assume cubic symmetry): _(~k) = D _ k2, where D is the so-called
spin-wave sti_ness. Therefore, by the substitution x := _D _ k2 and
the replacement d3k = 4_k2dk one gets the famous result Ms(T) =
Ms(0)_ const0: _ ( kBT
D )3=2, where (up to exponentially-small terms) the
constant const:0 = const: _ 2_
1R
0
x1=2dx
exp x_1 .
For Anderson's "poor man", [2], instead of the usual derivation,
one can also give the following simpler argument: exp _Dk2 _ 1 is
approximated for long enough wavelengths and/or high enough T by
the 'quasi-classical thermal-energy approximation' _D_ k2, so that one
simply gets Ms(T) = Ms(0)_const:( kBT
D )_PhsR(T), where PhsR(T)
means a typical phase-space radius in k-space, replacing the integral
2
0_=a0
R0
4_k2
k2 dk. Here 0_=a0 represents a (very large) wavenumber-cuto_
corresponding to the upper edge of the Brillouin zone, which is replaced
by a sphere as in typical renormalization group arguments. But
it would be wrong, if at this place one would perform directly the integration
(after having made the above-mentioned quasi-classical approximation
leading to the 'thermal-energy prefactor' ( kBT
D )); instead, one
gets the correct PhsR(T) by a Pippard-type argument, i.e. simply by
equating the dominating energy- resp. temperature-ranges: _(~kdom:) =
D _ k2dom: _ kBT, i.e. PhsR(T) = kdom:(T) (_= _k(T)) / ( kBT
D )1=2.
What would be di_erent in a planar system magnetized in-theplane
?
One would again expect a quasi-classical approximation exp __(~k) _
1 ! __(~k), but instead of _(~k) = D _ k2 one would have _(~k) =
q_a(~k) _ _b(~k), i.e. a pronounced elliptical precession instead of the
circular one. For example, if one is dealing with a _lm of in_nite
extension in the x- and y-directions, with _nite thickness in the zdirection,
then (if the _lm is magnetized in the x-direction) spin-wave
deviations in the z-direction are strongly disfavoured energetically, due
to the demagnetizing _eld HDM
z = _4_Mz. Thus, one would have
_b(~k) _ _z(~k) = Cb + D _ k2 (_= Cb in the long-wavelength limit, where
Cb = 2_M2
z _ V is the e_ective anisotropy energy corresponding to
the demagnetizing _eld (V is the volume of the system)). In contrast,
spin deviations in the y-direction would not be disfavoured, i.e.
_a(~k) _ _y(~k) = D _ k2, as before.
Thus for k2 _ l2
exch <_ 1, where lexch = qCb
D is an e_ective 'exchangelength',
[3], one would not have _(~k) = D_k2, but instead _(~k) = D0 _k1,
where D0 = pCb _ D.
This change of the spin-wave dispersion at small wavenumbers by
the magnetostatic _elds was already noted years ago in an early paper
of P. Bruno, [4], who stated that exactly in this way the famous
Mermin-Wagner theorem (which says that in the Heisenberg model
there cannot be magnetic long-range order in d=2 dimensions, [5]) is
invalidated by a 'cut-o_ e_ect' related to the magnetostatic interactions,
which are not considered in the theorem, [6]. However, note that
for the existence of an elliptical precession and of the planar uniaxial
3
anisotropy, we do not need a _nite thickness of the magnetic _lm, i.e.
it can also just be a monolayer.
So I suggest that in planar systems (magnetized 'in the plane' by
the simultaneous inuence of the exchange interaction and an e_ective
in-plane uniaxial anisotropy), Ms(T) should behave as
Ms(T) = Ms(0) _ const:0 _ (
kBT
D0 ) _ PhsR(T) : (2)
Here I assumed a similar 'quasi-classical thermal-energy factor' ( kBT
D0 )
as before, but now the phase-space radius PhsR(T) replaces the integral
0pi=a0
R l_1
exch
2_k1dk
k1 . Here the changes of the exponents of the nominator
and of the denominator, both exponents changing from 2 to 1, result
(for the denominator) from the fact that for k2 _ l2
exch <_ 1 the exponent
in the dispersion relation has changed fom 2 to 1, whereas the change in
the nominator comes from d2k _ k1dk. But for temperatures above a
crossover value T_ corresponding to the crossover from long-wavelength
to short-wavelength magnons (i.e. for kB T >_ kBT_ := D0 _ (_=a)),
the dominating modes are spin waves with _(~k) = Dk2, since for
k2 l2
exch >_ 1 one gets this quadratic dispersion, and so PhsR(T) should
once more be proportional to ( kBT
D )
1
2 , i.e. also PhsR(T) is essentially
unchanged. (The fact that here the system 'remembers' the long-wave
behaviour while behaving thermodynamically according to the shortwave
branch of the magnon dispersion becomes understandable in the
renormalization-group (i.e. 'ow line') scenario of Appendix A.)
As a consequence, one has again Bloch's T3=2-law: Ms(T) = Ms(0)_ const00: _ T3=2, although probably with a larger constant and only for
temperatures T > T_. (Note that T_ is very small compared with Tc.)
In fact, the T3=2 Bloch's law is not only measured for three-dimensional
amorphous ferromagnets, [8], but also for two-dimensional ultrathin
_lms, [9], which has already been noted by many people, e.g. by [10],
and now by [1] for nanostructured planar systems.
In any case, it would again be wrong to replace our simple scaling
arguments by a direct integration, which would lead to the 'Doringtype
behaviour' Ms(T) = Ms(0) _ const:00 _ kBT _
0_=a0
R l_1
exch
d2k 1
k2 , [7],
4
i.e. to Ms(T) = Ms(0) _ const:000 _ T _ ln T
T0
, where T0 is a constant
temperature unit. As far as I know, this has never been observed.
Is all this related to quantum phase transitions ?
Obviously, the detailed behaviour earns a more thorough study; fortunately,
there is a recent careful analysis of an extremely large set of
experimental results by U. Kobler, [10], which leads to an apparently
universal classi_cation for the behaviour of Ms(T), although the arguments
look quite complicated. The universality of the classi_cation
reminds to second-order phase transitions (i.e. for thermal phase transitions
there is the well-known 'Gri_ths Universality Hypothesis', see e.g.
[11]). But since here the universality is in the low- and intermediatetemperature
region and not in the vicinity of the Curie temperature,
the 'Gri_ths universality' does not apply, and one is probably dealing
with 'Quantum Phase Transitions', i.e. one is perhaps in the _nitetemperature
region of a weakly unstable 'quantum _xed point' and
should consider the renormalization-group ow-lines in a diagram joining
that 'quantum _xed point' and the thermal 'Curie' _xed point (see
the appendices). According to the recent book of S. Sachdev, [12], in
this _nite-T-region one should have a very wide range of scaling behaviour,
which would explain the wide range of applicability of Bloch's
law.
The experimental analysis of Kobler is additionally remarkable because
of the fact that his generalized 'Bloch's law exponent' _, de_ned
by the behaviour Ms(T) = Ms(0)_ const: T_, depends on whether the
spin-quantum number s of the magnetic atoms is integer or half-integer.
In a quasi-classical approximation this distinction makes no sense, but
in the context of a quantum phase transition it does. E.g. for an atom
with s = 1=2 a change m ! (m_1) would only correspond to a ip of
the sign of the local magnetization, whereas for s = 1, the corresponding
change would lead to vanishing of all local magnetic properties, see
[13].
Therefore one should reconsider the behaviour of Ms(T) in the context
of renormalization-group ow lines of quantum phase transitions;
perhaps in this way one can better understand and unify the Bloch-like
behaviour in thin _lms and related (patterned) nanostructures, [1], also
for more complicated multispin models as those advocated by U. Kobler,
see [10], by simple crossover-scaling laws for various classes of models,
5
arising from such an approach.
Some ideas concerning these points are already sketched in the appendices,
which should be considered as an integral part of this letter
and should be read before the following 'Conclusions'.
Conclusions: As a consequence of the preceding paragraphs and the
appendices, concerning the temperature range considered, we stress
that the simple arguments of the present letter, combined with semiquantitative
renormalization group scenarios in Appendix A and Appendix
B, do not consider the immediate vicinity of the quantum phase
transion at T = 0 nor that of the thermal phase transition at Tc but
rather apply for the (wide) temperature range between the (very low)
crossover temperature T_ (_ qCb _ D _ (_=a)2=kB in the case of Appendix
A ) and a (much higher) crossover temperature in the range of
~ T = D _ (_=a)2=kB. So the quantum _xed point is signi_cant, but at
the same time it only plays a secondary role.
All statements in the appendices A to C apply to half-integer s
in Kobler's classi_cation, [10]. Additional complications for integer s
(outside the region of overwelming importance of the thermal uctuations)
will be shifted to future work and to a more thorough analysis.
Acknowledgements
The author acknowledges stimulating communications from U. Kobler,
U. Gradmann and S. Krompiewski, and fruitful discussions with C.
Back, G. Bayreuther, R. Hollinger, A. Killinger, W. Kipferl, and J.
Siewert.
References
[1] W. Kipferl, Spin-wave excitations in epitaxial Fe(001)-nanostructures
on GaAs, unpublished seminar, University of Regensburg,
October 2002
[2] P.W. Anderson, A poor man's derivation of scaling laws ...,
J. Phys. C: Solid St. Phys. 3 (1970) 2439
6
[3] E.g. in a magnetic vortex structure of a at circular nano-dot the
vortex core has a radius, which is of the order of lexch:, see R.
Hollinger, A. Killinger, U. Krey, J. Magn. Magn. Mater. , in press.
[4] P. Bruno, Phys. Rev. B 43 (1991) 8015
[5] N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 (1966) 1133
[6] Actually, although rarely discussed, the square-root dispersion was
already known much earlier, e.g. to Ch. Kittel, or to W. Doring,
but they did apparently not yet see the consequences, e.g. those
calculated in [4].
[7] W. Doring, Z. Naturforsch. 16a (1961), 1008 (in German)
[8] C.L. Chien, R. Hasegawa, Phys. Rev. B 16 (1977) 2115; 3024;
S.N. Kaul, Phys. Rev. B 24 (1981) 6550
[9] J. Korecki, U. Gradmann, Europhys. Lett. 2 (1986) 651
[10] U. Kobler, e.g. J. Phys.: CM 14 (2002) 8861
[11] W. Gebhardt, U. Krey, Phasenubergange und Kritische Panomene,
Vieweg, Wiesbaden, (1981, in German)
[12] S. Sachdev, Quantum Phase Transitions, Cambridge University
Press (1999)
[13] In one-dimensional Heisenberg systems at T = 0 this di_erence
gives rise to the famous 'Haldane conjecture', [12].
[14] This ow line is obtained by eliminating from the original 3d partition
function Z(T;Cb) successively the high-wavenumber degrees
of freedom between kmax _ exp(_l) and kmax, where l is a positive
in_nitesimal. In this way one obtains a renormalized Hamiltonian,
i.e. _H ! _0H0, with renormalized anisotropy and renormalized
temperature, leading to the same value of Z(T;Cb), see e.g. [11].
[15] The value of T_ does practically not change if one starts the ow
not at the _xed point value Cmin
q , but at larger anisotropies.
7
[16] E.g. in d=3 dimensions the Curie temperature of a Heisenberg
model with a large value of s is / s2, whereas the largest magnon
energy is only / s. Here we only need T_ _ ~ T, where ~ T (rsp.
T_) is the upper (rsp. lower) crossover temperature of the 'Bloch
region' T_ < T < ~ T, see the remarks at the end of Appendix B.
[17] M. Przybilski, U. Gradmann, Phys. Rev. Lett. 59 (1987) 1152
Appendix A: T3=2 crossover-scaling from the quantum _xed point to
the thermal _xed point for in-plane magnetized _lms
This is a semi-quantitative 'scenario': We consider a planar cartesian
coordinate system, where along the horizontal axis ('x-axis') the (nonnegative)
reciprocal uniaxial anisotropy (Cb)_1 is plotted, whereas on
the perpendicular axis ('y-axis') the temperature T is plotted. On
the horizontal axis one has an 'ordered line segment' ranging from
(Cb)_1 = 0 up to a critical value (corresponding to a minimal necessary
uniaxial anisotropy Cmin
b , where the quantum uctuations just destroy
the magnetic order); we equate this minimal anisotropy to an (extremely
low) e_ective temperature Tx, i.e. kBTx = Cmin
b . On the outer
segment of the 'x- axis', i.e. from (Cb)_1 _ (Cmin
b )_1 to (Cb)_1 = 1,
the quantum uctuations, which are always present in the case of elliptical
spin precession, make the system paramagnetic , i.e. a 'quantum
paramagnet' in the sense of [12].
From the quantum _xed point on the 'x-axis' a ow line emerges,
[14], the 'separatrix' between the paramagnetic outer region and the
magnetic inner region of the system, i.e. this ow line joins the 'quantum
_xed point' with the 'thermal _xed point' on the y-axis, i.e. at the
temperature Tc the thermal uctuations are just su_ciently strong to
destroy the long-range order. The separatrix begins at _rst with a positive
vertical slope, but soon it turns to the left, resembling a at elliptical
segment running almost parallel to the x-axis in the direction of the
y-axis, where it extrapolates, almost horizontally, to a small crossover
temperature T_, see [15] and below. On the other hand, at the vertical
axis one has a similar scenario, namely long-range order up to the thermal
critical temperature (Curie temperature) Tc, which is of the order
of ~ T := D_(_
a )2=kB or typically even larger, [16]. From this Curie _xed
point on the 'y-axis' our (reciprocal) ow line emerges, starting at _rst
8
horizontally, but soon turning downwards almost vertically, until it extrapolates
to the horizontal axis roughly at (kBT_)_1 (_ (kBTx)_1 ),
i.e. roughly at the crossover point, where kBT_ _ qCb _ D _ (_=a)2.
Now, what counts in the above-mentioned Bloch behaviour, is on the
one-hand the thermal energy factor / T, and on the other hand the
e_ective phase space radius / T1=2 corresponding to only the vertical
segment T > T_ of the ow line between T_ and ~ T (in this region the
quasi-classical 'poor man's scaling' considerations apply). Thus one kspace
dimension has been e_ectively 'renormalized away'. For starting
points with smaller values of x the ow is not much di_erent, i.e. all
ow lines must turn around the 'edge' between the crossover point and
the origin and then turn upwards towards ~ T with the same T3=2- behaviour;
this is the essential point of the apparent universality of the
behaviour of the systems in d=2 for T_ < T < ~ T, i.e. the described
'channeling' of the ow does not appear in d=3.
Appendix B: T5=2 crossover-scaling from the quantum _xed point to
the thermal _xed point for magnetic wires (d=1)
Here the arguments are essentially the same as in Appendix A. We
consider a homogeneous ferromagnetic wire stretching from x = _1 to x = +1 along the x-axis, with constant circular cross section,
magnetized longitudinally, i.e. in the x-direction. Now the e_ective
anisotropy acts with equal strength along the y-axis and the z-axis of
the ferromagnetic wire, and the crossover point, which has an ordinatevalue
T_ (_ Tc) as before, is now at the smaller anisotropy C__
b =
(C_
b )2, where C__
b refers to the present Appendix B and C_
b to Appendix
A (we use dimensionless quantities). This means that now one has
a quadratic 'energy factor' / T2, but the same 'phase-space radius'
/ T1=2 as before in the general formula Ms(T) = Ms(0) _ const: _0
Energy Factor(T)0 _ 0Phase Space Radius(T)0, although now in the
November 21, 2002
Abstract
Simple arguments are given, related to the apparent universality with
which Bloch's famous T3=2-law, and generalizations thereof, are not
only found in d=3-dimensional ferromagnetic systems, but astonish-
ingly also in lower dimensions. It is argued that
_ one should not simply apply the usual isotropic dispersion re-
lation !(~k) = D _ ~k2 known to almost everyone but only valid
for circular precession of the spins (i.e. where only the exchange
interaction is taken into account), but instead one should con-
sider also the other interactions and use the less-known relation
!(~k) = q!a(~k) _ !b(~k) for elliptical precession,
_ one might consider the apparent universality of Bloch's T3=2-law
as some kind of 'apparent quantum universality' in a certain
_nite-temperature crossover region from a quantum phase tran-
sition at T=0,
_ one should use certain simple crossover-scaling arguments for bet-
ter understanding of the phenomena, instead of the usual more
complicated derivations by direct integration.
This is also exempli_ed for more general cases in three appendices on
the signi_cance of a crossover from a quantum phase transition.
PACS numbers: 75. Magnetic properties;
05.50 Fh Phase Transitions: General Studies;
Keywords: Bloch's law; Quantum Phase Transitions; Universality
Introduction: This letter grew out of discussions following a recent presentation
of certain experimental results in our institute, which showed
1
the T3=2 Bloch law for Ms(T) for a nanostructured planar system, [1];
and although I originally thought that the considerations presented below
were too simple for publication, some of the participants of those
discussions suggested that I should write them down. So here I do so,
just hoping to broaden and intensify more personal discussions in this
way.
Bloch's law: According to this law (which is derived in any textbook
on solid-state magnetism and is one of the most prominent results of
theoretical physics) the temperature dependence of the magnetization
Ms(T) of a three-dimensional ferromagnet is simply given by
Ms(T) _ Ms(0) _ const: _ Z
BZ
d3k
1
exp __(~k) _ 1
; (1)
because each excited magnon reduces the magnetic moment of a ferromagnetic
sample by 2 Bohr magnetons. Here T is the Kelvin temperature,
_ = (kBT)_1 with the Boltzmann constant kB, and the fraction
1
exp(__(~k)_1
represents the thermal expectation value of the number of
magnetic excitations ('magnons') with excitation energy _(~k), where
the wave-vector ~k has its usual meaning. The integration is over the
Brillouin zone BZ of the crystal.
For the excitation energy _(~k) of a magnon in a Heisenberg ferromagnet
(and also in itinerant ferromagnets when the spin-orbit interaction,
and also the Stoner excitations, are neglected against the
collective magnon-like spin excitations) one simply has (for simplicity
we assume cubic symmetry): _(~k) = D _ k2, where D is the so-called
spin-wave sti_ness. Therefore, by the substitution x := _D _ k2 and
the replacement d3k = 4_k2dk one gets the famous result Ms(T) =
Ms(0)_ const0: _ ( kBT
D )3=2, where (up to exponentially-small terms) the
constant const:0 = const: _ 2_
1R
0
x1=2dx
exp x_1 .
For Anderson's "poor man", [2], instead of the usual derivation,
one can also give the following simpler argument: exp _Dk2 _ 1 is
approximated for long enough wavelengths and/or high enough T by
the 'quasi-classical thermal-energy approximation' _D_ k2, so that one
simply gets Ms(T) = Ms(0)_const:( kBT
D )_PhsR(T), where PhsR(T)
means a typical phase-space radius in k-space, replacing the integral
2
0_=a0
R0
4_k2
k2 dk. Here 0_=a0 represents a (very large) wavenumber-cuto_
corresponding to the upper edge of the Brillouin zone, which is replaced
by a sphere as in typical renormalization group arguments. But
it would be wrong, if at this place one would perform directly the integration
(after having made the above-mentioned quasi-classical approximation
leading to the 'thermal-energy prefactor' ( kBT
D )); instead, one
gets the correct PhsR(T) by a Pippard-type argument, i.e. simply by
equating the dominating energy- resp. temperature-ranges: _(~kdom:) =
D _ k2dom: _ kBT, i.e. PhsR(T) = kdom:(T) (_= _k(T)) / ( kBT
D )1=2.
What would be di_erent in a planar system magnetized in-theplane
?
One would again expect a quasi-classical approximation exp __(~k) _
1 ! __(~k), but instead of _(~k) = D _ k2 one would have _(~k) =
q_a(~k) _ _b(~k), i.e. a pronounced elliptical precession instead of the
circular one. For example, if one is dealing with a _lm of in_nite
extension in the x- and y-directions, with _nite thickness in the zdirection,
then (if the _lm is magnetized in the x-direction) spin-wave
deviations in the z-direction are strongly disfavoured energetically, due
to the demagnetizing _eld HDM
z = _4_Mz. Thus, one would have
_b(~k) _ _z(~k) = Cb + D _ k2 (_= Cb in the long-wavelength limit, where
Cb = 2_M2
z _ V is the e_ective anisotropy energy corresponding to
the demagnetizing _eld (V is the volume of the system)). In contrast,
spin deviations in the y-direction would not be disfavoured, i.e.
_a(~k) _ _y(~k) = D _ k2, as before.
Thus for k2 _ l2
exch <_ 1, where lexch = qCb
D is an e_ective 'exchangelength',
[3], one would not have _(~k) = D_k2, but instead _(~k) = D0 _k1,
where D0 = pCb _ D.
This change of the spin-wave dispersion at small wavenumbers by
the magnetostatic _elds was already noted years ago in an early paper
of P. Bruno, [4], who stated that exactly in this way the famous
Mermin-Wagner theorem (which says that in the Heisenberg model
there cannot be magnetic long-range order in d=2 dimensions, [5]) is
invalidated by a 'cut-o_ e_ect' related to the magnetostatic interactions,
which are not considered in the theorem, [6]. However, note that
for the existence of an elliptical precession and of the planar uniaxial
3
anisotropy, we do not need a _nite thickness of the magnetic _lm, i.e.
it can also just be a monolayer.
So I suggest that in planar systems (magnetized 'in the plane' by
the simultaneous inuence of the exchange interaction and an e_ective
in-plane uniaxial anisotropy), Ms(T) should behave as
Ms(T) = Ms(0) _ const:0 _ (
kBT
D0 ) _ PhsR(T) : (2)
Here I assumed a similar 'quasi-classical thermal-energy factor' ( kBT
D0 )
as before, but now the phase-space radius PhsR(T) replaces the integral
0pi=a0
R l_1
exch
2_k1dk
k1 . Here the changes of the exponents of the nominator
and of the denominator, both exponents changing from 2 to 1, result
(for the denominator) from the fact that for k2 _ l2
exch <_ 1 the exponent
in the dispersion relation has changed fom 2 to 1, whereas the change in
the nominator comes from d2k _ k1dk. But for temperatures above a
crossover value T_ corresponding to the crossover from long-wavelength
to short-wavelength magnons (i.e. for kB T >_ kBT_ := D0 _ (_=a)),
the dominating modes are spin waves with _(~k) = Dk2, since for
k2 l2
exch >_ 1 one gets this quadratic dispersion, and so PhsR(T) should
once more be proportional to ( kBT
D )
1
2 , i.e. also PhsR(T) is essentially
unchanged. (The fact that here the system 'remembers' the long-wave
behaviour while behaving thermodynamically according to the shortwave
branch of the magnon dispersion becomes understandable in the
renormalization-group (i.e. 'ow line') scenario of Appendix A.)
As a consequence, one has again Bloch's T3=2-law: Ms(T) = Ms(0)_ const00: _ T3=2, although probably with a larger constant and only for
temperatures T > T_. (Note that T_ is very small compared with Tc.)
In fact, the T3=2 Bloch's law is not only measured for three-dimensional
amorphous ferromagnets, [8], but also for two-dimensional ultrathin
_lms, [9], which has already been noted by many people, e.g. by [10],
and now by [1] for nanostructured planar systems.
In any case, it would again be wrong to replace our simple scaling
arguments by a direct integration, which would lead to the 'Doringtype
behaviour' Ms(T) = Ms(0) _ const:00 _ kBT _
0_=a0
R l_1
exch
d2k 1
k2 , [7],
4
i.e. to Ms(T) = Ms(0) _ const:000 _ T _ ln T
T0
, where T0 is a constant
temperature unit. As far as I know, this has never been observed.
Is all this related to quantum phase transitions ?
Obviously, the detailed behaviour earns a more thorough study; fortunately,
there is a recent careful analysis of an extremely large set of
experimental results by U. Kobler, [10], which leads to an apparently
universal classi_cation for the behaviour of Ms(T), although the arguments
look quite complicated. The universality of the classi_cation
reminds to second-order phase transitions (i.e. for thermal phase transitions
there is the well-known 'Gri_ths Universality Hypothesis', see e.g.
[11]). But since here the universality is in the low- and intermediatetemperature
region and not in the vicinity of the Curie temperature,
the 'Gri_ths universality' does not apply, and one is probably dealing
with 'Quantum Phase Transitions', i.e. one is perhaps in the _nitetemperature
region of a weakly unstable 'quantum _xed point' and
should consider the renormalization-group ow-lines in a diagram joining
that 'quantum _xed point' and the thermal 'Curie' _xed point (see
the appendices). According to the recent book of S. Sachdev, [12], in
this _nite-T-region one should have a very wide range of scaling behaviour,
which would explain the wide range of applicability of Bloch's
law.
The experimental analysis of Kobler is additionally remarkable because
of the fact that his generalized 'Bloch's law exponent' _, de_ned
by the behaviour Ms(T) = Ms(0)_ const: T_, depends on whether the
spin-quantum number s of the magnetic atoms is integer or half-integer.
In a quasi-classical approximation this distinction makes no sense, but
in the context of a quantum phase transition it does. E.g. for an atom
with s = 1=2 a change m ! (m_1) would only correspond to a ip of
the sign of the local magnetization, whereas for s = 1, the corresponding
change would lead to vanishing of all local magnetic properties, see
[13].
Therefore one should reconsider the behaviour of Ms(T) in the context
of renormalization-group ow lines of quantum phase transitions;
perhaps in this way one can better understand and unify the Bloch-like
behaviour in thin _lms and related (patterned) nanostructures, [1], also
for more complicated multispin models as those advocated by U. Kobler,
see [10], by simple crossover-scaling laws for various classes of models,
5
arising from such an approach.
Some ideas concerning these points are already sketched in the appendices,
which should be considered as an integral part of this letter
and should be read before the following 'Conclusions'.
Conclusions: As a consequence of the preceding paragraphs and the
appendices, concerning the temperature range considered, we stress
that the simple arguments of the present letter, combined with semiquantitative
renormalization group scenarios in Appendix A and Appendix
B, do not consider the immediate vicinity of the quantum phase
transion at T = 0 nor that of the thermal phase transition at Tc but
rather apply for the (wide) temperature range between the (very low)
crossover temperature T_ (_ qCb _ D _ (_=a)2=kB in the case of Appendix
A ) and a (much higher) crossover temperature in the range of
~ T = D _ (_=a)2=kB. So the quantum _xed point is signi_cant, but at
the same time it only plays a secondary role.
All statements in the appendices A to C apply to half-integer s
in Kobler's classi_cation, [10]. Additional complications for integer s
(outside the region of overwelming importance of the thermal uctuations)
will be shifted to future work and to a more thorough analysis.
Acknowledgements
The author acknowledges stimulating communications from U. Kobler,
U. Gradmann and S. Krompiewski, and fruitful discussions with C.
Back, G. Bayreuther, R. Hollinger, A. Killinger, W. Kipferl, and J.
Siewert.
References
[1] W. Kipferl, Spin-wave excitations in epitaxial Fe(001)-nanostructures
on GaAs, unpublished seminar, University of Regensburg,
October 2002
[2] P.W. Anderson, A poor man's derivation of scaling laws ...,
J. Phys. C: Solid St. Phys. 3 (1970) 2439
6
[3] E.g. in a magnetic vortex structure of a at circular nano-dot the
vortex core has a radius, which is of the order of lexch:, see R.
Hollinger, A. Killinger, U. Krey, J. Magn. Magn. Mater. , in press.
[4] P. Bruno, Phys. Rev. B 43 (1991) 8015
[5] N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 (1966) 1133
[6] Actually, although rarely discussed, the square-root dispersion was
already known much earlier, e.g. to Ch. Kittel, or to W. Doring,
but they did apparently not yet see the consequences, e.g. those
calculated in [4].
[7] W. Doring, Z. Naturforsch. 16a (1961), 1008 (in German)
[8] C.L. Chien, R. Hasegawa, Phys. Rev. B 16 (1977) 2115; 3024;
S.N. Kaul, Phys. Rev. B 24 (1981) 6550
[9] J. Korecki, U. Gradmann, Europhys. Lett. 2 (1986) 651
[10] U. Kobler, e.g. J. Phys.: CM 14 (2002) 8861
[11] W. Gebhardt, U. Krey, Phasenubergange und Kritische Panomene,
Vieweg, Wiesbaden, (1981, in German)
[12] S. Sachdev, Quantum Phase Transitions, Cambridge University
Press (1999)
[13] In one-dimensional Heisenberg systems at T = 0 this di_erence
gives rise to the famous 'Haldane conjecture', [12].
[14] This ow line is obtained by eliminating from the original 3d partition
function Z(T;Cb) successively the high-wavenumber degrees
of freedom between kmax _ exp(_l) and kmax, where l is a positive
in_nitesimal. In this way one obtains a renormalized Hamiltonian,
i.e. _H ! _0H0, with renormalized anisotropy and renormalized
temperature, leading to the same value of Z(T;Cb), see e.g. [11].
[15] The value of T_ does practically not change if one starts the ow
not at the _xed point value Cmin
q , but at larger anisotropies.
7
[16] E.g. in d=3 dimensions the Curie temperature of a Heisenberg
model with a large value of s is / s2, whereas the largest magnon
energy is only / s. Here we only need T_ _ ~ T, where ~ T (rsp.
T_) is the upper (rsp. lower) crossover temperature of the 'Bloch
region' T_ < T < ~ T, see the remarks at the end of Appendix B.
[17] M. Przybilski, U. Gradmann, Phys. Rev. Lett. 59 (1987) 1152
Appendix A: T3=2 crossover-scaling from the quantum _xed point to
the thermal _xed point for in-plane magnetized _lms
This is a semi-quantitative 'scenario': We consider a planar cartesian
coordinate system, where along the horizontal axis ('x-axis') the (nonnegative)
reciprocal uniaxial anisotropy (Cb)_1 is plotted, whereas on
the perpendicular axis ('y-axis') the temperature T is plotted. On
the horizontal axis one has an 'ordered line segment' ranging from
(Cb)_1 = 0 up to a critical value (corresponding to a minimal necessary
uniaxial anisotropy Cmin
b , where the quantum uctuations just destroy
the magnetic order); we equate this minimal anisotropy to an (extremely
low) e_ective temperature Tx, i.e. kBTx = Cmin
b . On the outer
segment of the 'x- axis', i.e. from (Cb)_1 _ (Cmin
b )_1 to (Cb)_1 = 1,
the quantum uctuations, which are always present in the case of elliptical
spin precession, make the system paramagnetic , i.e. a 'quantum
paramagnet' in the sense of [12].
From the quantum _xed point on the 'x-axis' a ow line emerges,
[14], the 'separatrix' between the paramagnetic outer region and the
magnetic inner region of the system, i.e. this ow line joins the 'quantum
_xed point' with the 'thermal _xed point' on the y-axis, i.e. at the
temperature Tc the thermal uctuations are just su_ciently strong to
destroy the long-range order. The separatrix begins at _rst with a positive
vertical slope, but soon it turns to the left, resembling a at elliptical
segment running almost parallel to the x-axis in the direction of the
y-axis, where it extrapolates, almost horizontally, to a small crossover
temperature T_, see [15] and below. On the other hand, at the vertical
axis one has a similar scenario, namely long-range order up to the thermal
critical temperature (Curie temperature) Tc, which is of the order
of ~ T := D_(_
a )2=kB or typically even larger, [16]. From this Curie _xed
point on the 'y-axis' our (reciprocal) ow line emerges, starting at _rst
8
horizontally, but soon turning downwards almost vertically, until it extrapolates
to the horizontal axis roughly at (kBT_)_1 (_ (kBTx)_1 ),
i.e. roughly at the crossover point, where kBT_ _ qCb _ D _ (_=a)2.
Now, what counts in the above-mentioned Bloch behaviour, is on the
one-hand the thermal energy factor / T, and on the other hand the
e_ective phase space radius / T1=2 corresponding to only the vertical
segment T > T_ of the ow line between T_ and ~ T (in this region the
quasi-classical 'poor man's scaling' considerations apply). Thus one kspace
dimension has been e_ectively 'renormalized away'. For starting
points with smaller values of x the ow is not much di_erent, i.e. all
ow lines must turn around the 'edge' between the crossover point and
the origin and then turn upwards towards ~ T with the same T3=2- behaviour;
this is the essential point of the apparent universality of the
behaviour of the systems in d=2 for T_ < T < ~ T, i.e. the described
'channeling' of the ow does not appear in d=3.
Appendix B: T5=2 crossover-scaling from the quantum _xed point to
the thermal _xed point for magnetic wires (d=1)
Here the arguments are essentially the same as in Appendix A. We
consider a homogeneous ferromagnetic wire stretching from x = _1 to x = +1 along the x-axis, with constant circular cross section,
magnetized longitudinally, i.e. in the x-direction. Now the e_ective
anisotropy acts with equal strength along the y-axis and the z-axis of
the ferromagnetic wire, and the crossover point, which has an ordinatevalue
T_ (_ Tc) as before, is now at the smaller anisotropy C__
b =
(C_
b )2, where C__
b refers to the present Appendix B and C_
b to Appendix
A (we use dimensionless quantities). This means that now one has
a quadratic 'energy factor' / T2, but the same 'phase-space radius'
/ T1=2 as before in the general formula Ms(T) = Ms(0) _ const: _0
Energy Factor(T)0 _ 0Phase Space Radius(T)0, although now in the
November 21, 2002
Abstract
Simple arguments are given, related to the apparent universality with
which Bloch's famous T3=2-law, and generalizations thereof, are not
only found in d=3-dimensional ferromagnetic systems, but astonish-
ingly also in lower dimensions. It is argued that
_ one should not simply apply the usual isotropic dispersion re-
lation !(~k) = D _ ~k2 known to almost everyone but only valid
for circular precession of the spins (i.e. where only the exchange
interaction is taken into account), but instead one should con-
sider also the other interactions and use the less-known relation
!(~k) = q!a(~k) _ !b(~k) for elliptical precession,
_ one might consider the apparent universality of Bloch's T3=2-law
as some kind of 'apparent quantum universality' in a certain
_nite-temperature crossover region from a quantum phase tran-
sition at T=0,
_ one should use certain simple crossover-scaling arguments for bet-
ter understanding of the phenomena, instead of the usual more
complicated derivations by direct integration.
This is also exempli_ed for more general cases in three appendices on
the signi_cance of a crossover from a quantum phase transition.
PACS numbers: 75. Magnetic properties;
05.50 Fh Phase Transitions: General Studies;
Keywords: Bloch's law; Quantum Phase Transitions; Universality
Introduction: This letter grew out of discussions following a recent presentation
of certain experimental results in our institute, which showed
1
the T3=2 Bloch law for Ms(T) for a nanostructured planar system, [1];
and although I originally thought that the considerations presented below
were too simple for publication, some of the participants of those
discussions suggested that I should write them down. So here I do so,
just hoping to broaden and intensify more personal discussions in this
way.
Bloch's law: According to this law (which is derived in any textbook
on solid-state magnetism and is one of the most prominent results of
theoretical physics) the temperature dependence of the magnetization
Ms(T) of a three-dimensional ferromagnet is simply given by
Ms(T) _ Ms(0) _ const: _ Z
BZ
d3k
1
exp __(~k) _ 1
; (1)
because each excited magnon reduces the magnetic moment of a ferromagnetic
sample by 2 Bohr magnetons. Here T is the Kelvin temperature,
_ = (kBT)_1 with the Boltzmann constant kB, and the fraction
1
exp(__(~k)_1
represents the thermal expectation value of the number of
magnetic excitations ('magnons') with excitation energy _(~k), where
the wave-vector ~k has its usual meaning. The integration is over the
Brillouin zone BZ of the crystal.
For the excitation energy _(~k) of a magnon in a Heisenberg ferromagnet
(and also in itinerant ferromagnets when the spin-orbit interaction,
and also the Stoner excitations, are neglected against the
collective magnon-like spin excitations) one simply has (for simplicity
we assume cubic symmetry): _(~k) = D _ k2, where D is the so-called
spin-wave sti_ness. Therefore, by the substitution x := _D _ k2 and
the replacement d3k = 4_k2dk one gets the famous result Ms(T) =
Ms(0)_ const0: _ ( kBT
D )3=2, where (up to exponentially-small terms) the
constant const:0 = const: _ 2_
1R
0
x1=2dx
exp x_1 .
For Anderson's "poor man", [2], instead of the usual derivation,
one can also give the following simpler argument: exp _Dk2 _ 1 is
approximated for long enough wavelengths and/or high enough T by
the 'quasi-classical thermal-energy approximation' _D_ k2, so that one
simply gets Ms(T) = Ms(0)_const:( kBT
D )_PhsR(T), where PhsR(T)
means a typical phase-space radius in k-space, replacing the integral
2
0_=a0
R0
4_k2
k2 dk. Here 0_=a0 represents a (very large) wavenumber-cuto_
corresponding to the upper edge of the Brillouin zone, which is replaced
by a sphere as in typical renormalization group arguments. But
it would be wrong, if at this place one would perform directly the integration
(after having made the above-mentioned quasi-classical approximation
leading to the 'thermal-energy prefactor' ( kBT
D )); instead, one
gets the correct PhsR(T) by a Pippard-type argument, i.e. simply by
equating the dominating energy- resp. temperature-ranges: _(~kdom:) =
D _ k2dom: _ kBT, i.e. PhsR(T) = kdom:(T) (_= _k(T)) / ( kBT
D )1=2.
What would be di_erent in a planar system magnetized in-theplane
?
One would again expect a quasi-classical approximation exp __(~k) _
1 ! __(~k), but instead of _(~k) = D _ k2 one would have _(~k) =
q_a(~k) _ _b(~k), i.e. a pronounced elliptical precession instead of the
circular one. For example, if one is dealing with a _lm of in_nite
extension in the x- and y-directions, with _nite thickness in the zdirection,
then (if the _lm is magnetized in the x-direction) spin-wave
deviations in the z-direction are strongly disfavoured energetically, due
to the demagnetizing _eld HDM
z = _4_Mz. Thus, one would have
_b(~k) _ _z(~k) = Cb + D _ k2 (_= Cb in the long-wavelength limit, where
Cb = 2_M2
z _ V is the e_ective anisotropy energy corresponding to
the demagnetizing _eld (V is the volume of the system)). In contrast,
spin deviations in the y-direction would not be disfavoured, i.e.
_a(~k) _ _y(~k) = D _ k2, as before.
Thus for k2 _ l2
exch <_ 1, where lexch = qCb
D is an e_ective 'exchangelength',
[3], one would not have _(~k) = D_k2, but instead _(~k) = D0 _k1,
where D0 = pCb _ D.
This change of the spin-wave dispersion at small wavenumbers by
the magnetostatic _elds was already noted years ago in an early paper
of P. Bruno, [4], who stated that exactly in this way the famous
Mermin-Wagner theorem (which says that in the Heisenberg model
there cannot be magnetic long-range order in d=2 dimensions, [5]) is
invalidated by a 'cut-o_ e_ect' related to the magnetostatic interactions,
which are not considered in the theorem, [6]. However, note that
for the existence of an elliptical precession and of the planar uniaxial
3
anisotropy, we do not need a _nite thickness of the magnetic _lm, i.e.
it can also just be a monolayer.
So I suggest that in planar systems (magnetized 'in the plane' by
the simultaneous inuence of the exchange interaction and an e_ective
in-plane uniaxial anisotropy), Ms(T) should behave as
Ms(T) = Ms(0) _ const:0 _ (
kBT
D0 ) _ PhsR(T) : (2)
Here I assumed a similar 'quasi-classical thermal-energy factor' ( kBT
D0 )
as before, but now the phase-space radius PhsR(T) replaces the integral
0pi=a0
R l_1
exch
2_k1dk
k1 . Here the changes of the exponents of the nominator
and of the denominator, both exponents changing from 2 to 1, result
(for the denominator) from the fact that for k2 _ l2
exch <_ 1 the exponent
in the dispersion relation has changed fom 2 to 1, whereas the change in
the nominator comes from d2k _ k1dk. But for temperatures above a
crossover value T_ corresponding to the crossover from long-wavelength
to short-wavelength magnons (i.e. for kB T >_ kBT_ := D0 _ (_=a)),
the dominating modes are spin waves with _(~k) = Dk2, since for
k2 l2
exch >_ 1 one gets this quadratic dispersion, and so PhsR(T) should
once more be proportional to ( kBT
D )
1
2 , i.e. also PhsR(T) is essentially
unchanged. (The fact that here the system 'remembers' the long-wave
behaviour while behaving thermodynamically according to the shortwave
branch of the magnon dispersion becomes understandable in the
renormalization-group (i.e. 'ow line') scenario of Appendix A.)
As a consequence, one has again Bloch's T3=2-law: Ms(T) = Ms(0)_ const00: _ T3=2, although probably with a larger constant and only for
temperatures T > T_. (Note that T_ is very small compared with Tc.)
In fact, the T3=2 Bloch's law is not only measured for three-dimensional
amorphous ferromagnets, [8], but also for two-dimensional ultrathin
_lms, [9], which has already been noted by many people, e.g. by [10],
and now by [1] for nanostructured planar systems.
In any case, it would again be wrong to replace our simple scaling
arguments by a direct integration, which would lead to the 'Doringtype
behaviour' Ms(T) = Ms(0) _ const:00 _ kBT _
0_=a0
R l_1
exch
d2k 1
k2 , [7],
4
i.e. to Ms(T) = Ms(0) _ const:000 _ T _ ln T
T0
, where T0 is a constant
temperature unit. As far as I know, this has never been observed.
Is all this related to quantum phase transitions ?
Obviously, the detailed behaviour earns a more thorough study; fortunately,
there is a recent careful analysis of an extremely large set of
experimental results by U. Kobler, [10], which leads to an apparently
universal classi_cation for the behaviour of Ms(T), although the arguments
look quite complicated. The universality of the classi_cation
reminds to second-order phase transitions (i.e. for thermal phase transitions
there is the well-known 'Gri_ths Universality Hypothesis', see e.g.
[11]). But since here the universality is in the low- and intermediatetemperature
region and not in the vicinity of the Curie temperature,
the 'Gri_ths universality' does not apply, and one is probably dealing
with 'Quantum Phase Transitions', i.e. one is perhaps in the _nitetemperature
region of a weakly unstable 'quantum _xed point' and
should consider the renormalization-group ow-lines in a diagram joining
that 'quantum _xed point' and the thermal 'Curie' _xed point (see
the appendices). According to the recent book of S. Sachdev, [12], in
this _nite-T-region one should have a very wide range of scaling behaviour,
which would explain the wide range of applicability of Bloch's
law.
The experimental analysis of Kobler is additionally remarkable because
of the fact that his generalized 'Bloch's law exponent' _, de_ned
by the behaviour Ms(T) = Ms(0)_ const: T_, depends on whether the
spin-quantum number s of the magnetic atoms is integer or half-integer.
In a quasi-classical approximation this distinction makes no sense, but
in the context of a quantum phase transition it does. E.g. for an atom
with s = 1=2 a change m ! (m_1) would only correspond to a ip of
the sign of the local magnetization, whereas for s = 1, the corresponding
change would lead to vanishing of all local magnetic properties, see
[13].
Therefore one should reconsider the behaviour of Ms(T) in the context
of renormalization-group ow lines of quantum phase transitions;
perhaps in this way one can better understand and unify the Bloch-like
behaviour in thin _lms and related (patterned) nanostructures, [1], also
for more complicated multispin models as those advocated by U. Kobler,
see [10], by simple crossover-scaling laws for various classes of models,
5
arising from such an approach.
Some ideas concerning these points are already sketched in the appendices,
which should be considered as an integral part of this letter
and should be read before the following 'Conclusions'.
Conclusions: As a consequence of the preceding paragraphs and the
appendices, concerning the temperature range considered, we stress
that the simple arguments of the present letter, combined with semiquantitative
renormalization group scenarios in Appendix A and Appendix
B, do not consider the immediate vicinity of the quantum phase
transion at T = 0 nor that of the thermal phase transition at Tc but
rather apply for the (wide) temperature range between the (very low)
crossover temperature T_ (_ qCb _ D _ (_=a)2=kB in the case of Appendix
A ) and a (much higher) crossover temperature in the range of
~ T = D _ (_=a)2=kB. So the quantum _xed point is signi_cant, but at
the same time it only plays a secondary role.
All statements in the appendices A to C apply to half-integer s
in Kobler's classi_cation, [10]. Additional complications for integer s
(outside the region of overwelming importance of the thermal uctuations)
will be shifted to future work and to a more thorough analysis.
Acknowledgements
The author acknowledges stimulating communications from U. Kobler,
U. Gradmann and S. Krompiewski, and fruitful discussions with C.
Back, G. Bayreuther, R. Hollinger, A. Killinger, W. Kipferl, and J.
Siewert.
References
[1] W. Kipferl, Spin-wave excitations in epitaxial Fe(001)-nanostructures
on GaAs, unpublished seminar, University of Regensburg,
October 2002
[2] P.W. Anderson, A poor man's derivation of scaling laws ...,
J. Phys. C: Solid St. Phys. 3 (1970) 2439
6
[3] E.g. in a magnetic vortex structure of a at circular nano-dot the
vortex core has a radius, which is of the order of lexch:, see R.
Hollinger, A. Killinger, U. Krey, J. Magn. Magn. Mater. , in press.
[4] P. Bruno, Phys. Rev. B 43 (1991) 8015
[5] N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 (1966) 1133
[6] Actually, although rarely discussed, the square-root dispersion was
already known much earlier, e.g. to Ch. Kittel, or to W. Doring,
but they did apparently not yet see the consequences, e.g. those
calculated in [4].
[7] W. Doring, Z. Naturforsch. 16a (1961), 1008 (in German)
[8] C.L. Chien, R. Hasegawa, Phys. Rev. B 16 (1977) 2115; 3024;
S.N. Kaul, Phys. Rev. B 24 (1981) 6550
[9] J. Korecki, U. Gradmann, Europhys. Lett. 2 (1986) 651
[10] U. Kobler, e.g. J. Phys.: CM 14 (2002) 8861
[11] W. Gebhardt, U. Krey, Phasenubergange und Kritische Panomene,
Vieweg, Wiesbaden, (1981, in German)
[12] S. Sachdev, Quantum Phase Transitions, Cambridge University
Press (1999)
[13] In one-dimensional Heisenberg systems at T = 0 this di_erence
gives rise to the famous 'Haldane conjecture', [12].
[14] This ow line is obtained by eliminating from the original 3d partition
function Z(T;Cb) successively the high-wavenumber degrees
of freedom between kmax _ exp(_l) and kmax, where l is a positive
in_nitesimal. In this way one obtains a renormalized Hamiltonian,
i.e. _H ! _0H0, with renormalized anisotropy and renormalized
temperature, leading to the same value of Z(T;Cb), see e.g. [11].
[15] The value of T_ does practically not change if one starts the ow
not at the _xed point value Cmin
q , but at larger anisotropies.
7
[16] E.g. in d=3 dimensions the Curie temperature of a Heisenberg
model with a large value of s is / s2, whereas the largest magnon
energy is only / s. Here we only need T_ _ ~ T, where ~ T (rsp.
T_) is the upper (rsp. lower) crossover temperature of the 'Bloch
region' T_ < T < ~ T, see the remarks at the end of Appendix B.
[17] M. Przybilski, U. Gradmann, Phys. Rev. Lett. 59 (1987) 1152
Appendix A: T3=2 crossover-scaling from the quantum _xed point to
the thermal _xed point for in-plane magnetized _lms
This is a semi-quantitative 'scenario': We consider a planar cartesian
coordinate system, where along the horizontal axis ('x-axis') the (nonnegative)
reciprocal uniaxial anisotropy (Cb)_1 is plotted, whereas on
the perpendicular axis ('y-axis') the temperature T is plotted. On
the horizontal axis one has an 'ordered line segment' ranging from
(Cb)_1 = 0 up to a critical value (corresponding to a minimal necessary
uniaxial anisotropy Cmin
b , where the quantum uctuations just destroy
the magnetic order); we equate this minimal anisotropy to an (extremely
low) e_ective temperature Tx, i.e. kBTx = Cmin
b . On the outer
segment of the 'x- axis', i.e. from (Cb)_1 _ (Cmin
b )_1 to (Cb)_1 = 1,
the quantum uctuations, which are always present in the case of elliptical
spin precession, make the system paramagnetic , i.e. a 'quantum
paramagnet' in the sense of [12].
From the quantum _xed point on the 'x-axis' a ow line emerges,
[14], the 'separatrix' between the paramagnetic outer region and the
magnetic inner region of the system, i.e. this ow line joins the 'quantum
_xed point' with the 'thermal _xed point' on the y-axis, i.e. at the
temperature Tc the thermal uctuations are just su_ciently strong to
destroy the long-range order. The separatrix begins at _rst with a positive
vertical slope, but soon it turns to the left, resembling a at elliptical
segment running almost parallel to the x-axis in the direction of the
y-axis, where it extrapolates, almost horizontally, to a small crossover
temperature T_, see [15] and below. On the other hand, at the vertical
axis one has a similar scenario, namely long-range order up to the thermal
critical temperature (Curie temperature) Tc, which is of the order
of ~ T := D_(_
a )2=kB or typically even larger, [16]. From this Curie _xed
point on the 'y-axis' our (reciprocal) ow line emerges, starting at _rst
8
horizontally, but soon turning downwards almost vertically, until it extrapolates
to the horizontal axis roughly at (kBT_)_1 (_ (kBTx)_1 ),
i.e. roughly at the crossover point, where kBT_ _ qCb _ D _ (_=a)2.
Now, what counts in the above-mentioned Bloch behaviour, is on the
one-hand the thermal energy factor / T, and on the other hand the
e_ective phase space radius / T1=2 corresponding to only the vertical
segment T > T_ of the ow line between T_ and ~ T (in this region the
quasi-classical 'poor man's scaling' considerations apply). Thus one kspace
dimension has been e_ectively 'renormalized away'. For starting
points with smaller values of x the ow is not much di_erent, i.e. all
ow lines must turn around the 'edge' between the crossover point and
the origin and then turn upwards towards ~ T with the same T3=2- behaviour;
this is the essential point of the apparent universality of the
behaviour of the systems in d=2 for T_ < T < ~ T, i.e. the described
'channeling' of the ow does not appear in d=3.
Appendix B: T5=2 crossover-scaling from the quantum _xed point to
the thermal _xed point for magnetic wires (d=1)
Here the arguments are essentially the same as in Appendix A. We
consider a homogeneous ferromagnetic wire stretching from x = _1 to x = +1 along the x-axis, with constant circular cross section,
magnetized longitudinally, i.e. in the x-direction. Now the e_ective
anisotropy acts with equal strength along the y-axis and the z-axis of
the ferromagnetic wire, and the crossover point, which has an ordinatevalue
T_ (_ Tc) as before, is now at the smaller anisotropy C__
b =
(C_
b )2, where C__
b refers to the present Appendix B and C_
b to Appendix
A (we use dimensionless quantities). This means that now one has
a quadratic 'energy factor' / T2, but the same 'phase-space radius'
/ T1=2 as before in the general formula Ms(T) = Ms(0) _ const: _0
Energy Factor(T)0 _ 0Phase Space Radius(T)0, although now in the
November 21, 2002
Abstract
Simple arguments are given, related to the apparent universality with
which Bloch's famous T3=2-law, and generalizations thereof, are not
only found in d=3-dimensional ferromagnetic systems, but astonish-
ingly also in lower dimensions. It is argued that
_ one should not simply apply the usual isotropic dispersion re-
lation !(~k) = D _ ~k2 known to almost everyone but only valid
for circular precession of the spins (i.e. where only the exchange
interaction is taken into account), but instead one should con-
sider also the other interactions and use the less-known relation
!(~k) = q!a(~k) _ !b(~k) for elliptical precession,
_ one might consider the apparent universality of Bloch's T3=2-law
as some kind of 'apparent quantum universality' in a certain
_nite-temperature crossover region from a quantum phase tran-
sition at T=0,
_ one should use certain simple crossover-scaling arguments for bet-
ter understanding of the phenomena, instead of the usual more
complicated derivations by direct integration.
This is also exempli_ed for more general cases in three appendices on
the signi_cance of a crossover from a quantum phase transition.
PACS numbers: 75. Magnetic properties;
05.50 Fh Phase Transitions: General Studies;
Keywords: Bloch's law; Quantum Phase Transitions; Universality
Introduction: This letter grew out of discussions following a recent presentation
of certain experimental results in our institute, which showed
1
the T3=2 Bloch law for Ms(T) for a nanostructured planar system, [1];
and although I originally thought that the considerations presented below
were too simple for publication, some of the participants of those
discussions suggested that I should write them down. So here I do so,
just hoping to broaden and intensify more personal discussions in this
way.
Bloch's law: According to this law (which is derived in any textbook
on solid-state magnetism and is one of the most prominent results of
theoretical physics) the temperature dependence of the magnetization
Ms(T) of a three-dimensional ferromagnet is simply given by
Ms(T) _ Ms(0) _ const: _ Z
BZ
d3k
1
exp __(~k) _ 1
; (1)
because each excited magnon reduces the magnetic moment of a ferromagnetic
sample by 2 Bohr magnetons. Here T is the Kelvin temperature,
_ = (kBT)_1 with the Boltzmann constant kB, and the fraction
1
exp(__(~k)_1
represents the thermal expectation value of the number of
magnetic excitations ('magnons') with excitation energy _(~k), where
the wave-vector ~k has its usual meaning. The integration is over the
Brillouin zone BZ of the crystal.
For the excitation energy _(~k) of a magnon in a Heisenberg ferromagnet
(and also in itinerant ferromagnets when the spin-orbit interaction,
and also the Stoner excitations, are neglected against the
collective magnon-like spin excitations) one simply has (for simplicity
we assume cubic symmetry): _(~k) = D _ k2, where D is the so-called
spin-wave sti_ness. Therefore, by the substitution x := _D _ k2 and
the replacement d3k = 4_k2dk one gets the famous result Ms(T) =
Ms(0)_ const0: _ ( kBT
D )3=2, where (up to exponentially-small terms) the
constant const:0 = const: _ 2_
1R
0
x1=2dx
exp x_1 .
For Anderson's "poor man", [2], instead of the usual derivation,
one can also give the following simpler argument: exp _Dk2 _ 1 is
approximated for long enough wavelengths and/or high enough T by
the 'quasi-classical thermal-energy approximation' _D_ k2, so that one
simply gets Ms(T) = Ms(0)_const:( kBT
D )_PhsR(T), where PhsR(T)
means a typical phase-space radius in k-space, replacing the integral
2
0_=a0
R0
4_k2
k2 dk. Here 0_=a0 represents a (very large) wavenumber-cuto_
corresponding to the upper edge of the Brillouin zone, which is replaced
by a sphere as in typical renormalization group arguments. But
it would be wrong, if at this place one would perform directly the integration
(after having made the above-mentioned quasi-classical approximation
leading to the 'thermal-energy prefactor' ( kBT
D )); instead, one
gets the correct PhsR(T) by a Pippard-type argument, i.e. simply by
equating the dominating energy- resp. temperature-ranges: _(~kdom:) =
D _ k2dom: _ kBT, i.e. PhsR(T) = kdom:(T) (_= _k(T)) / ( kBT
D )1=2.
What would be di_erent in a planar system magnetized in-theplane
?
One would again expect a quasi-classical approximation exp __(~k) _
1 ! __(~k), but instead of _(~k) = D _ k2 one would have _(~k) =
q_a(~k) _ _b(~k), i.e. a pronounced elliptical precession instead of the
circular one. For example, if one is dealing with a _lm of in_nite
extension in the x- and y-directions, with _nite thickness in the zdirection,
then (if the _lm is magnetized in the x-direction) spin-wave
deviations in the z-direction are strongly disfavoured energetically, due
to the demagnetizing _eld HDM
z = _4_Mz. Thus, one would have
_b(~k) _ _z(~k) = Cb + D _ k2 (_= Cb in the long-wavelength limit, where
Cb = 2_M2
z _ V is the e_ective anisotropy energy corresponding to
the demagnetizing _eld (V is the volume of the system)). In contrast,
spin deviations in the y-direction would not be disfavoured, i.e.
_a(~k) _ _y(~k) = D _ k2, as before.
Thus for k2 _ l2
exch <_ 1, where lexch = qCb
D is an e_ective 'exchangelength',
[3], one would not have _(~k) = D_k2, but instead _(~k) = D0 _k1,
where D0 = pCb _ D.
This change of the spin-wave dispersion at small wavenumbers by
the magnetostatic _elds was already noted years ago in an early paper
of P. Bruno, [4], who stated that exactly in this way the famous
Mermin-Wagner theorem (which says that in the Heisenberg model
there cannot be magnetic long-range order in d=2 dimensions, [5]) is
invalidated by a 'cut-o_ e_ect' related to the magnetostatic interactions,
which are not considered in the theorem, [6]. However, note that
for the existence of an elliptical precession and of the planar uniaxial
3
anisotropy, we do not need a _nite thickness of the magnetic _lm, i.e.
it can also just be a monolayer.
So I suggest that in planar systems (magnetized 'in the plane' by
the simultaneous inuence of the exchange interaction and an e_ective
in-plane uniaxial anisotropy), Ms(T) should behave as
Ms(T) = Ms(0) _ const:0 _ (
kBT
D0 ) _ PhsR(T) : (2)
Here I assumed a similar 'quasi-classical thermal-energy factor' ( kBT
D0 )
as before, but now the phase-space radius PhsR(T) replaces the integral
0pi=a0
R l_1
exch
2_k1dk
k1 . Here the changes of the exponents of the nominator
and of the denominator, both exponents changing from 2 to 1, result
(for the denominator) from the fact that for k2 _ l2
exch <_ 1 the exponent
in the dispersion relation has changed fom 2 to 1, whereas the change in
the nominator comes from d2k _ k1dk. But for temperatures above a
crossover value T_ corresponding to the crossover from long-wavelength
to short-wavelength magnons (i.e. for kB T >_ kBT_ := D0 _ (_=a)),
the dominating modes are spin waves with _(~k) = Dk2, since for
k2 l2
exch >_ 1 one gets this quadratic dispersion, and so PhsR(T) should
once more be proportional to ( kBT
D )
1
2 , i.e. also PhsR(T) is essentially
unchanged. (The fact that here the system 'remembers' the long-wave
behaviour while behaving thermodynamically according to the shortwave
branch of the magnon dispersion becomes understandable in the
renormalization-group (i.e. 'ow line') scenario of Appendix A.)
As a consequence, one has again Bloch's T3=2-law: Ms(T) = Ms(0)_ const00: _ T3=2, although probably with a larger constant and only for
temperatures T > T_. (Note that T_ is very small compared with Tc.)
In fact, the T3=2 Bloch's law is not only measured for three-dimensional
amorphous ferromagnets, [8], but also for two-dimensional ultrathin
_lms, [9], which has already been noted by many people, e.g. by [10],
and now by [1] for nanostructured planar systems.
In any case, it would again be wrong to replace our simple scaling
arguments by a direct integration, which would lead to the 'Doringtype
behaviour' Ms(T) = Ms(0) _ const:00 _ kBT _
0_=a0
R l_1
exch
d2k 1
k2 , [7],
4
i.e. to Ms(T) = Ms(0) _ const:000 _ T _ ln T
T0
, where T0 is a constant
temperature unit. As far as I know, this has never been observed.
Is all this related to quantum phase transitions ?
Obviously, the detailed behaviour earns a more thorough study; fortunately,
there is a recent careful analysis of an extremely large set of
experimental results by U. Kobler, [10], which leads to an apparently
universal classi_cation for the behaviour of Ms(T), although the arguments
look quite complicated. The universality of the classi_cation
reminds to second-order phase transitions (i.e. for thermal phase transitions
there is the well-known 'Gri_ths Universality Hypothesis', see e.g.
[11]). But since here the universality is in the low- and intermediatetemperature
region and not in the vicinity of the Curie temperature,
the 'Gri_ths universality' does not apply, and one is probably dealing
with 'Quantum Phase Transitions', i.e. one is perhaps in the _nitetemperature
region of a weakly unstable 'quantum _xed point' and
should consider the renormalization-group ow-lines in a diagram joining
that 'quantum _xed point' and the thermal 'Curie' _xed point (see
the appendices). According to the recent book of S. Sachdev, [12], in
this _nite-T-region one should have a very wide range of scaling behaviour,
which would explain the wide range of applicability of Bloch's
law.
The experimental analysis of Kobler is additionally remarkable because
of the fact that his generalized 'Bloch's law exponent' _, de_ned
by the behaviour Ms(T) = Ms(0)_ const: T_, depends on whether the
spin-quantum number s of the magnetic atoms is integer or half-integer.
In a quasi-classical approximation this distinc
 
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