arXiv:cond-mat/0211479 v1 21 Nov 2002

On the signi_cance of quantum phase transitions for

the apparent universality of Bloch laws for Ms(T)

U. Krey_

Inst. fur Physik II, Universitat Regensburg, 93040 Regensburg, Germany

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

relation _(kdom

= D _ k2dom: = kBT, de_ning the phase-space radius,

the vector ~kdom: has only one independent component instead of two in

Appendix A.

In any case, the resulting T5=2 dependence is in agreement with

the classi_cation of U. Kobler for half-integer s, [10], and with experiments

by U. Gradmann and coworkers, [17]. (The _lms in [17] are

two-dimensional, as those in [9], but whereas in [9] one has the T3=2

9

behaviour of Appendix A, for the _lms of [17] one has the T5=2 behaviour

of Appendix B, which means that the anisotropies now make

the magnetic behaviour of the system after the crossover e_ectively

one-dimensional. In the critical region around Tc the e_ective dimension

should return to the value 2).

Actually the magnetic behaviour of the longitudinally magnetized

wire is even more complicated. On the one hand, due to the fact

that now the dispersion is circular, there are no quantum uctuations

to destroy the long-range order at T = 0, so the minimal anisotropy

Cmin

b is 0+, and [15] plus the remark at the end of Appendix A comes

into play for the (ordered) quantum phase. On the other hand, i.e.

concerning the 'thermal behavior' near Tc, it is known that there is

no long range order at _nite T in a one-dimensional Ising-class model,

because along the wire there are too many 'kink excitations', i.e. too

many domain walls separating x-regions with opposite orientation of

the magnetization. Here our previous remark, [16], on the di_erence

between the upper value of the spin wave dispersion and the critical

value of kBTc comes into play: the ow line considered should really

move almost vertically upwards from the (lower) crossover point T_ up

to the (upper) crossover temperature, which should be of the order of

D_ (_

a )2=kB, as already mentioned. From the 'upper crossover point' it

should turn steeply downwards to the origin, i.e. to the genuine critical

temperature Tc = 0+. But this 'downturn region' (or 'upturn region' in

case of U. Gradmann's T5=2-_lms) is experimentally outside the range

of the 'Bloch behaviour', where one observes this T5=2 law.

Also two-dimensional systems with a strong uniaxial perpendicular

anisotropy should belong the this 'quasi-universality class' with Bloch

exponent 5/2. In contrast, in the particular case of a 'reorientation'

transition at T = 0, corresponding to a situation, where the e_ective

uniaxial anisotropy is exactly zero, if no additional anisotropies are

present, the original Mermin-Wagner theorem becomes applicable and

the magnetic order disappears: In practice, due to the magnetostatic

interaction, a nontrivial domain structure appears.

Appendix C: T2 crossover-scaling from the quantum _xed point to

the thermal _xed point for itinerant magnets in d=3

The following arguments are especially simple. We consider a crystalline

metallic ferromagnet in d=3 dimensions. This system has sepa-

10

rate energy bands fE"(~k)g rsp. fE#(~k)g, and at T = 0 (among others)

the 'single spin-ip Stoner excitations', i.e. electron-hole excitations

with spin ip, where e.g. an electron with energy 'immediately below

the Fermi energy EF ', with initial-state wave-vector ~k and spin ", is

moved to a state 'immediately above EF ', with a di_erent _nal-state

wave-vector ~k 0 and spin #. Now at _nite T, the expression 'immediately

above or below EF ' means: 'within an interval of width _ kBT around

EF . Every such excitation reduces the magnetic moment of the sample

by two Bohr magnetons; i.e. in the above-mentioned 'general formula'

one has a 'phase-space factor' / (kBT)2, where one of the two powers of

kBT stands for the e_ective phase-space of the initial states, while the

other one represents the _nal states. There is no 'energy factor', since

all this happens in the above-mentioned interval immediately near EF .

Similar arguments also apply to itinerant antiferromagnets, in accordance

with the observations of U. Kobler, [10], who always observed

the same Bloch power for ferromagnets and antiferromagnets.

(But hithertoo I have not found similar arguments for semiconducting

magnets, e.g. for EuO.) In contrast, the collective magnon excitations,

which are also present in itinerant magnets and which have the same

quadratic dispersion as before, seem to give a neglegible contribution to

the behaviour of Ms(T) in the temperature range considered, although

at very low temperatures a T3=2-contribution would always be larger

than a T2-one. So this means again that one should remain above a

certain crossover-temperature, in this time that one to the dominance

of the T3=2 behaviour.

11arXiv:cond-mat/0211479 v1 21 Nov 2002

On the signi_cance of quantum phase transitions for

the apparent universality of Bloch laws for Ms(T)

U. Krey_

Inst. fur Physik II, Universitat Regensburg, 93040 Regensburg, Germany

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

relation _(kdom

= D _ k2dom: = kBT, de_ning the phase-space radius,

the vector ~kdom: has only one independent component instead of two in

Appendix A.

In any case, the resulting T5=2 dependence is in agreement with

the classi_cation of U. Kobler for half-integer s, [10], and with experiments

by U. Gradmann and coworkers, [17]. (The _lms in [17] are

two-dimensional, as those in [9], but whereas in [9] one has the T3=2

9

behaviour of Appendix A, for the _lms of [17] one has the T5=2 behaviour

of Appendix B, which means that the anisotropies now make

the magnetic behaviour of the system after the crossover e_ectively

one-dimensional. In the critical region around Tc the e_ective dimension

should return to the value 2).

Actually the magnetic behaviour of the longitudinally magnetized

wire is even more complicated. On the one hand, due to the fact

that now the dispersion is circular, there are no quantum uctuations

to destroy the long-range order at T = 0, so the minimal anisotropy

Cmin

b is 0+, and [15] plus the remark at the end of Appendix A comes

into play for the (ordered) quantum phase. On the other hand, i.e.

concerning the 'thermal behavior' near Tc, it is known that there is

no long range order at _nite T in a one-dimensional Ising-class model,

because along the wire there are too many 'kink excitations', i.e. too

many domain walls separating x-regions with opposite orientation of

the magnetization. Here our previous remark, [16], on the di_erence

between the upper value of the spin wave dispersion and the critical

value of kBTc comes into play: the ow line considered should really

move almost vertically upwards from the (lower) crossover point T_ up

to the (upper) crossover temperature, which should be of the order of

D_ (_

a )2=kB, as already mentioned. From the 'upper crossover point' it

should turn steeply downwards to the origin, i.e. to the genuine critical

temperature Tc = 0+. But this 'downturn region' (or 'upturn region' in

case of U. Gradmann's T5=2-_lms) is experimentally outside the range

of the 'Bloch behaviour', where one observes this T5=2 law.

Also two-dimensional systems with a strong uniaxial perpendicular

anisotropy should belong the this 'quasi-universality class' with Bloch

exponent 5/2. In contrast, in the particular case of a 'reorientation'

transition at T = 0, corresponding to a situation, where the e_ective

uniaxial anisotropy is exactly zero, if no additional anisotropies are

present, the original Mermin-Wagner theorem becomes applicable and

the magnetic order disappears: In practice, due to the magnetostatic

interaction, a nontrivial domain structure appears.

Appendix C: T2 crossover-scaling from the quantum _xed point to

the thermal _xed point for itinerant magnets in d=3

The following arguments are especially simple. We consider a crystalline

metallic ferromagnet in d=3 dimensions. This system has sepa-

10

rate energy bands fE"(~k)g rsp. fE#(~k)g, and at T = 0 (among others)

the 'single spin-ip Stoner excitations', i.e. electron-hole excitations

with spin ip, where e.g. an electron with energy 'immediately below

the Fermi energy EF ', with initial-state wave-vector ~k and spin ", is

moved to a state 'immediately above EF ', with a di_erent _nal-state

wave-vector ~k 0 and spin #. Now at _nite T, the expression 'immediately

above or below EF ' means: 'within an interval of width _ kBT around

EF . Every such excitation reduces the magnetic moment of the sample

by two Bohr magnetons; i.e. in the above-mentioned 'general formula'

one has a 'phase-space factor' / (kBT)2, where one of the two powers of

kBT stands for the e_ective phase-space of the initial states, while the

other one represents the _nal states. There is no 'energy factor', since

all this happens in the above-mentioned interval immediately near EF .

Similar arguments also apply to itinerant antiferromagnets, in accordance

with the observations of U. Kobler, [10], who always observed

the same Bloch power for ferromagnets and antiferromagnets.

(But hithertoo I have not found similar arguments for semiconducting

magnets, e.g. for EuO.) In contrast, the collective magnon excitations,

which are also present in itinerant magnets and which have the same

quadratic dispersion as before, seem to give a neglegible contribution to

the behaviour of Ms(T) in the temperature range considered, although

at very low temperatures a T3=2-contribution would always be larger

than a T2-one. So this means again that one should remain above a

certain crossover-temperature, in this time that one to the dominance

of the T3=2 behaviour.

11arXiv:cond-mat/0211479 v1 21 Nov 2002

On the signi_cance of quantum phase transitions for

the apparent universality of Bloch laws for Ms(T)

U. Krey_

Inst. fur Physik II, Universitat Regensburg, 93040 Regensburg, Germany

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

r