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Breakdown of the static approximation in
itinerant-electron magnetism
J.H. Samson
To cite this version:
J.H. Samson. Breakdown of the static approximation in itinerant-electron magnetism. Journal
de Physique, 1984, 45 (10), pp.1675-1680. <10.1051/jphys:0198400450100167500>. <jpa00209908>
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Submitted on 1 Jan 1984
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J.
Physique 45 (1984)
1675-1680
OCTOBRE
1984,
1675
Classification
Physics Abstracts
75 . lOL - 65.40
Breakdown of the static
J. H. Samson
approximation in itinerant-electron magnetism
(*)
LASSP, Clark Hall, Cornell University, Ithaca, NY 14853, U.S.A.
(Reçu le 29 mars 1984, accepte le 7 juin 1984)
Résumé.
L’approximation statique est largement utilisée pour décrire la mécanique statistique du magnétisme
itinérant. Il est démontré ici que la fonction de partition qui en résulte ne correspond à aucun Hamiltonien ce qui
peut donc produire des résultats anormaux. En particulier, bien qu’elle donne correctement l’état à haute température, elle conduit à un état fondamental self-consistant ou variationnel, dont l’énergie constitue une limite supérieure à l’énergie réelle. La chaleur spécifique s’en trouve donc sous-estimée. Deux modèles précis pour lesquels la
chaleur spécifique est négative dans l’approximation statique sont ici présentés.
2014
The static approximation is widely used to treat the statistical mechanics of itinerant magnets. It is
Abstract.
shown that the resulting partition function is not that of any Hamiltonian, and can therefore lead to anomalous
results. In particular, it gives a self-consistent or variational ground state, whose energy is an upper bound to the
true energy, although it gives the correct energy in the high-temperature state. It therefore underestimates the heat
capacity. Two specific models are presented in which the heat capacity in the static approximation is negative.
2014
1. Introduction.
The functional integral technique [1] is a useful
method for the treatment of the statistical mechanics
of many-body systems. The two-body interaction
between particles is replaced by an interaction between
particles and an auxiliary field ; the partition function
is then that of particles in the field, averaged over all
configurations of the field with a Gaussian weight.
For example, in a classical plasma with Coulomb
interactions the auxiliary field is the electrostatic
potential Ø(r); the partition function of the plasma
is then that of particles and field, normalized by that
of a free field. An advantage of the method is that
it is often possible to restrict the functional integration
over the auxiliary field to some class of configurations
regarded as important on physical grounds. It is the
purpose of this work to show that this restriction can
sometimes lead to anomalies, specifically in the case
of the static approximation.
The functional integral method has been applied
to quantum many-body systems. A major application
is to magnetic impurities [2] and magnetic transition
metals [3-8], described by the Anderson and Hubbard
Hamiltonians respectively. The local spin-spin and
charge-charge interactions are replaced by an exchange
field Ai(T) and a Coulomb field wi(,r). The method has
also been applied to the spin s Heisenberg model [9],
and to nuclear dynamics [10].
A serious difficulty in the quantum problem, as
opposed to the classical problem, is that the auxiliary
fields Ai(,r) are « time »-dependent, and the partition
function becomes a functional integral of a timeordered exponential over all such paths, for imaginary
time i from 0 to P
llkt. Even the contribution
from a single arbitrary path is difficult to compute.
It is at this point that most authors [4, 5, 8] take the
static approximation (SA), in which the integral is
restricted to time-independent auxiliary fields, as in
the classical problem. This generates a classical
effective Hamiltonian for the auxiliary field, whose
energy is given by the quantum mechanical free
energy of the system in the field. This approximation
is believed to be valid when the temperature is higher
than the excitation energies of the auxiliary field, e.g.,
spin wave energies.
The separation into classical and quantum parts
=
that arises from the SA however can lead to anomalies ; the resulting free energy is not that of any Hamiltonian, classical or quantum, and therefore need not
have the usual convexity properties. In particular,
the specific heat capacity C in the SA is not necessarily
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450100167500
1676
in fact be negative at all temperatures,
specific example will show. The basic reason for
this is that the SA is correct in the high-temperature
limit, but at low temperatures gives a self-consistent
mean field solution, which neglects zero-point fluctuations and correlations, and may therefore be
higher in energy than the true ground state. Less
energy is available than in the exact solution; this
can lead to a negative C.
It has long, been recognized that the SA gives
quantitatively incorrect results. Evenson et al. [3]
comment that the SA is exact only in the limit of
zero bandwidth (with a scalar exchange field) or in
the limit of zero interaction, and that the leading
corrections from either limit are wrong. Quantum
fluctuations, treated in their RPA’, correct this.
Hamann [11], in a treatment of the Kondo problem,
finds that the static paths dominate only above the
Kondo temperature; the SA fails at low temperatures.
Prange [12] argues that the SA overestimates the
weight of short-wavelength fluctuations, and therefore
underestimates the short-range order. This « ultraviolet catastrophe )) arises because the magnons are
positive ; it
can
as a
acting on a spin s, with J &#x3E; 0. (This is related to the
large-U limit of the Hubbard model.) This Hamiltonian obviously has the exact partition function
and
a
temperature-independent
internal energy
It is therefore of interest to show explicitly how the
static approximation breaks down in this case. The
exact functional integral expression for the partition
function (specializing Leibler and Orland’s [9] result
for the Heisenberg model) is
not quantized.
The negative C problem has been noted by Murata
and Doniach in a spin-fluctuation theory of weak
itinerant ferromagnets. A constant Dulong-Petit (or
equipartition theorem) contribution to the heat capacity is present in their classical phenomenological
model [13], but is absent in Murata’s microscopic
calculation [14], which corresponds to our equations
(14-15). They point out that the latter theory gives a
negative magnetic contribution to C above T c.
It appears that this negative term is similarly an
artefact of the SA. The effect has been noted in some
recent work that uses the SA [15]. (It is however
possible that the effect could be real. Callaway [16]
finds a negative magnetic contribution at high temperature from a virial expansion for the Hubbard
model in some cases).
The plan of this paper is as follows. Firstly, we
introduce two models and define the SA. These
models isolate two different contributions to the
statistical mechanics of the Hubbard model. Although
these models describe limiting cases, they explicitly
demonstrate the failure of the SA, and suggest how
it may fail in more general cases. The first model is
trivial and exactly soluble, and describes the quantization of the local magnetization. The second describes longitudinal spin fluctuations, specifically in
an enhanced paramagnet. We then demonstrate
that the SA can give negative C in both cases. Physical
reasons for this problem are discussed. Finally, we
consider how the problem might be circumvented.
where
is the « partition function » of the spin in the imaginarytime-dependent field A(i), T being the time-ordering
symbol.
We will also investigate the
bard model in the form
where I is
The simplest system that shows the
trivial « one-spin Heisenberg model »
anomaly
is the
Hub-
sufficiently small that the ground state
paramagnetic.
Here Hband describes tight-binding bands, and
is
and
the number and spin operators respectively for
electrons on site i. The index m labels orbitals, and a
labels spin states. The motivation for this explicitly
rotationally invariant form is given elsewhere [8].
The partition function is [1, 2].
are
2. Model systems.
degenerate-band
1677
where we
integrate over an N ( =
1
or
3)-component exchange field, and
To obtain the static approximation (SA), we simply restrict the integrations in (4) and (9) to time-independent fields (or equivalently drop the time-ordering symbol in (5) and (10)). This gives the partition function
as
with
for the
one-spin Heisenberg model,
and
with
for the Hubbard model.
The presence of the denominator in (11) and (14)
is controversial. It is included by some authors [2,
8, 14] and omitted by others [4, 5]. It is certainly
necessary as an infinite normalization of the functional
integrals in (4) and (9). In the SA partition function,
degrees of freedom of a fictitious auxiliary field have
been added to the system; the denominator subtracts
the free energy of these fields. Thus (14) gives the
correct result that the specific heat capacity C -&#x3E; 0
in the low- and high-temperature limits, and that
ZSA -+ Tr efJHband in the U -+ 0, 1 -+ 0 limits.
3. Anomalous heat capacity.
We now demonstrate how the anomaly comes about.
We deal first with the one-spin model. In this case
the SA in
this
(11-13) gives
simplifies
to
for spin 1/2. This does not correspond to the partition
function
1678
of any system with a positive-definite density of states
p(E) ; we find instead that (18) is satisfied by the
unphysical function
The internal energy falls with temperature; the specific
heat capacity C in the SA is negative.
which is not
positive-definite.
The internal energy
(for 21
spin
is
The reason for this anomaly is easy to see. The SA
works at high temperatures (above the characteristic
excitation energies) and therefore gives the correct
energy (22) in this limit. At low temperatures it finds
a « self-consistent » state, with a classical energy (21).
This neglects zero-point fluctuations, and is higher
in energy than the true ground state.
One will
give rise
now
suspect that the
same
effect will
negative contribution to C in the SA
to the Hubbard model (6). There is in fact an additional negative contribution [14] from longitudinal
fluctuations in the local magnetic moments, which
is still present if a scalar exchange field (N
1)
This then is the anomaly ; USA falls monotonically
than a vector field is used in (9). These contrirather
with temperature from
butions are usually, but not always, compensated by
larger positive contributions from the disordering
of the moments and from the single-particle excitations.
As a simple illustration of this, we consider a nearly
We find analogous results for larger spins :
ferromagnetic metal at low temperatures. For simplicity we set U 0, thereby neglecting the charge
fluctuations. We then expand the effective Hamiltonian
to fourth order in the exchange field :
to
a
=
=
where
and X and A have similar temperature dependences. We assume that x(q) I -1 and that À( q - qq’ - q’) and
permutations are positive for all q and q’, so that the ground state is paramagnetic. We take the quartic term
in (24) to first order in the cumulant expansion. The partition function is then
with
1679
is
positive, giving
and the heat
the
capacity
grand potential
as
as
The longitudinal fluctuations give a negative contribution to the linear coefficient of the heat capacity
in the static approximation, which may dominate
over the electronic contribution if the system is near
an instability towards ordering. Higher order terms
in the exchange field, and the temperature dependence
of x and A, will contribute only to the 0(T2) terms in
(30), Charge fluctuations contribute an effect of the
same sign if U is non-zero. The leading-order coupling
O(wA’) between spin and charge fluctuations will
be small if the band is approximately half-filled [17].
The negative contribution arises because the potential
of the interacting exchange field is stiffer than quadratic ; at finite T there is less entropy available
than for the free exchange field. The specific heat
capacity is therefore less than the equipartition value.
The problem also arises in the magnetic contribution to the partition function of the symmetric Anderson model in the SA, obtained by Evenson, Wang
and Schrieffer [2, 3]. In the present notation they find
which has a positive quartic term and thus gives a
negative linear term in C. They then go beyond the
SA to the RPA’, which introduces finite frequency
components to second order; this renormalizes ZSA (Ll )
and may remove the anomaly.
We can understand this anomaly in two ways.
Firstly, as in the one-spin case, the SA is good in the
high temperature limit. At low temperatures it gives
the self-consistent Hartree-Fock state, whose energy
is a variational upper bound to the true ground state
energy. Thus the SA is again allowed too little energy,
and can therefore lead to negative C.
Secondly, the heat capacity is that of a coupled
exchange field and electron gas minus that of a free
exchange field. One might expect that adding degrees
of freedom to
a
system would increase the heat capa-
city. However, the electron gas has only O(T/TF)
effective degrees of freedom, and the coupling can
therefore reduce the heat capacity of the exchange
field.
4. Discussion.
We have seen how the SA can give unphysical results.
In particular, it can underestimate the heat capacity,
so that the magnetic contribution may be negative.
We have identified the terms that cause the anomaly
by demonstrating that it occurs in certain limits.
In the general case there will be positive terms as well,
which will often dominate. (The SA does give good
energetics of the transition in the ferromagnetic
transition metals [8] ; presumably it provides a better
approximation for the difference between the energies
of the ordered and disordered states, in which case
the zero-point energies largely cancel.)
There are several possible remedies. One possibility
is to include some quantum fluctuations. The RPA’
[2, 3] includes finite-frequency components of the
exchange field to second order. In the one-spin case
this does compensate for the anomaly. However,
it adds enormously to the complexity of numerical
calculations for the Hubbard model, since for each
static configuration of the exchange field the dynamic
response function is needed. Also, quartic terms
may be needed for the partition function to exist.
Another possibility is that the denominator in (11)
and (4) could be adjusted. It is correct in the hightemperature limit; however, at low temperatures it
may be absent, since high-q modes of the exchange
field are not excited. A temperature-dependent q-space
cutoff [12,13] may be of use here. Hasegawa [5] argues
that omitting the denominator gives a better classical
approximation, since the low-temperature heat capa-
1 Nk rather
than
would then need to
interpolate
between 1 and
(4 nlkT)Nl2 (4 nUkT)1/2
with
city
spin
only
per atom is then
wave
a crossover
energies. Fortunately
O(T). One
at
typical
the denominator
affects thermal properties and not expectations
of electronic quantities.
Finally, we note that the SA is correct for the Heisenberg model in the large-spin (classical) limit [18],
1680
for temperatures higher than spin wave energies,
T &#x3E; Tc/s. Heuristically one would expect the corresponding limit for the Hubbard model to be the
large-degeneracy (many-band) limit, where the local
moments are large and can be treated classically.
Quantum fluctuations become less important in the
large-degeneracy limit; one can then expand about
this limit. The leading corrections do indeed provide
a positive contribution to the heat capacity [19].
Acknowledgments.
This work was supported by the Cornell Materials
Science Center, Report # 5124. I would like to thank
S. Chakravarty, A. J. Leggett and S. Leibler for helpful
discussions, and R. Pandit and S. A. Trugman for a
critical reading of the manuscript.
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