Breakdown of the static approximation in itinerant-electron magnetism

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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�. �jpa-00209908�

Breakdown of the static approximation ⁱⁿ itinerant-electron magnetism

J. H. Samson (*)

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 tempé- rature, elle conduit à un état fondamental self-consistant ou variationnel, ^dont l’énergie ^{constitue une limite} supé-

rieure 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.

Abstract.

The static approximation ^is widely ^{used to treat} the statistical mechanics of itinerant magnets. ^{It is} 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.

Classification

Physics ^Abstracts

75 . lOL - 65.40

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 time-}

ordered 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 anoma-

lies ; ^the resulting ^free energy is not that of _any Hamil-

tonian, ^{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

positive ; ^{it can} in fact be negative ^{at all} temperatures,

as a 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 ^fluc-

tuations 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 « ultra- violet catastrophe )) arises because the _magnons are 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 _capa-

city ^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 ^tem-

perature ^{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 quanti-

zation of the local magnetization. The second des- cribes 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.

2. Model systems.

The simplest system that shows the anomaly ^{is the}

trivial ^« one-spin Heisenberg ^{model »}

acting ^{on a} spin s, ^{with J} > 0. (This is related to the

large-U limit of the Hubbard model.) This Hamilto- nian 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

where

is the « partition function » of the spin ^{in the} imaginary- time-dependent ^field A(i), ^T being ^the time-ordering symbol.

We will also investigate ^the degenerate-band ^Hub-

bard model in the form

where I is sufficiently small that the ground ^{state is} paramagnetic.

Here Hband ^describes tight-binding bands, ^and

and

are 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].

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-inde-} pendent ^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 -> 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 (11-13) gives

this 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

which is not positive-definite.

The internal _energy

(for ^spin 21 ^is

This then is the anomaly ; USA ^falls monotonically

with temperature ^from

We find analogous results for larger spins :

The internal _energy falls with temperature; ^the specific

heat capacity C in the SA is negative.

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 now suspect ^{that the same} effect will

give ^{rise to a} negative contribution to C in the SA to the Hubbard model (6). There is in fact an addi- tional negative contribution [14] ^from longitudinal

fluctuations in the local magnetic moments, ^which is still present ^{if a scalar} exchange ^field (N

1)

rather than a vector field is used in (9). These contri- 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 ferromagnetic ^{metal at low} temperatures. ^For simpli- city ^{we set U}

0, thereby neglecting ^the charge

fluctuations. We then expand ^{the effective} Hamiltonian to fourth order in the exchange ^{field :}

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

is positive, giving ^the grand potential ^as

and the heat capacity ^as

The longitudinal fluctuations give ^a negative ^contri-

bution 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 _qua-

dratic ; ^{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 ^contribu-

tion to the partition function of the symmetric ^Ander-

son 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} high- temperature 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-

city per ^atom is then 1 Nk rather than O(T). ^One

would then need to interpolate ^{between 1 and} (4 nlkT)Nl2 (4 nUkT)1/2 ^{with a crossover at typical} spin ^wave energies. Fortunately ^the denominator

only affects thermal properties ^{and not} expectations

of electronic quantities.

Finally, ^{we note} that the SA is correct for the Hei-

senberg model in the large-spin (classical) ^limit [18],

1680

for temperatures higher ^than spin ^wave energies, T > Tc/s. Heuristically ^one would expect ^{the cor-}

responding 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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