~ Solid State Communications, Vol. 81, No. 11, pp. 883-886, 1992. Printed in Great Britain.

0038-1098/9255.00+ .00 Pergamon Press plc

DYN/~CAL ASPECTS OF THE FERROMAGNETIC TRANSITION IN M HUBBARD MODEL

M. L. Lyra Departamento de Fisica -Universidade Federal de Alagoas

Macel6 - AL, 57061, Brazil

M. D. Coutinho-Filho Departamento de Fisica - Universidade Federal de Pernambuco

Recife - PE, 50739, Brazil

(Received 16 October 1991 b~l C.E.T. Gon~d'ueJ da S~a) (In revised form 7 January, 1992)

Using field-theoretic sad renormaiization-group methods we analyze some dynsmlcai aspects of the ferromagnetic transition in the one-bsad Hubbard model,such as its nature at T=0, where quantum fluctuations are relevant, sad the effect of constraining the zero-mode charge fluctuations on the dynamical susceptibility at To. Moreover, using phenomenological arguments, we obtain the conditions under which spin-waves propagate aboveTc in metals.

Introd.ctio~

One of the fundamental questions re~arding the one-bsad Hubbard model concerns the emstence sad nature of a ferromagnetic transition in three dimensions. Recently, within a field theoretic sad renormalization group framework, the finite- temperature ferromagnetic critical behavior of this model in the presence of constraints was studiedt. Contrary to preliminary expectations 2, it was shown that, if constraints are imposed on the zero-mode charge fluctuations, the continuous ferromagnetic transition is strongly inhibited and the system may be compelled to undergo a first-order transition. A fluctuation-induced tricritical point was predicted with tricr/tical exponents directly related to the Heisenherg critical ones, with surprisinsly interesting results t.

In this work, we will analyze some dynamics] aspects related to the ferromagnetic transition under the above conditions. First, we will study the critical behavior of the system at T=0, where quantum fluctuations are relevant. HertsS frst showed that the latter fluctuations move the critical dimension down to dc=1, and thus the critical behavior in three dimensions is gaussian. However in his analysis, he did not include the spin-charge coupling, but rather the sd-hoc argument that the non-critical charge field could not chsa~e the masnetic critical behavior of the system. Now that it is knownl that the spin-charge coupling plays a relevant role in the determination of the nature of the ferromagnetic transition st finite temperatures, we have to check the Hertz hypothesis by including this coupling in the description of the transition at T=0.

We will also analyze how these constraints affect the interpretation of the controversial experiments on the existence or not of spin-wave excitations in the paramagnetic phase of Fe and Ni. Some groups reported peaks at finite energy transfer, for

q > 0.3 )~'~ in Ni 4 taken as an evidence for spin-wave

like excitations above Tc and for a breakdown of 883

dynamical scalinsS. However, no such evidence was found by Shirsae sad co-workeree, am one expects from the dynamical critical theoryT. The ide~ here is to exarniue the possibility that constraints imposed on the charge fluctuatio~ might be respondble for a change in the nature (propagative or dissipative) of the magnetic collective excitatlons.

In most studies the existence of propagating spin waves above Tc in metals is, in general, taken as a consequence of the strong short-range correlations in these nmtedals in the paramJq~netic phme', csnsin 8 the baud splitting persistence above Tc 0. However KirshnerS0 verified in Fe and Ni that some bands collapse and others do not as the transition is neared, indicating that statistical correlst.ions may not be the ilhportsat mechanism underlying the p h e ~ n e n ~ Instead, one may argue that if the band structure characteristics are implemented in the theory through a proper bsad density of states, the p r o p ~ t i N | spln waves, ff existent, should appear already at the R.P.A. (random phase approximation) level. The conditions for such an occurence will be reported below.

The F e n o n u ~ c Trm/ t i c~

The basic feature involved in the Hubbard model is the competition between a hoP . l~ term (kinetic energy) sad a local Coulomb repulsion interaction, i.e

i~ j,O" t

where tij is the hopping integral between sites i sad j,

U is the Coulomb coupling strength, ci, o (c~,o) is the annihilation (creation) operator for a fermion of spin q

at site i sad fit,q is the ~.~Ion number operator.

Using field-theoretic methods, the lmldtion function can ~oe written u a functions] integral over the Fourier components of the auxil/ary field fluctuations

884 FERROMAGNETIC TRANSITION IN THE HUBBARD MODEL Vol. 81, No. 11

conjugated to the spin and charge operators

Z -- Zo ID t~q D ~q exp [---~ F {1Oq, ~q}] , (2)

where an expansion around the paramagnetic uniform saddle point was made, Zo is the partition function of the non-interacting system, q is the quadrivector

q = (~,~) composed by the Matsubara frequencies

(uffi 2rn) and wave vectors, le and ~ are the charge and spin field fluctuations, respectively, and

F {toq, ~q) is the free energy functional. At criticality

F (l~q, ~q} can be expanded as a power series of/0q and

~q, and, after integrating over the charge field

fluctuations, we have the following relevant terms

q

+ J,, ~ . ~tql '~q~ ~q~ 9-(q1+q~+q+ ) q l

q;ito (3)

'E where ~q = ~k ~q-k" Here the uniform component

k of the qua-tic charge mediated term is singled out to permit distinct boundary conditions of interest 1.

If the temperature transition satisfies the condition 0<<kBT c <<El, so that quantum effects are negligible, the Matsubara frequencies will form a discrete espectrum and only the wffi0 mode is relevant

. . . . . . . . - 4

at cntlcality. The statlstlcai susceptibility /(q, wffio) can thus be expanded in powers o f q andonly terms up to q~ are relevant. The propagator thus will take

-1 2 the form Go -- r, + q , where rs ffi 1 - UN(Ef)/2. A renormalisation program has been implemented, and the fixed points and related critical exponents have been determinedL In particular, for total charge fixed

(~°)=0) we found a fluctuation-induced tricritical point, with Heisenherg exponents renormalized a la Fisher. On the other hand, when the total charge

B ~

fluctuates (j~0)ffi j~d), the free energy functional is reduced to a $4 theory with effective qu r t i c coupling Jc-Jc and no fluctuation-induced first-order transition is predicted.

At T=0, whege quantum fluctuations are relevant3, the M&tsubara frequencies form a continuum and these must he explicitly considered in the free energy functional. The expansion of the dynamical susceptibility in a power series of q and u, have the following relevant terms:

X(~,u) --- a + b q2 + c uY qS-~, (4)

where for paramagnous excitations y--1 and sffi3 3. In this case, the criterion to determine the relevance of the texms in the functional, (Eq 3) ,must be redefined because now the integrals involve both frequency and momentum. If the momenta are scaled as q -, ~1', the frequencies scale as w-~ gKq, in order to retain the

dynamical propagator in its form after renormMization. As the free energy functional have dimension L -'(d+s) the fields must scale as

-, s(d+s-~) 3', and the qusrtic couplings as

,~t-~n 4--{d+s) Ji'- So, the critical dimension, below which corrections to the ganssian theory are needed, is dc = 4--¢, and in the case of parsmagnons excitations one obtains dc = 1, as predicted by HertzS.

Therefore, in d = 3 , the qusrtic couplings are irrelevant at T=0, and the system has a mean-field critical behavior with gaussian critical exponents (affi0, v=l/2). Statistical fluctuations are not relevant, and no tricriticality induced by fluctuations is found. The quantum dynamical degrees of freedom make neither spin nor charge fluctuations important near criticality in d--3 and Tffi0.

Constraint effects on the Dynamical Susceptibility

We have already mentioned that constraints on the charge fluctuations can change the nature o f the ferromagnetic transition.Here we verity its influence on the dyn~mlcal susceptibility, and, consequently, on the magnetic excitations of the system.

The dynamical stocastic equation for the order parameter in a Heisenberg ferromagnet, in which the Heisenberg fixed point is the most stable, has a Larmor precessive term that generates a $3 vertex, and so the dynamical critical dimension is dcffi6 11. To properly study the critical dynamics in d=3 a double expansion, around ~1ffi6-d and ~3=4--d~ is needed. For simplicity lets study the constraint effects in another dG~ninzamics, namely the time dependent

burg-Landau model (TDGL) 13 described by the following equations:

8Sct r b'H = - + ,

1 6H

where Sot is the a component of the order parameter, P is a constant kinetic coefficient, h is a time dependent magnetic field and I/c t is s noise. In this

dynamics only •4 vertex appears and the dynamic critical dimension is dcffi4, m in the static case. To analyze the dynamic critical behavior, lets calculate the dynamical susceptibility at T¢

x'l(~,u) ro -I- q3_ i~ / r + E (q,u) , (6)

here Oo(q,u)ffi ro - I -q3- iw/F is the free dynamical propagator and ~ (q,~) are the self-energy diagrams. The graphs" that contribute to ~. (q,u) are ihown in fig. 1. Among them, those with external legs coming from a same vertex only renonnali~ To, and will not contribute to the susceptibility at To. The contributing diagrmns that have at least one g~0) vertex have internal legs with well-defined momenta and thus represent a null set. Finally, the diagrams with vertices g, and gl I) will contribute to ~.(q,,~), to order t3, in the form

E l q , u ) ffi [ g ~ - g I ' ) l ' I ( q , w ) , (7)

Vol. 81, No. 11 FERROMAGNETIC TRANSITION IN THE HUBBARD MODEL 885

@ (o] (b) (c) (d)

contribute to renormalize To. Graphs (c) and

with at least one gc (°) vertex have null contribution due to the existence of internal lines with well defined momenta.

where I (q,~) is an integral given by Eq. 8, Ref. 12.

Since at Te, X (q#) depends only on gm-'gl 0, the system will have the same dyamun/c critical behavior both in the Heisenber 8 and in the Renormalized Heisenber g fixed points. Consequently,the dispersive nature" of the magnetic excitations at Tc does not change even if constraints are added to the theory.

Spin Waves above Tc: Phenomenological approach

Several authors have asserted that the magnetic excitations at T¢ remain dispersive when statistical

fluctuations are considered 6 '" '" . We are thus impelled to think that in itinerant electron systems the band density of states might provide the basic mechanism to make possible the occurrence of

propagating spin waves above and at Tc. Using a phenomenological approach lets derive the conditions under which propagative spin-wave solutions axe found at the R.P.A. level.

The real part of the dynamic transversal susceptibility can be written in a power series, of q and w, asz4

X,~" (q#) = I- Aq'-B [+]= + ...+

Xo Co,o) [ ( ' ) ] + UM V~ --~ + D= +... +... , (8)

where M is the magnetization sad the coefficients are determined from the baud structure near the Fermi level, with A > 0. In the ILP.A. approximtion, the dispersion relation of spin waves is obtained from the equation 1 - UP~ {X~" (q,o~q)} ffi 0,and criticality is

reached when 1 -U/*'(O,O) = 0. If t l

7ffi UIo'(0,0 ) > I, the system is ferromagneticaHy

ordered. On the other hand, if 7 < 1 the system is paramsl[netic. Retalniug o,ly the first terms in Eq. 8, we obtain that the spin wave energies are the solution to the following equation

For ? > 1 (ordered phase) we have ¢ a q=. At criticality (Tffi I), u ffi q3 ~--"A~ and, for B > 0, the dispersion relation is di~pative at Tc. However, the experimental obsmTttions 4 indicate that for l~rge momenta the dispersion relation may become propagative. By considering higher-order terms in the susceptibility expansion, we will have the following spin-wave dispersion in the paramagnetic phase

T~<T©

Fig. 2: Typical dispersion relation curves, showing the possibility of propngating spin waves above Tc in metals for not too small momenta. Note that they vanish for T >> To.

w = q J (? - 1) - "r (Aq2 + Cq4 + Dq') ,(10) B

and at criticality (7 -- 1) the system will display propagating spin-waves for A + Cq= + Dq4 < 0. As A > 0, we must have C < 0 in order to satisfy the latter condition, and D > 0 to Limit the dispersion curve. Therefore, spin-waves will appear at Tc in the range of momenta

- C - I CL4AD<q<- C + J CL4AD , (Ii) 2A 2A

with C 2 > 4AD, C < 0. Typical dispersion relation curves are shown in fig. 2. Note that spin waves can propagate for T > To, for not too small momenta, but vanish for higher temperatures, as experimentally observed. In conclusion, although the present approach does not specify the actual physical conditions, as, for example, the shape of the bare density of states, in order that inequality (11) be

• satisfied, it clearly points out the pouibiLity of propagative spin-wave solutions in the disordered phue, even at the R.P.A. level.

This work wu pmzti~lly supported by CNPq, CAPES, FINEP (Brm~dlian Agenda).

Relmmom

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886 FERROMAGNETIC TRANSITION IN THE HUBBARD MODEL Vol. 81, No. II

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