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Dynamical aspects of the ferromagnetic transition in the Hubbard model
~
SolidStateCommunications,Vol. 81, No. 11, pp. 883-886,1992.
0038-1098/9255.00+ .00 Pergamon Press plc
Printed in Great Britain.
DYN/~CAL
ASPECTS O F T H E F E R R O M A G N E T I C 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'ueJda 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 finitetemperature 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
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', csnsin8 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/tic~
The basic feature involved in the Hubbard model is the competition between a h o P . 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,
883
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
FERROMAGNETIC TRANSITION IN THE HUBBARD MODEL
884
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 a n d 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+ ) ql
(3)
q;ito
'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<
.
.
.
B
.
.
.
.
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-~n4--{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
+
,
-4
~
X(~,u) --- a + b q2 + c uY qS-~,
Vol. 81, No. 11
(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
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/ct 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 r o - I - q 3 - i w / 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)
T~
(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-'gl0, the system will have the same dyamun/c critical behavior both in the Heisenber8 and in the Renormalized Heisenberg fixed points. Consequently,the dispersive nature" of the magnetic excitations at Tc does not change even if constraints are added to the theory.
Fig. 2: Typical dispersion relation curves, showing the possibility of propngating spin waves above T c in metals for not too small momenta. Note that they vanish for T >> To.
Spin Waves above Tc: Phenomenological approach w=qJ
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#)
Xo
Co,o)
= I- Aq'-B [ + ] = + ...+
[(')
]
+ 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 tl 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
(? - 1) - "r (Aq2 + Cq4 + Dq') B
,(10)
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
C + J CL4AD 2A
, (Ii)
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 w u pmzti~lly supported by CNPq, CAPES, FINEP (Brm~dlianAgenda). Relmmom 1.
M.D. Coutinho--FiLho, M.L. Lyra and A. Nemirovmky, Sol. StiLt. Commmn 7.4, 1175
2.
t1~.M.~)Chaves,P. Lederer mad A.A. Gomes, J. Phr, L,~ ~67 (I077). J. S e t , , P ~ . ~ v . B 14. 118S (1,7e). H.A. Mook and J.W. Lynn, Phys. Rev. Left. 150 (1986); D.Mck. Paul, H.A. Mook, P.W. Mitchell and S.M. Hayden, .i. Physique, 49, ca--so ( l m ) . J.W. Lynn, Phys. Rev. Left. 52, 775 (!984).
3. 4.
5.
886
6. 7.
8. 9.
FERROMAGNETIC TRANSITION IN THE HUBBARD MODEL G. Shirane, P. B6ni and J.L. Martinez, Phys. Rev. B 36) 881 (1987) and references therein. B.I. Halperin, P.C. Hohenberg and S.-K. Ma, Phys. Rev. B 13, 4119 (1976); R. Folk and H. Iro, Phys. Rev. B 32, 1880 (1985); ibid 34, 6571 (1986). See, e.g., M.U. Luchini and V. Heine, Europhys. Left. 14, 609 (1991). J.B. ~]~oloff, J. Phys. F 5, 528 (1975); ibid 5, 1946 (1975); V. Korenman and R.E. Prange, Phys. Rev. Lett. 53, 186 (1984).
Vol. 81, No. II
10. J. Kirschner, M. Globl, V. Dose and H. Scheidt, Phys. Rev. Lett. 53, 612 (1984) 11. Shang-Keng Ms and G.F. Mazenko, Phys. Rev. B 11, 4077 (1975) 12. P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977) 13. B.I. Halperin and P.C. Hohenberg, Phys. Rev. 188, 898 (1969) 14. T. Moriya, Spin Fluctuations m Itinerant Electron Magnetism (Springer-Verlag. Berlin 1985).
SolidStateCommunications,Vol. 81, No. 11, pp. 883-886,1992.
0038-1098/9255.00+ .00 Pergamon Press plc
Printed in Great Britain.
DYN/~CAL
ASPECTS O F T H E F E R R O M A G N E T I C 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'ueJda 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 finitetemperature 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
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', csnsin8 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/tic~
The basic feature involved in the Hubbard model is the competition between a h o P . 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,
883
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
FERROMAGNETIC TRANSITION IN THE HUBBARD MODEL
884
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 a n d 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+ ) ql
(3)
q;ito
'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<
.
.
.
B
.
.
.
.
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-~n4--{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
+
,
-4
~
X(~,u) --- a + b q2 + c uY qS-~,
Vol. 81, No. 11
(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
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/ct 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 r o - I - q 3 - i w / 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)
T~
(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-'gl0, the system will have the same dyamun/c critical behavior both in the Heisenber8 and in the Renormalized Heisenberg fixed points. Consequently,the dispersive nature" of the magnetic excitations at Tc does not change even if constraints are added to the theory.
Fig. 2: Typical dispersion relation curves, showing the possibility of propngating spin waves above T c in metals for not too small momenta. Note that they vanish for T >> To.
Spin Waves above Tc: Phenomenological approach w=qJ
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#)
Xo
Co,o)
= I- Aq'-B [ + ] = + ...+
[(')
]
+ 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 tl 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
(? - 1) - "r (Aq2 + Cq4 + Dq') B
,(10)
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
C + J CL4AD 2A
, (Ii)
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 w u pmzti~lly supported by CNPq, CAPES, FINEP (Brm~dlianAgenda). Relmmom 1.
M.D. Coutinho--FiLho, M.L. Lyra and A. Nemirovmky, Sol. StiLt. Commmn 7.4, 1175
2.
t1~.M.~)Chaves,P. Lederer mad A.A. Gomes, J. Phr, L,~ ~67 (I077). J. S e t , , P ~ . ~ v . B 14. 118S (1,7e). H.A. Mook and J.W. Lynn, Phys. Rev. Left. 150 (1986); D.Mck. Paul, H.A. Mook, P.W. Mitchell and S.M. Hayden, .i. Physique, 49, ca--so ( l m ) . J.W. Lynn, Phys. Rev. Left. 52, 775 (!984).
3. 4.
5.
886
6. 7.
8. 9.
FERROMAGNETIC TRANSITION IN THE HUBBARD MODEL G. Shirane, P. B6ni and J.L. Martinez, Phys. Rev. B 36) 881 (1987) and references therein. B.I. Halperin, P.C. Hohenberg and S.-K. Ma, Phys. Rev. B 13, 4119 (1976); R. Folk and H. Iro, Phys. Rev. B 32, 1880 (1985); ibid 34, 6571 (1986). See, e.g., M.U. Luchini and V. Heine, Europhys. Left. 14, 609 (1991). J.B. ~]~oloff, J. Phys. F 5, 528 (1975); ibid 5, 1946 (1975); V. Korenman and R.E. Prange, Phys. Rev. Lett. 53, 186 (1984).
Vol. 81, No. II
10. J. Kirschner, M. Globl, V. Dose and H. Scheidt, Phys. Rev. Lett. 53, 612 (1984) 11. Shang-Keng Ms and G.F. Mazenko, Phys. Rev. B 11, 4077 (1975) 12. P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977) 13. B.I. Halperin and P.C. Hohenberg, Phys. Rev. 188, 898 (1969) 14. T. Moriya, Spin Fluctuations m Itinerant Electron Magnetism (Springer-Verlag. Berlin 1985).