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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> HAL Id: jpa-00209908 https://hal.archives-ouvertes.fr/jpa-00209908 Submitted on 1 Jan 1984 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. 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 > 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 -> 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 > 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. References [1] STRATONOVICH, R. L., Dokl. Akad. Nauk SSSR 115 (1957) 1097, Sov. Phys. Dokl. 2 (1958) 416 ; HUBBARD, J., Phys. Rev. Lett. 3 (1959) 77 ; SHERRINGTON, D., J. Phys. C 4 (1971) 401. [2] SCHRIEFFER, J. R., Lecture notes, Banff summer school, 1969 (unpublished); HAMANN, D. R. and SCHRIEFFER, J. R., in Magnetism, vol. 5, ed. Rado and Suhl, (Academic, New York) 1973, p. 237. [3] EVENSON, W. E., SCHRIEFFER, J. R. and WANG, S. Q., J. Appl. Phys. 41 (1970) 1199. [4] HUBBARD, J., Phys. Rev. B 19 (1979) 2626 ; 20 (1979) 4584. [5] HASEGAWA, H., J. Phys. Soc. Japan 49 (1980) 178 ; 49 (1980) 963. [6] MORIYA, T., J. Mag. Mag. Mat. 14 (1979) 1 and references (NY) 148 (1983) 436. [11] HAMANN, D. R., Phys. Rev. B 2 (1970) 1373. [12] PRANGE, R. E., in Electron correlation and Magnetism in Narrow-Band Systems, edited by T. Moriya (Springer, Berlin) 1981, p. 55. [13] MURATA, K. K. and DONIACH, S., Phys. Rev. Lett. 29 (1972) 285. [14] MURATA, K. K. Phys. Rev. B 12 (1975) 282. [15] KAKEHASHI, Y., private communication. [16] CALLAWAY, J., Phys. Rev. B 5 (1972) 106; SINGHAL, S. P. and CALLAWAY, J., Phys. Rev. B 7 (1973) 1125. [17] HEINE, V. and SAMSON, J. H., J. Phys. F 10 (1980) 2609. [18] RUSHBROOKE, G. S., BAKER, G. A. and WOOD, P. J., therein. [7] PRANGE, R. E. and KORENMAN, V., Phys. Rev. B 19 (1979) 4691; 19 (1979) 4698. [8] SAMSON, J. H., Phys. Rev. B 28 (1983) 6387. [9] LEIBLER, S., ORLAND, H., Ann. Phys. (NY) 132 (1981) 227. Rev. C 21 (1980) 1594; KERMAN, A. K., LEVIT, S., TROUDET, T., Ann. Phys. [10] LEVIT, S., Phys. [19] in Phase transitions and Critical Phenomena III, edited by C. Domb and M. S. Green (Academic, London) 1974, p. 245. SAMSON, J. H., Phys. Rev. B 30 (1984) 1437.