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1、arXiv:cond-mat/9906231v2 cond-mat.str-el 8 Nov 2000 Theory of the metamagnetic crossover in CeRu2Si2 Hiroyuki Satoh and Fusayoshi J. Ohkawa Division of Physics, Hokkaido University, Sapporo 060-0810, Japan (November 8, 2000) Based on the periodic Anderson model, it is shown that the competition betw
2、een the quenching of magnetic moments by local quantum spin fl uctuations and a magnetic exchange interaction caused by the virtual exchange of pair excitations of quasiparticles in spin channels is responsible for the metamagnetic crossover in CeRu2Si2, cooperated with the electron-lattice interact
3、ion. The strength of the exchange interaction is proportional to the bandwidth of quasiparticles and its sign changes with increasing magnetizations; it is antiferromagnetic in the absence of magnetizations, whereas it is ferromagnetic in the metamagnetic crossover region. Experimental results of st
4、atic quantities are well reproduced. 1999 PACS: 71.10.-w, 71.27.+a, 75.30.Et, 75.30.Kz I. INTRODUCTION The metamagnetic crossover in CeRu2Si2is an im- portant issue.The compound has a large electronic specifi c heat coeffi cient of 360 mJ/mol K2,1and it shows a sharp increase of magnetization at the
5、 fi eld of HM 7.7 T.2,3Other physical properties such as magnetostriction4,5 and specifi c heat6are also anomalous in this fi eld region. One of the most crucial experimental results to be explained is the single-parameter scaling;4,7 independent experimental quantities for diff erent pres- sures ar
6、e scaled with a single energy parameter kBTK, where TKis called the Kondo temperature. The Kondo temperature is the energy scale of local quantum spin fl uctuations and is approximately equal to the bandwidth of quasiparticles. The present authors showed in a previous paper8that an exchange interact
7、ion caused by the virtual exchange of pair excitations of quasiparticles in spin channels plays a critical role in the metamagnetic crossover as well as the electron-lattice interaction called the Kondo volume- collapse eff ect.9This exchange interaction has the fol- lowing two novel properties. Fir
8、st, it changes from being antiferromagnetic at zero fi elds to being ferromagnetic in the metamagnetic crossover region because of a pseudo- gap structure in the density of quasiparticle states, which is characteristic of heavy-electron compounds. Second, the strength of the exchange interaction is
9、proportional to the bandwidth of quasiparticles.Then, the single- parameter scaling can be easily explained. After the previous paper was submitted, a detailed measurement of the fi eld dependence of specifi c heat was reported by Aoki et al.10 At suffi ciently low tempera- tures, a single sharp pea
10、k was observed at HM, while a double-peak structure was found at higher tempera- tures. They argued that such a result can be explained if a sharp peak exists in the density of states. A similar ar- gument applies straightforwardly to our previous model; however, if the pseudogap is enough deep, the
11、 magneti- zation process shows a fi rst-order transition within the theoretical framework of the previous paper. Recently, two theories were proposed.11,12They claimed that an anisotropic c-f mixing plays an important role in the metamagnetic behavior. In both theories, however, the peak structure i
12、n the density of states is too sharp be- cause the k dependence of hybridization matrices is im- properly treated; one-dimensional van Hove singularity is irrelevant. Such an extremely sharp peak is inconsistent with the experimental result of specifi c heat. Further- more, it should give a disconti
13、nuous transition instead of the metamagnetic crossover if the novel exchange inter- action mentioned above is properly taken into account. To reproduce the experimental results of magnetization and specifi c heat simultaneously has not been achieved so far. A main purpose of this paper is to reformu
14、late and improve the previous theory based on the periodic An- derson model so as to explain static properties of the system. We will also give a brief comment on intersite spin fl uctuations around the zone center. II. MODEL The periodic Anderson model is written as H = X k E(k)a kak+ X k (Ef H)f k
15、fk + X k V(k)a kfk+ h.c. + 1 2U X i nini, (2.1) with the band index of conduction electrons, ni= f ifi, and H = m0H, where m0is the saturation mag- netization per f electron. Other notations are standard. The kinetic energy of conduction electrons, E(k), and the f electron level, Ef, are measured fr
16、om the chemical potential, respectively. 1 When multiple conduction bands are assumed, a pseu- dogap structure is expected in the density of states be- cause of the hybridization between the f band and con- duction bands.13Since the property of the magnetic ex- change interaction that plays a critic
17、al role in the mag- netization process depends on the shape of the density of quasiparticle states as shown in subsequent sections, a phenomenological model for the density of quasiparticle states is employed instead of considering explicit forms of E(k) and V(k) for simplicity. Furthermore, in orde
18、r to make our treatment easy, we assume that the system is symmetrical. In other words, the Hamiltonian (2.1) is assumed to be invariant under the particle-hole transfor- mation. One of the most crucial issues in constructing a theory of strongly correlated electron systems is how properly lo- cal q
19、uantum spin fl uctuations, which are responsible for the quenching of magnetic moments, are treated. They can be correctly taken into account in the single-site ap- proximation (SSA)14that is rigorous for paramagnetic states in infi nite dimensions. Consider the zero-fi eld case at fi rst; H = 0. Wi
20、thin the SSA, Greens functions for f electrons and conduction electrons are respectively given by Gff(in,k) = 1 in Ef(in) X |V(k)|2 in E(k) , (2.2) with the single-site self-energy function, and G(in,k) = g(in,k) +g(in,k)V(k)Gff(in,k)V (k)g(in,k), (2.3) with g(in,k) = inE(k)1. Here, inis an imagi- n
21、ary fermion energy with n an integer. To obtain the single-site self-energy function is reduced to solving a single-impurity Anderson model14that has the same lo- calized electron level Efand on-site Coulomb repulsion U as those in Eq. (2.1). Call this Anderson model a mapped Anderson model (MAM). O
22、ther parameters in the MAM are determined through the mapping condition: Gff(in) = 1 N X k Gff(in,k),(2.4) where N is the number of unit cells and Gffis the Greens function of the MAM written as Gff(in) = 1 in Ef(in) L(in) ,(2.5) with L(in) = (1/) R d ()/(in ). Here, () is the hybridization energy o
23、f the MAM. Once a trial function for () is given, (in) is obtained by solving the MAM numerically.Therefore, Eq. (2.4) is a self- consistency condition for (). However, we do not per- form this self-consistent calculation in this paper.In- stead, several approximations will be taken in subsequent se
24、ctions with the use of well-known results for the Kondo problem. Consider the spin susceptibility of the MAM, s(il), where ilis a boson energy with l an integer. The Kondo temperature for the periodic Anderson model, which is the energy scale of local quantum spin fl uctuations, is defi ned by lim T
25、0 s(+i0) 1 kBTK .(2.6) In the same way as the previous paper, the electron- lattice interaction is taken into account simply through the volume dependence of the Kondo temperature: TK(x) = TK(0)ex,(2.7) with x = V/V0.Here, 190 is the Gr uneisen constant of TK. For the sake of simplicity, the argumen
26、t x will be omitted unless particularly required hereafter. III. FERMI LIQUID DESCRIPTION The self-energy function is expanded as (in) = (+i0) + 1 min+ ,(3.1) for small |n|, where mis a mass enhancement factor in the SSA. Note that (+i0) = Efin the symmetrical case. Then the coherent part of Eq. (2.
27、2) is written as G(c) ff(in,k) = 1 min X |V(k)|2 in E(k) ,(3.2) and correspondingly, G(c) is given by Eq. (2.3) with replacing Gffby G(c) ff . Quasiparticles are defi ned as the poles of Eq. (3.2), namely, the dispersion relation of quasiparticles is obtained by solving the following equa- tion mz X
28、 |V(k)|2 z E(k) = 0.(3.3) We write the solutions as z = (k) with representing the branch of quasiparticles. By dividing the right-hand side of Eq. (3.2) into partial fractions, it follows that G(c) ff(in,k) = X 1 Zf(k) 1 in (k), (3.4) where the renormalization factor is given by 2 1 Zf(k) = 1 m+ X (
29、k) ,(3.5) with (k) = |V(k)|2/(k) E(k)2. Similarly, we have G(c) (in,k) = X 1 Z (k) 1 in (k), (3.6) with 1 Z (k) = (k) m+ X (k) .(3.7) It immediately follows from Eqs. (3.5) and (3.7) that the renormalization factors satisfy the relation m Zf(k) + X 1 Z (k) = 1.(3.8) With the use of Eq. (3.8), the de
30、nsity of quasiparticle states can be written in the form () 1 N X k ( (k) = 1 N X k Im ( mG(c) ff(+,k) + X G(c) (+,k) ) , (3.9) with += + i0. The energy scale for quasiparticles, TQ , is defi ned by (0) 1 4kBTQ .(3.10) On the other hand, Zf (k) obeys the following condition X 1 Zf(k) = 1 m ,(3.11) w
31、hich can be proved by comparing Eqs. (3.2) and (3.4) for the limiting case of |n| +. It should be noted that Zf (k) mfor the quasiparticle band that is closest to the Fermi level and Zf (k) mfor other branches for a given k, that is, m Zf(k) 1for|(k)| 2kBTQ 0for|(k)| 2kBTQ .(3.12) Consider a spectra
32、l function of renormalized f electrons, f , defi ned by 1 N X k G(c) ff(in,k) = 1 m Z d f() in , (3.13) or equivalently f() = 1 N X k m Zf(k) ( (k).(3.14) Equation (3.12) tells that f() ()for| 2kBTQ 0for| 2kBTQ ,(3.15) and it follows from Eq. (3.11) that f satisfi es the nor- malization condition of
33、 R d f() = 1. The Luttingers argument15applies straightforwardly to the periodic Anderson model. Consider the number of electrons per unit cell: n= 1 N X nk e+in0 ( Gff(in,k) + X G(in,k) ) , (3.16) with = 1/kBT. At T = 0 K, Eq. (3.16) can be trans- formed to n= Z 0 d (),(3.17) which is an exact rela
34、tion and is identical to the Lut- tingers theorem.Let us derive an expression for the specifi c heat. Recently an exact expression for the en- tropy of an interacting system was established.16For the periodic Anderson model, it is written as S = 1 N X k Z d f() T (X Im ln ?g1 (+,k) ? +Im ln h G1 ff(
35、+,k) i + Re Gff(+,k) Im (+) ) , (3.18) where f() = 1/(e+ 1).In Eq. (3.18), Gffcan be replaced by its coherent part because the derivative of f() is non-zero only for small | at low tempera- tures. In diff erentiating the entropy with respect to T, we introduce two assumptions; Im (+) = 0, and m= T-i
36、ndependent, that is, quasiparticles have an in- fi nite lifetime and temperature-independent mass. Since the chemical potential does not change as a function of temperature in a symmetrical model, the specifi c heat at constant volume is given by CV= T S T = X Z d f() T (),(3.19) where the identity
37、T2f/T2= /(f/T) has been made use of. 3 The zero-temperature limit of Eq. (3.19) is given by CV= T with = (22k2 B/3) (0). On the other hand, a combination of Eqs. (2.4), (3.13) and (3.15) gives (0) f(0) = m/(0). When () is constant, one can prove that8 m = 1 4kBTK ,(3.20) and then it follows TK TQin
38、the SSA, where TKand TQ are defi ned by Eqs.(2.6) and (3.10), respectively. We take this approximation in order to estimate the Kondo temperature. From the experimental value of , we have TK(0) 38 K. Let us consider the fi nite-fi eld case; H 6= 0.In the previous paper,8we constructed a perturbation
39、 method to derive a microscopic Landau free energy, one of whose independent variables is the magnetization m = P hf ifii. In the presence of magnetizations, physi- cal quantities, such as self-energy and polarization func- tions, can be expressed as a function of m instead of the magnetic fi eld H.
40、 The equation that defi nes quasiparti- cles becomes mz (m) X |V(k)|2 z E(k) = 0,(3.21) with (m) a magnetic part of the self-energy.In Eq. (3.21), m dependence of the mass enhancement fac- tor has been ignored.17The solutions for Eq. (3.21) are denoted by (k,m). All arguments for the fi nite-fi eld
41、case can be developed in parallel with those for the zero- fi eld case with replacing (k) by (k,m). For example, m/Zf (k,m) satisfi es a similar property as Eq. (3.12). Therefore, it follows by comparing Eqs. (3.3) and (3.21) that (k,m) (k) E(m),(3.22) where E(m) (m)/m.With the use of Eq. (3.22), th
42、e Luttingers theorem gives m = X Z 0 d f( + E(m), (3.23) where the polarization of conduction electrons has been ignored.In Eq. (3.23), we have replaced with f; because the right-hand side of Eq. (3.23) is a diff er- ence between the contributions from up and down spin directions, only the low-energ
43、y part is relevant.From Eq. (3.23), the magnetic part of the self-energy, E(m), can be determined as a function of m. A similar treatment for the MAM is also possible. Con- sider the coherent part of Eq. (2.5), which is defi ned by G(c) ff(in) (1/N) P kG (c) ff(in,k) and written as G(c) ff(in) = min
44、 L(c)(in)1 at zero fi elds. In the presence of magnetizations, it becomes G(c) ff(in,m) = 1 min+ EA(m) L(c)(in) . (3.24) Note that EA (m) is diff erent from E(m) because the MAM is determined in the absence of magnetizations by Eq (2.4) and then magnetic one-body potentials are added to both the per
45、iodic Anderson model and MAM separately.8In other words, EA(m) is a single-site term even with respect to m, whereas E(m) includes multi- site magnetizations. Because the magnetization in the MAM is also given by m to leading order in kBTK/U,8 EA(m) can be approximately evaluated from m = X Z 0 d ?
46、1 Im ? m G(c) ff(+,m). (3.25) In case of a constant hybridization energy, Eq.(3.25) is a rigorous relation and is nothing but the Friedel sum rule. It should be noted that the evaluation of E and EA from Eqs. (3.23) and (3.25) is one of the most essential improvements. In the previous paper, we assu
47、med that E(m) = EA(m) = (4kBTK/)tan(m/2). Before closing this section, it is helpful to mention the volume dependence of the physical quantities that have appeared above.First, it follows from Eqs. (2.7) and (3.20) that m ex. Taking notice of Eq. (3.12), we have f(;x) = e x f(e x;0) and L(c)(in;x) = L(c)(inex;0). Therefore, Eqs. (3.23) and (3.25) give E(m) exand EA(m) ex, respectively. IV. MAGNETIC EXCHANGE INTERACTION In this section, we study magnet