Eur. Phys. J. B 68, 201–208 (2009) DOI: 10.1140/epjb/e2009-00102-y THE EUROPEAN PHYSICAL JOURNAL B Regular Article
Quasiparticle properties of strongly correlated electron systems with itinerant metamagnetic behavior
J. Bauera Max-Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany
Received 14 August 2008 / Received in final form 25 February 2009 Published online 18 March 2009 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2009
Abstract. A brief account of the zero temperature magnetic response of a system of strongly correlated electrons in strong magnetic field is given in terms of its quasiparticle properties. The scenario is based on the paramagnetic phase of the half-filled Hubbard model, and the calculations are carried out with the dynamical mean field theory (DMFT) together with the numerical renormalization group (NRG). As well known, in a certain parameter regime one finds a magnetic susceptibility which increases with the field strength. Here, we analyze this metamagnetic response based on Fermi liquid parameters, which can be calculated within the DMFT-NRG procedure. The results indicate that the metamagnetic response can be driven by field-induced effective mass enhancement. However, also the contribution due to quasiparticle interactions can play a significant role. We put our results in context with experimental studies of itinerant metamagnetic materials.
PACS. 71.10.Fd Lattice fermion models – 71.27.+a Strongly correlated electron systems; heavy fermions – 71.30.+h Metal-insulator transitions and other electronic transitions – 75.20.-g Diamagnetism, param- agnetism, and superparamagnetism – 71.10.Ay Fermi-liquid theory and other phenomenological models
1 Introduction be relevant for the interpretation of experiments for itin- erant metamagnets where the magnetic response is mea- The interplay of strong correlation physics and magnetic sured simultaneously with the field dependence of the spe- behavior in itinerant electronic systems has been a fasci- cific heat. nating subject for many years. At low temperature it is In a naive single electron picture itinerant metam- often possible to describe the response of such systems agnetism is not intuitive as with increasing polarization in terms of the low energy excitations and quasiparticle the magnetic response usually decreases. For instance, in properties such as in a Fermi liquid picture. The ratio of weakly interacting systems, such as a Hubbard model with the spin susceptibility of the interacting system χs and small U, with a featureless concave density of states meta- 0 that of the non-interacting system χs is then given by the magnetic behavior does not occur. RPA based calcula- expression tions yield a decreasing susceptibility with increasing field ∗ χs m /m0 as spin fluctuations are suppressed. On the other hand, a = , (1) convex density of states, i.e. with positive curvature at the χ0 1+F a s 0 Fermi energy, such as in the Wohlfahrt and Rhodes [1]the- ∗ where m /m0 is the ratio of effective and bare electronic ory, can lead to metamagnetic behavior. This is exploited a mass, and F0 is the lowest order asymmetric Landau pa- in a number of works, where the Hubbard model with such rameter, which accounts for quasiparticle interactions. A convex density of states is analyzed [2,3]. Metamagnetic special kind of response is metamagnetism, which we de- behavior is shown to also occur in situations where the fine here as the existence of a regime where the system’s Fermi energy lies close to a van Hove singularity [4,5], or differential susceptibility, χs = dM/dH,increaseswith where a Pomeranchuk Fermi surface deformation instabil- magnetic field H, i.e. dχs/dH > 0, for H ∈ [H1,H2]with ity occurs [6]. It has been shown by calculations based on H1 > 0. The subject of this paper is the analysis of the the Gutzwiller approximation by Vollhardt [7] and Spalek metamagnetic response in correlated electron systems in and coworkers [8–10] that for a generic concave density of terms of the Fermi liquid description (1). For this we cal- states metamagnetic behavior is also found in the interme- culate the effective mass and the term due to quasiparticle diate coupling regime of the Hubbard model. The metam- interactions from a microscopic model. This allows us to agnetic scenario is then that of correlated electrons, with understand what drives the magnetic response. This can a (Mott) localization tendency due to the interaction. Our calculations are based on the half filled single a e-mail: [email protected] band Hubbard model which has been used frequently 202 The European Physical Journal B to describe itinerant metamagnetism for correlated condition reads electrons [2–5,9–12] due to its relative formal sim- −1 loc −1 plicity. We employ the dynamical mean field theory G0,σ(ω)=Gσ (ω) + Σσ(ω). (4) (DMFT) [11,13] combined with the numerical renormal- loc ization group (NRG) [14,15]tosolvetheeffectiveimpu- The Green’s function Gσ (ω) can be identified with the rity problem. We focus on the case of zero temperature, Green’s function Gσ(ω) of an effective Anderson model, −1 where sharp features are most clearly visible. We follow and G0,σ(ω) expressed as these earlier approaches here and restrict ourselves to the −1 response of the paramagnetic solutions of the Hubbard G0,σ(ω)=ω + μσ − Kσ(ω). (5) model, which is possible for mean field-like approaches. The half filled Hubbard model in a magnetic field has The function Kσ(ω) plays the role of a dynamical mean already been investigated by detailed DMFT studies by field describing the effective medium surrounding the im- Laloux et al. [11] and Bauer and Hewson [16]. Low tem- purity. Kσ(ω)andΣσ(ω) have to be calculated self- perature magnetization curves and field induced metal in- consistently using equations (3)–(5). Our calculations are sulator transitions have been investigated by Laloux et based on the numerical NRG [14,15] to solve the effec- al. Metamagnetic response based on correlated electron tive impurity problem. As in earlier work [16]wecalculate physics, seen in the Gutzwiller approach, was confirmed in spectral functions from a complete basis set [22,23] and use such calculations. Our analysis extends previous work [11] higher Green’s functions to obtain the self-energy [24]. For as we investigate the T = 0 magnetic response with a numerical calculations within the DMFT-NRG approach Fermi liquid interpretation based on the field dependent for ρ0(ε) we take the semi-elliptical form√ for the non- renormalized parameter approach [16–19]. This, together sem 2 2 2 interacting density of states ρ0 (ε)=2 D − ε /πD , with results for the spectral functions, allows us to identify where W =2D is the band width with D =2t for the what gives rise to the magnetic response in the system. Hubbard model. t = 1 sets the energy scale in the follow- The paper is organized as follows. In a brief Section 2 ing. we give details about the model and method. The Fermi liquid interpretation and the relation between Fermi liq- uid parameters and the field dependent renormalized pa- rameters are described in Section 3. Section 4 reports the 3 Field dependent renormalized parameters results for magnetization, susceptibilities and the inter- and Fermi liquid theory pretation in terms of effective mass and quasiparticle in- teractions. We conclude by putting our results in context The response of a metallic system of correlated electrons with itinerant metamagnetism studied experimentally. can often be described in terms of Fermi liquid theory. The ratio of the spin susceptibility of the interacting sys- 0 tem χs and that of the non-interacting system χs is given 2 Model and method in equation (1). Thus, when strongly interacting fermions have a large paramagnetic susceptibility, it can be inter- The basis for our calculation forms the Hubbard preted as due to quasiparticles with large effective masses. Hamiltonian in a magnetic field, which in the grand canon- It is, however, also possible that the susceptibility is ad- ical formulation reads ditionally enhanced due to the quasiparticle interaction † term 1/[1 + F a], which is for instance the case in liq- Hμ = (tij c cj,σ + h.c) − μσniσ + U ni,↑ni,↓. 0 i,σ 3 ∗ 0 i,j,σ iσ i uid He, where m /m0 5 but χs/χs 20 [25]. This (2) is usually described by the dimensionless Sommerfeld or † Wilson ratio R of the magnetic susceptibility and the lin- c creates an electron at site i with spin σ,andni,σ = i,σ ear specific heat coefficient γ.Wewilluseitintheform † − 0 ∗ ci,σci,σ. tij = t for nearest neighbors is the hopping am- R =(χs/χs)/(γ/γ0), where γ/γ0 = m /m0. plitude and U is the on-site interaction; μσ = μ + σh, Here we are interested in analyzing the behavior in where μ is the chemical potential of the interacting sys- finite field, and it is possible to calculate corrections of tem, and the Zeeman splitting term with external mag- higher order in H to equation (1)[26]. We will, however, netic field H is given by h = gμBH/2 with the Bohr mag- follow a different approach here, and assume that expres- neton μB. In the DMFT approach the proper self-energy sion (1) remains valid for finite field with field dependent is a function of ω only [20,21]. In this case the local lattice effective mass m∗(H) and Landau parameter F a(H). This loc 0 Green’s function Gσ (ω) can be expressed in the form, is in the spirit of the field dependent quasiparticle param- eters introduced in earlier work [16,18,19]. Notice that for loc ρ0(ε) 0 Gσ (ω)= dε , (3) the case considered the field dependence of χs,whichis ω + μσ − Σσ(ω) − ε given by the non-interacting density of states, varies very little in the relevant field range. In this picture with field where ρ0(ε) is the density of states for the non-interacting dependent parameters, metamagnetism can occur when model (U = 0). It is possible to convert this lattice prob- the effective mass increases with the magnetic field. Gen- lem into an effective impurity one [13], introduce the dy- erally, however, also the field dependence of the quasipar- −1 namical Weiss field G0,σ(ω). The DMFT self-consistency ticle interaction plays a role. One hypothesis tested in this J. Bauer: Quasiparticle properties for itinerant metamagnetic behavior 203 paper is that itinerant metamagnetic behavior is always η(h)=˜μ0(h)/z(h) − h. At half filling we have z↑ = z↓ ≡ z accompanied by a field induced localization and a sharp andμ ˜0,↑ = −μ˜0,↓ ≡ μ˜0. We define the function increase of the effective mass near the metamagnetic tran- ∗ sition. g(h):=h + η(h)=˜μ0(h)/z(h)=˜μ0(h)m (h)/m0, (9) ∗ −1 In order to calculate the microscopic Fermi liquid pa- as m /m0 = z in DMFT. In terms of the quasiparti- rameters, we expand Σσ(ω)inpowersofω for small ω, cles it is the product of the effective mass enhancement ∗ and retain terms to first order in ω only.Thisisusedto m /m0 and the shift of the quasiparticle bandμ ˜0.With define renormalized parameters [16], the applicability of Luttinger’s theorem the magnetization is then given by μ˜0,σ = zσ[μσ − Σσ(0)], and zσ =1/ [1 − Σσ(0)], (6) ∞ 1 1 and from (3) a normalized quasiparticle propagator, m(h)= (n↑ − n↓)= dε ρ0(ε)θ[g(h) − ε] − . (10) 2 2 −∞ ˜loc 1 ρ0(ε/zσ) G0,σ(ω)= dε . (7) For a local self-energy this is an exact expression for the zσ ω +˜μ0,σ − ε magnetization, which only depends on the field dependent renormalized parameters via g(h). For certain bare densi- Note that this ω-expansion can also be carried out in fi- ties of state, for instance, for the semi-elliptical density of sem nite magnetic field. Then the renormalized parameters be- states ρ0 (ε), it can be evaluated analytically, come field dependent, zσ = zσ(h)and˜μ0,σ =˜μ0,σ(h). 0,σ 0,σ 1 sem 1 The density of statesρ ˜ (ε) derived from (7),ρ ˜ (ε)= m(h)= g(h)ρ0 (g(h)) + arcsin(g(h)). (11) −ImG˜0,σ(ε + iδ)/π = ρ0[(ε +˜μ0,σ)/zσ]/zσ, is referred to 2 π as the free quasiparticle density of states. zσ is interpreted Differentiating (10) with respect to h yields the local static as the weight of the quasiparticle resonance andμ ˜0,σ gives spin susceptibility the position of the quasiparticle band. All energies are dm measured from the chemical potential μ. χs = = g (h)ρ0(g(h)) (12) To obtain the renormalized parameters zσ andμ ˜0,σ,we dh use two different methods based on the NRG approach. where here and in the following primes indicate derivatives The first method is a direct one where we use the self- with respect to h. A similar expression had already been energy Σσ(ω) determined by NRG and the chemical po- derived by Luttinger [28]. The metamagnetic condition tential μσ, and then substitute into equation (6)forzσ and χs(h) > 0isthen μ˜0,σ. The second method is indirect, and it is based on the 2 quasiparticle interpretation of the NRG low energy fixed g (h)ρ0(g(h)) + ρ0(g(h))g (h) > 0. (13) point of the effective impurity [17]. This approach has been The occurrence of metamagnetic behavior can be analyzed used earlier for the Hubbard model [16,27] and for the depending on the functional form of g(h)andρ0(ε). For Anderson impurity model in a magnetic field [18,19]. As a simple analysis let us assume h>0andthepowerlaw shown before the results of both methods usually agree form for g(h)=chα, c>0. The first term in (13)is within a few percent, and we use an average value of both then positive if α>1. For a convex density of states, methods for the numerical results. It is important to cal- ρ0 (ε) > 0, the second term is also positive and metamag- culate these parameters accurately, since for the following netic behavior occurs as mentioned earlier. For a concave results also their derivatives are needed. density of states, ρ0 (ε) < 0, the two terms in (13)compete. We can calculate static expectation values and re- If we also assume the power law form for the density of sponse functions in terms of the renormalized parameters. γ sem states, ρ0(ε)=r0 − dε ,(e.g.forρ0 one has r0 =2/πD The quasiparticle occupation numbern ˜0 is given by inte- σ d = r0/2andγ = 2) condition (13) becomes grating the quasiparticle density of states up to the Fermi level, r0 α − 1 αγ >h . (14) cγ d α(1 + γ) − 1 0 ∞ 0 Since the right hand side is positive, we can infer that for n˜ = dε ρ˜0,σ(ε)= dε ρ0,σ(ε)θ(μσ − Σσ − ε). (8) σ α>1andγ>(1 − α)/α metamagnetic behavior occurs. −∞ −∞ The actual field dependence of g(h) can be calculated from the renormalized parameters and it depends on the inter- Luttinger’s theorem [28] holds for each spin component for action strength. As we will see for the half filled Hubbard the Hubbard model in magnetic field [16], hence we have 0 model and intermediate U, g(h) grows faster than linear n˜ = nσ,wherenσ is the value of the occupation number σ with h, i.e. α>1. in the interacting system at T =0. In the limit of zero field the ratio of the susceptibility of To calculate the magnetic response we focus for the the interacting and non-interacting system has a simplified rest of this paper on the case with particle-hole symmetry expression in terms of the renormalized parameters, where μ = U/2, and we can write Σσ(0,h)=U/2−ση(h). ∗ We can calculate η(h) directly from the self-energy, e.g. χs m (0) = g(0) = μ˜ (0), (15) ↓ − ↑ 0 0 η(h)=(Σ Σ )/2, or from the renormalized parameters χs m0 204 The European Physical Journal B
0.5 For weak coupling it can be related to the rigid band shift U=2 ∼ U =2.5 and a large field h D is needed, but for larger interaction ≥ 0.4 U =3 strength hpol is reduced substantially. For U 3ameta- U =4 magnetic curvature in the magnetization can be observed, U =4.5 and we see that in the Hubbard model at zero tempera- 1 0.3 ture the metamagnetic transition field hm coincides with
0.5 hpol, which is not necessarily the case for T>0. Laloux
m(h) 0.45 U=4 et al. [11] have compared results from low temperature 0.2 0.4 DMFT calculations with the Gutzwiller approximation 0.35 and it was found that the occurrence of metamagnetic be- 0.1 0.3 havior is overestimated by the Gutzwiller approximation 0.25 (see also Fig. 3). 0.2 0.2 0.22 0.24 0.26 0.28 0.3 Earlier work [11] showed that the transition is a dis- 0 continuous first order one at low temperature. Our results 0 0.2 0.4 0.6 0.8 1 h show jumps in the magnetization curve at the transition field hm U U Fig. 1. (Color online) The local magnetization m(h) as a func- ,e.g.for =3and =4inFigure1, however, tion of the magnetic field h for different values of U.Wecan we can not exclude a very steep continuous increase which see that a metamagnetic curvature sets in at U =3.Inset: can not be resolved numerically. We have also found hys- Hysteresis curve for U = 4 (triangle up increasing h, triangle teresis, shown for U = 4 as an inset in Figure 1 (triangle down decreasing h). up increasing h, triangle down decreasing h). This suggests that the transition is also of first order for zero temper- ature. For larger interaction U ≥ 4.5thereexistsasmall forμ ˜0(0) = 0. Comparing with the Fermi liquid expres- field range near hm, where we have not found unique, well a sion (1)wecanidentify1/(1 + F0 )=˜μ0. This quantity converged DMFT solutions, so no definite statement can corresponds to the Wilson ratio R. In the general case, be made. the field dependent enhancement due to the quasiparticle The half filled repulsive Hubbard model in magnetic interactions reads field can be mapped to the attractive one [29], in which m∗ the chemical potential is related to the field in the original ∗ ρ0 μ˜0 1 m m0 model, μ = U/2+h. The attractive model has been studied R(h)= a = μ˜0 +˜μ0 ∗ . (16) by the DMFT in situations, where superconducting order 1+F0 (h) m ρ0(h) was not allowed for [30,31]. A first order transition from a So far the considerations have been independent of our metallic to a pairing state for fixed density was found at a DMFT-NRG approach. In the following section we will critical interaction. The occurrence of the transition can compare results for the magnetic susceptibility obtained be related to the metamagnetic transition here. A nearly from the static expectation values of integrating the polarized system corresponds to a low density limit, and Green’s functions, with the results based on the field to estimate when the transition sets in, one can analyze dependent parameters. We determine them as described the two-body problem in the attractive model and calcu- above. Alternatively they can be calculated by other meth- late the critical Uc for bound state formation. For a three ods, such as the Gutzwiller (GW) approach, and we will dimensional cubic lattice the result is Uc ≈ 0.659W [29]. make comparison as appropriate. Results are obtained as With the given bandwidth this corresponds to a value of in reference [7], where the critical interaction for the metal Uc ≈ 2.64, which is a reasonable estimate for the interac- GW sem insulator transition is Uc =16W/3π ≈ 6.79 for ρ0 (ε) tion strengths, where the metamagnetic behavior is found with W =4. here.
4Results 4.2 Magnetic susceptibilities and quasiparticle properties 4.1 Magnetization and metamagnetic transition From the initial slope of the magnetization curves in Fig- For a first overview we present results for the magnetiza- ure 1 we observe an increase of the magnetic susceptibility tion m(h) as a function of field h in Figure 1 for various with the interaction strength U. This increase can also be values of U. The magnetization m(h) was computed from seen in the following Figure 2 where we show the ratio 0 the static NRG expectation value (EV) for the occupation of zero field susceptibility to the non-interacting value χs number as well as from integrating the spectral function as function of U deduced from differentiating the EV for to the Fermi level, both of which agree very well. The m(h) in the limit h → 0. results for m(h) based on the field dependent renormal- For comparison we have also included the susceptibil- ized parameters (RP) and equation (11) are also in good ity calculated from equation (15) with the renormalized agreement, but not included in the figure. 1 The plot gives a clear picture of the field strength hpol The metamagnetic transition field is the field where the necessary to polarize the metal completely to m =1/2. susceptibility is maximal. J. Bauer: Quasiparticle properties for itinerant metamagnetic behavior 205
10 3.5 4 from EV 3 from RP 2 3 from GW 8 R(U) 5 1 4 15 3 0 10 2.5 0 s 6 2 (0) R(h)/R(0) χ s (U)/m * 5 1 m
(U)/ 0
s 0 (h)/ χ s χ 0 1 2 3 4 5 2 1
4 U χ (0) *
(h)/m 0.8 * 1.5 m 2 from EV 0.6 0 0.1 0.2 0.3 0.4 0.5 from RP h from GW 1 0 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 U h Fig. 2. (Color online) The U-dependence of the magnetic sus- 15 from EV ceptibility χs. We compare results deduced from the EV of m(h) with ones obtained from the RP and from the Gutzwiller from RP (GW) approximation. The inset shows the effective mass from GW ∗ m (U)/m0 and the Wilson ratio R(U) as a function of U. 4 10 3 2 (0) R(h)/R(0)
s 1 0
parameters (RP) and their derivatives, as well as the re- (h)/ χ s χ (0) sults obtained from the Gutzwiller (GW) approximation. * 3 5 (h)/m
EV and RP results agree very well, confirming the ap- * 2 m plicability of Fermi liquid results in this metallic regime. 1 The GW results follow a similar trend but overestimate 0 0.05 0.1 0.15 0.2 h the value for the susceptibility, which becomes more pro- nounced for larger U. 0 0 0.05 0.1 0.15 0.2 The inset plot shows the U-dependence of the effective h mass and the Wilson ratio. In terms of Fermi liquid theory Fig. 3. (Color online) The h-dependence of the ratio of the and the expression (1) the increase of χs with U can be finite and zero field magnetic susceptibility χs for U = 3 (upper understood by the behavior of the effective mass and the panel) and U =4.5 (lower panel). We compare results deduced progressive localization tendency, which brings out more from the EV for m(h) with ones obtained from the RP and the the spin degrees of freedom of the electrons. We can see, ones from the GW approach. The inset shows the ratio of finite ∗ however, that the effective mass ratio is larger than that and zero field effective mass m (h)/m0(0) and the Wilson ratio of the magnetic susceptibility. This difference can be at- R(h)/R(0) as a function of h. a −1 tributed to the factor R =˜μ0 =[1+F0 ] , which is due to the quasiparticle interaction. This factor is larger than one for smaller values of U, but decreases to values below one We can see that also in finite field the results for the for stronger interaction. This indicates a sign change of susceptibility calculated from the EV for m(h)andthe a the parameter F0 from negative to positive. The compar- field dependent RP agree fairly well with a deviation of ison with the corresponding quantities calculated in the less than 5%. For the case U = 3 (upper panel) the re- GW approximation shows a qualitatively similar behavior sults for χ(h) based on the field dependent RP are al- ∗ for both m /m0 and R,whenU is small. For larger values ways smaller. In both cases we find first a period where of U in Figure 2, however, the effective mass enhancement the susceptibility is nearly constant, but then starts to ∗ GW 2 in the GW approach, m /m0 =1− (U/Uc ) ,ismuch increase rapidly as h approaches hm.ForU =3theval- smaller and R increases with U in contrast to the DMFT ues obtained from the RP initially decrease slightly with result. the field, which is however incorrect, and comes about We return to the finite field response and focus on through numerical inaccuracies when determining the pa- the metamagnetic behavior which is found for interme- rameters and the numerical differentiation. As hm = hpol diate values of U. Results for the ratio of the magnetic the magnetic susceptibility is zero for h>hm. At finite susceptibility in finite and zero field deduced from dif- temperature a susceptibility maximum is expected. The ferentiating the magnetization (EV) are compared to the results for χs from the GW approximation show generally ones obtained from the quasiparticle parameters (RP) and a similar trend, but as mentioned earlier the metamag- equation (12). For completeness, we have also included re- netic behavior sets in at lower field strengths. sults from the GW approximation. This is shown in Fig- A difference in the behavior between the two cases is ure 3 for U = 3 in the upper panel and U =4.5inthe visible in the two insets where the ratios of field depen- lower panel. dent effective masses to their zero field values and the 206 The European Physical Journal B
0.35 0.35 h =0 h =0 0.3 h =0.5 0.3 h =0.15 h =0.8 h =0.22 h =1 h =0.229 0.25 0.25 ) 0.2 ) 0.2 ω ω ( ( ↑ ↑ ρ 0.15 ρ 0.15
0.1 0.1
0.05 0.05
0 0 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 ω ω
Fig. 4. (Color online) The majority spin density of states for 0.35 h =0 U = 2 and various field strengths in comparison. 0.3 h =0.15 h =0.22 h =0.229 field dependent Wilson ratios R(h)/R(0) are plotted. For 0.25 the U = 3 case the effective mass decreases with the field
) 0.2 ω
which is typical behavior in the weak coupling regime. It ( ↑ can be understood by RPA approximations where spin ρ 0.15 fluctuations, which give an effective mass enhancement, are suppressed in finite field. The metamagnetic increase 0.1 of the susceptibility, however, can not be explained by this. In terms of Fermi liquid theory it is related to the 0.05 magnetic field dependence of the quasiparticle interaction 0 rather than the localization tendency encoded in the ef- −0.4 −0.2 0 0.2 0.4 fective mass. R(h)/R(0) indeed is increasing sharply close ω to hm.Inequation(16) we have two competing terms Fig. 5. (Color online) The majority spin density of states for ∗ ∗ for this enhancement factor, m /m < 0, but one finds U =4.5 and various field strengths in comparison: upper panel ∗ ∗ μ˜0 > |μ˜0 m /m | which leads to the observed enhance- full frequency range, lower panel low frequency behavior. ment. The drive for the metamagnetic behavior is there- fore due to the shift of the quasiparticle band from the Fermi level with increasing field. This contrasts to the h or even decrease for larger fields. Such a behavior is also weak coupling situation, such as U =2,whereR(h)de- found within the GW approach for larger U near the metal creases with the field strength and no metamagnetic re- insulator transition. sponse is observed. The effective mass in the case of U =4.5(lowerpanel in Fig. 3) shows different behavior. We can see a sharp 4.3 Spectral functions increase with the field. However, the magnitude the ratio ∗ m /m0 increases is less than that of the susceptibility. The The behavior of the quasiparticle band can be seen di- difference again can be related to the Fermi liquid factor rectly in the local spectral function. For the cases with a R =1/[1 + F0 ], which is larger than one and increasing smaller coupling the field dependent response shows a con- with h as can be seen in the inset of the lower panel in Fig- tinuous shift of spectral weight to lower energies for the ure 3. In this case the second term in equation (16)ispos- majority spin (see Fig. 4 for U =2). ∗ ∗ itive and the first term negative, but |μ˜0| < |μ˜0 m /m |. Note that the minority spin density of states ρ↓(ω)is The results from the GW approach for the effective mass given by ρ↑(−ω) at half filling. To illustrate the behavior and R are in line with the DMFT calculations for the case of the quasiparticle peak for the stronger interacting case U = 3, however, for U =4.5, the GW result for m∗/m∗ with U =4.5 in more detail, we plot the local spectral only increases very little with the field, whereas R(h)in- function for the majority spin ρ↑(ω)inFigure5. creases sharply to yield the metamagnetic response. In the upper panel we can see how the lower Hubbard For larger interactions than the ones discussed here peak in the spectral density acquires weight when the field (5
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