Two very different methods -- exact diagonalization on finite chains and a
variational method -- are used to study the possibility of a metal-insulator
transition in the symmetric half-filled periodic Anderson-Hubbard model. With
this aim we calculate the density of doubly occupied $d$ sites as a function of
various parameters. In the absence of on-site Coulomb interaction ($U_f$)
between $f$ electrons, the two methods yield similar results. The double
occupancy of $d$ levels remains always finite just as in the one-dimensional
Hubbard model. Exact diagonalization on finite chains gives the same result for
finite $U_f$, while the Gutzwiller method leads to a Brinkman-Rice transition
at a critical value ($U_d^c$), which depends on $U_f$ and $V$.Comment: 10 pages, 5 figure

F Woynarovich

I Hagymási

I. Hagymási

J. Sólyom

K Itai

K. Itai

M C Gutzwiller

W F Brinkman

English

arXiv

Crossref

Journal of the Korean Physical Society

Hubbard physics in the symmetric half-filled periodic anderson-hubbard model

Hagymasi, I

Itai, K

Solyom, J

ELTE Digital Institutional Repository (EDIT)

arXiv:1207.0381v1  [cond-mat.str-el]  2 Jul 2012Hubbard physics in the symmetric half-filled periodicAnderson-Hubbard modelI. Hagymási∗Institute for Solid State Physics and Optics,Wigner Research Centre for Physics,Hungarian Academy of Sciences, Budapest,H-1525 P.O. Box 49, Hungary andInstitute of Physics, Eötvös University, Budapest,Pázmány Péter sétány 1/A, H-1117, HungaryK. Itai and J. SólyomInstitute for Solid State Physics and Optics,Wigner Research Centre for Physics, Hungarian Academy of Sciences,Budapest, H-1525 P.O. Box 49, Hungary0AbstractTwo very different methods – exact diagonalization on finite chains and a variational method –are used to study the possibility of a metal-insulator transition in the symmetric half-filled periodicAnderson-Hubbard model. With this aim we calculate the density of doubly occupied d sites (νd)as a function of various parameters. In the absence of on-site Coulomb interaction (Uf ) between felectrons, the two methods yield similar results. The double occupancy of d levels remains alwaysfinite just as in the one-dimensional Hubbard model. Exact diagonalization on finite chains givesthe same result for finite Uf , while the Gutzwiller method leads to a Brinkman-Rice transition ata critical value (U cd), which depends on Uf and V .PACS numbers: 71.10.Fd, 71.27.+a, 75.30.MbKeywords: strongly correlated system, periodic Anderson model, exact diagonalization, Gutzwiller method∗E-mail: hagymasi.imre@mta.wigner.hu1I. INTRODUCTIONThe periodic Anderson-Hubbard model defined by the HamiltonianH =∑k,σεd(k)d̂†k,σd̂k,σ + Ud∑jn̂dj↑n̂dj↓ + εf∑j,σn̂fj,σ + Uf∑jn̂fj↑n̂fj↓−V∑j,σ(f̂ †j,σd̂j,σ + d̂†j,σf̂j,σ)(1)is meant to describe the physics of systems in which two types of electrons, one filling arelatively broad conduction band, the other a narrow band, are allowed to hybridize. Inwhat follows we call them d and f electrons. The interactions within the bands are denotedby Ud and Uf , respectively. In the present work we restrict ourselves to the half-filledparamagnetic case, that is, when there are N↑ = N↓ = N up- and down-spin electrons in anarbitrary dimensional lattice with N lattice sites, each of which has one d and one f orbital.Moreover, we restrict ourselves to the symmetric case, where an equal number of d and felectrons is present on the average. This is realized when εf = (Ud − Uf )/2, if the energy ismeasured from the center of the d band [1].In our previous study of this model [1], in which we used two very different methods: avariational calculation using the Gutzwiller type wave function and exact diagonalization,we were mainly interested in the effect of the on-site interaction Ud between conductionelectrons on the f -electron physics. In the present study we examine the effect of d-fhybridization (V ) and of the interaction Uf between f electrons on the Hubbard physics,that is on the eventual metal-insulator transition at half filling.In the Gutzwiller-type treatment of the half-filled Hubbard model, the metal-insulatortransition, which is known in this case as the Brinkman-Rice transition [2], occurs at a finiteUd, where the number of doubly occupied d sites becomes zero. A similar transition wasobtained by the Gutzwiller method in the half-filled periodic Anderson-Hubbard model, too[1], when the f electrons in the narrow band are strongly correlated.In contrast to that, exact diagonalization of the half-filled periodic Anderson-Hubbardmodel on finite chains gave a finite number of doubly occupied d sites for any Ud, just as inthe one-dimensional half-filled Hubbard model, where this number is finite for arbitrary Ud[3], even though the ground state is conducting only for Ud = 0 and insulating for Ud > 0.In this paper we will consider the half-filled symmetric Anderson-Hubbard model inthe paramagnetic regime in the full Ud ≥ 0, Uf ≥ 0 sector, when it is not necessarily2in the strongly correlated Kondo region, and will study the possibility of metal-insulatortransition. We should note here that in the present model a “metallic” phase is in fact aband insulator with hybridization gap. Therefore, we should be speaking about insulator-insulator transition, though its physics is the same as a metal-insulator transition due tothe interactions between electrons. We calculate the number of doubly occupied d sites as afunction of Ud for various values of V and Uf by both methods at the symmetric Andersonpoint, where the average number of f and d electrons per site (denoted by nf and nd,respectively) is exactly 1, and examine the effects of these couplings on the one- and higherdimensional Hubbard physics.II. CALCULATION BY EXACT DIAGONALIZATIONFirst, we perform exact diagonalization of the model on finite chains, where the kineticenergy of conduction electrons moving along the chain is described by hopping betweennearest-neighbor d orbitals with hopping rate t.We consider the periodic Anderson-Hubbard model on a chain consisting of six sites withperiodic boundary conditions. The results are shown in Figs. 1 and 2, where the densityof doubly occupied d sites, νd, is shown as a function of Ud for V = 0.1W and 0.3W ,respectively (W = 4t is the bandwidth), and for Uf = 0 and 5W . The values calculatedwith the Bethe Ansatz for the pure Hubbard model are also shown in the figure by a solidline.One can see that when the conduction electrons of the d band are hybridized with non-interacting electrons in the f band (Uf = 0), the larger the hybridization the more thenumber of doubly occupied sites. The curves are reasonably close to the results obtainedby the Gutzwiller method. The agreement gets better for stronger hybridization, while forweak hybridization it holds for small Ud values only.The values of νd decrease for finite Uf and get close to those of the pure Hubbard modelfor large Uf . This suggests that the d-electron subsystem becomes decoupled from the felectrons when the f electrons are strongly correlated.The results in this section are valid for a chain, for a one-dimensional model. In the nextsection we discuss a variational method, which might be relevant for higher dimensionalmodels.3ν d00.050.10.150.20.25Ud/W0 1 2 3 4 5 6Fig. 1. νd as a function of Ud for V = 0.1W . The empty and filled circles indicate the resultsof exact diagonalization for Uf/W = 0 and 5, respectively. The dashed line is the result of theGutzwiller method for Uf/W = 0. The solid line is the exact solution of the one dimensionalHubbard-model.ν d00.050.10.150.20.25Ud/W0 1 2 3 4 5 6Fig. 2. The same as Fig. 1 except that V = 0.3W .III. VARIATIONAL CALCULATIONWe summarize the main steps of the variational calculation following Ref. [4]. The trialwave function is chosen in the form|Ψ〉 = P̂ dGP̂ fG∏k∏σ[ukf̂†kσ + vkd̂†kσ]|0〉, (2)4where the Gutzwiller projectors, P̂ dGand P̂ fG, which contain the variational parameters ηdand ηf , are written asP̂ dG =∏j[1− (1− ηd)n̂dj↑n̂dj↓], (3)P̂ fG=∏j[1− (1− ηf)n̂fj↑n̂fj↓]. (4)The variational parameters, ηd and ηf , depend on Ud and Uf , respectively. Performing theoptimization with respect to the mixing amplitudes, uk and vk, we getE = 1N∑k∈FS[qdεd(k) + ε̃f −√[qdεd(k)− ε̃f]2+ 4Ṽ 2]+ (εf − ε̃f)nf + Udνd + Ufνf(5)for the ground-state energy density, where qd denotes the kinetic energy renormalizationfactor of d electrons given byqd =1(1− nd2)nd2[√(nd2− νd)νd +√(nd2− νd)(1− nd + νd)]2, (6)which is formally identical to the expression found in the Hubbard model [5]. The renor-malized hybridization amplitude is now Ṽ = V√qdqf ; the other notations are the same asin our previous paper [4], and the self-consistency condition is given bynf =1N∑k∈FS1 +qdεd(k)− ε̃f√[qdεd(k)− ε̃f]2+ 4Ṽ 2 = 1. (7)The summation over k can be carried out assuming a constant density of states, ρ(ε) = 1/W ,in the interval ε ∈ [−W/2,W/2]. The values of νf , and νd are obtained by optimizing theenergy density with respect to these parameters numerically. For V ≪ W and nd = nf =1 the optimization condition with respect to νd and νf results in the following coupledequations:UdW−[14+ 2(VW)2qfqd]8(1− 4νd) = 0, (8)UfW+ 2(VW)2ln[4qfqd(VW)2]8(1− 4νf ) = 0. (9)When either of Ud or Uf is zero, the equations are decoupled. Note that in the absence ofhybridization Eq. (8) reduces to the result for the ordinary Hubbard model.5We now turn to the discussion of the d-electron subsystem. We calculate the density ofdoubly occupied d sites, νd, as a function of Ud. The results are displayed in Fig. 3 forseveral values of Uf for a relatively weak hybridization, V = 0.1W , and in Fig. 4 for astronger hybridization.ν d00.050.10.150.20.25Ud/W0 1 2 3 4 5 6Fig. 3. νd as a function of Ud. The solid, dashed, dotted and dot-dashed lines correspond toUf/W = 0, 0.3, 0.5, 5 respectively. V = 0.1W in all cases.ν d00.050.10.150.20.25Ud/W0 2 4 6 8 10Fig. 4. νd as a function of Ud. The solid, dashed and dotted lines correspond to Uf/W = 0, 1, 10respectively. V = 0.3W in all cases.First, we find that for Uf = 0, νd never becomes zero, just as it was obtained by the exactdiagonalization for a finite chain, that is, the Brinkman-Rice transition does not occur. We6can calculate the asymptotic behavior of νd for large value of Ud from the following analysis.When Uf = 0, qf = 1 irrespective of Ud, and Eq. (8) can be solved for νd. For Ud ≫ W inleading order we arrive at:νd ∝2V 2Ud. (10)Second, we find that the Gutzwiller method leads to a Brinkman-Rice transition for anyUf > 0, that is, there exists a finite value, Ucd , where νd becomes zero. For Ud > 0 and Uf > 0the optimization conditions, Eqs. (8) and (9) for νf and νd are coupled, and thus both νdand νf decrease when either of the interactions increases. Therefore, even for very small Uf ,when Ud is large enough, νf also becomes small, and finally both νd and νf simultaneouslybecome zero at U cd . As a matter of fact, Ud and Uf play a rather similar role. If we fix Udat a certain value larger than 2W , a Brinkman-Rice transition occurs at a certain U cf . Thedifference in the condition (Ud > 2W and Uf > 0) necessary for occurrence of a transitionis due to the different widths of the d and f bands (W and 0, respectively) in the presentmodel.Third, when Uf is large enough and νf is exponentially small even for small Ud, that is,when the system is the Kondo regime, the νd−Ud curves become straight lines and coincideto the known behavior of the Hubbard-model. In the limit Uf ≫ W we obtain:νd =14− Ud8(W + 4EK), (11)whereEK =W2exp{− Uf16V 2/W}. (12)The scenario is the same for weak or strong hybridizations.The results described above indicate that a phase boundary can be defined in the three-dimensional parameter space of Ud, Uf and V , which separates the region where νd = νf = 0from that where both νd and νf are finite. The former is a Mott insulator region and thelatter is a hybridized band insulator region (for small Uf) or a Kondo insulator region (forlarge Uf). The phase boundaries in the Ud–Uf plain are shown in Fig. 5. (About theboundary between a hybridized band insulator and a Kondo one, see Ref. [1].)IV. CONCLUSIONS7Ud/W05101520Uf/W0 1 2 3 4 5Fig. 5. The phase boundaries in the Ud–Uf plain separating the metallic and insulating regimes.The solid and dashed lines correspond to V = 0.1W and 0.3W , respectively.In this paper we discussed the periodic Anderson-Hubbard model focusing our attentionon the physics of the conduction electron subsystem, using two different methods: exactdiagonalization on finite chains and a variational method of the Gutzwiller-type. We studiedthe effects of the d-f hybridization (V ) and of the on-site interaction between f electrons(Uf ) on the number of doubly occupied d sites, νd. When Uf = 0, both methods gave similarresults. For larger Uf , however, the results of exact diagonalization in the one-dimensionalmodel showed that νd approaches that obtained from the Bethe-Ansatz solution of the pureHubbard model, while the Gutzwiller method indicates a Brinkman-Rice-type scenario fora metal-insulator transition.It is interesting from theoretical point of view that νd ∝ 1/U2d according to the Bethe-Ansatz solution for Ud ≫ W , while the Gutzwiller method gives a slower asymptotic behav-ior, νd ∝ 1/Ud for Uf = 0.ACKNOWLEDGMENTSThis work was supported in part by the Hungarian Research Fund (OTKA) throughGrant No. T 68340. One of the authors (I. H.) acknowledges the support of TÁMOP GrantNo. 4.2.2/B-10/1-2010-0030.8REFERENCES[1] I. Hagymási, K. Itai and J. Sólyom, to be published in Phys. Rev. B, cond-mat/1106.4999[2] W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970).[3] F. Woynarovich, J. Phys. C: Solid State Phys. 16, 6593 (1983).[4] K. Itai and P. Fazekas, Phys. Rev. B 54, R752 (1996).[5] M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963), Phys. Rev. 134, A923 (1964), Phys. Rev.137, A1726 (1965); see also D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984).9

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Hubbard physics in the symmetric half-filled periodic Anderson-Hubbard
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