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A study of the Hubbard–Holstein model on a four-site chain
Physica B 259—261 (1999) 725—726
A study of the Hubbard—Holstein model on a four-site chain M. Acquarone *, M. Cuoco, C. Noce, A. Romano G.N.S.M.-C.N.R., I.N.F.M., Dipartimento di Fisica, Universita% di Parma, I-43100 Parma, Italy Unita% I.N.F.M. di Salerno, and Dipartimento di Scienze Fisiche **E.R. Caianiello++, Universita% di Salerno, I-84081 Baronissi (Sa), Italy
Abstract Using the displacement and the squeezing transformations, we obtain the polaronic Hamiltonian for a chain of four sites including all the electronic one- and two-body terms for a single orbital, the free phonon term for a single frequency X, and an electron—phonon coupling g of the Holstein type. The ground state was studied for up to four electrons by exact diagonalization and variational optimization of the displacement and squeezing parameters. For any filling we find an abrupt polaron localization transition, with strong renormalization of all the electronic interactions. 1999 Elsevier Science B.V. All rights reserved. Keywords: Holstein—Hubbard Hamiltonian; Polarons; Superconductivity
Structural effects influence the physical properties of many important classes of new materials. Among those most intensively studied in the last few years, we mention, for instance, the manganites exhibiting giant magnetoresistance and the high-temperature superconducting perovskites. In many cases, the analysis of these effects has been performed by generalizing purely electronic Hamiltonians, such as those defining the Hubbard and the t—J models, with the inclusion of various kind of electron-phonon interactions. Less attention, however, has been devoted to how the specific features of the lattice structure affect the electronic interactions. This analysis can be of great usefulness in the description of several kinds of experiments, such as, for instance, all measurements performed on samples under either external or chemical pressure. The simplest way to take into account these effects is to introduce an explicit lattice constant dependence of the electronic interaction parameters, expressing the latter as integrals of suitable combinations of Wannier functions centered on the lattice sites. This
procedure will be followed here, referring to model systems for which (i) the electronic part of the Hamiltonian contains all possible one- and two-body interactions; (ii) the free phonon term describes dispersionless phonons with a single Einstein frequency X; (iii) the coupling between electrons and phonons is provided by an interaction term of strength g of the Holstein type. This approach has already been considered [1,5] in a previous study focusing on the properties of a simple dimer with a single orbital per site. We present here an extension of this study to the case of a chain of four sites, equally spaced by a. The bare electronic interaction parameters are evaluated by using normalized Gaussian “orbitals”
(r)"(2C/p) exp[!C(r!R )] and assuming that G G the modifications needed to pass from the dimer to the present case were negligible (we refer to Refs. [1,5] for the details). According to the above considerations, we refer to the Hamiltonian H"e n ! [t!X(n #n )](cR c #h.c.) GN G \N H \N GN HN GN 6GH7 N
* Correspondence address: Dipartimento di Fisica, Universita di Parma, Viale delle Scienze, I-43100 Parma, Italy. Fax: #39521-905223; e-mail: [email protected].
#º n n #» n n !2J SXSX Gj Gi G H X G H G 6GH7 6GH7
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 2 1 - 8
726
M. Acquarone et al. / Physica B 259—261 (1999) 725—726
!J (S>S\#h.c.)#P (cR cR c c #h.c.) VW G H Gj Gi Hi Hj 6GH7 6GH7 MX 1 u#g n u , # P# G G GN G 2 2M G G GN where standard notation is used. The effective polaronic Hamiltonian is obtained by first performing on the electron—phonon term a displaced-oscillator transformation through the operator e0,exp[c d l (bR!b )], N N N N \N where c,g / X, l ,(N) n exp(ipR ), j labels N HN HN H the sites, and p"0,$p/2a, p/a is one of the wavevectors of the chain. Then the “displaced” Hamiltonian is averaged over the squeezed phonon wavefunction exp[! a (bRbR !b b )]"02, with "02 being the N N N \N N \N phonon vacuum [2,6]. Both +d , and +a , are variational N N parameters, whose value is set by optimizing the ground state energy. The eigenstates and the eigenvalues of the Hamiltonian have been determined by means of a previously developed exact diagonalization method [3,4], fully taking into account the SU(2) spin symmetry and the translational invariance. The application of this method leads to a significative reduction of the size of the Hamiltonian blocks, thus considerably simplifying the diagonalization procedure. For N"2, 3, 4, i.e. from the quarter-filled to the halffilled case, and selecting X"0.1 eV and g "0.44 eV, we have evaluated the spectrum of the energy eigenvalues and the renormalized electronic interactions, evaluated with the optimal values of the displacement and squeezing parameters. For different fillings there are quantitative differences in the result, but the overall trends are common to all cases. Therefore it is enough to show for the N"4 case only the most significant renormalized electronic parameters (Fig. 1, top panel), the displacement parameters d (middle panel), and the squeezing N related quantity exp (!2a ) (bottom panel), all versus N aC, which is the quantity defining the shape of the Wannier functions. The ground state has vanishing total spin for any value of aC. The bottom panel indicates that at a critical aC value there is a sharp localization transition, where the renormalized hopping t* vanishes. Notice that, well before the localization transition takes place, the on-site repulsion º* is hugely renormalized by the phonons, becoming negative. This is due to the increase of d shown in the top panel. Notice that the hoppingp ? depressive effect of d is, at least initially, balanced by p ? the increased squeezing at the same wavevector, as shown in the middle panel. Only for larger aC values (corresponding to more localized Wannier functions), the increase of the phononic energy +Sh(2q ) due to N the squeezing becomes too large to be compensated by the kinetic energy, and the system passes abruptly to the localized state. The inter-site repulsion »* is also renormalized appreciably, but does not show any precursory effect.
Fig. 1. Dependence on the Wannier function shape parameter aC of the main renormalized electronic interactions (top panel), the displacement parameters d (middle panel), and the squeezN ing quantities exp(!2a ) (bottom panel). N
For lower fillings, the localization transition occurs at a value of aC which increases as the filling decreases. However, the precursory behaviour of º* disappears, as well as the accompanying gradual variation of d and a . p p Preliminary results for smaller g values show that, above a rather large value of aC, the localization transition is substituted by a decoupling of the electron and phonon subsystems, with no reduction of the bare hopping amplitude, in substantial agreement with the results for the dimer case [1,5]. Work for a complete study of the phase diagram is in progress.
References [1] M. Acquarone, J.R. Iglesias, M.A. Gusma o, C. Noce, A. Romano, J. Superconductivity 10 (1997) 305. [2] Zheng Hang, Solid State Commun. 65 (1988) 731. [3] C. Noce, M. Cuoco, Phys. Rev. B 54 (1996) 13047. [4] M. Cuoco, A. Romano, C. Noce, Int. J. Mod. Phys. B 11 (1997) 2511. [5] M. Acquarone, J.R. Iglesias, M.A. Gusma o, C. Noce, A. Romano, Phys. Rev. B 58 (1998) 7626. [6] Zheng Hang, Z. Phys. B 82 (1991) 363.
A study of the Hubbard—Holstein model on a four-site chain M. Acquarone *, M. Cuoco, C. Noce, A. Romano G.N.S.M.-C.N.R., I.N.F.M., Dipartimento di Fisica, Universita% di Parma, I-43100 Parma, Italy Unita% I.N.F.M. di Salerno, and Dipartimento di Scienze Fisiche **E.R. Caianiello++, Universita% di Salerno, I-84081 Baronissi (Sa), Italy
Abstract Using the displacement and the squeezing transformations, we obtain the polaronic Hamiltonian for a chain of four sites including all the electronic one- and two-body terms for a single orbital, the free phonon term for a single frequency X, and an electron—phonon coupling g of the Holstein type. The ground state was studied for up to four electrons by exact diagonalization and variational optimization of the displacement and squeezing parameters. For any filling we find an abrupt polaron localization transition, with strong renormalization of all the electronic interactions. 1999 Elsevier Science B.V. All rights reserved. Keywords: Holstein—Hubbard Hamiltonian; Polarons; Superconductivity
Structural effects influence the physical properties of many important classes of new materials. Among those most intensively studied in the last few years, we mention, for instance, the manganites exhibiting giant magnetoresistance and the high-temperature superconducting perovskites. In many cases, the analysis of these effects has been performed by generalizing purely electronic Hamiltonians, such as those defining the Hubbard and the t—J models, with the inclusion of various kind of electron-phonon interactions. Less attention, however, has been devoted to how the specific features of the lattice structure affect the electronic interactions. This analysis can be of great usefulness in the description of several kinds of experiments, such as, for instance, all measurements performed on samples under either external or chemical pressure. The simplest way to take into account these effects is to introduce an explicit lattice constant dependence of the electronic interaction parameters, expressing the latter as integrals of suitable combinations of Wannier functions centered on the lattice sites. This
procedure will be followed here, referring to model systems for which (i) the electronic part of the Hamiltonian contains all possible one- and two-body interactions; (ii) the free phonon term describes dispersionless phonons with a single Einstein frequency X; (iii) the coupling between electrons and phonons is provided by an interaction term of strength g of the Holstein type. This approach has already been considered [1,5] in a previous study focusing on the properties of a simple dimer with a single orbital per site. We present here an extension of this study to the case of a chain of four sites, equally spaced by a. The bare electronic interaction parameters are evaluated by using normalized Gaussian “orbitals”
(r)"(2C/p) exp[!C(r!R )] and assuming that G G the modifications needed to pass from the dimer to the present case were negligible (we refer to Refs. [1,5] for the details). According to the above considerations, we refer to the Hamiltonian H"e n ! [t!X(n #n )](cR c #h.c.) GN G \N H \N GN HN GN 6GH7 N
* Correspondence address: Dipartimento di Fisica, Universita di Parma, Viale delle Scienze, I-43100 Parma, Italy. Fax: #39521-905223; e-mail: [email protected].
#º n n #» n n !2J SXSX Gj Gi G H X G H G 6GH7 6GH7
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 2 1 - 8
726
M. Acquarone et al. / Physica B 259—261 (1999) 725—726
!J (S>S\#h.c.)#P (cR cR c c #h.c.) VW G H Gj Gi Hi Hj 6GH7 6GH7 MX 1 u#g n u , # P# G G GN G 2 2M G G GN where standard notation is used. The effective polaronic Hamiltonian is obtained by first performing on the electron—phonon term a displaced-oscillator transformation through the operator e0,exp[c d l (bR!b )], N N N N \N where c,g / X, l ,(N) n exp(ipR ), j labels N HN HN H the sites, and p"0,$p/2a, p/a is one of the wavevectors of the chain. Then the “displaced” Hamiltonian is averaged over the squeezed phonon wavefunction exp[! a (bRbR !b b )]"02, with "02 being the N N N \N N \N phonon vacuum [2,6]. Both +d , and +a , are variational N N parameters, whose value is set by optimizing the ground state energy. The eigenstates and the eigenvalues of the Hamiltonian have been determined by means of a previously developed exact diagonalization method [3,4], fully taking into account the SU(2) spin symmetry and the translational invariance. The application of this method leads to a significative reduction of the size of the Hamiltonian blocks, thus considerably simplifying the diagonalization procedure. For N"2, 3, 4, i.e. from the quarter-filled to the halffilled case, and selecting X"0.1 eV and g "0.44 eV, we have evaluated the spectrum of the energy eigenvalues and the renormalized electronic interactions, evaluated with the optimal values of the displacement and squeezing parameters. For different fillings there are quantitative differences in the result, but the overall trends are common to all cases. Therefore it is enough to show for the N"4 case only the most significant renormalized electronic parameters (Fig. 1, top panel), the displacement parameters d (middle panel), and the squeezing N related quantity exp (!2a ) (bottom panel), all versus N aC, which is the quantity defining the shape of the Wannier functions. The ground state has vanishing total spin for any value of aC. The bottom panel indicates that at a critical aC value there is a sharp localization transition, where the renormalized hopping t* vanishes. Notice that, well before the localization transition takes place, the on-site repulsion º* is hugely renormalized by the phonons, becoming negative. This is due to the increase of d shown in the top panel. Notice that the hoppingp ? depressive effect of d is, at least initially, balanced by p ? the increased squeezing at the same wavevector, as shown in the middle panel. Only for larger aC values (corresponding to more localized Wannier functions), the increase of the phononic energy +Sh(2q ) due to N the squeezing becomes too large to be compensated by the kinetic energy, and the system passes abruptly to the localized state. The inter-site repulsion »* is also renormalized appreciably, but does not show any precursory effect.
Fig. 1. Dependence on the Wannier function shape parameter aC of the main renormalized electronic interactions (top panel), the displacement parameters d (middle panel), and the squeezN ing quantities exp(!2a ) (bottom panel). N
For lower fillings, the localization transition occurs at a value of aC which increases as the filling decreases. However, the precursory behaviour of º* disappears, as well as the accompanying gradual variation of d and a . p p Preliminary results for smaller g values show that, above a rather large value of aC, the localization transition is substituted by a decoupling of the electron and phonon subsystems, with no reduction of the bare hopping amplitude, in substantial agreement with the results for the dimer case [1,5]. Work for a complete study of the phase diagram is in progress.
References [1] M. Acquarone, J.R. Iglesias, M.A. Gusma o, C. Noce, A. Romano, J. Superconductivity 10 (1997) 305. [2] Zheng Hang, Solid State Commun. 65 (1988) 731. [3] C. Noce, M. Cuoco, Phys. Rev. B 54 (1996) 13047. [4] M. Cuoco, A. Romano, C. Noce, Int. J. Mod. Phys. B 11 (1997) 2511. [5] M. Acquarone, J.R. Iglesias, M.A. Gusma o, C. Noce, A. Romano, Phys. Rev. B 58 (1998) 7626. [6] Zheng Hang, Z. Phys. B 82 (1991) 363.