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Effect of the intersite Coulomb interaction in the Hubbard–Holstein model on a four-site chain
Physb=21777=Shankar=Venkatachala=BG
Physica B 284}288 (2000) 1561}1562
E!ect of the intersite Coulomb interaction in the Hubbard}Holstein model on a four-site chain M. Acquarone , M. Cuoco, C. Noce *, A. Romano GNSM-CNR, Unita% INFM di Parma, Dipartimento di Fisica, Universita% di Parma, 43100 Parma, Italy Unita% INFM di Salerno, Dip. di Scienze Fisiche **E.R. Caianiello++, Universita% di Salerno, 84081 Baronissi (SA), Italy
Abstract We derive an e!ective polaronic Hamiltonian by applying the displacement and the squeezing transformations to an extended version of the Hubbard}Holstein model, which includes all the electronic one- and two-body interactions for electrons coupled to dispersionless optical phonons. This e!ective Hamiltonian is then exactly solved on a four-site chain and the e!ect of the intersite Coulomb interaction on the spin and charge ordering in the ground state at half-"lling is investigated. 2000 Elsevier Science B.V. All rights reserved. Keywords: Antiferromagnetism; Charge density wave; Hubbard}Holstein Hamiltonian; Polarons
In several important classes of new materials the lattice structure plays a relevant role in the determination of the physical properties. We focus here our attention on how the electronic coupling constants are a!ected by speci"c features of the underlying lattice. The study is performed via a parametrization of the bare electronic interactions in terms of the lattice constant a and the width C\ of the Gaussian functions (r)"(2C/n) exp[!C(r!R )] G G which approximate the orbitals on each site. We start from a general Hamiltonian H containing all one- and two-body electronic interactions, as well as an Holstein coupling term between electrons and optical phonons [1]. A displaced oscillator transformation is then applied to the electron}phonon term, leading to an e!ective Hamiltonian that is subsequently averaged over a squeezed phonon wave function. In this way one gets a purely electronic e!ective Hamiltonian with coupling constants renormalized by the electron}phonon interaction [1] HH" eHn ! tH cR c #;H n n H HN HJ HN JN Ht Hs HN H6J7N H #
* Corresponding author. E-mail address: [email protected] (C. Noce)
# JH S ) S # PH (cR cR c c #h.c.) GH G H GH Gt Hs Hs Gt G6H7 G6H7 # X [sinh(a )#]. O N O
(1)
The explicit expressions of the e!ective coupling constants in terms of the displacement parameters d O and the squeezing parameters a are given elsewhere O [2]. Let us only remark here that a relevant e!ect introduced by the electron}phonon coupling is that the intersite density}density interaction, which in H is assumed to be e!ective only between nearest-neighbor sites, acquires a phonon-dependent long-range contribution, which is non-vanishing if the wave vector dependence of the displacement parameters is accounted for. The e!ective Hamiltonian (1) has been exactly diagonalized [3] on a four-site chain and the optimal values of the variational parameters d and a have been deterO O mined via minimization of the total energy in the ground state, under the assumption of dispersionless phonons with "xed frequency u. This procedure has been followed for several values of the electron "lling, but for brevity we will only discuss here the case N "4 (half-"lling). In order to study the possible occurrence of spin and charge order in the ground state, let us introduce the
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 7 9 9 - 4
Physb=21777=KGM=VVC=BG
1562
M. Acquarone et al. / Physica B 284}288 (2000) 1561}1562
Fig. 1. Phase diagram for the four-site chain at half-filling.
corresponding form factors 1 S(q)" exp [iq(R !R )] 1S S 2, (2) G H G H ¸ GH 1 N(q)" exp [iq(R !R )] 1n n 2, (3) G H G H ¸ GH where, for a number of lattice sites ¸ equal to 4, the wave vector q takes the values +0, n/2, n, 3n/2,. In Fig. 1 we report the ground state phase diagram for N "4, ex pressed in terms of the rescaled electron}phonon coupling g / u and of the quantity aC accounting for the degree of localization of the electronic orbitals. The phase boundaries have been determined by looking at the behavior of the correlation functions (2) and(3) evaluated in the variationally determined ground state. We can identify two regions characterized by the predominance of antiferromagnetic (AF) and charge density wave (CDW)
correlations, signalled by enhanced values at q"n of S(q) and N(q), respectively, plus a third region where the system exhibits a frustrated or reduced magnetization (RM). The boundaries are shown in the two cases where the next-nearest-neighbor Coulomb repulsion
References [1] M. Acquarone et al., Phys. Rev. B 58 (1998) 7626. [2] M. Acquarone et al., Int. J. Mod. Phys. B 13 (1999) 3331. [3] C. Noce, M. Cuoco, Phys. Rev. B 56 (1996) 13 047.
Physica B 284}288 (2000) 1561}1562
E!ect of the intersite Coulomb interaction in the Hubbard}Holstein model on a four-site chain M. Acquarone , M. Cuoco, C. Noce *, A. Romano GNSM-CNR, Unita% INFM di Parma, Dipartimento di Fisica, Universita% di Parma, 43100 Parma, Italy Unita% INFM di Salerno, Dip. di Scienze Fisiche **E.R. Caianiello++, Universita% di Salerno, 84081 Baronissi (SA), Italy
Abstract We derive an e!ective polaronic Hamiltonian by applying the displacement and the squeezing transformations to an extended version of the Hubbard}Holstein model, which includes all the electronic one- and two-body interactions for electrons coupled to dispersionless optical phonons. This e!ective Hamiltonian is then exactly solved on a four-site chain and the e!ect of the intersite Coulomb interaction on the spin and charge ordering in the ground state at half-"lling is investigated. 2000 Elsevier Science B.V. All rights reserved. Keywords: Antiferromagnetism; Charge density wave; Hubbard}Holstein Hamiltonian; Polarons
In several important classes of new materials the lattice structure plays a relevant role in the determination of the physical properties. We focus here our attention on how the electronic coupling constants are a!ected by speci"c features of the underlying lattice. The study is performed via a parametrization of the bare electronic interactions in terms of the lattice constant a and the width C\ of the Gaussian functions (r)"(2C/n) exp[!C(r!R )] G G which approximate the orbitals on each site. We start from a general Hamiltonian H containing all one- and two-body electronic interactions, as well as an Holstein coupling term between electrons and optical phonons [1]. A displaced oscillator transformation is then applied to the electron}phonon term, leading to an e!ective Hamiltonian that is subsequently averaged over a squeezed phonon wave function. In this way one gets a purely electronic e!ective Hamiltonian with coupling constants renormalized by the electron}phonon interaction [1] HH" eHn ! tH cR c #;H n n H HN HJ HN JN Ht Hs HN H6J7N H #
* Corresponding author. E-mail address: [email protected] (C. Noce)
# JH S ) S # PH (cR cR c c #h.c.) GH G H GH Gt Hs Hs Gt G6H7 G6H7 # X [sinh(a )#]. O N O
(1)
The explicit expressions of the e!ective coupling constants in terms of the displacement parameters d O and the squeezing parameters a are given elsewhere O [2]. Let us only remark here that a relevant e!ect introduced by the electron}phonon coupling is that the intersite density}density interaction, which in H is assumed to be e!ective only between nearest-neighbor sites, acquires a phonon-dependent long-range contribution, which is non-vanishing if the wave vector dependence of the displacement parameters is accounted for. The e!ective Hamiltonian (1) has been exactly diagonalized [3] on a four-site chain and the optimal values of the variational parameters d and a have been deterO O mined via minimization of the total energy in the ground state, under the assumption of dispersionless phonons with "xed frequency u. This procedure has been followed for several values of the electron "lling, but for brevity we will only discuss here the case N "4 (half-"lling). In order to study the possible occurrence of spin and charge order in the ground state, let us introduce the
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 7 9 9 - 4
Physb=21777=KGM=VVC=BG
1562
M. Acquarone et al. / Physica B 284}288 (2000) 1561}1562
Fig. 1. Phase diagram for the four-site chain at half-filling.
corresponding form factors 1 S(q)" exp [iq(R !R )] 1S S 2, (2) G H G H ¸ GH 1 N(q)" exp [iq(R !R )] 1n n 2, (3) G H G H ¸ GH where, for a number of lattice sites ¸ equal to 4, the wave vector q takes the values +0, n/2, n, 3n/2,. In Fig. 1 we report the ground state phase diagram for N "4, ex pressed in terms of the rescaled electron}phonon coupling g / u and of the quantity aC accounting for the degree of localization of the electronic orbitals. The phase boundaries have been determined by looking at the behavior of the correlation functions (2) and(3) evaluated in the variationally determined ground state. We can identify two regions characterized by the predominance of antiferromagnetic (AF) and charge density wave (CDW)
correlations, signalled by enhanced values at q"n of S(q) and N(q), respectively, plus a third region where the system exhibits a frustrated or reduced magnetization (RM). The boundaries are shown in the two cases where the next-nearest-neighbor Coulomb repulsion
References [1] M. Acquarone et al., Phys. Rev. B 58 (1998) 7626. [2] M. Acquarone et al., Int. J. Mod. Phys. B 13 (1999) 3331. [3] C. Noce, M. Cuoco, Phys. Rev. B 56 (1996) 13 047.