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arXiv:cond-mat/0309295 · Physical Review Letters

Exact Bond Ordered Ground State for the Transition between the Band and the Mott Insulator

Cristian D. Batista, Armando A. Aligia

Published Jun 18, 2004• 66 citations
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Abstract

We derive an effective Hamiltonian ${H}_{\mathrm{eff}}$ for an ionic Hubbard chain, valid for $t\ensuremath{\ll}U,\ensuremath{\Delta}$, where $t$ is the hopping, $U$ is the Coulomb repulsion, and $\ensuremath{\Delta}$ is the charge-transfer energy. ${H}_{\mathrm{eff}}$ is the minimal model for describing the transition between the band insulator (BI) ($\ensuremath{\Delta}\ensuremath{-}U\ensuremath{\gg}t$) and the Mott insulator (MI) ($U\ensuremath{-}\ensuremath{\Delta}\ensuremath{\gg}t$). Using spin-particle transformations [Phys. Rev. Lett. 86, 1082 (2001)], we map ${H}_{\mathrm{eff}}(U=\ensuremath{\Delta})$ into an SU(3) antiferromagnetic Heisenberg model whose exact ground state is known. In this way, we show rigorously that a spontaneously dimerized insulating ferroelectric phase appears in the transition region between the BI and the MI.

DOI
10.1103/physrevlett.92.246405

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We derive an effective Hamiltonian Heff{H}_{\mathrm{eff}}Heff​ for an ionic Hubbard chain, valid for t\ensuremath≪U,\ensuremathΔt\ensuremath{\ll}U,\ensuremath{\Delta}t\ensuremath≪U,\ensuremathΔ, where ttt is the hopping, UUU is the Coulomb repulsion, and \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is the charge-transfer energy. Heff{H}_{\mathrm{eff}}Heff​ is the minimal model for describing the transition between the band insulator (BI) (\ensuremathΔ\ensuremath−U\ensuremath≫t\ensuremath{\Delta}\ensuremath{-}U\ensuremath{\gg}t\ensuremathΔ\ensuremath−U\ensuremath≫t) and the Mott insulator (MI) (U\ensuremath−\ensuremathΔ\ensuremath≫tU\ensuremath{-}\ensuremath{\Delta}\ensuremath{\gg}tU\ensuremath−\ensuremathΔ\ensuremath≫t). Using spin-particle transformations [Phys. Rev. Lett. 86, 1082 (2001)], we map Heff(U=\ensuremathΔ){H}_{\mathrm{eff}}(U=\ensuremath{\Delta})Heff​(U=\ensuremathΔ) into an SU(3) antiferromagnetic Heisenberg model whose exact ground state is known. In this way, we show rigorously that a spontaneously dimerized insulating ferroelectric phase appears in the transition region between the BI and the MI.
We investigate the ground state phase diagram of the half-filled repulsive Hubbard model in two dimensions in the presence of a staggered potential Delta, the so-called ionic Hubbard model, using cluster dynamical mean-field theory. We find that for large Coulomb repulsion, U >> Delta, the system is a Mott insulator (MI). For weak to intermediate values of Delta, on decreasing U, the Mott gap closes at a critical value Uc1(Delta) beyond which a correlated insulating phase with possible bond order is found. Further, this phase undergoes a first-order transition to a band insulator (BI) at Uc2(Delta) with a finite charge gap at the transition. For large Delta, there is a direct first-order transition from a MI to a BI with a single metallic point at the phase boundary.
We investigate the ground-state phase diagram of the half-filled $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ repulsive Hubbard model in the presence of a staggered ionic potential $\ensuremath{\Delta}$ using the continuum-limit bosonization approach. We find that with increasing on-site repulsion UUU, depending on the value of the next-nearest-neighbor hopping amplitude ${t}^{\ensuremath{'}}$, the model shows three different versions of the ground-state phase diagram. For ${t}^{\ensuremath{'}}<{t}_{c}^{{*}^{\ensuremath{'}}}$, the ground-state phase diagram consists of the following three insulating phases: band insulator at U<UcU<{U}_{c}U<Uc​, ferroelectric insulator at Uc<U<Us{U}_{c}<U<{U}_{s}Uc​<U<Us​, and correlated (Mott) insulator at U>UcU>{U}_{c}U>Uc​. For ${t}^{\ensuremath{'}}>{t}_{c}^{\ensuremath{'}}$ there is only one transition from a spin-gapped metallic phase at U<UcU<{U}_{c}U<Uc​ to a ferroelectric insulator at U>UcU>{U}_{c}U>Uc​. Finally, for intermediate values of the next-nearest-neighbor hopping amplitude ${t}_{c}^{{*}^{\ensuremath{'}}}<{t}^{\ensuremath{'}}<{t}_{c}^{\ensuremath{'}}$ we find that with increasing on-site repulsion, at Uc1{U}_{c1}Uc1​ the model undergoes a second-order commensurate-incommensurate-type transition from a band insulator into a metallic state and at larger Uc2{U}_{c2}Uc2​ there is a Kosterlitz-Thouless-type transition from a metal to a ferroelectric insulator.
We derive an effective Hamiltonian for the ionic Hubbard model at half filling, extended to include nearest-neighbor repulsion. Using a spin-particle transformation, the effective model is mapped onto simple spin-1 models in two particular cases. Using another spin-particle transformation, a slightly modified model is mapped into an SU(3) antiferromagnetic Heisenberg model whose exact ground state is known to be spontaneously dimerized. From the effective models several properties of the dimerized phase are discussed, like ferroelectricity and fractional charge excitations. Using bosonization and recent developments in the theory of macroscopic polarization, we show that the polarization is proportional to the charge of the elementary excitations.
We investigate the ground-state phase diagram of the Hubbard model for the ${\mathrm{AB}}_{N\ensuremath{-}1}$ chain with filling 1∕N1∕N1∕N, where NNN is the number of atoms per unit cell. In the strong-coupling limit, a charge transition takes place from a band insulator (BI) to a correlated insulator for increasing on-site repulsion UUU and positive on-site energy difference $\ensuremath{\Delta}$ (energy at A sites lower than at B sites). In the weak-coupling limit, a bosonization analysis suggests that for N>2N>2N>2 the physics is qualitatively similar to the case N=2N=2N=2 which has already been studied: an intermediate phase emerges, which corresponds to a bond-ordered ferroelectric insulator (FI) with spontaneously broken inversion symmetry. We have determined the quantum phase diagram for the cases N=3N=3N=3 and N=4N=4N=4 from the crossings of energy levels of appropriate excited states, which correspond to jumps in the charge and spin Berry phases, and from the change of sign of the localization parameter zLc{z}_{L}^{c}zLc​. From these techniques we find that, quantitatively, the BI and FI phases are broader for N>2N>2N>2 than when N=2N=2N=2, in agreement with the bosonization analysis. Calculations of the Drude weight and zLc{z}_{L}^{c}zLc​ indicate that the system is insulating for all parameters, with the possible exception of the boundary between the BI and FI phases.
An unusual metallic phase is argued to develop in the one-dimensional ionic Hubbard model, at half-filling and zero magnetization, at intermediate electron-electron repulsion UUU when second-neighbors hopping is allowed and tuned close to a topological Lifshitz transition (connected with a change of the Fermi surface in the noninteracting system). The metallic state lies between a band insulator phase at low repulsion and a correlated (Mott-like) insulator phase at high repulsion. In approaching the latter, the model supports short-range antiferromagnetic order and spontaneous dimerization of both bond charge and nearest-neighbor antiferromagnetic correlations. A combination of mean-field and effective-field theory (bosonization) provides an analytical understanding of the physical processes underlying the argued phase transitions. The ground and low-energy excited states of finite-length chains are explored by density-matrix renormalization-group (DMRG) calculations, providing numerical evidence for the intermediate gapless phase. Such finite systems are attainable by cold atoms in optical lattices for a wide range of the parameter UUU.
We report a systematic study of the transition from a band insulator (BI) to a Mott insulator (MI) in a one-dimensional Hubbard model at half-filling with an on-site Coulomb interaction U and an alternating periodic site potential V. We employ both the zero-temperature density matrix renormalization group (DMRG) method to determine the gap and critical behavior of the system and the finite-temperature transfer matrix renormalization group method to evaluate the thermodynamic properties. We find two critical points at U=Uc{U=U}_{c}U=Uc​ and U=Us{U=U}_{s}U=Us​ that separate the BI and MI phases for a given V. A charge-neutral spin-singlet exciton band develops in the BI phase (U<Uc)(U<{U}_{c})(U<Uc​) and drops below the band gap when U exceeds a special point Ue.{U}_{e}.Ue​. The exciton gap closes at the first critical point Uc{U}_{c}Uc​ while the charge and spin gaps persist and coincide between Uc<U<Us{U}_{c}<U<{U}_{s}Uc​<U<Us​ where the system is dimerized. Both the charge and spin gaps collapse at U=Us{U=U}_{s}U=Us​ when the transition to the MI phase occurs. In the MI phase (U>Us)(U>{U}_{s})(U>Us​) the charge gap increases almost linearly with U while the spin gap remains zero. These findings clarify earlier published results on the same model, and offer insights into several important issues regarding an appropriate scaling analysis of DMRG data and a full physical picture of the delicate nature of the phase transitions driven by electron correlation. The present work provides a comprehensive understanding for the critical behavior and phase diagram for the transition from BI to MI in one-dimensional correlated electron systems with a periodic alternating site potential.