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arXiv:0911.0810 · Physical Review B

Classical analogue of the ionic Hubbard model

Mohsen Hafez, S. A. Jafari, Sh. Adibi, Farhad Shahbazi

Published Jun 29, 2010• 7 citations
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Abstract

In our earlier work [M. Hafez et al., Phys. Lett. A 373, 4479 (2009)] we employed the flow equation method to obtain a classical effective model from a quantum mechanical parent Hamiltonian called, the ionic Hubbard model. The classical ionic Hubbard model (CIHM) obtained in this way contains solely Fermionic occupation numbers of two species corresponding to particles with $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ spin, respectively. In this paper, we employ the transfer matrix method to analytically solve the CIHM at finite temperature in one dimension. In the limit of zero temperature, we find two insulating phases at large and small Coulomb interaction strength, $U$, mediated with a gapless phase, resulting in two continuous metal-insulator transitions. Our results are further supported with Monte Carlo simulations.

DOI
10.1103/physrevb.81.245131

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In our earlier work [M. Hafez et al., Phys. Lett. A 373, 4479 (2009)] we employed the flow equation method to obtain a classical effective model from a quantum mechanical parent Hamiltonian called, the ionic Hubbard model. The classical ionic Hubbard model (CIHM) obtained in this way contains solely Fermionic occupation numbers of two species corresponding to particles with \ensuremath↑\ensuremath{\uparrow}\ensuremath↑ and \ensuremath↓\ensuremath{\downarrow}\ensuremath↓ spin, respectively. In this paper, we employ the transfer matrix method to analytically solve the CIHM at finite temperature in one dimension. In the limit of zero temperature, we find two insulating phases at large and small Coulomb interaction strength, UUU, mediated with a gapless phase, resulting in two continuous metal-insulator transitions. Our results are further supported with Monte Carlo simulations.
The Quantum Monte Carlo simulations of the ionic Hubbard model on a two-dimensional square lattice at half filling were performed. The method based on the direct-space, proposed by Suzuki and al., Hirsch and al., was used. Cycles of increasing and decreasing values of the Coulomb interaction $ U $ were performed for fixed temperature ($ kT=0.01 $). Results indicate that, at low temperature, the two insulator phases are separated by a metallic phase for weak to intermediate values of the staggered potential $ Δ$. For large Coulomb repulsion the system is in a Mott insulator with an antiferromagnetism order. On increasing and decreasing the Coulomb interaction $ U $ the metal-Mott insulator transition shows an hysteresis phenomenon while the metal-band insulator transition is continue. For large $ Δ$ it seems that the metallic region shrinks to a single metallic point. However, the band insulator to the Mott insulator transition is not direct for the studied model. A phase diagram is drawn for the temperature $ kT=0.01 $. For $ Δ=0.5 $ cycles of increasing and decreasing temperature were programmed for different values of the Coulomb interaction $ U $ . A behaviour change appears for $ U\simeq 1.75 $. This suggests that a crossover line divides the metallic region of the phase diagram.
The interaction-driven insulator-to-metal transition has been reported in the ionic Hubbard model (IHM) for moderate interaction UUU, while its metallic phase only occupies a narrow region in the phase diagram. To explore the enlargement of the metallic regime, we extend the ionic Hubbard model to two coupled layers and study the interplay of interlayer hybridization VVV and two types of intralayer staggered potentials $\mathrm{\ensuremath{\Delta}}$: one with the same (in-phase) and the other with a $\ensuremath{\pi}$-phase shift (antiphase) potential between layers. Our determinant quantum Monte Carlo (DQMC) simulations at lowest accessible temperatures demonstrate that the interaction-driven metallic phase between Mott and band insulators expands in the $\mathrm{\ensuremath{\Delta}}\text{\ensuremath{-}}V$ phase diagram of bilayer IHM only for in-phase ionic potentials; while antiphase potential always induces an insulator with charge density order. This implies possible further extension of the ionic Hubbard model from the bilayer case here to a realistic three-dimensional model.
We study an ionic Hubbard model in one dimension by using a quantum Monte Carlo method. The ionic Hubbard model we study is a strongly correlated model for electrons in alternating potentials with a lattice period of 2. By modifying the Hirsch-Fye algorithm, we incorporate the local variation of the potential in the weight of the partition function at each imaginary-time segment. For a given alternating potential of strength Δ = 1.0, as we change the local Coulomb repulsion, U , between spin-up and spin-down electrons, we calculate various physical quantities, including kinetic energies, correlation energies, and band energies. We try to find the signature of the phase transitions from the behavior of the kinetic energy and that of double occupancies at finite temperatures.
We investigate the neutral-to-ionic insulator-insulator transition in one-dimensional materials by treating a strong-coupling effective model based on the ionic Hubbard model using the density-matrix renormalization group and finite-size scaling. The effective model, formulated in a spin-one representation, contains a single parameter. We carry out an extensive finite-size scaling analysis of the relevant gaps and susceptibilities to characterize the two zero-temperature transitions. We find that the transition from the ionic band-insulating phase to an intermediate spontaneously dimerized phase is Ising, and the transition from the dimerized phase to the Mott-insulating phase is Kosterlitz-Thouless, in agreement with the field-theory-based predictions.
A detailed study of the one-dimensional ionic Hubbard model with interaction UUU is presented. We focus on the band insulating (BI) phase and the spontaneously dimerized insulating (SDI) phase which appears on increasing UUU. By a recently introduced continuous unitary transformation [H. Krull et al., Phys. Rev. B 86, 125113 (2012)] we are able to describe the system even close to the phase transition from BI to SDI although the bare perturbative series diverges before the transition is reached. First, the dispersion of single fermionic quasiparticles is determined in the full Brillouin zone. Second, we describe the binding phenomena between two fermionic quasiparticles leading to an S=0S=0S=0 and to an S=1S=1S=1 exciton. The latter corresponds to the lowest spin excitation and defines the spin gap which remains finite through the transition from BI to SDI. The former becomes soft at the transition, indicating that the SDI corresponds to a condensate of these S=0S=0S=0 excitons. This view is confirmed by a BCS mean-field theory for the SDI phase.
We study quantum phase transitions of an ionic Hubbard model in two dimensions. The ionic Hubbard model explains the quantum states of strongly correlated electrons under the influence of checkerboard-type alternating chemical potentials. For a given amplitude of the alternating potentials Δ, we obtain quantum ground states as we tune the local repulsive energy U between a spin-up electron and a spin-down electron by using an exact diagonalization method of a modified Lanczos algorithm. The system undergoes a quantum phase transition from a band insulator to a Mott insulator as U increases at half-filling. We find the signature of a quantum phase transition by investigating the behavior of ground-state energies and that of double occupancies for the size of L × L = 4 × 4, which was the largest possible lattice in this work. We compare our results with those of quantum Monte Carlo simulations employing the Hirsch-Fye algorithm.