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arXiv:2211.10543

Mean-field ground-state phase diagram of the ${t-t^\prime}$ ionic-Hubbard chain

Sekania, Mikheil, Shota Garuchava, Jamal Berakdar, G. I. Japaridze

Published Nov 19, 2022• 0 citations
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Abstract

Ground state (GS) phase diagram of the one dimensional repulsive Hubbard model with both nearest neighbor ($t$) and next-nearest-neighbor ($t^{\prime}$) hopping and a staggered potential ($Δ$) is determined in the case of half-filled band and zero net magnetization within the mean-field theory. The model may be realized by cold atoms in engineered optical lattices. Connection between the peculiarities of the GS phase diagram of the interacting system and the topological Lifshitz transition in GS of free particle chain is discussed. The mean-field Hamiltonian is given by six order parameters, which are determined self-consistently. It is shown that the GS phase diagram {\em predominantly} consists of insulating phases characterized by coexistence of the long-range ordered modulations of the charge- (I) and antiferromagnetic (AF) spin-density with wavelengths equal to two and/or four lattice units. Below the Lifshitz transition, the GS phase diagram is qualitatively similar to that of the standard (${t^{\prime}=0}$) ionic Hubbard model. After the Lifshitz transition, charge- and spin-density modulations emerge within the sublattices and have wavelengths equal to four lattice units. For moderate values of the on-site repulsion the insulating phases are characterized by the presence of I and AF modulations coexisting with ferrimagnetic spin- and charge-density modulations within the one sublattice, or by the coexistence of I order and additional AF spin density modulation only within the sublattice with high ionic potential. At large repulsion the insulating phase with AF order inside both sublattices and remaining small ionicity left between the sublattices is realized.

DOI
10.48550/arxiv.2211.10543

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Mean-field ground-state phase diagram of the t−t′{t-t^\prime}t−t′ ionic-Hubbard chain
Ground state (GS) phase diagram of the one dimensional repulsive Hubbard model with both nearest neighbor (ttt) and next-nearest-neighbor (t′t^{\prime}t′) hopping and a staggered potential (ΔΔΔ) is determined in the case of half-filled band and zero net magnetization within the mean-field theory. The model may be realized by cold atoms in engineered optical lattices. Connection between the peculiarities of the GS phase diagram of the interacting system and the topological Lifshitz transition in GS of free particle chain is discussed. The mean-field Hamiltonian is given by six order parameters, which are determined self-consistently. It is shown that the GS phase diagram {\em predominantly} consists of insulating phases characterized by coexistence of the long-range ordered modulations of the charge- (I) and antiferromagnetic (AF) spin-density with wavelengths equal to two and/or four lattice units. Below the Lifshitz transition, the GS phase diagram is qualitatively similar to that of the standard (t′=0{t^{\prime}=0}t′=0) ionic Hubbard model. After the Lifshitz transition, charge- and spin-density modulations emerge within the sublattices and have wavelengths equal to four lattice units. For moderate values of the on-site repulsion the insulating phases are characterized by the presence of I and AF modulations coexisting with ferrimagnetic spin- and charge-density modulations within the one sublattice, or by the coexistence of I order and additional AF spin density modulation only within the sublattice with high ionic potential. At large repulsion the insulating phase with AF order inside both sublattices and remaining small ionicity left between the sublattices is realized.
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.
A repulsive Hubbard model with both spin-asymmetric hopping $({t}_{\ensuremath{\uparrow}}\ensuremath{\ne}{t}_{\ensuremath{\downarrow}})$ and a staggered potential (of strength $\mathrm{\ensuremath{\Delta}})$ is studied in one dimension. The model is a compound of the mass-imbalanced $({t}_{\ensuremath{\uparrow}}\ensuremath{\ne}{t}_{\ensuremath{\downarrow}},\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Delta}}=0)$ and ionic $({t}_{\ensuremath{\uparrow}}={t}_{\ensuremath{\downarrow}},\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Delta}}>0)$ Hubbard models, and may be realized by cold atoms in engineered optical lattices. We use mostly mean-field theory to determine the phases and phase transitions in the ground state for a half-filled band (one particle per site). We find that a period-two modulation of the particle (or charge) density and an alternating spin density coexist for arbitrary Hubbard interaction strength, $U\ensuremath{\ge}0$. The amplitude of the charge modulation is largest at U=0U=0U=0, decreases with increasing UUU and tends to zero for $U\ensuremath{\rightarrow}\ensuremath{\infty}$. The amplitude for spin alternation increases with UUU and tends to saturation for $U\ensuremath{\rightarrow}\ensuremath{\infty}$. Charge order dominates below a value Uc{U}_{c}Uc​, whereas magnetic order dominates above. The mean-field Hamiltonian has two gap parameters, ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\uparrow}}$ and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\downarrow}}$, which have to be determined self-consistently. For U<UcU<{U}_{c}U<Uc​ both parameters are positive, for U>UcU>{U}_{c}U>Uc​ they have different signs, and for U=UcU={U}_{c}U=Uc​ one gap parameter jumps from a positive to a negative value. The weakly first-order phase transition at Uc{U}_{c}Uc​ can be interpreted in terms of an avoided criticality (or metallicity). The system is reluctant to restore a symmetry that has been broken explicitly.
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.
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We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/81/81/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/51/51/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.