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arXiv:1704.07459 · Physical review. B./Physical review. B

Mass-imbalanced ionic Hubbard chain

Michael Sekania, D. Baeriswyl, Luka Jibuti, G. I. Japaridze

Published Jul 11, 2017• 10 citations
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Abstract

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 $U=0$, decreases with increasing $U$ and tends to zero for $U\ensuremath{\rightarrow}\ensuremath{\infty}$. The amplitude for spin alternation increases with $U$ and tends to saturation for $U\ensuremath{\rightarrow}\ensuremath{\infty}$. Charge order dominates below a value ${U}_{c}$, 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 $U<{U}_{c}$ both parameters are positive, for $U>{U}_{c}$ they have different signs, and for $U={U}_{c}$ one gap parameter jumps from a positive to a negative value. The weakly first-order phase transition at ${U}_{c}$ can be interpreted in terms of an avoided criticality (or metallicity). The system is reluctant to restore a symmetry that has been broken explicitly.

DOI
10.1103/physrevb.96.035116

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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.
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.
The Mott - Hubbard metal - insulator transition in the half-filled mass imbalanced ionic Hubbard model is investigated using the two-site dynamical mean field theory. We find that for a fixed mass imbalanced parameter r the critical interaction Uc increases when the ionic energy Δ\DeltaΔ is increased. In the other hand, for a fixed Δ\DeltaΔ, UcU_cUc​ decreases with increasing the mass imbalance. We also show the existence of BI phase in the system for the case Δ≠0\Delta \ne 0Δ=0, U=0U=0U=0 and calculate the staggered charge density nB−nAn_B − n_AnB​−nA​ as a function of the interaction for different values of the mass imbalance. Our results in the limiting cases (r=1r = 1r=1; Δ≠0\Delta \ne 0Δ=0 or/and Δ=0\Delta = 0Δ=0; r≠1r\ne 1r=1) are in good agreement with those obtained from full dynamical mean field theory.
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 study the phase diagram of the ionic Hubbard model (IHM) at half filling on a Bethe lattice of infinite connectivity using dynamical mean-field theory (DMFT), with two impurity solvers, namely, iterated perturbation theory (IPT) and continuous time quantum Monte Carlo (CTQMC). The physics of the IHM is governed by the competition between the staggered ionic potential $\mathrm{\ensuremath{\Delta}}$ and the on-site Hubbard UUU. We find that for a finite $\mathrm{\ensuremath{\Delta}}$ and at zero temperature, long-range antiferromagnetic (AFM) order sets in beyond a threshold U=UAFU={U}_{AF}U=UAF​ via a first-order phase transition. For UUU smaller than UAF{U}_{AF}UAF​ the system is a correlated band insulator. Both methods show a clear evidence for a quantum transition to a half-metal (HM) phase just after the AFM order is turned on, followed by the formation of an AFM insulator on further increasing UUU. We show that the results obtained within both methods have good qualitative and quantitative consistency in the intermediate-to-strong-coupling regime at zero temperature as well as at finite temperature. On increasing the temperature, the AFM order is lost via a first-order phase transition at a transition temperature ${T}_{AF}(U,\mathrm{\ensuremath{\Delta}})$ [or, equivalently, on decreasing UUU below ${U}_{AF}(T,\mathrm{\ensuremath{\Delta}})$], within both methods, for weak to intermediate values of U/tU/tU/t. In the strongly correlated regime, where the effective low-energy Hamiltonian is the Heisenberg model, IPT is unable to capture the thermal (Neel) transition from the AFM phase to the paramagnetic phase, but the CTQMC does. At a finite temperature T,DMFT+CTQMCT, \mathrm{DMFT}+\mathrm{CTQMC}T,DMFT+CTQMC shows a second phase transition (not seen within DMFT+IPT\mathrm{DMFT}+\mathrm{IPT}DMFT+IPT) on increasing UUU beyond UAF{U}_{AF}UAF​. At UN>UAF{U}_{N}>{U}_{AF}UN​>UAF​, when the Neel temperature TN{T}_{N}TN​ for the effective Heisenberg model becomes lower than TTT, the AFM order is lost via a second-order transition. For $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}, {T}_{N}\ensuremath{\sim}{t}^{2}/U(1\ensuremath{-}{x}^{2})$, where $x=2\mathrm{\ensuremath{\Delta}}/U$ and thus TN{T}_{N}TN​ increases with increase in $\mathrm{\ensuremath{\Delta}}/U$. In the three-dimensional parameter space of $(U/t,\phantom{\rule{0.28em}{0ex}}T/t,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\mathrm{\ensuremath{\Delta}}/t)$, as TTT increases, the surface of first-order transition at ${U}_{AF}(T,\mathrm{\ensuremath{\Delta}})$ and that of the second-order transition at ${U}_{N}(T,\mathrm{\ensuremath{\Delta}})$ approach each other, shrinking the range over which the AFM order is stable. There is a line of tricritical points that separates the surfaces of first- and second-order phase transitions.
We revisit the ground-state phase diagram of the one-dimensional half-filled extended Hubbard model with on-site (U) and nearest-neighbor (V) repulsive interactions. In the first half of the paper, using the weak-coupling renormalization-group approach (g(g(g-ology) including second-order corrections to the coupling constants, we show that bond-charge-density-wave (BCDW) phase exists for $U\ensuremath{\approx}2V$ in between charge-density-wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin electrons disfavors the BCDW state and leads to a bicritical point where the CDW-BCDW and SDW-BCDW continuous-transition lines merge into the CDW-SDW first-order transition line. In the second half of the paper, we investigate the phase diagram of the extended Hubbard model with either additional staggered site potential $\ensuremath{\Delta}$ or bond alternation $\ensuremath{\delta}.$ Although the alternating site potential $\ensuremath{\Delta}$ strongly favors the CDW state (that is, a band insulator), the BCDW state is not destroyed completely and occupies a finite region in the phase diagram. Our result is a natural generalization of the work by Fabrizio, Gogolin, and Nersesyan [Phys. Rev. Lett. 83,83,83, 2014 (1999)], who predicted the existence of a spontaneously dimerized insulating state between a band insulator and a Mott insulator in the phase diagram of the ionic Hubbard model. The bond alternation $\ensuremath{\delta}$ destroys the SDW state and changes it into the BCDW state (or Peierls insulating state). As a result the phase diagram of the model with $\ensuremath{\delta}$ contains only a single critical line separating the Peierls insulator phase and the CDW phase. The addition of $\ensuremath{\Delta}$ or $\ensuremath{\delta}$ changes the universality class of the CDW-BCDW transition from the Gaussian transition into the Ising transition.