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arXiv:0802.3880 · Physical Review B

Doped AB2 Hubbard chain: Spiral, Nagaoka and resonating-valence-bond states, phase separation, and Luttinger-liquid behavior

R. R. Montenegro-Filho, Mauricio D. Coutinho-Filho

Published Sep 28, 2006• 23 citations
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Abstract

We present an extensive numerical study of the Hubbard model on the doped $A{B}_{2}$ chain, both in the weak coupling and the infinite-$U$ limit. Due to the special unit-cell topology, this system displays a rich variety of phases as a function of hole doping $(\ensuremath{\delta})$ away from half-filling. Near half-filling spiral states develop in the weak-coupling regime, while Nagaoka itinerant ferromagnetism is observed in the infinite-$U$ limit. For higher doping, the system phase-separates before reaching a Mott insulating phase of short-range resonating-valence-bond states at $\ensuremath{\delta}=1∕3$. Moreover, for $\ensuremath{\delta}>1∕3$ we observe a crossover, which anticipates the Luttinger-liquid behavior for $\ensuremath{\delta}>2∕3$.

DOI
10.1103/physrevb.74.125117

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We present an extensive numerical study of the Hubbard model on the doped AB2A{B}_{2}AB2​ chain, both in the weak coupling and the infinite-UUU limit. Due to the special unit-cell topology, this system displays a rich variety of phases as a function of hole doping (\ensuremathδ)(\ensuremath{\delta})(\ensuremathδ) away from half-filling. Near half-filling spiral states develop in the weak-coupling regime, while Nagaoka itinerant ferromagnetism is observed in the infinite-UUU limit. For higher doping, the system phase-separates before reaching a Mott insulating phase of short-range resonating-valence-bond states at \ensuremathδ=1∕3\ensuremath{\delta}=1∕3\ensuremathδ=1∕3. Moreover, for \ensuremathδ>1∕3\ensuremath{\delta}>1∕3\ensuremathδ>1∕3 we observe a crossover, which anticipates the Luttinger-liquid behavior for \ensuremathδ>2∕3\ensuremath{\delta}>2∕3\ensuremathδ>2∕3.
We present a functional-integral formalism suitable to describe the strong-coupling regime below half filling of AB2A{B}_{2}AB2​ Hubbard chains, with experimental realizations in inorganic and organic polymeric compounds. At half filling (one electron per site: $\ensuremath{\delta}=0$), we obtain the long-range-ordered ferrimagnetic ground state and correctly reproduce the associated quantum nonrelativistic nonlinear $\ensuremath{\sigma}$ model, with presence of topological Wess-Zumino terms, which has been also found in a coherent-state representation, restricted however to treat the case of localized spins. A fully polarized (Nagaoka) ferromagnetic state is obtained in the infinite-coupling regime near half filling, and at doping $\ensuremath{\delta}=1/3$ (two electrons per unit cell) insulating short-range resonating-valence-bond (RVB) singlet states take place. Competition between Nagaoka and RVB mechanisms can lead to phase separation for $0<\ensuremath{\delta}<1/3$. By increasing the hole doping, a crossover regime for $1/3<\ensuremath{\delta}<2/3$ anticipates the emergence of Luttinger-liquid behavior for $\ensuremath{\delta}\ensuremath{\ge}2/3$. In particular, at $\ensuremath{\delta}=2/3$ (one electron per cell) the system behaves similarly to a critical spin-1/2 antiferromagnetic linear chain. All these results find support in numerical studies both in the half filling and doped regimes.
We discuss the rich phase diagram of doped $A{B}_{2}\phantom{\rule{4pt}{0ex}}t\text{\ensuremath{-}}J$ chains by using data from density matrix renormalization group and exact diagonalization techniques. The JJJ vs $\ensuremath{\delta}$ (hole doping) phase diagram exhibits regions of itinerant ferrimagnetism, incommensurate, resonating valence bond and Nagaoka states, phase separation, and Luttinger liquid (LL) physics. Several features are highlighted, such as the modulated ferrimagnetic structure, the occurrence of Nagaoka spin polarons in the underdoped regime and small values of J=4t2/UJ=4{t}^{2}/UJ=4t2/U, where ttt is the first-neighbor hopping amplitude and UUU is the on-site repulsive Coulomb interaction, incommensurate structures with nonzero magnetization, and strong-coupling LL physics in the high-doped regime. We also verify that relevant findings are in agreement with the corresponding findings in square and nnn-leg ladder lattices. In particular, we mention the instability of Nagaoka ferromagnetism against JJJ and $\ensuremath{\delta}$.
Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation in large-U AB2AB_{2}AB2​ Hubbard chains
The ground state (GS) properties of the quasi-one-dimensional AB2AB_2AB2​ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, we derive a low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and Lieb total spin per unit cell of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding t-Jt\textrm{-}Jt-J Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonating-valence-bond states. We also calculate the doping-dependent GS energy and total spin per unit cell, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure.
Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, a functional integral representation allows us to derive a low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and total spin per unit cell in the Lieb ferrimagnetic GS of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding [Formula: see text] Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonating-valence-bond states. We also calculate the doping-dependent GS energy and total spin per unit cell, including both Zeeman and orbital contributions, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure.
The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the μ−Uμ-Uμ−U plane, restricted to phases with no charge density modulation, extending the results presented in cond-mat/9511116. This allows antiferromagnetism, saturated ferromagnetism, spiral spin density waves and a collinear structure with unit cell ↑↑↓↓\uparrow \uparrow \downarrow \downarrow↑↑↓↓. The spiral phase is unstable against phase separation near quarter-, half- and three-quarter-filling. For large UUU this occurs at hole (or electron) doping of (3t/π2U)1/3(3t/π^{2}U)^{1/3}(3t/π2U)1/3 from half filling.
A wide variety of experimental platforms, ranging from semiconductor quantum-dot arrays to moiré materials, have recently emerged as powerful quantum simulators for studying the Hubbard model and its variants. Motivated by these developments, here, we investigate a generalization of the Hubbard model which includes the effects of long-range Coulomb interactions. Working on finite-sized two-dimensional square and triangular lattices, we use exact diagonalization and density-matrix renormalization group calculations to probe the magnetic structure of the ground state in the strong-coupling regime, where UUU (the onsite repulsion) ≫\gg≫ ttt (the nearest-neighbor hopping). For small electron dopings above the half-filled antiferromagnet, we numerically uncover a rich variety of magnetically ordered states, and in conjunction with theoretical arguments, infer the phase diagram of the system as a function of doping and interaction strengths. In particular, we find that the inclusion of long-range Coulomb interactions induces an instability of high-spin states$\unicode{x2014}$such as the saturated Nagaoka ferromagnet$\unicode{x2014}$towards phase separation and stripe ordering. We also present proposals for the observation of some of our key findings in experiments that would shed further light on this paradigmatic strongly correlated system.