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.
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.
Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation in large-U AB2 Hubbard chains
The ground state (GS) properties of the quasi-one-dimensional AB2 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-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.
In the first part of the present Thesis, the ground state (GS) properties of the quasi-one-dimensional AB₂ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. Using a functional integral approach, combined with a perturbative expansion in the strong-coupling regime, we obtain the Lagrangian density associated with the charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. In the strong-coupling regime, we derive a perturbative low-energy theory 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 t-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 resonatingvalence-bond (RVB) 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 (DMRG) 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. In the second part, motivated by analogy with photonic lattices, we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a single edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-André-Harper model, which allows us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers.We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices.
Field-theoretic methods are used to investigate the large-U Hubbard model on the honeycomb lattice at half-filling and in the hole-doped regime. Within the framework of a functional-integral approach, we obtain the Lagrangian density associated with the charge and spin degrees of freedom. The Hamiltonian related to the charge degrees of freedom is exactly diagonalized. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the quantum antiferromagnetic phase (AF) as a function of hole doping. At half-filling, we deal with the underlying spin degrees of freedom of the quantum AF Heisenberg model by employing a second-order spin-wave analysis, in which case we have calculated the ground-state energy and the staggered magnetization; the results are in very good agreement with previous studies. Further, in the continuum, we derive a nonlinear $\ensuremath{\sigma}$ model with a topological Hopf term that describes the AF-VBS (valence bond solid) competition. Lastly, in the challenging doped regime, our approach allows the derivation of a $t\text{\ensuremath{-}}J$ Hamiltonian, and the analysis of the role played by charge and spin quantum fluctuations on the ground-state energy and, particularly, on the breakdown of the AF order at a critical hole doping; the results are benchmarked against recent Grassmann tensor product state simulations.
We present a functional-integral formalism suitable to describe the strong-coupling regime below half filling of 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 present an extensive numerical study of the Hubbard model on the doped AB2 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 $\ensuremath{\delta}>2∕3$.
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 J 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/U, where t is the first-neighbor hopping amplitude and U 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 n-leg ladder lattices. In particular, we mention the instability of Nagaoka ferromagnetism against J and $\ensuremath{\delta}$.