We present an extensive study of the
AB2 polymeric Hubbard chains with experimental motivation in inorganic and organic compounds. In the half-filled strong-coupling limit, these systems can be mapped onto a quantum antiferromagnetic spin-
1/2AB2 Heisenberg model that presents a zero-temperature quantum phase transition to a ferrimagnetic ground state. We derive the spin-wave modes with the same correct ferromagnetic dispersion relation calculated through two different techniques. We also build an Euclidean integral formulation in the coherent state representation which naturally gives rise to Wess-Zumino terms of topological origin. The low-energy infrared behavior of the
AB2 chains is properly described through a renormalization group analysis by the semiclassical fixed point of a one-dimensional quantum nonrelativistic nonlinear $\ensuremath{\sigma}$ model with critical dynamical exponent
z=2. Their low-temperature behavior is shown to be the same of a quantum ferromagnetic spin-1/2 Heisenberg chain, as evidenced by the temperature dependence of the correlation length, susceptibility, and specific heat. On the other hand, a quantum critical fixed point is also found, with the same zero-temperature critical behavior of a classical Heisenberg system in
D=d+z=3 dimensions. Its low-temperature critical properties are related to the latter through the crossover exponent $\ensuremath{\varphi}=z{\ensuremath{\nu}}_{3},$ where ${\ensuremath{\nu}}_{3}$ is the correlation length exponent of the
D=3 classical Heisenberg model. Sufficiently strong quantum critical fluctuations destroy the ordered ground state of the quantum
z=2 nonlinear $\ensuremath{\sigma}$ model (but not that of the quantum
AB2 Heisenberg spin systems, in agreement with a theorem by Lieb), giving rise to a quantum disordered phase at zero temperature. These results are supported by a connection between the quantum
AB2 Heisenberg and generalized quantum rotor models.
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.
Using the density-matrix renormalization-group technique we study the long-wavelength properties of the spin
S=3/2 nearest-neighbor Heisenberg chain. We obtain an accurate value for the spin velocity $v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.87\ifmmode\pm\else\textpm\fi{}0.02$, in agreement with experiment. Our results show conclusively that the model belongs to the same universality class as the
S=1/2 Heisenberg chain, with a conformal central charge
c=1 and critical exponent $\ensuremath{\eta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.
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.
The ground-state phase diagram of the one-dimensional asymmetric Hubbard model at half-filling is studied by means of a perturbation expansion in the large-U limit and White's density matrix renormalization-group method for various values of the couplings. We argue that the introduction of asymmetry in the hopping parameters is a relevant perturbation for any repulsive U and drives the system into an ordered N\'eel antiferromagnet. The transition belongs to the Kosterlitz-Thouless universality class.
We show that a chain of Heisenberg spins interacting with long-range dipolar forces in a magnetic field
h perpendicular to the chain exhibits a quantum critical point belonging to the two-dimensional Ising universality class. Within linear spin-wave theory, the magnon dispersion for small momenta
k is ${[{\ensuremath{\Delta}}^{2}+{v}_{k}^{2}{k}^{2}]}^{1/2}$, where ${\ensuremath{\Delta}}^{2}\ensuremath{\propto}|h\ensuremath{-}{h}_{c}|$ and ${v}_{k}^{2}\ensuremath{\propto}|\phantom{\rule{-0.16em}{0ex}}\mathrm{ln}k|$. For fields close to
hc linear spin-wave theory breaks down, and we investigate the system using density-matrix and functional renormalization group methods. The Ginzburg regime where non-Gaussian fluctuations are important is found to be rather narrow on the ordered side of the transition and very broad on the disordered side.