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Physical review. B, Condensed matter

Field theory of ferrimagnetic Hubbard chains

Ernesto P. Raposo, Mauricio D. Coutinho-Filho

Published Jun 1, 1999• 33 citations
Publisher

Abstract

We present an extensive study of the ${\mathrm{AB}}_{2}$ 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-$1/2 {\mathrm{AB}}_{2}$ 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 ${\mathrm{AB}}_{2}$ 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 ${\mathrm{AB}}_{2}$ 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 ${\mathrm{AB}}_{2}$ Heisenberg and generalized quantum rotor models.

DOI
10.1103/physrevb.59.14384

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We study inorganic and organic ferrimagnetic Hubbard chains at half-filling and in the strong coupling limit and show that, in the long-wavelength regime, they are described by a one-dimensional (1D) nonlinear sigma model with an extra Wess-Zumino term. We then argue, based on a renormalization-group analysis, that these spin-1/2 AB2{\mathrm{AB}}_{2}AB2​ chains have quantum critical properties and belong to the same universality class as constrained classical 3D Heisenberg systems. For higher spins, the constraint is lifted but the critical behavior remains three dimensional.
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 study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear σσσ-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (hc1<h<hc2h_{c1} < h < h_{c2}hc1​<h<hc2​) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.
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.
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.
A quantum spherical spin model with an antiferromagnetic coupling JJJ on the AB2A{B}_{2}AB2​ chain is studied, whose topology is of interest in the context of ferrimagnetic polymers and oxocuprates. A ferrimagnetic long-range order is found at the only critical point g=T=h=0g=T=h=0g=T=h=0, where TTT denotes the temperature, hhh the magnetic field, and ggg the quantum coupling constant in energy units. The approach to the critical point, with diverging correlation length $\ensuremath{\xi}$ and cell susceptibility ${\ensuremath{\chi}}_{\text{cell}}$, is characterized through several paths in the g,T,h{g,T,h}g,T,h parameter space: for $(T∕J)\ensuremath{\rightarrow}0$ and g=h=0g=h=0g=h=0, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕{T}^{2}$, as also found in several classical and quantum spherical and Heisenberg models; for $(h∕J)\ensuremath{\rightarrow}0$ and g=T=0g=T=0g=T=0, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕h$; and for $(g∕J)\ensuremath{\rightarrow}0$ and T=h=0T=h=0T=h=0, $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕\sqrt{g}$, thus evidencing an essential singularity due to quantum fluctuations. In any path chosen the relation ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}{\ensuremath{\xi}}^{2}$ is satisfied. For finite ggg and TTT a field-induced short-range ferrimagnetism occurs to some extent in the g,T,h{g,T,h}g,T,h space, as confirmed by the analysis of the local spin averages, cell magnetization with a rapid increase for very low fields, and spin-spin correlation functions. The asymptotic limits of the correlation functions are also provided with respect to ggg, TTT, hhh, and spin distance. The analysis of the entropy and specific heat reveals that the quantum fluctuations fix the well-known drawback of classical spherical models concerning the third law of thermodynamics.