Frustrated Quantum Heisenberg Antiferromagnets Research Articles

Strongly correlated flat-band systems: The route from Heisenberg spins to Hubbard electrons

On a large class of lattices (such as the sawtooth chain, the kagome and the pyrochlore lattices), the quantum Heisenberg and the repulsive Hubbard models may host a completely dispersionless (flat) energy band in the single-particle spectrum. The flat-band states can be viewed as completely localized within a finite volume (trap) of the lattice and allow for construction of many-particle states, roughly speaking, by occupying the traps with particles. If the flat-band happens to be the lowest-energy one, the manifold of such many-body states will often determine the ground-state and low-temperature physics of the models at hand even in the presence of strong interactions. The localized nature of these many-body states makes possible the mapping of this subset of eigenstates onto a corresponding classical hard-core system. As a result, the ground-state and low-temperature properties of the strongly correlated flat-band systems can be analyzed in detail using concepts and tools of classical statistical mechanics (e.g., classical lattice-gas approach or percolation approach), in contrast to more challenging quantum many-body techniques usually necessary to examine strongly correlated quantum systems. In this review, we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. The main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems, we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat-band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temperature behavior of several flat-band Hubbard models, the emergence of ground-state ferromagnetism in the square-lattice Tasaki–Hubbard model and the related Pauli-correlated percolation problem, as well as the dispersion-driven ground-state ferromagnetism in flat-band Hubbard systems. Closely related studies and possible experimental realizations of the flat-band physics are also described briefly.

Frustrated quantum Heisenberg antiferromagnets at high magnetic fields: Beyond the flat-band scenario

We consider the spin-1/2 antiferromagnetic Heisenberg model on three frustrated lattices (the diamond chain, the dimer-plaquette chain, and the two-dimensional square-kagome lattice) with almost dispersionless lowest magnon band. Eliminating high-energy degrees of freedom at high magnetic fields, we construct low-energy effective Hamiltonians, which are much simpler than the initial ones. These effective Hamiltonians allow a more extended analytical and numerical analysis. In addition to the standard strong-coupling perturbation theory, we also use a localized-magnon-based approach leading to a substantial improvement of the strong-coupling approximation. We perform extensive exact diagonalization calculations to check the quality of different effective Hamiltonians by comparison with the initial models. Based on the effective-model description, we examine the low-temperature properties of the considered frustrated quantum Heisenberg antiferromagnets in the high-field regime. We also apply our approach to explore thermodynamic properties for a generalized diamond spin chain model suitable to describe azurite at high magnetic fields. Interesting features of these highly frustrated spin models consist in a steep increase of the entropy at very small temperatures and a characteristic extra low-temperature peak in the specific heat. The most prominent effect is the existence of a magnetic-field-driven Berezinskii-Kosterlitz-Thouless phase transition occurring in the two-dimensional model.

Quantum Heisenberg antiferromagnet on low-dimensional frustrated lattices

Using a lattice-gas description of the low-energy degrees of freedom of the quantum Heisenberg antiferromagnet on the frustrated two-leg ladder and bilayer lattices we examine the magnetization process at low temperatures for these spin models. In both cases the emergent discrete degrees of freedom implicate a close relation of the frustrated quantum Heisenberg antiferromagnet to the classical lattice gas with finite nearest-neighbor repulsion or, equivalently, to the Ising antiferromagnet in a uniform magnetic field. Using this relation we obtain analytical results for thermodynamically large systems in the one-dimensional case. In the two-dimensional case we perform classical Monte Carlo simulations for systems of up to $100 \times 100$ sites.

Magnetic Order and Disorder in the Frustrated Quantum Heisenberg Antiferromagnet in Two Dimensions

We have performed a numerical investigation of the ground state properties of the frustrated quantum Heisenberg antiferromagnet on the square lattice (``$J_1-J_2$ model''), using exact diagonalization of finite clusters with 16, 20, 32, and 36 sites. Using a finite-size scaling analysis we obtain results for a number of physical properties: magnetic order parameters, ground state energy, and magnetic susceptibility (at $q=0$). For the unfrustrated case these results agree with series expansions and quantum Monte Carlo calculations to within a percent or better. In order to assess the reliability of our calculations, we also investigate regions of parameter space with well-established magnetic order, in particular the non-frustrated case $J_2<0$. We find that in many cases, in particular for the intermediate region $0.3 < J_2/J_1 < 0.7$, the 16 site cluster shows anomalous finite size effects. Omitting this cluster from the analysis, our principal result is that there is N\'eel type order for $J_2/J_1 < 0.34$ and collinear magnetic order (wavevector $\bbox{Q}=(0,\pi)$) for $J_2/J_1 > 0.68$. There thus is a region in parameter space without any form of magnetic order. Including the 16 site cluster, or analyzing the independently calculated magnetic susceptibility we arrive at the same conclusion, but with modified values for the range of existence of the nonmagnetic region. We also find numerical values for the spin-wave velocity and the spin stiffness. The spin-wave velocity remains finite at the magnetic-nonmagnetic transition, as expected from the nonlinear sigma

Spin-liquid phase for the frustrated quantum Heisenberg antiferromagnet on a square lattice.

The existance of a spin disordered ground state for the frustrated Quantum Heisenberg Antiferromagnet on a square lattice is reconsidered. It is argued that there is a unique action which is continuous through the whole phase diagram, except at the Lifshitz point, so that the Neel and helicoidal states can not coexist and there has to be an intermediate spin liquid state. To show it, a detailed study combining Spin-Wave theory, Schwinger Bosons Mean Field Theory and a scaling analysis of the appropriate hydrodynamic action is performed. When done carefully, all these theories agree and strongly support the existance of the spin liquThe manuscript has eight figures, which are available upon request to the author. e-mail address is ferrer@physics.rutgers.edu

Zero-temperature magnetic ordering in the inhomogeneously frustrated quantum Heisenberg antiferromagnet on a square lattice

We examine the influence of homogeneous and inhomogeneous frustration on the ground-state ordering of the spin-1/2 ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ Heisenberg antiferromagnet on a square lattice with N=16 and N=20 sites. For a critical value of frustration ${\mathit{J}}_{2}^{\mathrm{crit}}$\ensuremath{\approxeq}0.4${\mathit{J}}_{1}$ the conventional collinear antiferromagnetic long-range order is expected to break down in the homogeneously frustrated system. This critical frustration is drastically decreased by inhomogeneities simulating doping; we find ${\mathit{J}}_{2}^{\mathrm{crit}}$\ensuremath{\approxeq}0.15${\mathit{J}}_{1}$, which is quite realistic for the situation in slightly doped cuprate superconductors. In the region of strong frustration a quantum spin-liquid state is realized without conventional collinear ordering. The properties of this quantum spin liquid are still under controversial discussion. Three different possibilities of a noncollinear ordering in the strongly frustrated region (${\mathit{J}}_{2}$\ensuremath{\approxeq}0.5${\mathit{J}}_{1}$) are discussed: A dimerized (or spin-Peierls) state, a state with scalar chiral ordering, and a state with enhanced vector chiral correlations. In particular, owing to the possibility of excitations with fractional statistics and owing to experimental investigations of broken reflection symmetry and parity the symmetry properties of the chiral order parameters are of interest. The scalar chirality is odd under time reversal and two-dimensional reflection, whereas the vector chirality conserves the time-reversal symmetry. We find evidence for a vector chiral ordering as well as for a spin-Peierls state. These vector chiral correlations can be enhanced by local inhomogeneities (holes), whereas the spin-Peierls state is suppressed.

Dimer stability region in a frustrated quantum Heisenberg antiferromagnet.

We study the stability region for the columnar dimer state proposed as a candidate ground state for the square-lattice quantum antiferromagnet with first- and second-neighbor antiferromagnetic couplings (${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ model). We use a boson representation of the spin operators suited to the perturbative expansion around a dimer ground state. At lowest order, the columnar dimer is found to be stable only at the classical critical value ${\mathit{J}}_{2}$/${\mathit{J}}_{2}$=1/2. However, we show that the leading anharmonic corrections stabilize the dimerized phase in a region of a finite width around ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$=1/2. A comparison of the ground-state energies shows that among the possible dimerized states the columnar dimer is the most favorable candidate to separate the two ordered states in the S=1/2 antiferromagnetic with first- and second-neighbor exchange.

Bond-operator representation of quantum spins: Mean-field theory of frustrated quantum Heisenberg antiferromagnets.

We introduce a new representation of S=1/2 quantum spins in terms of bond operators. The bond operators create and annihilate singlet and triplet bonds between a pair of spins. The representation is useful in describing the transition between dimerized and magnetically ordered phases of quantum antiferromagnets. It is used to obtain a mean-field theory of the two-dimensional frustrated quantum Heisenberg antiferromagnets considered recently by Gelfand, Singh, and Huse. The method should also be useful in the analysis of random quantum antiferromagnets.

Zero-temperature ordering in two-dimensional frustrated quantum Heisenberg antiferromagnets.

La phase magnetiquement desordonnee du modele d'Heisenberg pour un reseau carre, accessible aux developpements des dimers, se revele spontanement dimerisee dans la configuration «en colonne» predite par Read et Sachdev. Les estimations du parametre d'ordre de dimerisation sont une fraction substantielle de celui de l'etat entierement dimerise