We present a class of exactly solvable quantum spin models which consist of two Heisenberg subsystems coupled via a long-range Lieb-Mattis interaction. The total system is exactly solvable whenever the individual subsystems are solvable and allows to study the effects of frustration. We consider i) the antiferromagnetic linear chain and ii) the Lieb-Mattis antiferromagnet for the subsystem Hamiltonians and present i) the complete ground-state phase diagram and ii) the full thermodynamic phase diagram. We find a novel phase which exhibits order-from-disorder phenomena.
We use series expansion methods to investigate the phase diagram of a frustrated spin-
21 Heisenberg antiferromagnet on the simple cubic lattice, with interactions extending to third neighbors. Our results do not support a recent prediction of a rather large paramagnetic phase at zero temperature, but do show behavior which may be indicative of a small disordered region near the point of maximum frustration.
We present an exact diagonalization study on layered
J1−J2 antiferromagnetic Heisenberg spin model to examine the role of frustration induced by inplane next-nearest neighbor coupling
J2, in presence of interlayer antiferromagnetic coupling
J⊥. A finite lattice of 24 spins in layered geometry of
(4×3)⊕(4×3) is considered and the resulting Hamiltonian matrix diagonalized using Davidson iterative algorithm to obtain the ground and a few low-lying excited states. The lattice size (
24 spins with Hilbert space dimensionality of
2704156 in
Sztot=0 subspace) has been kept relatively small because of the large number of runs required to sample
J1−J2−J⊥ parameter space. Quantities like spin-gap, Shannon entropy, spin-spin correlation, static spin structure-factor, magnetic specific-heat and magnetic susceptibility are calculated for various values of spin-spin coupling parameters. With increase in interlayer coupling, the system is driven to states with long range order and the interval of quantum paramagnetically disordered state, sandwiched between Néel and collinear ordered states, narrows on the scale of inplane frustration parameter.
The phase diagram of a spin-1/2
J−J′−J2 model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the
J1−J2 model on the square lattice with two different nearest-neighbor couplings
J,J′ and may be also viewed as an array of coupled Heisenberg chains. The results suggest that the resonnating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B {\bf 67}, 024422 (2003)] for
J2=0.5J′≪J is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both side by a columnar dimer long range order. This columnar order could extends for
J′→J which correspond to the
J1−J2 model.
Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-
21 J1-
J2 model on the square lattice with plaquette structure. We consider antiferromagnetic (
J1>0) as well as ferromagnetic (
J1<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction
J2>0. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform
J1-
J2 model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic
J1) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case
s=1/2. For the antiferromagnetic case (
J1>0) N\'eel antiferromagnetic long-range order at small
J2/J1 and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large
J2/J1 and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (
J1<0) we have the trivial ferromagnetic ground state at small
J2/∣J1∣. By increasing
J2 this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length
s=2 are antiferromagnetically long-range ordered. Further increasing of
J2 then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.
The phase diagram of a spin-1/2 $J\ensuremath{-}{J}^{\ensuremath{'}}\ensuremath{-}{J}_{2}$ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the
J1−−J2 model on the square lattice with two different nearest-neighbor couplings ${J,J}^{\ensuremath{'}}$ and may also be viewed as an array of coupled Heisenberg chains. The results suggest that the resonating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B 67, 024422 (2003)] for ${J}_{2}{=0.5J}^{\ensuremath{'}}\ensuremath{\ll}J$ is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both sides by a columnar dimer long range order. This columnar order could extend for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}J$ which correspond to the
J1−−J2 model.