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arXiv:cond-mat/9411035 · Europhysics Letters (EPL)

Integrable Models of Interacting Quantum Spins with Competing Interactions

Johannes Richter, Sven E. Krüger, A Voigt, Claudius Gros

Published Nov 10, 1994• 6 citations
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Abstract

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.

DOI
10.1209/0295-5075/28/5/012

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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.
Stimulated by the two-dimensional frustrated Heisenberg antiferromagnet with first-, second-, and third-neighbor couplings (J1-J2-J3 model) we consider a corresponding three-parameter model with a long-range antiferromagnetic Lieb-Mattis interaction.This model can be solved exactly and leads to a better understanding of the role of frustration in the J1 -J2-J3 model.We calculate the correłations in the groundstate and consider their finite size behavior.Furthermore we present the full thermodynamic phase diagram.We find the possibility of a disordered phase at T = 0.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
A mixed Ising-Heisenberg spin system consisting of triangular XXZXXZXXZ-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model is given in terms of the generalized decoration-iteration map and within the transfer-matrix technique. Exact expressions for thermodynamic functions are derived. Ground-state phase diagrams and thermodynamic and magnetic properties of the system are examined.
A distinct integrable quantum-spin model is proposed. The model has a biaxial magnetic anisotropy of alternating coupling between spins together with multiple spin interactions. Our model gives the possibility to exactly find thermodynamic characteristics of the considered spin chain. The ground state of the model can reveal spontaneous values of the total magnetic and antiferromagnetic moments caused by multiple spin couplings. Also, in the ground state, depending on the strength of multiple spin couplings, our model manifests several quantum critical points, some of which are governed by the external magnetic field.
The spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices is exactly solved using a star-triangle transformation, which affords an exact mapping correspondence to an effective spin-1/2 Ising model on a triangular lattice. The ground-state phase diagram of both investigated quantum spin models display two spontaneously ordered ferromagnetic phases and one macroscopically degenerate disordered phase. In contrast to a classical ferromagnetic phase where the spontaneous magnetization of the Ising as well as Heisenberg spins acquire fully saturated values, the spontaneous magnetization of the Heisenberg spins is subject to a quantum reduction to one-third of its saturated value within a quantum ferromagnetic phase. The spontaneous magnetization and logarithmic divergence of the specific heat as the most essential features of both ferromagnetic phases disappear whenever the investigated quantum spin model is driven to the highly degenerate disordered phase. The disordered phase with nonzero residual entropy originates either from a geometric spin frustration caused by antiferromagnetic interactions or more strikingly it may also alternatively arise from a competition of the ferromagnetic Ising and Heisenberg interactions of easy-axis and easy-plane type, respectively. All three available ground states coexist together at a single triple point, around which anomalous magnetic and thermodynamic behavior can be detected.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.