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Journal of Physics C Solid State Physics

On the influence of competing interactions on physical properties of spin systems: numerical exact results for finite Ising and Heisenberg systems

Johannes Richter, S. Kobe

Published Apr 10, 1982• 17 citations
Publisher

Abstract

The ground-state properties and low-lying excitations of spin systems with competing interactions are studied for various finite two-dimensional S= 1 / 2 Heisenberg systems of 8 spins and corresponding Ising systems of up to 60 spins. The response of the spin systems on the conflict in the interactions is studied using numerical exact results. In the Ising model the competition of the interactions influences the ground state and the excitations essentially. Contrary to conventional Ising systems in disordered strongly frustrated Ising systems the low-energy density of states is smooth, the gap between the ground state and the first excitations vanishes. The low-lying excitations are characterised by a large portion of multiple spin excitations. Thermodynamic consequences are indicated. In the Heisenberg case the influence of the conflict in the interactions is less obvious, the character of the ground state and of the first excitation has similarities with that of conventional antiferromagnets, the spins have an enlarged number of possibilities to evade the conflict in the interactions.

DOI
10.1088/0022-3719/15/10/020

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The ground-state properties and low-lying excitations of spin systems with competing interactions are studied for various finite two-dimensional S= 1 / 2 Heisenberg systems of 8 spins and corresponding Ising systems of up to 60 spins. The response of the spin systems on the conflict in the interactions is studied using numerical exact results. In the Ising model the competition of the interactions influences the ground state and the excitations essentially. Contrary to conventional Ising systems in disordered strongly frustrated Ising systems the low-energy density of states is smooth, the gap between the ground state and the first excitations vanishes. The low-lying excitations are characterised by a large portion of multiple spin excitations. Thermodynamic consequences are indicated. In the Heisenberg case the influence of the conflict in the interactions is less obvious, the character of the ground state and of the first excitation has similarities with that of conventional antiferromagnets, the spins have an enlarged number of possibilities to evade the conflict in the interactions.
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.
We study a spin- 1/2 Heisenberg model H=J 1 Sigma i=2 N S 1 S i +J 2 Sigma i=2 N S i S i+1 ; J 1 , J 2 >or=0 (Heisenberg star) which may be considered either as an essential structure element of a lattice with frustration or, alternatively, as an antiferromagnetic linear chain with a perturbation. We discuss general relations for the energy eigenvalues, the eigenstates as well as the spin correlation of the system in dependence on J 2 /J 1 . The correlation function (S 1 S i ) scales with 1/Z, where Z =N-1 is the number of nearest neighbours i of the central spin 1. In the mean-field limit (Z to infinity ) (S 1 S i ) goes to -0.25 for small J 2 /J 1 , but vanishes for strong frustration J 2 /J 1 . Adding exact numerical results for N=5, 7, ..., 23 we discuss the ground-state phase diagram, in particular, the ground-state spin correlations versus J 2 /J 1 . Analysing the spin correlation of frustrated and non-frustrated stars we suggest an upper bound -0.25 for the ground-state correlator (S i S j )0 of two antiferromagnetically interacting spins i and j in a non-frustrated Heisenberg spin- 1/2 antiferromagnet. We argue that any measured nearest-neighbour spin correlation (S i S j )0 larger than this bound indicates of frustration in quantum spin Heisenberg antiferromagnets.
Ground-state properties are studied for the one-dimensional isotropic spin-1 Heisenberg magnet with antiferromagnetic nearest-neighbor (nn) and next-nearest-neighbor (nnn) interactions. The energy, the singlet-triplet energy gap, the two-spin correlation function, and the den Nijs and Rommelse string correlation function in the ground states of finite-size systems of up to 16 spins are calculated exactly. By extrapolating these results, the ground-state properties at the infinite-size limit are discussed. It is found, in particular, that as the ratio of the nnn interaction constant to the nn one increases, the string order parameter (the limiting value of the longest-distance string correlation function) as well as the Haldane gap (the limiting value of the singlet-triplet energy gap) divided by the sum of both interaction constants increases. This result implies that the competition between the antiferromagnetic nn and nnn interactions stabilizes the Haldane phase.
Ground-state properties of spin 1 / 2 Heisenberg systems are calculated for finite clusters up to nine spins. By a comparison of systems with different degrees of competition in the interactions, the frustration effects in the spin configuration of the ground state are investigated. A minimum value for the strength of the spin correlation mod <S i .S j > 0 mod of antiferromagnetically interacting spins i and j in systems without competition is estimated to be 0.25. The occurrence of values for mod <S i .S j < 0 mod lower than this minimum value indicates frustration. By varying the degree of competition, abrupt changes in the ground state are found. Furthermore the competition for certain systems causes ground states with much lower symmetry than that of the Hamiltonian.
We examine the ground state and the excitations of an one-dimensional Heisenberg spin 1/2 antiferromagnet with alternating dimers and four-spin plaquettes (dimer-plaquette chain). The properties of the system depend on the competing dimer and plaquette bonds. Several exact, exact-numerical and perturbational results are presented. We find that the system is gapped for all parameter values. For the latter one we find an effective S=1 Haldane like behaviour in the limit of dominating dimer bonds. The model considered is related to the recently found 1/5-depleted square-lattice Heisenberg system CaV4O9CaV_4O_9CaV4​O9​.
Small Ising lattices with both ferromagnetic (F) and antiferromagnetic (AF) exchange interactions (or bonds) and increasing numbers of spins are studied by means of two independent methods: computational solutions to the Hamiltonian problem and topological counting of frustration paths. Equal magnitudes and concentrations are assumed for both types of bonds. Two different geometries are considered: square lattices (SL's) with coordination number 4 and triangular lattices (TL's) with coordination number 6. Two-dimensional samples with a total number of spins N between 4 and 64 are considered for SL's, while N is varied between 4 and 44 for TL's. They are distributed in two-dimensional arrays where periodic boundary conditions are imposed. After an array is selected, bond distributions (samples) are independently and randomly generated in fixed positions. The physical parameters are then calculated exactly for each sample. The emphasis here is on the ground-state properties and their dependence with size and shape for the two kinds of lattices. All magnitudes correspond to a basic statistics over a large number of samples for each array. The following magnitudes are reported: ground-state energy per bond, frustration segment, abundance of first excited states, remnant entropy, low-temperature specific heat, and site order parameters q, p, and h. Parameters p and h are introduced here, showing advantages over other similar magnitudes. The results are in good correspondence with analytic studies for the thermodynamic limit. This means that the spin site correlation (p) tends to vanish as N grows. However, we have found that the shape dependence modulates the behavior of these systems toward the thermodynamic limit. There is no tendency to vanish for the bond correlation parameter (h). For both kinds of lattices h might be a constant independent of size and shape.