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Journal of Physics A Mathematical and General

The spin- 1/2 Heisenberg star with frustration: numerical versus exact results

Johannes Richter, A Voigt

Published Feb 21, 1994• 28 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/27/4/010

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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.
We investigate the spin- Heisenberg star introduced by Richter and Voigt. The model is defined by ; , . In extension to the work of Richter and Voigt we consider a more general describing the properties of the spins surrounding the central spin . The Heisenberg star may be considered as an essential structure element of a lattice with frustration (namely a spin embedded in a magnetic matrix ) or, alternatively, as a magnetic system with a perturbation by an extra spin. We present some general features of the eigenvalues, the eigenfunctions as well as the spin correlation of the model. For being a linear chain, a square lattice or a Lieb - Mattis-type system we present the ground-state properties of the model in dependence on the frustration parameter . Furthermore, the thermodynamic properties are calculated for being a Lieb - Mattis antiferromagnet.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J(1)(F)) and antiferromagnetic (J(1)(A)) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (J(2)(F)). In this model frustration is present due to the non-zero J(2)(F). The model with site spin s behaves like a Haldane spin chain, with site spin 2s in the limit of vanishing J(2)(F) and large J(1)(F)/J(1)(A). We show that the exact ground state of the model can be found along a line in the parameter space. For fixed J(1)(F), the phase diagram in the space of J(1)(A)-J(2)(F) is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J(F)(1)) and antiferromagnetic (J(A)(1)) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (J(F)(2)). In this model frustration is present due to the non-zero J(F)(2). The model with site spin s behaves like a Haldane spin chain, with site spin 2s in the limit of vanishing J(F)(2)and large J(F)(1)/J(A)(1). We show that the exact ground state of the model can be found along a line in the parameter space. For fixed J(F)(1), the phase diagram in the space of J(A)(1)-J(F)(2) is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
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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.