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arXiv:cond-mat/0003207 · Acta Physica Polonica A

Marshall-Peierls Sign Rule in Frustrated Heisenberg Chains

A. Voigt, Johannes Richter

Published May 1, 2000• 3 citations
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Abstract

We consider the frustrated antiferromagnetic S = 1 Heisenberg quantum spin chain with regard to the Marshall-Peierls sign rule.Using exact diagonalization data we investigate the breakdown of the Marshall-Peierls sign rule in dependence on frustration and compare our findings with data for s = 1/2.We calculate a critical value of frustration J21" , where the Marshall-Peierls sign rule is violated.The extrapolation of this value to the infinite chain limit holds J?' s: 0.016, lower than in the case of s = 1/2 (J2crit 0.027).This points to a stronger influence of frustration in the case of s = 1.Nevertheless, the calculation of the weight of the Ising-states violating the Marshall-Peierls sign rule shows that the latter holds approximately even for a quite large frustration and may be used for numerical techniques.

DOI
10.12693/aphyspola.97.979

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We consider the frustrated antiferromagnetic S = 1 Heisenberg quantum spin chain with regard to the Marshall-Peierls sign rule.Using exact diagonalization data we investigate the breakdown of the Marshall-Peierls sign rule in dependence on frustration and compare our findings with data for s = 1/2.We calculate a critical value of frustration J21" , where the Marshall-Peierls sign rule is violated.The extrapolation of this value to the infinite chain limit holds J?' s: 0.016, lower than in the case of s = 1/2 (J2crit 0.027).This points to a stronger influence of frustration in the case of s = 1.Nevertheless, the calculation of the weight of the Ising-states violating the Marshall-Peierls sign rule shows that the latter holds approximately even for a quite large frustration and may be used for numerical techniques.
Using a Lanczos technique we study the frustrated spin-(1/2 Heisenberg model on square lattices of 16 and 20 sites. Frustration is introduced by an interaction along the diagonals of the plaquettes with coupling J2{J}_{2}J2​\ensuremath{\ge}0. For large J2{J}_{2}J2​ we found that the ground state breaks (spontaneously) the lattice rotational symmetry. For intermediate values of J2{J}_{2}J2​, the squares of order parameters associated with spin-Peierls and ``twisted'' states have a peak while a similar quantity for a chiral state shows no interesting structure.
Finite-size scaling calculations for deformable Heisenberg chains with spin value S=(1/2, 1, (3/2, and 2 indicate that as far as the spin-Peierls transition is concerned, the model with any spin S>(1/2 behaves differently from the S=(1/2 situation. For S=(1/2 the ground state of the isotropic antiferromagnetic or planar models is dimerized, irrespective of the strength of the elastic constant. On the other hand, from the extrapolation of finite-chain results it seems that for any higher-spin value the dimerized and uniform phases are separated by a finite critical value of the elastic constant, and the transition is discontinuous.
We calculate the spin stiffness ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$ for the frustrated spin-1/2 Heisenberg antiferromagnet on a square lattice by exact diagonalizations on finite clusters of up to 36 sites followed by extrapolations to the thermodynamic limit. For the nonfrustrated case, we find that ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$=(0.183\ifmmode\pm\else\textpm\fi{}0.003)J1{\mathit{J}}_{1}J1​, in excellent agreement with the best results obtained by other means. Turning on frustration, the extrapolated stiffness vanishes for 0.4\ensuremath{\lesssim}J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​\ensuremath{\lesssim}0.6. In this intermediate region, the finite-size scaling works poorly---an additional sign that there is neither N\'eel nor collinear magnetic order. Using a hydrodynamic relation, and previous results for the transverse susceptibility, we also estimate the spin-wave velocity in the N\'eel-ordered region.
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.
The spin stiffness of frustrated spin-1/2 Heisenberg models in one and two dimensions is computed by exact diagonalizations on small clusters that implement spin-dependent twisted boundary conditions. Finite-size extrapolation to the thermodynamic limit yields a value of 0.14\ifmmode\pm\else\textpm\fi{}0.01 for the spin stiffness of the unfrustrated planar antiferromagnet. We also present a general discussion of the linear-response theory for spin twists, which ultimately leads to the moment sum rule.