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Journal of Physics Condensed Matter

Critical behaviour of coupled spin chains

JUSSI TIMONEN, Jenő Sólyom, John B. Parkinson

Published May 13, 1991• 5 citations
Publisher

Abstract

The authors investigate, using numerical computation of the eigenvalues of short chains, the critical behaviour of two composite spin models, which interpolate smoothly between isotropic Heisenberg chains with different values of S. For the first model which interpolates between S=1/2 and S=3/2 they find that the model is critical over the whole range and the values of the central charge and critical exponents (scaling dimensions) appear to be constant in the thermodynamic limit. In the second model, which interpolates between S=1/2 and S=1 they find that, except at S=1/2, the central charge is effectively zero, implying a non-critical behaviour.

DOI
10.1088/0953-8984/3/19/013

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The authors investigate, using numerical computation of the eigenvalues of short chains, the critical behaviour of two composite spin models, which interpolate smoothly between isotropic Heisenberg chains with different values of S. For the first model which interpolates between S=1/2 and S=3/2 they find that the model is critical over the whole range and the values of the central charge and critical exponents (scaling dimensions) appear to be constant in the thermodynamic limit. In the second model, which interpolates between S=1/2 and S=1 they find that, except at S=1/2, the central charge is effectively zero, implying a non-critical behaviour.
We consider in detail the ground-state properties of recently introduced generalized Heisenberg models which can have several spin operators at each site and which interpolate smoothly between Heisenberg chains of different spin lengths. We show that the mappings to field-theoretical models used to describe the critical properties of the Heisenberg model remain valid in the composite-spin model. In models which interpolate between the spin-(1/2 and the spin-1 behavior, these mappings predict an extended singlet phase around the isotropic antiferromagnetic point whenever the models move away from the spin-(1/2 point. Numerical calculations on finite chains seem to confirm the existence of this singlet phase. The phase boundaries are, however, found to be independent of the interpolation parameter, and thus in disagreement with the predictions of the continuum theory, indicating its limitation.
The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centres are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centred in the mixed Heisenberg chain where we have at even (odd) sites a spin- SU(2) operator interacting with a XXZ-like interaction (anisotropy ). Our results indicate universal properties for all these chains. The whole phase, , where the models change from ferromagnetic to ferrimagnetic behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions.
The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centers are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centered in the mixed Heisenberg chain where we have at even (odd) sites a spin-S (S ′) SU(2) operator interacting with a XXZ like interaction (anisotropy ∆). Our results indicate universal properties for all these chains. The whole phase, 1> ∆> −1, where the models change from ferromagnetic ( ∆ = 1) to ferrimagnetic ( ∆ = −1) behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions. 1
Using a Lanczos method, we study the spectra of Heisenberg chains with spin S=1, 3/2, 2, 5/2, and 3, as a function of the anisotropy parameter \ensuremath{\lambda} and the number of sites of the chain. We use periodic, twisted, and open boundary conditions. We found that for all values of the spin when -1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}0, these models are massless with critical behavior described by a conformal theory with central charge c=1. We discuss the whole operator content. In particular, we found that the critical exponent ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$ follows the behavior ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$=(\ensuremath{\pi}-\ensuremath{\gamma})/2\ensuremath{\pi}S and ${\mathrm{\ensuremath{\eta}}}_{\mathit{z}}$=2 where \ensuremath{\gamma}=${\mathrm{cos}}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\lambda}). For \ensuremath{\lambda}>0, our results indicate that the S=1 model is massive, while for S>1, the massless behavior characterized by the critical exponents described above continues up to \ensuremath{\lambda}=${\ensuremath{\lambda}}^{\mathrm{*}}$(S). For ${\ensuremath{\lambda}}^{\mathrm{*}}$\ensuremath{\le}1, our results are consistent with a massive phase for the integer-spin models and a massless phase with a rapid variation of the critical exponents for the half-integer-spin models.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.