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Physical review. B, Condensed matter

Critical behavior of anisotropic spin-SHeisenberg chains

Francisco Castilho Alcaraz, Adriana Moreo

Published Aug 1, 1992• 50 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.46.2896

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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 \ensuremathηx{\mathrm{\ensuremath{\eta}}}_{\mathit{x}}\ensuremathηx​ follows the behavior \ensuremathηx{\mathrm{\ensuremath{\eta}}}_{\mathit{x}}\ensuremathηx​=(\ensuremath{\pi}-\ensuremath{\gamma})/2\ensuremath{\pi}S and \ensuremathηz{\mathrm{\ensuremath{\eta}}}_{\mathit{z}}\ensuremathηz​=2 where \ensuremath{\gamma}=cos\ensuremath−1{\mathrm{cos}}^{\mathrm{\ensuremath{-}}1}cos\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}=\ensuremathλ∗{\ensuremath{\lambda}}^{\mathrm{*}}\ensuremathλ∗(S). For \ensuremathλ∗{\ensuremath{\lambda}}^{\mathrm{*}}\ensuremathλ∗\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.
Using a modified Lanczos method we have evaluated the conformal anomaly c of anisotropic Heisenberg models with S=(1/2 and (3/2 in the critical region (0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1). For the theory with S=(1/2, we obtain c=1 in the whole range of the parameter \ensuremath{\lambda} and we check the expression ${\ensuremath{\eta}}_{z}$=1/[1-arccos(\ensuremath{\lambda})/\ensuremath{\pi}] for the critical exponent of the Sz{S}^{z}Sz correlation function. Studying the model with S=(3/2, the conformal anomaly seems to converge to 1 for 0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1 but ${\ensuremath{\eta}}_{z}$ does not appear to follow the same qualitative behavior as for S=(1/2. .AE
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.
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
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.
Using the density-matrix renormalization-group technique we study the long-wavelength properties of the spin S=3/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2S=3/2 nearest-neighbor Heisenberg chain. We obtain an accurate value for the spin velocity $v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.87\ifmmode\pm\else\textpm\fi{}0.02$, in agreement with experiment. Our results show conclusively that the model belongs to the same universality class as the S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 Heisenberg chain, with a conformal central charge c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 and critical exponent $\ensuremath{\eta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.
The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.