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Physical review. B, Condensed matter

Conformal anomaly and critical exponents of Heisenberg spin models with half-integer spin

Adriana Moreo

Published Dec 1, 1987• 17 citations
Publisher

Abstract

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 ${S}^{z}$ 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

DOI
10.1103/physrevb.36.8582

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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 \ensuremathηz{\ensuremath{\eta}}_{z}\ensuremathηz​=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 \ensuremathηz{\ensuremath{\eta}}_{z}\ensuremathηz​ does not appear to follow the same qualitative behavior as for S=(1/2. .AE
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Using the density-matrix renormalization group, we investigate the critical behavior of the anisotropic Heisenberg chains with spins up to s=9/2s=9/2s=9/2. We show that through the relations arising from the conformal invariance and the DMRG technique it is possible to obtain accurate finite-size estimates of the conformal anomaly ccc, the sound velocity vs{v}_{s}vs​, the anomalous dimension xbulk{x}_{bulk}xbulk​, and the surface exponent xs{x}_{s}xs​ of the anisotropic spin-sss Heisenberg chains with relatively good accuracy without fitting parameters. Our results indicate that the entanglement entropy S(L,lA,s)S(L,{l}_{A},s)S(L,lA​,s) of the spin-sss Heisenberg chains satisfies the relation $S(L,{l}_{A},s)\ensuremath{-}S(L,{l}_{A},s\ensuremath{-}1)=1/(2s+1)$ for s>3/2s>3/2s>3/2 in the thermodynamic limit.
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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