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arXiv:cond-mat/9704063 · Journal of Physics A Mathematical and General

Critical behaviour of integrable mixed-spin chains

S R Aladim, Marcio Jose Martins

Published Jun 21, 1993• 39 citations
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Abstract

The authors construct a mixed spin-1/2 and spin-S integrable model and investigate its finite-size properties. For a certain conformal invariant mixed spin system the central charge can be decomposed in terms of the conformal anomaly of two single integrable models of spin 1/2 and spin (S-1/2). They also compute the ground-state energy and the sound velocity in the thermodynamic limit.

DOI
10.1088/0305-4470/26/12/009

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The authors construct a mixed spin-1/2 and spin-S integrable model and investigate its finite-size properties. For a certain conformal invariant mixed spin system the central charge can be decomposed in terms of the conformal anomaly of two single integrable models of spin 1/2 and spin (S-1/2). They also compute the ground-state energy and the sound velocity in the thermodynamic limit.
The ground-state properties of a quantum spin chain with three-spin couplings are investigated using finite-size calculations. The results are compared with the predictions of conformal invariance for finite systems at the critical point. The critical point. The critical exponents eta E and eta S of energy density and spin correlation functions are calculated and estimated to be eta E approximately=1.30 and eta S approximately=0.25. The results suggest that this model is conformally invariant with a central charge very close to or equal to one.
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.
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
The conformal anomaly c determines the universality class of a model system in statistical mechanics. The value of c characterizes both 2D classical models and their 1D quantum counterparts. The conformal anomaly may therefore be determined numerically for quantum spin chains using the relation: E0(N)≂E0(∞)−(NΔE/12)c(1/N2), where E0 (N) is the ground-state energy of an N-spin finite system, E0 (∞) is the ground-state energy in the thermodynamic limit, and ΔE is the energy gap between the ground state at k=0 and the first excited state of the dispersion curve at k=2π/N. The numerical approach is highly successful when tested on the integrable s= 1/2 Heisenberg antiferromagnetic XXZ chain and the integrable s=1 SU(2) model. The method gives c=1 to within 2% accuracy for the s=1 and (3)/(2) XY chains, placing them in the universality class of the 2D XY model. The result c=1 (2% accuracy) is obtained for the s= (3)/(2) Heisenberg antiferromagnetic chain, in agreement with the Haldane prediction. The s=1 pure antiferromagnetic biquadratic chain and the s=1 XXZ model with uniaxial anisotropy in the vicinity of the critical point Δ=Δ2 ∼1.15 −1.18 have also been studied.
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.