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Physical Review Letters

Excitation spectrum of the spin-(1/2 Heisenberg chain and conformal invariance

Ferenc Woynarovich

Published Jul 20, 1987• 88 citations
Publisher

Abstract

The spectrum of excitations of a spin-(1/2 Heisenberg chain in the region of anisotropy 1>\ensuremath{\Delta}>0 (where the model is critical and has no bound states) exhibits a ``tower structure'' in the energy-momentum plane as predicted by the hypothesis of conformal invariance. For a given magnetization the apices of the towers are located on a parabola. The spectrum shows remarkable degeneracies at some discrete values of \ensuremath{\Delta} given by the Tutte-Behara numbers.

DOI
10.1103/physrevlett.59.259

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The spectrum of excitations of a spin-(1/2 Heisenberg chain in the region of anisotropy 1>\ensuremath{\Delta}>0 (where the model is critical and has no bound states) exhibits a ``tower structure'' in the energy-momentum plane as predicted by the hypothesis of conformal invariance. For a given magnetization the apices of the towers are located on a parabola. The spectrum shows remarkable degeneracies at some discrete values of \ensuremath{\Delta} given by the Tutte-Behara numbers.
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 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.
The phase diagram of two coupled anisotropic spin-1/2 Heisenberg chains forming a ladder is studied via the finite-size machinery arising from conformal invariance. Our results show that this system has similar behaviour to the anisotropic spin-1 Heisenberg chain. Both models for low values of anisotropies have a massless phase of Gaussian type, governed by a c = 1 conformal field theory. The operator that destroys the massless phase in these models is the same as in the Gaussian picture. We also verify that the massive phase in the anisotropic Heisenberg ladder has the same type of nonlocal order as that in the spin-1 Heisenberg chain.
Using the phenomenological renormalization group (PRG), we evaluate the low-energy excitation spectrum along the critical line of a quantum spin chain having a local interaction between three Ising spins and longitudinal and transverse magnetic fields, i.e., a Turban model. The low-energy excitation spectrum found with the PRG agrees with the spectrum predicted for the (D(4),A(4)) conformal minimal model under a nontrivial correspondence between translations at the critical line and discrete lattice translations. Under this correspondence, the measurements confirm a prediction that the critical line of this quantum spin chain and the critical point of the two-dimensional three-state Potts model are in the same universality class.
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