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Revista brasileira de fisica

The relation between mass-gap amplitudes and critical exponents in the Heisenberg model

Francisco Castilho Alcaraz, J. R. Drugowich de Felício

Published Jan 1, 1985• 0 citations
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Abstract

We discuss a recent result concerning the universality of the ratio of mass-gap amplitudes using the well known 1-D Heisenberg mdel which is the quantum version of the two-dimensional eight-vertex model. We confirm the believed extended scaling relation (x =x /4) relatingthe polarization and energy anomalous dimensions. We alsg ogtain the ex- ponent, a, v, ym and y by usual phenomenological renormalization group methods . P

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We discuss a recent result concerning the universality of the ratio of mass-gap amplitudes using the well known 1-D Heisenberg mdel which is the quantum version of the two-dimensional eight-vertex model. We confirm the believed extended scaling relation (x =x /4) relatingthe polarization and energy anomalous dimensions. We alsg ogtain the ex- ponent, a, v, ym and y by usual phenomenological renormalization group methods . P
Recently it has been found that for several two-dimensional classical models the amplitude of the correlation length of a finite system at a critical point is universal and related to the correlation-function exponent $\ensuremath{\eta}$. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin S=12,1,32,and2S=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} 2S=21​,1,23​,and2. It is used to calculate $\ensuremath{\eta}$ in a simplified way.
The extended scaling relations between the critical exponents of the 8-vertex model can be derived by a mapping of this model onto the Luttinger model. The equivalence of this method to the one that connects the 8-vertex model to the Gaussian model is discussed. The Luttinger model is equivalent to the Gaussian model. Its operators are identified as vortex and spin-wave operators. The spin-wave operator $cos4\ensuremath{\varphi}$ is present in the critical 8-vertex Hamiltonian via an umklapp process. This explains the Kosterlitz-Thouless transition in the 6-vertex model, and resolves questions concerning the validity of the lattice continuum limit in the treatment by Luther and Peschel.
Sarker's real-space renormalization-group method, applied previously to the one-dimensional dimerized XY model, is used to study the one-dimensional pure and dimerized Heisenberg models. Comparison of critical exponents (the dynamical and the gap exponents) with exact or conjectured values indicates that the method, in its original version, does not yield an improved real-space technique, at least for the models studied for which the opening of the gap leads to essential singularity or logarithmic corrections.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.