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Physical review. B, Condensed matter

Scaling theory of conserved current and universal amplitudes at anisotropic critical points

Xiao-Gang Wen

Published Aug 1, 1992• 25 citations

Abstract

A scaling theory of conserved currents is studied at critical points with critical charge fluctuations. It is shown that the conserved current has no anomalous dimension even for anisotropic critical points. Many universal amplitudes are identified. One of the universal amplitudes at the critical points is ${\mathrm{lim}}_{(\mathit{k},\mathrm{\ensuremath{\omega}})\ensuremath{\rightarrow}0\mathrm{\ensuremath{\omega}}}^{2\mathrm{\ensuremath{-}}\mathit{d}}$${\mathit{k}}^{(\mathit{z}\mathrm{\ensuremath{-}}\mathit{d})}$ $^{(2\mathrm{\ensuremath{-}}\mathit{d})}\mathrm{\ensuremath{\sigma}}$(\ensuremath{\omega})${]}^{\mathit{z}}$[\ensuremath{\kappa}(k)${]}^{2\mathrm{\ensuremath{-}}\mathit{d}}$, which reduces to the universal conductance in 2+1 dimensions. We find that the ratio \ensuremath{\surd}\ensuremath{\kappa}\ensuremath{\chi} /\ensuremath{\sigma} always approaches a constant as we approach the critical point. We also determine exponents of many scaling relations, using the scaling properties of the current.

DOI: 10.1103/physrevb.46.2655

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arXiv:hep-th/0010164•Non-perturbative Quantum Effects•Sep 25, 2000•1 citation

A scaling theory of conserved currents is studied at critical points with critical charge fluctuations. It is shown that the conserved current has no anomalous dimension even for anisotropic critical points. Many universal amplitudes are identified. One of the universal amplitudes at the critical points is

{\mathrm{lim}}_{(\mathit{k},\mathrm{\ensuremath{\omega}})\ensuremath{\rightarrow}0\mathrm{\ensuremath{\omega}}}^{2\mathrm{\ensuremath{-}}\mathit{d}}$${\mathit{k}}^{(\mathit{z}\mathrm{\ensuremath{-}}\mathit{d})}

^{(2\mathrm{\ensuremath{-}}\mathit{d})}\mathrm{\ensuremath{\sigma}}

(\ensuremath{\omega})

{]}^{\mathit{z}}

[\ensuremath{\kappa}(k)

{]}^{2\mathrm{\ensuremath{-}}\mathit{d}}

, which reduces to the universal conductance in 2+1 dimensions. We find that the ratio \ensuremath{\surd}\ensuremath{\kappa}\ensuremath{\chi} /\ensuremath{\sigma} always approaches a constant as we approach the critical point. We also determine exponents of many scaling relations, using the scaling properties of the current.

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.

We compute current correlators of the

CP

${}^{N\ensuremath{-}1}$ field theory in 2+1 dimensions, both at the critical point and in the phase with spontaneously broken

\mathrm{SU}(N)

symmetry. Universal constants are obtained to next-to-leading order in the

1/N

expansion. Implications are noted for quantum critical points of antiferromagnets and their vicinity.

Power laws are distinctive features of critical phenomena and can be ascribed to diverging length-scales. One of the most important contributions of the studies of critical phenomena is the shift in point of view to a length-scale based analysis and classifying terms of a Hamiltonian as relevant or irrelevant rather than classifying them as numerically weak or strong. The idea of universality transcends the domain of critical phenomena. Critical points are classified by the number of relevant variables required to describe them. A universality class would be described by the exponents and also the amplitude ratios of the various singulax quantities on both sides of the critical point. In a correlation based approach, we need to compute the critical correlations of all possible combinations of the degrees of freedom. Scaling is the rule for criticality, originating from a diverging length scale, while anomalous dimension is seen for fluctuation dominated cases.

Near a continuous phase transition, there exist universal combinations of critical amplitudes. We present an estimation of such a universal quantity, in the framework of the ${\ensuremath{\varphi}}^{4}$ field theory in three dimensions, using information recently available in the literature, and already used to determine precise values for critical exponents. This leads to an accuracy only obtained, hitherto, in the framework of field theory, for critical exponents. The universal quantity calculated, often denoted by $R_{\ensuremath{\xi}}^{}{}_{}{}^{+}$, combines the amplitude of the correlation length and that of the singular part of the specific heat in the vicinity of the critical point.