Field Anomalous Dimension Research Articles

Quantum critical point in theQED3theory of underdoped high-temperature superconductors

We study the effect of gapless quasiparticles in a d-wave superconductor on the $T=0$ end point of the Kosterlitz-Thouless transition line in underdoped high-temperature superconductors. Starting from a lattice model that has gapless fermions coupled to three-dimensional $\mathrm{XY}$ phase fluctuations of the superconducting order parameter, we propose a continuum field theory to describe the quantum phase transition between the d-wave superconductor and the spin-density-wave insulator. Without fermions, the theory reduces to the standard Higgs scalar electrodynamics (HSE), which is known to have the critical point in the inverted $\mathrm{XY}$ universality class. Extending the renormalization group calculation for the HSE to include the coupling to fermions, we find that the qualitative effect of fermions is to increase the portion of the space of coupling constants where the transition is discontinuous. The critical exponents at the stable fixed point vary continuously with the number of fermion fields N, and we estimate the correlation length exponent $(\ensuremath{\nu}=0.65)$ and the vortex field anomalous dimension $({\ensuremath{\eta}}_{\ensuremath{\Phi}}=\ensuremath{-}0.48)$ at the quantum critical point for the physical case $N=2.$ The stable critical point in the theory disappears for the number of Dirac fermions $N>{N}_{c},$ with ${N}_{c}\ensuremath{\approx}3.4$ in our approximation. We discuss the relationship between the superconducting and the chiral (SDW) transitions, and point to some interesting parallels between our theory and the Thirring model.

Critical exponents of theN-vector model

Recently the series for two RG functions (corresponding to the anomalous dimensions of the fields phi and phi^2) of the 3D phi^4 field theory have been extended to next order (seven loops) by Murray and Nickel. We examine here the influence of these additional terms on the estimates of critical exponents of the N-vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within errors of the previous evaluation. Exponents like eta (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou--Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined. The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values. Finally, because an error has been discovered in the last order of the published epsilon=4-d expansions (order epsilon^5), we have also reanalyzed the determination of exponents from the epsilon-expansion. The conclusion is that the general agreement between epsilon-expansion and 3D series has improved with respect to Le Guillou--Zinn-Justin.

Progress with large Nf β-functions

We report on the progress in the computation of the β-functions of Φ 4 theory and QCD in the large N expansion. For the former we give an analytic formula for the critical exponent which encodes higher order coefficients in the series beyond those currently known in the MS scheme at O( 1 N 2 ) in a large N expansion. This allows us to deduce new coefficients in the field anomalous dimension which relate to Kreimer's knot theory of the divergences of a quantum field theory. A similar exponent is computed for the β-function of QCD at O( 1 N f ) .

Momentum scale expansion of sharp cutoff flow equations

We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organized by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations - the O( p M ) approximations - result from discarding from these parts, all terms of higher than the M th degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond O( p 0).