A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.
We have analyzed new twelfth-order high-temperature series for the susceptibility and correlation length of classical planar models on the triangular lattice using an
n-fit method of analysis tailored to the form of the singularity $A\mathrm{exp}(b{t}^{\ensuremath{-}\ensuremath{\nu}})$ predicted by Kosterlitz and Thouless. Test-function analysis shows that the
n-fit method is significantly more reliable in treating a number of possible corrections to the leading singularity than is the
D log Pad\'e analysis of the logarithm and logarithmic derivative used in earlier series work on tenth-order series. Our
n-fit analysis leads to the results $\ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1$ and $\ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03$ in good agreement with the Kosterlitz-Thouless predictions $\ensuremath{\nu}=\frac{1}{2}$ and $\ensuremath{\eta}=\frac{1}{4}$.