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Journal of Physics A Mathematical and General

Finite-size scaling functions and conformal invariance

P Reinicke

Published Sep 11, 1987• 19 citations
Publisher

Abstract

A method to calculate perturbative corrections to the spectrum of a one-dimensional quantum chain in the scaling variable z is developed from the conformal invariance at its fixed point. Then, from the knowledge of certain correlation functions, one can calculate the first coefficients of the scaling functions. The method is applied successfully to the Ising model, where the corrections are also known from the exact solution. The first coefficients of the magnetic scaling functions of the Ising model are also obtained.

DOI
10.1088/0305-4470/20/13/048

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A method to calculate perturbative corrections to the spectrum of a one-dimensional quantum chain in the scaling variable z is developed from the conformal invariance at its fixed point. Then, from the knowledge of certain correlation functions, one can calculate the first coefficients of the scaling functions. The method is applied successfully to the Ising model, where the corrections are also known from the exact solution. The first coefficients of the magnetic scaling functions of the Ising model are also obtained.
Statistical systems near a classical critical point have been intensively studied from both theoretical and experimental points of view. In particular, correlation functions are of relevance in comparing theoretical models with the experimental data of real systems. In order to compute physical quantities near a critical point, one needs to know the model at the critical (conformal) point. In this line, recent progress in the knowledge of conformal field theories, through the conformal bootstrap, gives the hope of getting some interesting results also outside of the critical point. In this paper, we will review and clarify how, starting from the knowledge of the critical correlators, one can calculate in a safe way their behavior outside the critical point. The approach illustrated requires the model to be just scale invariant at the critical point. We will clarify the method by applying it to different kind of perturbations of the 2D Ising model.
Numerical solutions of the Bethe-Ansatz equations for the eigenenergies of XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkin-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. The irrelevant operators governing the dominant finite-size corrections are also identified.
We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.
The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.