Off - diagonal density profiles and conformal invariance

Off-diagonal profiles φ od (v) of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, on a strip with transverse coordinate v, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by the non-vanishing matrix element 0|ˆφ(v)|φ of the appropriate operatorˆφ(v) between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model. Following the pioneering work of Fisher and de Gennes [1], the study of order parameter and energy density profiles near surfaces has been an active field of research during the past years. These profiles have been calculated at, and near the critical point, in the mean-field approximation [2], using field-theoretical approaches [3] and through exact solutions [4, 5]. Much progress has also been achieved in their calculation at bulk criticality in two-dimensional (2d) systems making use of conformal techniques [6–12]. In a semi-infinite 2d system, the profile φ(y) of a fluctuating quantity, such as the order parameter or the energy density, is obtained in the transfer matrix formalism as the diagonal matrix element 0|ˆφ(y)|0 of the corresponding operatorˆφ in the ground state |0 of the Hamiltonian H = − ln T , where T denotes the transfer operator along the surface. On a strip of infinite length and finite width L, one may also consider the off-diagonal profile φ od (v), where v is the transverse coordinate. The profile is then defined as the off-diagonal matrix element 0|ˆφ(v)|φ between the ground state |0 of the Hamiltonian H on the strip and its lowest excited state |φ leading to a nonvanishing matrix element.