Conformal off-diagonal boundary density profiles on a semi-infinite strip
چکیده
The off-diagonal profile φ od (v) associated with a local operator φ̂(v) (order parameter or energy density) close to the boundary of a semi-infinite strip with width L is obtained at criticality using conformal methods. It involves the surface exponent x φ and displays a simple universal behaviour which crosses over from surface finite-size scaling when v/L is held constant to corner finite-size scaling when v/L→ 0. The finite-size behaviour of order parameter or energy density profiles has been the subject of much interest during the last two decades following the work of Fisher and de Gennes [1]. These profiles have been studied in the vivinity of the critical point in the mean-field approximation [2], using field-theoretical methods [3] and through exact solutions [4, 5]. Such profiles display universal behaviour at criticality and in two-dimensional systems they can be deduced from ordinary scaling and covariance under conformal transformation [6–17]. A short review can be found in reference [18]. With symmetry-breaking boundary conditions, one may consider diagonal order parameter profiles [6], i.e., ground-state expectation values. Otherwise, off-diagonal profiles can be used with any type of boundary conditions [13]. For the order parameter with Dirichlet boundary conditions, off-diagonal matrix elements must be considered since a diagonal order parameter profile then vanishes for symmetry reasons. On a strip with fixed boundary conditions at v = 0 and v = L the diagonal order-parameter profile φ(v) associated with an operator φ̂ takes the following form at criticality [6] φ(v) = 〈0|φ̂(v)|0〉 = A [ L π sin (πv L ) ] −xφ 0 < v < L . (1) The exponent xφ is the bulk scaling dimension of φ̂, |0〉 is the ground state of the Hamiltonian H = − lnT where T denotes the row-to-row transfer operator on the strip. When L → ∞ with a fixed value of the ratio v/L, one obtains the bulk finitesize scaling behaviour φ(L) ∼ Lφ . When L → ∞ while keeping v fixed, one obtains the profile φ(v) ∼ vφ on the half-plane with fixed boundary conditions which is a consequence of ordinary scaling. Actually the profile on the strip in (1) follows from the profile on the half-plane through the logarithmic conformal transformation w = (L/π) ln z [6]. 1 2 Letter to the Editor