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Journal of Physics A Mathematical and General

Critical-off-critical interface in the Ising quantum chain and conformal invariance

Bertrand Berche, Loïc Turban

Published Jan 7, 1991• 12 citations
Publisher

Abstract

The low-energy excitation spectrum of an Ising quantum chain in a transverse field which is critical (h=J=1) for 1<or=n<or=L/2 and off-critical (h not=J) for L/2(n<or=L is calculated up to O(L -1 ). Interface critical exponents are determined numerically in the odd sector and exactly in the even sector using finite-size scaling. One gets an ordinary surface transition when h)J and an extraordinary surface transition when h(J. The gap-exponent relation is verified, the mass gaps and the level degeneracy are in agreement with conformal invariance.

DOI
10.1088/0305-4470/24/1/031

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The low-energy excitation spectrum of an Ising quantum chain in a transverse field which is critical (h=J=1) for 1<or=n<or=L/2 and off-critical (h not=J) for L/2(n<or=L is calculated up to O(L -1 ). Interface critical exponents are determined numerically in the odd sector and exactly in the even sector using finite-size scaling. One gets an ordinary surface transition when h)J and an extraordinary surface transition when h(J. The gap-exponent relation is verified, the mass gaps and the level degeneracy are in agreement with conformal invariance.
The quantum version of the Hilhorst-van Leeuwen model i.e. an inhomogeneous Ising model in a transverse field with a marginal perturbation of the bulk interaction varying like l -1 with the distance l from the surface is studied numerically on a finite chain with symmetric interactions using fermion techniques and finite-size scaling at the bulk critical point. The surface exponents for the energy and the magnetisation are respectively x e =2 alpha +2, x m = alpha +1/2 when alpha >or= alpha c and x e =- alpha +1/2, x m =0 and x' m =- alpha -1/2 when alpha <or= alpha c where alpha c =-1/2 is the critical value of the perturbation amplitude below which the surface becomes ordered at the bulk critical point.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
We study the critical and off-critical (Griffiths-McCoy) regions of the random transverse-field Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents are exactly calculated. Analyzing the structure of low-energy excitations we present a phenomenological theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the Griffiths-McCoy singularities are characterized by a single, varying exponent, the value of which is related through duality in the paramagnetic and ferromagnetic phases.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems ( L ⩽ 128). The spin-spin autocorrelation function is found to decay proportionally to (log t ) −2 x m and (log t ) −2 x m s in the bulk and on the surface, respectively, with x m and x m s the bulk and surface magnetization exponents, respectively. On the other hand, the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents η e ≈ 2.2 in the bulk and η e s ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory.