Critical behavior of the aperiodic quantum Ising chain in a transverse magnetic field

Abstract

We consider the quantum spin-1/2 Ising chain in a uniform transverse magnetic field, with an aperiodic sequence of ferromagnetic exchange couplings. This system is a limiting anisotropic case of the classical two-dimensional Ising model with an arbitrary layered modulation. Its formal solution via a Jordan-Wigner transformation enables us to obtain a detailed description of the influence of the aperiodic modulation on the singularity of the ground-state energy at the critical point. The key concept is that of thefluctuation of the sums of any number of consecutive couplings at the critical point. When the fluctuation isbounded, the model belongs to the “Onsager universality class” of the uniform chain. The amplitude of the logarithmic divergence in the specific heat is proportional to the velocity of the fermionic excitations, for which we give explicit expressions in most cases of interest, including the periodic and quasiperiodic cases, the Thue-Morse chain, and the random dimer model. When the couplings exhibit anunbounded fluctuation, the critical singularity is shown to be generically similar to that of the disordered chain: the ground-state energy has finite derivatives of all orders at the critical point, and an exponentially small singular part, for which we give a quantitative estimate. In themarginal case of a logarithmically divergent fluctuation, e.g., for the period-doubling sequence or the circle sequence, there is a negative specific heat exponentα, which varies continuously with the strength of the aperiodic modulation.