Gradient critical phenomena in the Ising quantum chain

TL;DR

This paper develops a scaling theory for gradient-induced interface phenomena in critical systems and validates it through analytical and numerical studies of the Ising quantum chain with a linearly varying transverse field.

Contribution

It introduces a new scaling framework for gradient perturbations in critical systems and applies it to the exactly solvable Ising quantum chain, confirming predictions with analytical and numerical results.

Findings

Scaling predictions match numerical results

Energy density and magnetization profiles follow the proposed scaling laws

Exact diagonalization confirms the theoretical scaling behavior

Abstract

We consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings. In the direction of the gradient an interface region separates the ordered phase from the disordered one. We develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region. The scaling predictions are tested in the framework of the mean-field Ginzburg-Landau theory. Then we consider the Ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two-dimensional Ising model. The quantum Hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator. The energy density, the magnetization profile and the two-point correlation function are studied either analytically or by exact numerical calculations. Their scaling behaviour are in agreement with the predictions of the scaling theory.