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Communications in Mathematical Physics

Localization in the ground-state of the one dimensionalX−Y model with a random transverse field

Abel Klein, J. Fernando Perez

Published Mar 1, 1990• 30 citations
Publisher
DOI
10.1007/bf02097047

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We consider the ground state of the one-dimensional quantum Ising model with transverse field hx in one dimension depending on the site x∈Z in a finite volume. We assume hx = λξx, where (ξx),x∈Z is a family of strictly positive i.i.d. random variables with a positive density at 0 satisfying the condition. E(ξ0−δ)<∞ for some δ > 0 and λ > 0 is the disorder parameter. We prove that if λ is sufficiently large, the entanglement entropy of the ground state has a stochastic bound uniform with respect to the volume.
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In many-body systems with quenched disorder, dynamical observables can be singular not only at the critical point, but in an extended region of the paramagnetic phase as well. These Griffiths singularities are due to rare regions, which are locally in the ordered phase and contribute to a large susceptibility. Here, we study the geometrical properties of rare regions in the transverse Ising model with dilution or with random couplings and transverse fields. In diluted models, the rare regions are percolation clusters, while in random models the ground state consists of a set of spin clusters, which are calculated by the strong disorder renormalization method. We consider the so called energy cluster, which has the smallest excitation energy and calculate its mass and linear extension in one-, two- and three-dimensions. Both average quantities are found to grow logarithmically with the linear size of the sample. Consequently, the energy clusters are not compact: for the diluted model they are isotropic and tree-like, while for the random model they are quasi-one-dimensional.