It is shown that, for a class of Hamiltonians of XXZ chains in an external,
longitudinal magnetic field that are small perturbations of an Ising
Hamiltonian, the spectral gap above the ground-state energy remains strictly
positive when the perturbation is turned on, uniformly in the length of the
chain. The result is proven for both the ferromagnetic and the
antiferromagnetic Ising Hamiltonian; in the latter case the external magnetic
field is required to be small and the two-fold degenerate ground-state energy
of the unperturbed Hamiltonian may split into two energy levels whose
difference is bounded above by a fractional power of the (small) coupling
constant of the transverse terms. This result is proven by using a new, quite
subtle refinement of a method developed in earlier work and used to iteratively
block-diagonalize Hamiltonians of ever larger subsystems with the help of local
unitary conjugations. One novel ingredient of the method presented in this
paper consists of the use of Lieb-Robinson bounds.
Description
Low energy spectrum of the XXZ model coupled to a magnetic field
%0 Generic
%1 delvecchio2023energy
%A Del Vecchio, Simone
%A Fröhlich, Jürg
%A Pizzo, Alessandro
%A Ranallo, Alessio
%D 2023
%K xxz
%T Low energy spectrum of the XXZ model coupled to a magnetic field
%U http://arxiv.org/abs/2306.02772
%X It is shown that, for a class of Hamiltonians of XXZ chains in an external,
longitudinal magnetic field that are small perturbations of an Ising
Hamiltonian, the spectral gap above the ground-state energy remains strictly
positive when the perturbation is turned on, uniformly in the length of the
chain. The result is proven for both the ferromagnetic and the
antiferromagnetic Ising Hamiltonian; in the latter case the external magnetic
field is required to be small and the two-fold degenerate ground-state energy
of the unperturbed Hamiltonian may split into two energy levels whose
difference is bounded above by a fractional power of the (small) coupling
constant of the transverse terms. This result is proven by using a new, quite
subtle refinement of a method developed in earlier work and used to iteratively
block-diagonalize Hamiltonians of ever larger subsystems with the help of local
unitary conjugations. One novel ingredient of the method presented in this
paper consists of the use of Lieb-Robinson bounds.
@misc{delvecchio2023energy,
abstract = {It is shown that, for a class of Hamiltonians of XXZ chains in an external,
longitudinal magnetic field that are small perturbations of an Ising
Hamiltonian, the spectral gap above the ground-state energy remains strictly
positive when the perturbation is turned on, uniformly in the length of the
chain. The result is proven for both the ferromagnetic and the
antiferromagnetic Ising Hamiltonian; in the latter case the external magnetic
field is required to be small and the two-fold degenerate ground-state energy
of the unperturbed Hamiltonian may split into two energy levels whose
difference is bounded above by a fractional power of the (small) coupling
constant of the transverse terms. This result is proven by using a new, quite
subtle refinement of a method developed in earlier work and used to iteratively
block-diagonalize Hamiltonians of ever larger subsystems with the help of local
unitary conjugations. One novel ingredient of the method presented in this
paper consists of the use of Lieb-Robinson bounds.},
added-at = {2023-06-08T19:42:21.000+0200},
author = {Del Vecchio, Simone and Fröhlich, Jürg and Pizzo, Alessandro and Ranallo, Alessio},
biburl = {https://www.bibsonomy.org/bibtex/269849a41af2a51ac7f4ac553ab1abea7/gzhou},
description = {Low energy spectrum of the XXZ model coupled to a magnetic field},
interhash = {53afe249c8fa4167d469266c38887f4e},
intrahash = {69849a41af2a51ac7f4ac553ab1abea7},
keywords = {xxz},
note = {cite arxiv:2306.02772Comment: 6 figures},
timestamp = {2023-06-08T19:42:21.000+0200},
title = {Low energy spectrum of the XXZ model coupled to a magnetic field},
url = {http://arxiv.org/abs/2306.02772},
year = 2023
}