On the uniqueness of the energy density in the infinite volume limit for quantum field models

D. Ya. Petrina, Stanislav Ivanov

Martin Pokorny

Various results concerning the spectra of the Ising model in a strong transverse field and the anisotropic Heisenberg antiferromagnet are proved. It is proved that the ground state energy of the Ising model in a strong transverse field in all dimensions converges to its infinite volume limit exponentially with a specific power law correction. It is also proved that this model in all dimensions has continuous spectrum in the infinite volume limit. For the anisotropic Heisenberg antiferromagnet it is proved that in dimensions of at least two, the energy spectrum contains a continuous part in the infinite volume limit. All results are obtained by perturbation theory using polymer expansions.

Maxime Dugave

We establish several properties of the solutions to the linear integral equations describing the infinite volume properties of the XXZ spin-1/2 chain in the disordered regime.In particular, we obtain lower and upper bounds for the dressed energy, dressed charge and density of Bethe roots.Furthermore, we establish that given a fixed external magnetic field (or a fixed magnetization) there exists a unique value of the boundary of the Fermi zone.

D. Ya. Petrina, Stanislav Ivanov

Theoretical Physics

Yohei Kashima

We prove by means of a renormalization group method that in weakly interacting many-electron systems at half-filling on a periodic hyper-cubic lattice, the free energy density uniformly converges to an analytic function of the coupling constants in the infinite-volume, zero-temperature limit if the external magnetic field has a chessboard-like flux configuration. The spatial dimension is allowed to be any number larger than 1. The system covers the Hubbard model with a nearest-neighbor hopping term, on-site interactions, exponentially decaying density-density interactions and exponentially decaying spin-spin interactions. The magnetic field must be included in the kinetic term by the Peierls substitution. The flux configuration and the sign of the nearest-neighbor density-density/spin-spin interactions can be adjusted so that the free energy density is minimum among all the flux configurations. Consequently, the minimum free energy density is proved to converge to an analytic function of the coupling constants in the infinite-volume, zero-temperature limit. These are extension of the results on a square lattice in the preceding work ([Kashima, Y., "The special issue for the 20th anniversary", J. Math. Sci. Univ. Tokyo. 23 (2016), 1-288]). We refer to lemmas proved in the reference in order to complete the proof of the main results of this paper. So this work is a continuation of the preceding work.