Dressed energy of the XXZ chain in the complex plane

• Published in: Letters in mathematical physics Vol. 111; no. 5
• Main Authors: Faulmann, Saskia, Göhmann, Frank, Kozlowski, Karol K.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2021; Springer Nature B.V; Springer Verlag
• Subjects: Antiferromagnetism; Chains; Complex Systems; Complex Variables; Eigenvalues; Fredholm equations; Geometry; Group Theory and Generalizations; Integral equations; Low temperature; Lower bounds; Magnetic fields; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Physics; Physics and Astronomy; Real numbers; Theoretical; Transfer matrices
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-021-01473-3

We consider the dressed energy ε of the XXZ chain in the massless antiferromagnetic parameter regime at 0 < Δ < 1 and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers, it describes the energy of particle–hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures, the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve Re ε ( λ ) = 0 . We describe this curve and give lower bounds to the function Re ε in regions of the complex plane, where it is positive.