Abstract
We introduce effective form factors for one-dimensional lattice fermions with arbitrary phase shifts. We study tau functions defined as series of these form factors. On the one hand we perform the exact summation and present tau functions as Fredholm determinants in the thermodynamic limit. On the other hand simple expressions of form factors allow us to present the corresponding series as integrals of elementary functions. Using this approach we re-derive the asymptotics of static correlation functions of the XY quantum chain at finite temperature.
Highlights
We focus on the static correlation functions for which we demonstrate that after complete summation of the effective form factors series and taking the thermodynamic limit the answer can be presented in the form of Fredholm determinants
We focus on the following spin-spin correlation function at finite temperature
In this work we have introduced the form factors to simulate the static correlation functions for the states with finite entropy
Summary
Solvable one-dimensional quantum mechanical systems of interacting spins, bosons, and fermions provide a unique platform for studying non-perturbative effects. As we have mentioned above, this complexity is combinatorial in nature and reflects the fact that each form factor for the thermal states is exponentially small so the number of relevant form factors is exponentially large This makes direct computation of the corresponding sum for the correlation functions notoriously difficult and force researchers to focus at most on the two particle-hole excitations [34,35,36,37], consider semiclassical approximations [46] or develop other approximation schemes [41, 42]. By first taking the thermodynamic limit of the effective form factors and performing their summation we manage to present the Fredholm determinants as integrals of elementary functions This kind of asymptotic behavior for models in the continuum (not the lattice) arises from the solution of the Riemann–Hilbert problem for operators acting on the whole real line [58].
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