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Physics Letters A

Exact integral formulas and asymptotics for the correlations in the quantum many body system

Peter J. Forrester

Published Aug 1, 1993• 42 citations
Publisher
DOI
10.1016/0375-9601(93)90661-i

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The author shows that the finite-temperature equal-time field correlators of the exactly integrable (1+1)-dimensional gas of impenetrable fermions in a magnetic field can be expressed as the first Fredholm minor of a completely integrable linear operator. This result enables the author to derive differential equations for quantum correlators and to obtain analytical formulae for long-distance asymptotics. In particular, a formula for correlation length as a function of temperature and magnetic field is presented. His analysis reveals the presence of two length scales, which separate short-distance, intermediate and extreme asymptotic domains. The importance of the cross-over region is emphasized.
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For an integrable generalization of the Painleve'-II equation (P-II) to a system of coupled equations with symmetry breaking terms, an asymptotically exact WKB analysis is applied to obtain connection formulas for solutions at different infinities. The analysis relies on an exact solution of the quantum mechanical Demkov--Osherov model (DOM), revealing a possible deeper relation between classical integrable systems and solvable multistate Landau--Zener models. An application of the connection formulas to the problem of unstable vacuum decay during a second-order phase transition provides precise scaling of the number of excitations, including subdominant contributions.