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Physics-Uspekhi

Once again about analytical methods of calculating correlation functions in quantum statistical physics

Михаил Федорович Сарры

Published Sep 30, 1994• 0 citations
Publisher
DOI
10.1070/pu1994v037n09abeh001439

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A brief but clear and complete account is given for two analytical methods of calculating correlation functions in quantum statistical physics from first principles—the widely used method of two-time temperature Green's functions (the GF method) and a new, "direct algebraic" (DA) method. The mathematical and technical clarity and simplicity of the DA method and its resulting practical value are demonstrated for the five most widely used models in quantum statistical physics. Since the DA method is an exactly self-consistent method (in the sense that the expansion coefficients in the equations of motion are chosen from the requirement that the Jacobi operator identity be satisfied exactly), it in principle affords the possibility of an internal check, which is not possible in the GF method. Like the GF method, the DA method permits calculation of the spectra of possible elementary excitations and, hence, of the density of single-particle energy states corresponding to them.
The Lieb–Liniger model is a prototypical integrable model and has been turned into the benchmark physics in theoretical and numerical investigations of low-dimensional quantum systems. In this note, we present various methods for calculating local and nonlocal M -particle correlation functions, momentum distribution, and static structure factor. In particular, using the Bethe ansatz wave function of the strong coupling Lieb–Liniger model, we analytically calculate the two-point correlation function, the large moment tail of the momentum distribution, and the static structure factor of the model in terms of the fractional statistical parameter α = 1−2/ γ , where γ is the dimensionless interaction strength. We also discuss the Tan’s adiabatic relation and other universal relations for the strongly repulsive Lieb–Liniger model in terms of the fractional statistical parameter.