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Solid State Communications

Low temperature asymptotic behaviours of the correlation functions in the one-dimensional fermi gas

Akiomi Mishima, Minoru Kimura

Published Dec 1, 1983• 0 citations
Publisher
DOI
10.1016/0038-1098(83)90825-6

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A quantum field-theoretical model, which describes spatially non-homogeneous repulsive Bose gas in an external harmonic potential is considered. Two-point thermal correlation functions of the Bose gas are calculated in the framework of the functional integration approach. Successive integration over the ``high-energy'' functional variables first and then over the ``low-energy'' ones is used. The effective action functional for the low-energy variables is obtained in one loop approximation. The functional integral representations for the correlation functions are estimated by means of the stationary phase approximation. A power-law asymptotical behaviour of the correlators of the one-dimensional Bose gas is demonstrated in the limit, when the temperature is going to zero, while the volume occupied by the non-homogeneous Bose gas infinitely increases. The power-law behaviour is governed by the critical exponent dependent on the spatial arguments.
We summarize results on the asymptotics of the two-particle Green functions of interacting electrons in one dimension.Below a critical value of the chemical potential the Fermi surface vanishes, and the system can no longer be described as a Luttinger liquid.Instead, the non-relativistic Fermi gas with infinite point-like repulsion becomes the universal model for the long-wavelength, low temperature physics of the one-dimensional electrons.This model, which we call the impenetrable electron gas, allows for a rigorous mathematical treatment.In particular, a so-called determinant representation for the two particle Green function could be derived.This determinant representation is related to an integrable classical evolution equation and to a Riemann-Hilbert problem, that enable the exact calculation of the asymptotics of the two-particle Green functions.
We consider the one-dimensional delta-interacting electron gas in the case of infinite repulsion. We use determinant representations to study the long time, large distance asymptotics of correlation functions of local fields in the gas phase. We derive differential equations which drive the correlation functions. Using a related Riemann–Hilbert problem we obtain formulae for the asymptotics of the correlation functions, which are valid at all finite temperatures. At low temperatures these formulae lead to explicit asymptotic expressions for the correlation functions, which describe power law behavior and exponential decay as functions of temperature, magnetic field and chemical potential.