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arXiv:cond-mat/0310499 · Journal of Physics A Mathematical and General

Zero temperature correlation functions for the impenetrable fermion gas

Vadim Cheianov, Mikhail B. Zvonarev

Published Jan 28, 2004• 60 citations
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Abstract

We calculate the long time and distance asymptotics of the one-particle correlation functions in the model of impenetrable spin 1/2 fermions in 1 + 1 dimensions. We consider the spin disordered zero temperature regime, which occurs when the limit T → 0 is taken at a positive chemical potential. The asymptotic expressions are found from the asymptotic solution of the matrix Riemann–Hilbert problem related to the determinant representation of the correlation functions.

DOI
10.1088/0305-4470/37/6/021

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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.
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Introduction Impenetrable bosons in one space dimension are a special case (at coupling constant c → ∞) of the one-dimensional Bose gas (NS model) considered in detail in Chapter I. The results obtained there for the NS model can be specialized to the case of impenetrable bosons. It is to be said that all the formulæe are considerably simplified in this case. From the point of view of physics this is due to the fact that the δ-function potentials in the N -particle Hamiltonian (I.1.11) are now infinitely strong and the bosons cannot penetrate one another, so that the wave function should be equal to zero if the bosons' coordinates coincide. In this respect impenetrable bosons are rather similar to free fermions (see Appendix I.I). In this chapter correlation functions of impenetrable bosons are considered and their representations as Fredholm determinants of linear integral operators are given. The (very essential) simplification with respect to the general case is due to the fact that now all the auxiliary quantum fields can be set equal to zero (see IX.6), so that the kernels of these operators do not contain quantum fields. The determinant representation for the time-dependent correlator is easy to obtain in this case. Temperature correlation functions are considered as well as zero temperature ones.