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arXiv:1002.0772 · Journal of Mathematical Physics

Exponential decay of correlation functions in many-electron systems

Yohei Kashima

Published Jun 1, 2010• 7 citations
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Abstract

For a class of tight-binding many-electron models on hypercubic lattices, the equal-time correlation functions at nonzero temperature are proved to decay exponentially in the distance between the center of positions of the electrons and the center of positions of the holes. The decay bounds hold in any space dimension in the thermodynamic limit if the interaction is sufficiently small depending on temperature. The proof is based on the U(1)-invariance property and volume-independent perturbative bounds of the finite dimensional Grassmann integrals formulating the correlation functions.

DOI
10.1063/1.3409395

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Four point correlation functions for many electrons at finite temperature in periodic lattice of dimension d (≥1) are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower bound on the radius of convergence and an upper bound on the perturbation series are obtained by evaluating the Taylor expansion of logarithm of the finite dimensional Grassmann Gaussian integrals. The perturbation series up to second-order is numerically implemented along with the volume-independent upper bounds on the sum of the higher order terms in the 2-dimensional case.
We consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimension D<2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryan-Spencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric long-range orders.
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