Discrete Laplacian structure and kernel reduction of the gap equation in odd-dimensional large-N vector models at imaginary chemical potential

Abstract

We identify a mathematical structure in the gap equations of the large-N Gross–Neveu model at imaginary chemical potential in odd spacetime dimensions. Using this model as the primary example, the central observation is an elementary dimensional ladder: a single centred first difference with respect to σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document} relates the thermal part of the gap equation in dimension d to that in dimension d-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d-2$$\end{document}, for every odd d. Iterating this elementary step generates the higher-order finite-difference (discrete Laplacian) relations; for the distinguished sequence d=4k+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=4k+3$$\end{document}, whose kernel exponent is even, the closed stencil hierarchy connects d to 2d-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2d-3$$\end{document} through 2k iterates. We show that the gap equations can be written as the sum of two parts: one defined by higher-order finite-difference patterns and a cutoff-dependent part given by a truncated asymptotic expansion of a hypergeometric function. The same discrete finite-difference hierarchy is shown to arise in the thermal sector of large-N bosonic models, indicating that the underlying structure is not restricted to fermionic theories. We further argue that this picture suggests a relation between thermal field theories in these odd d dimensions and exactly solvable one-dimensional quantum problems. We find that the thermal mass at specific imaginary chemical potential values corresponds to points where internal-energy and entropy contributions balance, which is formally analogous to OPE inversion formula techniques where thermal mass values arise from sets of transcendental equations.