Algebraic Spectral Curves of Two-Channel Operator Pencils with Power-Law Response

Abstract

A finite matrix can acquire a nontrivial algebraic spectral curve when its entries contain a multivalued response. We study this mechanism for a two-channel pencil with scalar response $z^{\beta }$ and $J$-self-adjoint matrices. If $\beta =r/n$ is in lowest terms and the coefficients are generic, the normalized determinant curve in $(z,\lambda )$ has genus $n-1$ and is a double cover. It is rational for $n=1$, elliptic for $n=2$, and hyperelliptic for $n\ge 3$. The generic genus depends only on the denominator $n$. The proof reduces birationally to a classical hyperelliptic normal form. The numerator appears as the cyclic character of the distinguished coupling and controls its divisor and boundary singularity. Here $\lambda$ is a complexified coupling coordinate: the curve is the joint determinant locus, not a fixed-coupling spectrum. For irrational exponents the local monodromy has infinite order, while rational approximants converge uniformly on compact sets and preserve isolated zeros on a fixed branch. Negative exponents in $(-1,0)$ arise from causal Volterra kernels and cutoff continuum-bath Schur complements, separating finite channels from nonlocal memory. As an application, the exact leading infrared monomial model of a zero-temperature conformal response of dimension $h_{\mathrm{probe}}$ has, for rational $2h_{\mathrm{probe}}$, genus $\mathrm{den}(2h_{\mathrm{probe}})-1$. For the fundamental fermion in the even-$q>2$ Sachdev--Ye--Kitaev saddle, this becomes $q/2-1$. The core statements, including determinant algebra, branch counts, and causal-response identities, are machine-checked from raw matrix definitions.

35 pages, 3 figures. Updated with Lean 4 formalization