Landscape of scattering universality with general dispersion relations

TL;DR

This work reveals a rich landscape of low-energy universality in 1D scattering with general dispersion relations, showing that universal S-matrix values at zero energy emerge from the competition between a dispersion's infrared behavior and the structure of emitter interactions. By introducing a robust framework that includes the Jost function, a Gram–Schmidt-derived integer set $\mathcal{N}$, and a generalized Levinson's theorem, the authors classify universal scattering across interior and borderline regimes for antisymmetric and symmetric dispersions. The key findings are explicit universal zero-energy S-matrix values, parity-resolved behavior in the symmetric case, and a unifying criterion for criticality given by the det$\bm{J}^\perp$ condition, together with a complete zero-energy S-matrix classification. The results have broad relevance for synthetic quantum systems where engineered dispersion relations can be realized in atomic arrays, photonic crystals, and quantum circuits, enabling experimental observation of these new universality classes and topological phase-like transitions in scattering. Overall, the paper extends scattering universality beyond the conventional quadratic paradigm and provides a comprehensive theoretical toolkit for predicting low-energy scattering phases in a wide class of dispersive systems.

Abstract

Universality in physics describes how disparate systems can exhibit identical low-energy behavior. Here, we reveal a rich landscape of new universal scattering phenomena governed by the interplay between an interaction and a system's density of states. We investigate one-dimensional scattering with general dispersion relations of the form $ε(k) = |k|^m$ and $ε(k) = \text{sign}(k)|k|^m$ for any real $m \geq 1$. For key models such as emitter scattering and separable potentials, we prove that the low-energy S-matrix converges to universal forms determined solely by the dispersion exponent $m$ and a few integers defining the interaction. This establishes a broad classification of new universality classes, extending far beyond the standard quadratic dispersion paradigm. Furthermore, we derive a generalized Levinson's theorem relating the total winding of the scattering phase to the number of bound states. Our findings are directly relevant to synthetic quantum systems, where engineered dispersion relations in atomic arrays and photonic crystals offer a platform to explore these universal behaviors.

Figures (8)

• Figure 1: Universal scattering in a 1D waveguide with a diverging density of states (DOS). (a) Schematic of a single photon scattering from quantum emitters. (b), (c) The general dispersion relations studied: (b) antisymmetric, $\epsilon(k)=\mathrm{sign}(k)|k|^m$, and (c) symmetric, $\epsilon(k)=|k|^m$. For $m>1$, the group velocity vanishes at the band edge ($k=0$), causing the DOS to diverge. The red arrows indicate the low-energy limit ($E \to 0$) where this universal behavior emerges.
• Figure 2: Universal zero-energy scattering. Schematics for (a) antisymmetric and (b) symmetric dispersion, where arrows indicate incident waves and the numbers $1$ and $-1$ denote their amplitudes. (a) For antisymmetric dispersion, the transmission amplitudes are given by $S(0^+)$ for incidence from the left and $S(0^-)$ for incidence from the right; the bottom formula shown applies only to interior interactions. (b) For symmetric dispersion, scattering decouples into even and odd parity channels. The universal S-matrix formulas for both cases are parameterized by integers ($n_A, n_{\mathrm{e}}, n_{\mathrm{o}}$) and given in Theorems \ref{['theoremUniScatteringAnti']} and \ref{['theoremUniScatteringSym']}.
• Figure 3: Illustration of the trajectories of $\det[\boldsymbol{S}(E)]$ in the complex plane for a dissipative system as $E$ increases along the continuum spectrum for (a) antisymmetric dispersion and (b) symmetric dispersion. In (a), the trajectory starts and ends at $1$ with a discontinuity across $E = 0$. The segments of the trajectory for $E < 0$ and $E > 0$ are colored yellow and blue, respectively. In (b), the trajectory begins at $E(0^+)$ and ends at $E(+\infty) = 1$.
• Figure S1: Newton polygons for the characteristic polynomial of $\bm{J}(\omega)$, illustrating the conditions for criticality. The panels compare (a, c) interior interactions with (b, d) borderline interactions for system sizes $D=N$ (top row) and $D<N$ (bottom row). The black dots represent the points $(l, v(a_l))$, where $l$ is the power of the eigenvalue in the characteristic polynomial and $v(a_l)$ is the valuation (the leading power of $q_\omega$) of the corresponding coefficient $a_l$ in the limit $q_{\omega}\rightarrow 0$. The polygon is the lower convex hull of these points, and the valuations of the eigenvalues of $\bm{J}(\omega)$ correspond to the negative slopes of its segments. A system is critical if an eigenvalue vanishes as $q_{\omega}\rightarrow 0$, which requires a positive eigenvalue valuation. This condition is met when the polygon has a segment with a negative slope, as highlighted in red.
• Figure S2: (a) Illustration of the key result from Lemma \ref{['lemmaInvJmatrix']}: the diagonal blocks of $\bm{J}(\omega)^{-1}$ are asymptotically equivalent to the inverses of the corresponding blocks of $\bm{J}(\omega)$. (b) Diagram of the asymptotic hierarchy of the components of the interaction vector $|v_{\mathrm{e}}(k)\rangle$, showing that components in the even subspace ($i \le N_{\mathrm{e}}$) are of a leading asymptotic order. (c) Illustration of Eq. \ref{['eqEquiEff']}, showing how the combined asymptotic hierarchies of the vectors and matrix blocks reduce the calculation for the original system to that of the simpler effective system.

Theorems & Definitions (20)

• Lemma 1
• proof : Proof sketch
• Lemma 2
• Theorem 1
• Theorem 2
• Definition 1
• Lemma 3
• proof
• Lemma 4
• Lemma 5