Exact Continuum Representation of Long-range Interacting Systems and Emerging Exotic Phases in Unconventional Superconductors

TL;DR

This paper develops an exact continuum representation for long-range interacting systems, enabling analysis of exotic phases in unconventional superconductors and revealing how long-range interactions influence collective excitations and stability.

Contribution

It introduces a rigorous, general framework for representing long-range interactions in lattice systems, applicable across dimensions and lattice types, and applies this to superconductivity and collective mode analysis.

Findings

Derived a generalized BCS gap equation for long-range interactions

Identified exotic phases with topological transitions in 2D superconductors

Showed long-range interactions can stabilize Higgs modes in condensates

Abstract

Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, any lattice, any power-law interaction, and for linear, nonlinear, and multi-atomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen--Cooper--Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize non-equilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine-tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization.