On-Shell Methods for Quantum Matter: Strongly correlated Dirac materials

TL;DR

The paper proposes applying on-shell scattering amplitude methods to emergent relativistic quantum matter, enabling compact, gauge-invariant calculations of nonlinear transport such as the Hall response in Dirac/Weyl semimetals. It demonstrates that the nonlinear Hall conductivity $\sigma^{(2)}_{ijk}=e^3\,\epsilon_{ijm}D_{mk}$ can be obtained from the soft limit of a parity-odd three-photon amplitude in an emergent $\mathrm{QED}_4$ description, with the Berry curvature dipole $D_{mk}$ arising from the unitarity cut. It then extends to higher-order responses via BCFW recursion, building $(n+1)$-photon amplitudes from the parity-odd three-photon vertex and yielding $\chi^{(n)}$ through systematic soft limits, all while automatically preserving gauge invariance and crossing symmetry. This framework offers a unifying, analytic route to nonlinear and topological responses in strongly correlated Dirac materials and suggests broad applicability to edge CFTs and other emergent relativistic phases.

Abstract

We propose a framework for applying on-shell scattering amplitude methods to emergent relativistic phases of quantum matter. Many strongly correlated systems, from Dirac and Weyl semimetals to topological-insulator surfaces, exhibit low-energy excitations that are effectively massless relativistic spinors. We show that physical observables such as nonlinear optical and Hall responses can be obtained from compact on-shell amplitudes, bypassing the complexity of Feynman diagrams. As a concrete demonstration, we derive the nonlinear Hall conductivity of a Dirac semimetal from a single parity-odd three-photon amplitude, highlighting the analytic and conceptual power of amplitude-based approaches for strongly correlated condensed-matter systems.