Anomalous thermal broadening from an infrared catastrophe in two-dimensional quantum antiferromagnets

The nature of quasiparticles in two-dimensional quantum antiferromagnets at finite temperature remains an open question despite decades of theoretical work. In particular, it is not fully understood how long wavelength excitations contribute to significant broadening of the experimentally observable spectrum. Motivated by this problem, we consider the $XY$ model of easy-plane antiferromagnets, and compute the dynamic structure factor by direct summation of diagrams. In doing so, we find that noninteracting quasiparticles with infinite lifetimes can still lead to a broad response. This forms the basis for a new paradigm describing the interaction of experimental probes with a physical system, where broadening is due neither to the lifetime, nor to the emergence of fractional quasiparticles. Instead, strong fluctuations drive the probe to absorb and radiate an infinite number of arbitrarily low energy quasiparticles, leading us to draw parallels with the infrared catastrophe in quantum electrodynamics.

Figure 1

An external source (dashed line) emits three quasiparticles in a narrow cone centered around the direction of momentum transfer $\mathbf{k}$.

Figure 2

Diagrammatic expansion of the dynamic structure factor. Dashed lines denote the source, wavy lines denote quasiparticles, and vertical dashing denotes application of cut rules. Diagrams are grouped into columns according to the number of quasiparticles in each piece after cutting. $\left(S\right)$ denotes Stokes ($\omega >{\omega }_{\mathbf{k}}$ only), $\left(A\right)$ denotes anti-Stokes ($\omega <{\omega }_{\mathbf{k}}$ only), and $\left(R\right)$ denotes Raman scattering (any $\omega$) processes. The trivial elastic scattering contribution ($\omega =0$) is not shown.

Figure 3

The vertex correction can be expanded as a sum of loop diagrams. Each loop represents a factor of $\langle {\phi }^2\rangle$.

Figure 4

The dynamic structure factor at fixed source momentum transfer $\mathbf{k}$, as a function of energy transfer $\omega$. Continuation of (29) to $\omega <0$ follows from (20).