Thermally activated detection of dark particles in a weakly coupled quantum Ising ladder

The Ising$_h^2$ integrable field theory emerges when two quantum critical Ising chains are weakly coupled. This theory possesses eight types of relativistic particles, among which the lightest one ($B_1$) has been predicted to be a dark particle, which cannot be excited from the ground state through (quasi-)local operations. The stability on one hand highlights its potential for applications, and on the other hand makes it challenging to be observed. Here, we point out that the mass of the $B_1$ dark particle $m_{B_1}$ appears as a thermally activated gap extracted from local spin dynamical structure factor at low frequency ($\omega \ll m_{B_1}$) and low temperatures ($T \ll m_{B_1}$). We then further propose that this gapped behavior can be directly detected via the NMR relaxation rate measurement in a proper experimental setup. Our results provide a practical criterion for verifying the existence of dark particles.