Nonlinear optical response as a probe of emergent Lorentz symmetry violation in noncentrosymmetric materials

TL;DR

The work addresses how weak Lorentz-violating backgrounds could be detected in noncentrosymmetric solids through nonlinear optics. It derives a momentum-odd LV correction to the Bloch Hamiltonian from a Dirac equation with a fixed background vector, projecting onto a spinful Rice–Mele lattice to obtain $H_{\mathrm{eff}}(k)=H_{\mathrm{RM}}(k)+H_{\mathrm{LV}}(k)$ with $H_{\mathrm{LV}}(k)=\lambda_k\left[\tfrac{1}{2}(\tau_y\otimes\sigma_y)+(\tau_y\otimes\sigma_0)\right]$ and $\lambda_k=\alpha k$, $\alpha=\hbar\xi E_x/(m_e c)$, $\xi=g v$. This momentum-odd LV term perturbs the interband phase and yields a field-odd component in the second-order shift conductivity, enabling a directional photocurrent asymmetry when the static field orientation is reversed. The authors formulate a shift-conductivity calculation in the length gauge, connecting $r_{mn}(k)$ and the shift vector $R^{a}_{mn}(k)$ to $\sigma^{(2)}$ and showing that reversing the longitudinal field isolates the LV signal via $\Delta\sigma^{(2)}(\omega,\theta)$. They propose an experimentally viable 1D nanowire setup and estimate picoampere-scale LV-induced currents under realistic illumination, demonstrating a practical route to infer the LV coupling $\xi$ from nonlinear optical measurements in solids.

Abstract

We propose an electrically controlled protocol to detect weak Lorentz-violating (LV) backgrounds through the second-order shift photocurrent in noncentrosymmetric crystals. Using a spinful Rice--Mele model, we show that a stationary LV background induces a momentum-odd correction to the Bloch Hamiltonian, which generates an odd-in-field contribution to the shift current. This leads to a directional asymmetry, whereby the photocurrent distinguishes opposite orientations of an applied static field. The effect originates from an LV-induced deformation of the interband phase and can be isolated experimentally by comparing field-reversed configurations, with vanishing response at transverse orientations, providing an internal consistency check. Our results demonstrate that nonlinear optical responses offer a practical and symmetry-selective route for probing LV effects in solid-state systems.

Figures (3)

• Figure 1: Proposed angular-resolved setup for detecting LV signatures in the shift current. Opposite field orientations ($\theta$ and $\theta+\pi$) yield unequal photocurrents in the presence of a momentum-odd LV perturbation.
• Figure 2: Left: Difference $\Delta\sigma^{(2)}(\omega)=\sigma^{(2)}(\omega,\theta=\pi)-\sigma^{(2)}(\omega,\theta=0)$, showing a peak contrast of $1.53\ \mathrm{nm}\cdot\mathrm{nA}/\mathrm{V}^{2}$ due to the momentum-asymmetric contribution of the LVP. Right: Angular dependence of $\sigma^{(2)}(\omega)$ at $\omega=2\Delta$. The response remains $2\pi$-periodic but is no longer symmetric under $\theta\rightarrow\theta+\pi$, resulting in unequal lobe amplitudes.
• Figure 3: Shift conductivity $\sigma^{(2)}(\omega)$ for opposite field orientations $\theta=0$ and $\theta=\pi$. In the absence of LV the responses for $\theta$ and $\theta+\pi$ coincide, while the LVP lifts this equivalence and produces a directional asymmetry near the band-edge resonance.