Minimal Hamiltonian deformations as bulk probes of effective non-Hermiticity in Dirac materials

TL;DR

This work investigates how non-Hermiticity manifests in bulk observables of a two-dimensional Dirac semimetal in the weak-NH, real-spectrum regime. By introducing minimal deformations that either commute (tilt) or anticommute (velocity anisotropy) with the NH Hamiltonian, the authors distinguish observables that can be absorbed into renormalized velocities from those that retain irreducible NH structure. They find that the DOS slope is NH-sensitive for tilt but NH-insensitive under VAD, while the biorthogonal quantum geometric tensor and linear optical conductivity largely resist NH effects; second-order optics are only nonzero when inversion symmetry is broken by tilt, and the optical shear viscosity reveals NH through tensor anisotropy in VAD. Together, these results identify concrete bulk probes—tilt-sensitive DOS and viscosity-sensitive anisotropy—that reveal effective non-Hermiticity even when the spectrum remains real, with practical implications for topolectrical, photonic, and ultracold-atom platforms.

Abstract

Non-Hermitian (NH) Dirac semimetals describe open gain--loss systems, yet at charge neutrality models featuring real spectrum often look Hermitian-like, with NH effects absorbed into renormalized band parameters. Here we show that a response-based diagnostic of effective non-Hermiticity can be formulated using minimal pseudo-Lorentz-symmetry-breaking deformations, which separate observables that remain captured by parameter redefinitions from those that exhibit irreducible NH structure. For a two-dimensional NH Dirac semimetal in the weak-NH, real-spectrum regime, we analyze Dirac-cone tilt and velocity anisotropy and compute representative probes of spectral structure, quantum geometry, optical response, and viscoelasticity at zero temperature. We find that tilt yields an NH-dependent slope of the density of states that cannot be collapsed to a single effective velocity, while velocity anisotropy can be captured by effective-velocity reparametrization. Furthermore, the quantum metric and collisionless optical conductivities provide NH-insensitive benchmarks (with the nonlinear conductivity symmetry selected), whereas the shear viscosity offers a discriminator through its tensor structure. Our results identify minimal deformations and bulk response channels that enable access to effective non-Hermiticity even when the spectrum remains real.

Figures (5)

• Figure 1: Polarization diagram corresponding to Eq. \ref{['eq:Pij']}. Here, a solid line denotes the Dirac fermion propagator, a dashed line corresponds to the external electromagnetic potential, while the vertex corresponds to the current as dictated by the U(1) gauge invariance.
• Figure 2: Second-order susceptibility (triangle) diagram corresponding to Eq. \ref{['eq:chi2_def_main']} yielding the second-order optical conductivity. The labels are the same as in Fig. \ref{['fig:Polarization']}.
• Figure 3: Feynman diagram corresponding to the stress correlator in Eq. \ref{['eq:Cstress_def']} determining the optical shear viscosity. Solid lines denote Dirac fermion propagators, wavy lines represent external strain tensor, while the vertex corresponds to the stress tensor.
• Figure 4: Components of the shear viscosity tensor for the symmetric tilt realization with varying $\alpha$ at fixed $\beta=0.5$ at frequency $\Omega=1$. We here fix $v_H=1$ and $N_f=1$.
• Figure 5: Components of the shear viscosity tensor for the symmetric tilt realization with varying $\beta$ at fixed $\alpha=0.7$ at frequency $\Omega=1$. We here fix $v_H=1$ and $N_f=1$.