Erratum: Symmetry and Higher-Order Exceptional Points [Phys. Rev. Lett. 127, 186601 (2021)].

This corrects the article DOI: 10.1103/PhysRevLett.127.186601.

Symmetry and Higher-Order Exceptional Points.

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine-tuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. More saliently, third-order EPs generically require only two real tuning parameters in the presence of either a parity-time (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how third-order EPs with a generic ∼k^{1/3} dispersion are protected by PT symmetry, while third-order EPs with a ∼k^{1/2} dispersion are protected by the chiral symmetry emerging in non-Hermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetry-protected higher-order EPs, and tease out their concomitant phenomenology.

Transport properties in non-Fermi liquid phases of nodal-point semimetals.

In this review, we survey the current progress in computing transport properties in semimetals which harbour non-Fermi liquid (NFL) phases. We first discuss the widely-used Kubo formalism, which can be applied to the effective theory describing the stable NFL phase obtained via a renormalization group procedure and, hence, is applicable for temperatures close to zero (e.g. optical conductivity). For finite-temperature regimes, which apply to the computations of the generalized DC conductivity tensors, we elucidate the memory matrix approach. This approach is based on an effective hydrodynamic description of the system, and is especially suited for tackling transport calculations in strongly-interacting quantum field theories, because it does not rely on the existence of long-lived quasiparticles. As a concrete example, we apply these two approaches to find the response of the so-calledLuttinger-Abrikosov-Benelavskii phaseof isotropic three-dimensional Luttinger semimetals, which arises under the effects of long-ranged (unscreened) Coulomb interactions, with the chemical potential fine-tuned to cut exactly the nodal point. In particular, we focus on the electric conductivity tensors, thermal and thermoelectric response, Raman response, free energy, entropy density, and shear viscosity.

Direction-dependent conductivity in planar Hall set-ups with tilted Weyl/multi-Weyl semimetals.

We compute the magnetoelectric conductivity tensors in planar Hall set-ups, which are built with tilted Weyl semimetals (WSMs) and multi-Weyl semimetals (mWSMs), considering all possible relative orientations of the electromagnetic fields (EandB) and the direction of the tilt. The non-Drude part of the response arises from a nonzero Berry curvature in the vicinity of the WSM/mWSM node under consideration. Only in the presence of a nonzero tilt do we find linear-in-|B|terms in set-ups where the tilt-axis is not perpendicular to the plane spanned byEandB. The advantage of the emergence of the linear-in-|B|terms is that, unlike the various|B|2-dependent terms that can contribute to experimental observations, they have purely a topological origin, and they dominate the overall response-characteristics in the realistic parameter regimes. The important signatures of these terms are that they (1) change the periodicity of the response fromπto 2π, when we consider their dependence on the angleθbetweenEandB; and (2) lead to an overall change in sign of the conductivity depending onθ, when measured with respect to theB=0case.

Raman response and shear viscosity in the non-Fermi liquid phase of Luttinger semimetals.

Luttinger semimetals represent materials with strong spin-orbit coupling, harboring doubly-degenerate quadratic band touchings at the Brillouin zone center. In the presence of Coulomb interactions, such a system exhibits a non-Fermi liquid phase [dubbed as the Luttinger-Abrikosov-Beneslavskii (LAB) phase], at low temperatures and zero doping. However, a clear experimental evidence of this emergent state remains elusive to this date. Hence, we focus on extracting the Raman response as a complementary experimental signature. At frequencies much larger than the temperature, the Raman response exhibits a power-law behavior, which can be verified experimentally. On the other hand, at lower frequencies, the Raman response displays a quasi-elastic peak. We also compute the ratio of the shear viscosity and the entropy density, and the value obtained is a consequence of the hyperscaling violation that emerges in the LAB phase.

Floquet scattering of quadratic band-touching semimetals through a time-periodic potential well.

We consider tunneling of quasiparticles through a rectangular quantum well, subject to periodic driving. The quasiparticles are the itinerant charges in two-dimensional and three-dimensional semimetals having a quadratic bandtouching (QBT) point in the Brillouin zone. To analyze the time-periodic Hamiltonian, we assume a non-adiabatic limit where the Floquet theorem is applicable. By deriving the Floquet scattering matrices, we chalk out the transmission and shot noise spectra of the QBT semimetals. The spectra show Fano resonances, which we identify with the (quasi)bound states of the systems.