Quantitative materialization of black- and white-hole analogs at Weyl-semimetal junctions: microscopic band structure, horizon degree-of-freedom dynamics, material design, and measurable signatures

Kedem, Bergholtz, and Wilczek [Phys. Rev. Research 2, 043285 (2020)] showed that a junction between type-I and type-II Weyl semimetals is governed by equations identical to those for fermions near black- and white-hole horizons, proposed a quantization in which a horizon-localized degree of freedom with charging energy $E_c$ and coupling functions $g_i(\omega)$ regularizes the infinite propagation delay at the horizon, analyzed screening and causal structure qualitatively, and concluded that the material system exhibits no Hawking radiation. The paper's models are, however, idealized: a single linearized Weyl node, phenomenological regulator scales $\lambda$ and $\alpha$ , only qualitative assessment of realistic complications (multiple Weyl nodes, parasitic bands, Brillouin-zone periodicity, Fermi arcs), and no computed transport signatures. The open problem is to make these junction-horizon models quantitatively meaningful in real materials: derive the horizon Hamiltonian parameters from microscopic band structures, identify realizable material junctions with predicted length and steepening scales, treat the dynamics and equilibration of the horizon degree of freedom including dissipation and screening by additional bands, compute measurable transport and noise signatures that would distinguish material-interface behavior from genuine thermal (Hawking/Unruh-like) radiation, and determine when transverse-dimensional, Fermi-arc-mediated channels dominate over local horizon localization.

Background

A Weyl semimetal is a crystal whose low-energy electronic excitations behave as massless Weyl fermions: near isolated band-touching points (Weyl nodes) in momentum space, the effective Hamiltonian is $H=\pm\hbar\sigma\cdot p$ , with $\sigma$ the Pauli matrices acting on a pseudo-spin index; the two signs label nodes of opposite chirality. Because a crystal's momentum space is periodic (the Brillouin zone), the Nielsen-Ninomiya theorem forces Weyl nodes to occur in pairs, and on suitable surfaces the projections of nodes of opposite chirality are connected by surface states called Fermi arcs. Since crystals lack Lorentz symmetry, the Hamiltonian can acquire a 'tilt' term $\hbar\kappa\cdot p$ that slopes the Weyl cone. When the tilt exceeds the Fermi velocity ( $|\kappa|>1$ , a type-II Weyl semimetal), the cone is over-tilted and excitations propagate only one way along the tilt direction; type-I materials ( $|\kappa|<1$ ) support both directions. Tilted Weyl fermions realize analog gravity: the Lagrangian $\mathcal{L}=i\bar\psi\gamma^\mu\partial_\mu\psi+i\bar\psi\gamma^0\kappa(\mathbf{x})\cdot\nabla\psi$ is algebraically identical to that of a massless Dirac field in the curved metric $ds^2=(|\kappa|^2-1)dt^2-2\kappa\cdot d\mathbf{x}\,dt+d\mathbf{x}\cdot d\mathbf{x}$ , so a spatial tilt profile with $|\kappa(z)|$ crossing 1 mimics a black-hole event horizon (the surface beyond which nothing can escape), where $|\kappa|=1$ ; the oppositely oriented profile gives the time-reversed configuration, a white hole, a surface that nothing can enter. This connects to the broader analogue-gravity program (sonic black holes in Bose-Einstein condensates and fluids, optical fibers), which aims to observe quantum effects of horizons, notably Hawking radiation (the thermal radiation predicted to be emitted by black holes) and the Unruh effect (thermal response of uniformly accelerated detectors). Building on the curved-spacetime theory of inhomogeneous Weyl materials by Liang and Ojanen and on the identification of a natural horizon in the Dirac semimetal Zn2In2S5 by Huang, Jin, and Liu, Kedem, Bergholtz, and Wilczek (2020) analyzed junctions of type-I and type-II Weyl semimetals as material black and white holes. In their one-dimensional model the wave equation $[\partial_t+f(z)\partial_z+\frac{1}{2}(\partial_z f)]\psi=0$ with $f(z)=\kappa(z)\pm1$ is solved by the method of characteristics; the coordinate $r(z)=\int^z du/f(u)$ is the analogue of the tortoise coordinate of general relativity (a reparametrization that straightens out light-ray trajectories). As the horizon is approached, wave packets slow down and pile up without ever arriving, and the eigenmode normalization diverges logarithmically. The authors regularize this by steepening the horizon onset, replacing $f(z)\propto z$ near the horizon by $f(z)\propto z(\lambda+|z|)^\alpha/|z|^\alpha$ with a length scale $\lambda\ge0$ and a steepening exponent $0<\alpha<1$ , so that packets reach the horizon in finite time. Since propagation is unidirectional, the accumulated excitation cannot escape; the authors therefore endow the horizon with an explicit localized degree of freedom — in the spirit of the black-hole membrane paradigm (an effective description of a horizon as a material surface with properties such as conductivity), but representing what they argue is true local physics — which stores the accumulated charge $Q$ . The quantized Hamiltonian is $H=H_0+H_I+H_c$ : free regularized eigenmodes ( $H_0$ ), a coupling $H_I=\sum_i\int d\omega\, g_i(\omega)(A a^\dagger_{i,\omega}+A^\dagger a_{i,\omega})$ between the mode operators and the horizon charge ladder operators $A^{(\dagger)}$ with coupling functions $g_i(\omega)$ localized near the horizon, and a charging energy $H_c=E_c Q^2$ of the same form as in Coulomb-blockade physics (the mesoscopic regime where the electrostatic cost of adding charge to a small conductor dominates transport), with $E_c$ 'depending on microscopic details' and left undetermined. The authors analyze the response to voltages (charge transfer across the junction), to deposited charge (screened by particle-hole pairs generated via the Schwinger mechanism — pair creation in an electric field — which here propagate in the same direction), and the causal structure of heating and measurement, and conclude that the physical picture differs sharply from astrophysical expectations: the regularized horizon is a naked singularity, there is no Hawking radiation, and white holes, sources, and sinks arise on an equal footing. The paper's discussion explicitly flags the gap to reality: realistic Weyl semimetals host parasitic bands crossing the Fermi level (bands other than the linearized Weyl cone) and finite doping, though materials exist where these are avoided; the Nielsen-Ninomiya pairing of nodes, surface Fermi arcs, and the Brillouin-zone periodicity that forces the light cones to curve back are complications whose influence is assessed only qualitatively ('quantitative, but not qualitative alteration of the model'); the transverse spatial dimensions 'deserve further investigation'; and material-engineering prospects are delegated to earlier literature. A contrasting strand, the Floquet-engineered (periodically time-driven) type-II/type-I Dirac junction of Liu et al. (2018), even claims a Hawking-like emission spectrum, so whether material horizons radiate at all is itself contested. What is missing is the quantitative bridge between the idealized junction models and real, multi-node, multi-band materials with experimentally checkable predictions.

Problem Statement

The problem left open by Kedem, Bergholtz, and Wilczek (2020) is to convert their idealized single-node junction models of black- and white-hole analogs in Weyl semimetals into a quantitative, experimentally testable theory, with one common objective: determine whether, how, and with what measurable consequences the horizon phenomenology survives contact with realistic band structures and devices. This comprises five jointly supporting tasks. (1) Microscopic foundation: quantify how lattice-scale band structure — the class of realistic complications that defines the required scope, namely multiple Weyl nodes required by the Nielsen-Ninomiya theorem, parasitic bands at the Fermi level, finite doping, and Brillouin-zone periodicity with band curvature — modifies the regulated horizon dynamics, and derive the horizon Hamiltonian parameters, namely the coupling functions $g_i(\omega)$ and the charging energy $E_c$ (currently 'depending on microscopic details'), as well as the physical content of the regulator length scale $\lambda$ and steepening exponent $\alpha$ , from a realistic band structure rather than postulating them. (2) Material realization: identify concrete material systems or heterostructures that implement a type-I/type-II (black- or white-hole) junction with well-separated nodes and low doping, and predict the resulting horizon scales ( $\lambda$ , $\alpha$ , $E_c$ ) for candidate compounds. (3) Horizon degree-of-freedom dynamics: analyze the dynamics and equilibration of the horizon-localized charge, including dissipation, many-body interactions, and screening by additional bands. (4) Signatures: compute measurable transport and noise signatures — for example frequency-dependent conductance, shot noise (electrical current fluctuations arising from the discreteness of charge), or finite-temperature spectral functions of the junction — that would unambiguously distinguish the material-interface behavior (charge accumulation, screening, relaxation) from genuine thermal Hawking/Unruh-like radiation. (5) Dimensionality: include the transverse spatial dimensions and surface Fermi-arc physics systematically, determining when chiral-interface (wormhole-like) channels dominate charge accumulation versus local horizon localization. The five tasks are posed jointly: together they determine whether, how, and with what measurable consequences the horizon phenomenology survives realistic band structures. Answers to the tasks are well-supported quantitative results—positive or negative—stating their reference system, precision, and parameter regime; negative findings, such as a demonstration that no realizable low-doping type-I/type-II junction with well-separated nodes exists (task 2), or that no signature unambiguously separates interface behavior from Hawking/Unruh-like radiation (task 4), are part of the scientific content.

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