Hawking Radiation in Weyl Semimetals

• Hawking radiation in Weyl semimetals is an analogue-gravity phenomenon where inhomogeneous tilting of Weyl cones creates a type-I/type-II interface that mimics an event horizon.
• The effective metric, modeled via the Painlevé–Gullstrand spacetime, governs tunneling and scattering processes that yield thermal-like Hawking temperatures from horizon gradients.
• Experimental schemes leverage non-equilibrium transport and controlled boundary conditions to detect horizon-induced conductance anomalies and distinguish them from other emission effects.

Hawking radiation in Weyl semimetals denotes a class of analogue-gravity phenomena in which low-energy Weyl quasiparticles propagate in an effective spacetime generated by an inhomogeneous tilt of the Weyl cone. In the foundational formulation due to Volovik, a spatial crossover from a type-I cone to an overtilted type-II cone realizes an artificial event horizon: the type-II region plays the role of the black-hole interior, the type-I/type-II interface plays the role of the horizon, and Hawking-like emission is associated with the dynamics of quasiparticles at that interface rather than with Einstein gravity itself (Zhang et al., 2016, Volovik, 2016). Subsequent work refined this picture by separating three logically distinct notions: transient nonequilibrium refilling after horizon formation, thermal-like tunneling through a smooth horizon, and horizon-controlled scattering amplitudes in lattice models (Volovik, 2016, Beule et al., 2021, Sabsovich et al., 2021).

1. Analogue-gravity foundation and the type-I/type-II transition

The basic low-energy description near a Weyl point at momentum ${\bf p}^{(0)}$ is

$$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$

with Pauli matrices $\hat\sigma^k$ acting in the two-band Weyl subspace. In this formulation, $e_k^j$ and $e_0^j$ are emergent tetrad fields and ${\bf p}^{(0)}$ behaves as an emergent gauge field. The quasiparticle cone is encoded by

$$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$

so the band structure directly defines an effective Lorentzian geometry for Weyl quasiparticles (Volovik, 2016).

The type-I/type-II distinction is controlled by the tilt magnitude

$\left|e_0^j[e^{-1}]_j^k\right|.$

For

$\left|e_0^j[e^{-1}]_j^k\right|<1,$

the cone is type-I. For

$\left|e_0^j[e^{-1}]_j^k\right|>1,$

it is overtilted, and electron- and hole-like Fermi surfaces touch at the Weyl point, defining the type-II regime. In the isotropic choice

$$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$0

the condition reduces to

$$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$1

which is the direct analogue of the horizon condition in Painlevé–Gullstrand spacetime (Volovik, 2016).

This identification was framed from the outset as a Lifshitz transition. In the Weyl-semimetal language, horizon crossing is not merely a kinematic threshold; it is a reconstruction of the zero-energy manifold. Outside the horizon there is an isolated Weyl point, whereas inside the horizon the overtilted cone intersects zero energy and generates Fermi pockets attached to that point. In the ideal linear theory, the critical state can even support a Dirac line at the type-I/type-II boundary, characterized by the winding invariant $$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$2, although this critical structure is generally removed once ultraviolet corrections are included (Zhang et al., 2016).

2. Effective metric, horizon geometry, and Hawking temperature

The effective spacetime used in the Weyl-semimetal black-hole analogy is the Painlevé–Gullstrand metric

$$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$3

For a spherical black hole,

$$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$4

and the horizon is defined by $$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$5. In the condensed-matter interpretation, $$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$6 is the characteristic Weyl velocity and $$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$7 is the local cone tilt (Zhang et al., 2016, Volovik, 2016).

Two complementary horizon constructions appear in the literature. One is a spherical region of radius $$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$8 in which the interior is type-II and the exterior is type-I, so the boundary $$H= e_k^j(p_j-p^{(0)}_j)\hat\sigma^k + e_0^j(p_j-p^{(0)}_j)+\dots \,,$$9 is the analogue horizon. The other is a planar heterostructure with a spatially varying tilt parameter $\hat\sigma^k$0 in

$\hat\sigma^k$1

with horizon at

$\hat\sigma^k$2

The planar realization is the standard description for type-I/type-II Weyl-semimetal interfaces and heterostructures (Zhang et al., 2016).

The Hawking temperature is set by the horizon gradient. In the radial description,

$\hat\sigma^k$3

while in the flat-interface description,

$\hat\sigma^k$4

For the Schwarzschild-like profile $\hat\sigma^k$5, this gives

$\hat\sigma^k$6

up to the sign convention for the gradient. The direct implication is that smaller black-hole regions produce larger effective surface gravity. Volovik therefore argued that, for sufficiently small black-hole regions, the gradient of the system parameter can be large enough that the analogue Hawking temperature may reach room temperature (Volovik, 2016).

3. What counts as “Hawking radiation” in Weyl semimetals

The earliest Weyl-semimetal proposal did not identify Hawking radiation with perpetual equilibrium emission from a static analogue black hole. Instead, it identified a transient nonequilibrium relaxation process. When a type-II region is created inside an initially type-I system, the new Hamiltonian contains negative-energy states behind the horizon that were not occupied in the pre-formation state. Relaxation proceeds by filling these interior negative-energy states, accompanied by particle-hole pair creation near the horizon. The exterior excitations produced during this refilling process are interpreted as Hawking radiation. Once the interior Fermi pockets are filled, the configuration is stationary and no longer radiates (Volovik, 2016).

This distinction is central. In that formulation, the analogue black hole is non-radiating in equilibrium, and thermal behavior appears only as a channel within post-formation relaxation. The proposal is therefore strongest as an analogue of horizon-localized pair creation with a surface-gravity scale, and weaker as an analogue of steady black-hole evaporation (Zhang et al., 2016, Volovik, 2016).

Later microscopic work sharpened this point by studying static inhomogeneous tilts in a lattice model. In that setting there is no direct analogue of spontaneous Hawking radiation in equilibrium. Instead, the horizon manifests itself through scattering probabilities of prepared wavepackets. The central factor is

$\hat\sigma^k$7

with $\hat\sigma^k$8 the tilt gradient near the horizon. For white-hole analogues this produces Hawking fragmentation, in which an incoming packet splits into two outgoing packets in distinct channels; for black-hole analogues it produces Hawking attenuation, in which an injected packet emerges with reduced amplitude after partial mode conversion. These effects are controlled by an effective Hawking temperature but are explicitly not spontaneous vacuum emission (Sabsovich et al., 2021).

A plausible synthesis is that the literature uses “Hawking radiation” in two non-equivalent ways. One usage refers to transient quasiparticle emission generated when an event-horizon profile is formed and the occupations lag behind the new spectrum. The other refers to thermal-like horizon factors in scattering and tunneling amplitudes for states prepared in a stationary background. The first is a formation-and-relaxation effect; the second is a transport and mode-conversion effect.

4. Smooth-horizon scattering and non-equilibrium transport signatures

The most explicit transport theory for artificial event horizons in Weyl semimetals considers a type-I/type-II heterostructure with tilt $\hat\sigma^k$9 varying along the interface normal. The corresponding line element is

$e_k^j$0

and at normal incidence the null trajectories satisfy

$e_k^j$1

The horizon is located where one branch stalls,

$e_k^j$2

For a monotonic profile with $e_k^j$3, the interface acts as an artificial black-hole horizon (Beule et al., 2021).

In the analytically tractable “Hawking channel,”

$e_k^j$4

the low-energy scattering problem admits an exact treatment for both sharp and slowly varying horizons. In the smooth case, with near-horizon profile

$e_k^j$5

the two counterpropagating type-II modes $e_k^j$6 and $e_k^j$7 tunnel into a type-I mode $e_k^j$8 with probabilities

$e_k^j$9

corresponding to the effective Hawking temperature

$e_0^j$0

The same analysis yields poles at

$e_0^j$1

interpreted as quasi-bound states with lifetime $e_0^j$2 (Beule et al., 2021).

The transport consequence is subtle. In equilibrium, the two Hawking-like channels cancel: particle number is conserved, and the relation

$e_0^j$3

precludes a net equilibrium Hawking current. The horizon therefore does not reveal itself as spontaneous equilibrium emission. To expose the effect, one must prepare a stationary non-equilibrium population imbalance between $e_0^j$4 and $e_0^j$5 (Beule et al., 2021).

Two explicit protocols were proposed. Circularly polarized irradiation of the type-II region selectively enhances the occupation of the $e_0^j$6 branch, whereas coupling the type-II region to a magnetic lead with suitable spin polarization favors $e_0^j$7. In either case, the horizon-induced channel asymmetry generates a peak in the two-terminal differential conductance. The predicted sign of the peak is branch dependent: $e_0^j$8 peaks at positive $e_0^j$9, while ${\bf p}^{(0)}$0 peaks at negative ${\bf p}^{(0)}$1. The same work cited transport lifetimes up to

${\bf p}^{(0)}$2

and mean free path of about

${\bf p}^{(0)}$3

in TaAs as evidence that a prepared nonequilibrium population could reach the horizon before relaxing (Beule et al., 2021).

5. Microscopic constraints: boundaries, lattice doublers, and non-Hermitian generalizations

A major caveat is that the bulk effective metric does not by itself guarantee faithful black-hole kinematics. For a type-II Weyl semimetal with a flat boundary at ${\bf p}^{(0)}$4, the most general self-adjoint local boundary condition is a one-parameter family labeled by ${\bf p}^{(0)}$5,

${\bf p}^{(0)}$6

valid only for ${\bf p}^{(0)}$7. The associated edge modes can propagate opposite to the allowed bulk causal direction and therefore “escape” from the effective black-hole region when

${\bf p}^{(0)}$8

This implies that a literal black-hole interpretation requires boundary engineering, not just bulk overtilting. For the convention analyzed in that work, the range

${\bf p}^{(0)}$9

avoids edge escape, and $$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$0 was given as an explicit safe choice (Hashimoto et al., 2019).

The lattice also changes the analogy qualitatively. In the smooth-horizon tight-binding model, overtilting on a momentum cut forces the appearance of large-momentum low-energy doubler states. These are not additional three-dimensional Weyl nodes in the usual Nielsen–Ninomiya sense, but low-energy states along the one-dimensional scattering cut required by Brillouin-zone periodicity. They are absent in continuum general relativity but indispensable in the lattice theory: Hawking fragmentation and Hawking attenuation are mediated precisely by these doubler channels (Sabsovich et al., 2021). Relatedly, the original continuum proposals already required a nonlinear ultraviolet correction,

$$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$1

to regularize the overtilted spectrum and produce finite Fermi pockets behind the horizon (Zhang et al., 2016).

A separate line of work replaced Hermitian Weyl Hamiltonians with non-Hermitian Weyl-type models. One proposal used a non-PT-symmetric weakly pseudo-Hermitian two-band Hamiltonian and obtained a tunneling probability

$$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$2

interpreted through a Boltzmann form as an artificial Hawking temperature (Bagchi et al., 2022). Another studied $$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$3-symmetric dissipative Weyl-type Hamiltonians with exceptional cones and derived the thermal-like factor

$$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$4

which was interpreted as the purely thermal contribution to analogue Hawking radiation (Stålhammar et al., 2021). These constructions extend the Weyl-cone/horizon analogy, but they are better viewed as Weyl-type analogue systems than as transport theories for conventional Hermitian electronic Weyl semimetals.

6. Experimental prospects, adjacent platforms, and conceptual boundaries

The experimental program implied by the Weyl-semimetal literature is specific. It requires spatial control of the cone tilt so that a type-I region crosses continuously or abruptly into a type-II region, thereby creating a horizon with gradient large enough to set an observable $$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$5. In the transport formulation, the most robust signature is not equilibrium black-body emission but a non-equilibrium conductance anomaly associated with horizon-selected scattering channels (Beule et al., 2021). In the wavepacket formulation, the observables are branch-resolved fragmentation and attenuation controlled by the Fermi-function-like factor $$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$6 (Sabsovich et al., 2021).

Several common misconceptions are explicitly rejected by the literature. First, the analogue horizon is kinematic: it is an engineered inhomogeneous band-structure profile, not a solution of the Einstein equations. Second, a static Weyl-semimetal horizon need not radiate in equilibrium; in the original Volovik picture the analogue black hole is fully stationary after the interior Fermi pockets are filled (Volovik, 2016). Third, a thermal factor in tunneling or scattering amplitudes is not equivalent to a continuous Hawking flux. This is why later work emphasized cancellation in equilibrium and the need for nonequilibrium preparation (Beule et al., 2021).

The same cone-tilt logic has been exported to adjacent platforms. A photonic topological lattice based on a type-II $$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$7 type-III $$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$8 type-I Dirac/Weyl cone transition was proposed as a stationary, long-lived horizon with

$$g^{\mu\nu}(p_\mu-p_\mu^{(0)})(p_\nu-p_\nu^{(0)})=0,\qquad g^{\mu\nu}=e^\mu_a e^\nu_b\eta^{ab},$$9

underscoring that the effective metric construction is broader than electronic semimetals, even though that system is not itself a Weyl semimetal (Kang et al., 2019).

Finally, not all radiation phenomena in Weyl materials are Hawking-like. A moving magnetic domain wall in a magnetic Weyl semimetal generates axial fields $\left|e_0^j[e^{-1}]_j^k\right|.$0 and $\left|e_0^j[e^{-1}]_j^k\right|.$1, activates the axial anomaly through $\left|e_0^j[e^{-1}]_j^k\right|.$2, and produces oscillating electromagnetic radiation via the chiral magnetic effect. That mechanism is explicitly nonthermal and non-horizon-based, and it serves as a contrast class that any claimed Hawking signal in a Weyl system would need to be distinguished from (Hannukainen et al., 2019).

Taken together, the literature defines Hawking radiation in Weyl semimetals as an analogue-gravity phenomenon tied to overtilted Weyl cones and type-I/type-II interfaces, but it does not support a single universal experimental signature. The most conservative conclusion is that Weyl semimetals provide a precise platform for horizon kinematics and for Hawking-like thermal factors in relaxation, tunneling, and scattering, while the existence of a literal spontaneous equilibrium Hawking flux remains unsupported in static Hermitian realizations.