Geodesic geometry of 2+1-D Dirac materials subject to artificial, quenched gravitational singularities

The gist The spatial modulation of the Fermi velocity for gapless Dirac electrons in quantum materials is mathematically equivalent to the problem of massless fermions on a certain class of curved spacetime manifolds.

Model and Analogy to Gravitation

The study focuses on a 2+1-dimensional massless Dirac model in the presence of artificial quenched gravity, which involves perturbations that mimic gravity by coupling to the spatial-spatial components of the stress tensor and preserve time-reversal symmetry. The Hamiltonian is defined as H = -i/2 X a,b=1,2 Z d 2x v b a (x) h ψ¯(x) ˆσ a ←→∂b ψ(x) i, (1) and the action is given by S = i Z dt Z d 2x  ψ¯(t, x)∂tψ(t, x) + 1/2 X a,b=1,2 v b a (x) h ψ¯(t, x) ←→∂b ψ(t, x) i, (3a) = i Z dt Z d 2x ψ¯(t, x) ∂t + v1(x) · σˆ ∂1 + 1/2 ∂1v1(x) · σˆ + v2(x) · σˆ ∂2 + 1/2 ∂2v2(x) · σˆ ψ(t, x), (3b). The mapping to gravity involves using the vielbein formalism [9] to define the action for a system of massless Dirac fermions on a 2+1-dimensional spacetime as S = Z p g d 3x ψ¯(x)E µ A (x)ˆγ A i∂µ − 1/2 ω BC µ (x)SˆBC ψ(x), (5). Consistency between the Dirac Hamiltonian in Eq. (1) and the action in Eq. (5) requires setting E0 A6=0 = 0, due to the absence of time-space mixing terms. With this choice, temporal flatness is enforced by the condition pg E 0 0 ≡ 1. This leads to a quenched gravitational metric in Eq. (11c) which is defined by the mapping dictionary E 0 A = 1/p v1 × v2, and gµν → [E 0 A = 1/p v1 × v2 (1, 0, 0; 0 v 1/v 2; 0 v 2/v 2) (38).

Curvature and Singularities

The metric in Eq. (11c) becomes ill-defined at points where the cross-product v1×v2 vanishes, which corresponds to a failure of the temporal flatness condition and the non-invertibility of the inverse metric. The Ricci scalar curvature takes the form R = N v b a, ∂v d c v1 × v2 3, (12). Singularities are characterized in terms of Dirac cone geometry; nematic singularities correspond to nonzero v [Eq. (13)] and give rise to a local Dirac canyon, while isotropic singularities correspond to v = 0, and locally flatten the entire Dirac cone. Nematic singularities can only be formed by the breaking of rotational symmetry of the electron band structure.

Geodesics

The main focus is on null geodesics on the manifolds defined by Eq. (11c), which are governed by the geodesic equation d/2x µ ds2 = -Γ µ αβ[x(s)] dxα ds dxβ ds, (19). Null geodesics correspond to trajectories followed by massless particles, and they have a conserved spacetime interval gµν[x(s)]dxµ/ds dxν ds ≡ ∆, where null geodesics correspond to ∆ = 0. The temporal first integral for time along a geodesic is dt/ds = (E/m) v1[x(s)] × v2[x(s)] (24). A reparametrization by global time yields the gravitational time-dilation factor γ(x) ≡ 1/(v1(x) × v2(x)) (25). The geodesic equation in terms of global time coordinate is given by Eq. (27a).

Geodesic Collimation at Nematic Singularities

Null geodesics impinging on a nematic singularity are driven to pass through at the direction defined by the angle θˆ∗ and at the speed v (30a–c). This is because for φ to have a finite derivative at the singularity, it must be driven to ˆv so that at the singularity, φˆ is parallel to the vectors v1, v2. The geodesic equation thus locks the speed of the geodesic through the singularity to v (30a–c).

Submodels and Solvable Manifolds

The paper studies several solvable manifolds, including the isotropic power-law model with v(r) = rα for α > 0 (5.1). For this model, geodesics that start out heading towards the singularity are always captured and those that start out heading away always escape. The xy-factored model provides a class of models where all geodesics asymptote towards nematic singularity walls, and run along these walls until finally being captured by an isotropic singularity.

Quantum Treatments of Toy Models

The paper investigates the quantum dynamics for toy models, finding that in the dreibein wall geometries, while all geodesics are bounded for the linear wall, only some are bounded for the tanh wall. In contrast, in the circular nematic power-law model with α = 2, all quantum density asymptotically accumulates at the origin, mimicking the behavior of the geodesics.

Conclusion

The effects of “artificial” quenched gravity on 2D massless Dirac carriers could have important consequences for understanding and manipulating low-dimensional Dirac materials. Isotropic singularities can asymptotically capture geodesics that pass sufficiently close, while null geodesics can traverse nematic singularity domain walls but experience a geodesic collimation effect that fixes their transit velocity. These domain walls can exhibit a horizon effect, trapping null geodesics as bound states that perpetually lens back and forth across the nematic singularity line.

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SciPost Physics Submission

Geodesic geometry of 2+1-D Dirac materials subject to artificial, quenched gravitational singularities S. M. Davis1, M. S. Foster,1 Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA 2 Department of Physics and Astronomy and Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, June 13, 2022 Abstract The spatial modulation of the Fermi velocity for gapless Dirac electrons in quantum materials is mathematically equivalent to the problem of massless fermions on a certain class of curved spacetime manifolds. We study null geodesic lensing through these manifolds, which are dominated by curvature singularities, such as nematic singularity walls (where the Dirac cone flattens along one direction). Null geodesics lens across these walls, but do so by perfectly collimating to a local transit angle. Nevertheless, nematic walls can trap null geodesics into stable or metastable orbits characterized by repeated transits. We speculate about the role of induced one-dimensionality for such bound orbits in 2D dirty d-wave superconductivity Abstract The spatial modulation of the Fermi velocity for gapless Dirac electrons in quantum materials is mathematically equivalent to the problem of massless fermions on a certain class of curved spacetime manifolds. We study null geodesic lensing through these manifolds, which are dominated by curvature singularities, such as nematic singularity walls (where the Dirac cone flattens along one direction). Null geodesics lens across these walls, but do so by perfectly collimating to a local transit angle. Nevertheless, nematic walls can trap null geodesics into stable or metastable orbits characterized by repeated transits.

1. Improved AI can perform semiclassical approximations of quantum dynamics by treating null geodesics as bicharacteristics for both the curved-space Dirac equation and the associated wave equations. This allows for a geometric optics viewpoint of a generic field equation, which is crucial for understanding the semiclassical approach to QGD.

2. Improved AI can model and predict phenomena in 2D Dirac materials by simulating the effects of artificial quenched gravity (QG). Specifically, it can determine that Isotropic singularities are strongly attractive and asymptotically capture geodesics, while nematic singularities do not capture geodesics; instead, they drive all impinging geodesics to a unique velocity.

3. Improved AI can analyze the phase diagram of Dirac superconductors by linking nematic singularity walls to pairing enhancement. It can model how states gravitationally bound to domain wall horizons could provide a link to quasi-1D physics and how pairing enhancement due to nematic singularity walls could create a negative feedback loop, terminating when the gap has hardened sufficiently.

4. Improved AI can perform quantum state analysis in specific toy models by solving the time-dependent curved-space Dirac equation for geometries like the xy-factored orthogonal model, yielding solutions that are in extremely close correspondence with the geodesic picture.