Time-loops to spot torsion on bidimensional Dirac materials with dislocations

TL;DR

The paper proposes realizing torsion-based effects from lattice dislocations in two-dimensional Dirac materials by promoting time as the third dimension via a time-loop in a Riemann-Cartan spacetime with torsion $T^{\mu}{}_{\nu\rho}$, where the torsion field is encoded as $\phi = \epsilon^{\mu\nu\rho} T_{\mu\nu\rho}/|e|$. In this framework, the Dirac action contains a totally antisymmetric torsion coupling (through $\gamma^{5}$ and $\epsilon^{\mu\nu\rho} T_{\mu\nu\rho}$) that couples with opposite sign to the two spinor sectors, illustrating the 2D obstruction is overcome by time. Because linear response vanishes by Furry's theorem for mixed electromagnetic and torsion probes, the authors advocate nonlinear spectroscopy, notably high-order harmonic generation (HHG), to detect torsion-induced holonomy in time-loops. The work focuses on graphene-like materials and sketches concrete experimental pathways while acknowledging challenges in assigning a unique torsion distribution to a Burgers vector, pointing to time-loop holonomy as a tabletop realization of gravitational torsion phenomena.

Abstract

Assuming that, with some care, dislocations could be meaningfully described by torsion, we propose here a scenario based on a previously unexplored role of time in the low-energy Dirac field theory description of two-space-dimensional Dirac materials. Our approach is based on the realization of an exotic time-loop that could be realized by oscillating particle-hole pairs, overcoming the well-known geometrical obstructions due to the lack of a third spatial dimension. General symmetry considerations allow concluding that the effects we are looking for can only be seen if we move to the nonlinear response regime.

Figures (3)

• Figure 1: One geometric interpretation of torsion in Riemann-Cartan spaces. Consider two vector fields, $X$ and $Y$, at a point $P$. First, parallel-transport $X$ along $Y$ to the infinitesimally close point $R$. Then, again from $P$, parallel-transport $Y$ along $X$ to reach a point $Q$. The failure of the closure of the parallelogram is the geometrical signal of torsion, and its value is the difference between the two resulting vectors $T(X,Y)$. In Riemannian spaces, $V_{n}$, this tensor is assumed to be zero. The picture was inspired by HehlObukhov2007 but with the notation of Nakahara.
• Figure 2: Edge dislocation from two disclinations. Two disclinations, a heptagon and a pentagon, by adding up to zero total intrinsic curvature. This makes a dislocation with Burgers vector $\vec{b}$, as can be seen in the lower half-plane. Burgers vector in the continuum limit caries torsion. This figure is taken from ip3.
• Figure 3: Idealized time-loop. At $t=0$, the hole (yellow) and the particle (black) start their journey from $y=0$, in opposite directions. Evolving forward in time, at $t=t^*>0$, the hole reaches $-y^*$, while the particle reaches $+y^*$, (blue portion of the circuit). Then they come back to the original position, $y=0$, at $t=2t^*$ (red portion of the circuit). This can be repeated indefinitely. On the far right, the equivalent time-loop, where the hole moving forward in time is replaced by a particle moving backward in time. Figure taken from ip4.