Torsion in quantum field theory through time-loops on Dirac materials

Assuming dislocations could be meaningfully described by torsion, we propose here a scenario based on the role of time in the low-energy regime of two-dimensional Dirac materials, for which coupling of the fully antisymmetric component of the torsion with the emergent spinor is not necessarily zero. Appropriate inclusion of time is our proposal to overcome well-known geometrical obstructions to such a program, that stopped further research of this kind. In particular, our approach is based on the realization of an exotic time-loop, that could be seen as oscillating particle-hole pairs. Although this is a theoretical paper, we moved the first steps towards testing the realization of these scenarios, by envisaging Gedankenexperiments on the interplay between an external electromagnetic field (to excite the pair particle-hole and realize the time-loops), and a suitable distribution of dislocations described as torsion (responsible for the measurable holonomy in the time-loop, hence a current). Our general analysis here establishes that we need to move to a nonlinear response regime. We then conclude by pointing to recent results from the interaction laser-graphene that could be used to look for manifestations of the torsion-induced holonomy of the time-loop, e.g., as specific patterns of suppression/generation of higher harmonics.

To date, there is no experimental evidence of torsion of spacetime and the most prominent theory of gravity we have, Einstein's General Theory of Relativity, does not contemplate torsion. Nonetheless, torsion remains the focus of important research, both in fundamental and in condensed matter physics.
On the fundamental side, just like curvature is intimately connected with mass, torsion is intimately connected with spin, see, e.g., the pioneering work of Kibble [1]. Some argue that torsion manifests itself through the very existence of spinors, in an otherwise standard spacetime (see, e.g., [2]), while others continue to pursue the idea that torsion may as well be an actual physical property of our spacetime, within an extended theory of gravity (see, e.g., [3]) or of quantum gravity (see, e.g, [4]). Furthermore, both standard Supersymmetry (SUSY), in its curved space declination (supergravity, SUGRA) [5] and the more recent unconventional SUSY (USUSY) [6] make extensive use of torsion.
On the condensed matter side, the existence of two kinds of basic topological defects, disclinations and dislocations, related respectively to curvature and torsion, makes it natural to include the torsion in the geometrical description of the physical properties of materials [7,8]. This is not entirely free of ambiguities, in particular when it comes to associate a specific torsion to a given distribution of Burgers vector; but surely torsion is one of the two geometric entities at work there, along with curvature.
In the last years, due to their low energy spectrum structure, Dirac materials [9] have emerged as experimental playgrounds where both kinds of arenas, the fundamental research and the condensed matter one, met. In particular, the role of disclinations is under intense investigation to realize graphene analogs of Dirac quantum fields in curved spacetimes, see, e.g., [10][11][12][13][14][15][16] and recently the role of yet another kind of defects (grain boundaries) was also explored [17]. Investigations on how, in this context, dislocations could be used to construct an analog Dirac field theory coupled with torsion, rather than curvature, were of course carried on, see, e.g., [18].
If we were able to do so, it would be an invaluable help to shed light on some of the above mentioned mysteries on torsion. Let us mention, for instance, USUSY, especially in its (2 + 1)−dimensional formulation, that has been found to have many similarities with the Dirac field theory on graphene [19,20]. Unfortunately, the exploration of the role of torsion in this setting found a geometric obstacle, just due to the 2 + 1 dimensions [21][22][23]. These "no-go" results stopped research in this direction. It is the main goal of this letter to suggest a way to surmount this obstacle.
By definition, Dirac materials's π electrons 1 obey a low-energy dynamics near a Dirac point, governed by an emergent relativistic-like Hamiltonian with structure H D = v F σ · p, where v F is the Fermi velocity, and vectors are spatial two dimensional, see, e.g., [25]. To fully take into account this emergent relativistic-like structure [10], we include time as x 0 = v F t, hence turn to the (2 + 1)-dimensional action 2 Here, the Dirac spinor is not in the fundamental representation of the Lorentz group, it has four components ψ = (ψ + , ψ − ) T , with ψ ± = (α ± , β ± ) T . The variables α and β denote the sublattice * iorio@ipnp.troja.mff.cuni.cz † pais@ipnp.troja.mff.cuni.cz ‡ azampeli@utf.mff.cuni.cz 1 In the following we refer to two dimensional Dirac materials, with hexagonal lattice. Examples are graphene, germanene, silicene [9]. For χ3 ≡ t this configuration could give rise to nonzero temporal components of torsion, an instance to be investigated in the context of the "time crystals" of [24]. (ii) Edge dislocation in an hexagonal two-dimensional lattice, typical of a vast class of Dirac materials [9]. The Burgers vector, b, lies in the plane, while the dislocation line, L, is perpendicular to it, hence always orthogonal to b. To close the circuit, with this b = (1, 0), the number of steps (five here) is larger by one unit for the portion that includes the shaded area, with respect to portion running in the defect-free part (four steps here).
anticommuting operators, acting near the two inequivalent Dirac points labelled with "±" (see [17] for details on the role of the two Dirac points, and the various choices for the γ-algebra).
The natural generalization of (1) to a (2 + 1)-dimensional spacetime, equipped with a metric g µν = η ab e a µ e b ν and a metric-compatible connection Γ λ µν that includes torsion [26] T ] to be the Lorentz generators in the spinor space. The spin-connection, ω ab µ = e a ν (δ λ ν ∂ µ +Γ λ µν )e bρ , can be decomposed into torsion-free and contorsion contributions [27], ω ab µ =ω ab µ +κ ab µ . Standard manipulations of the action S, reported in detail in the Appendix A, lead to the form where the covariant derivative,D µ , is now based on the torsion-free connection,ω ab µ , only, and the contribution due to the torsion is all in the last term through its totally antisymmetric component [28]. From here, it is evident that the emergent fermions of Dirac materials, ψ can only to T 012 (or also with T 102 , or T 210 ). This is the above-mentioned geometric obstacle, that led earlier investigators to conclude that, for two dimensional Dirac materials, dislocations could not be accounted for by torsion [21][22][23].
In fact, the torsion tensor in crystals is related to the Burgers vector through the formula where Σ is a surface containing the dislocation, but otherwise arbitrary, a = 0, 1, 2. We clearly see that the only two possibilities that a nonzero Burgers vector can give rise to µνρ T µνρ = 0, necessary for the coupling in (3), are (cf Fig.1): (i) a time directed screw dislocation, i.e. b t ∝ T 012 dx ∧ dy or (ii) an edge dislocation spotted by a space-time circuit, e.g, b x ∝ T 102 dt ∧ dy. Here we took e a µ = δ a µ , in both circumstances. Our claim here is that both scenarios, are in fact not impossible. The first scenario 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.
could be explored in the context of the fascinating time crystals introduced by Wilczek [24,32], and the focus of intense experimental studies (see, e.g., [33] and the recent [34]). Such lattices, discrete in all dimensions, including time, would be an interesting playground to probe ideas of quantum gravity [35], although in 2 + 1 dimensions 4 . In what follows, we shall not focus on this, but rather on the second scenario.
In the Appendix B we show that we can take the Riemann curvature to be zero,R ab µν = 0, but κ ab µ = 0, and choose a frame whereω ab µ = 0. These settings make possible to isolate the effects of torsion on the system, and the corresponding action is where φ ≡ µνρ T µνρ .
To spot the effects of φ, we propose to make use of the particle-antiparticle structure, encoded in the action (5). Indeed, the regime of Dirac materials we describe, is the "half-filling" [25], whose vacuum state has the vacancies of the valence band (E < 0) completely filled, and the vacancies of the conductivity band (E > 0) empty. This is the analog of the Dirac sea of second quantization. If a pair particle-hole is excited out of this vacuum, and particle and antiparticle are made to oscillate, say, along y, as described in Fig.2, this amounts to a circuit of the particle-antiparticle pair in the (y, t)-plane. Fully exploiting the emergent relativistic-like structure of the model, the portion of the circuit described by the antiparticle moving forward in time, corresponds to the particle moving backward in time. This corresponds to the realization of a time-loop. The pictures in Fig.2 refer to a defect-free sheet. The presence of a dislocation, e.g., like the one in Fig.1, with Burgers vector b directed along x, would result in a failure to close the loop proportional to b. This is it, as long as the idealized situation is concerned. The challenge is to bring this idealized picture close to experiments. There are probably many ways to try that. One way, that involves an external electric field is described in the Appendix C, but it has some drawbacks. We propose here, instead, the use of an external magnetic field, so that the action governing this dynamics is It is then not hard to imagine a setting like the one depicted in Fig.3 , the antiparticle/hole travels through the shadowed region, that, although not necessarily so, can be thought of as buckling out of the plane, and deformed. The disturbance delays when the y-coordinate of particle and antiparticle is again the same (−ȳ here). This produces the deformed time-loop of (III). When the deformation is such that particle and antiparticle do not meet, as in (II), this produces a current, whose field theoretical description is represented in the depicted Feynman graph.
strength, excites a pair particle-hole out of the vacuum, and both particle and hole tunr around the dislocation line, in the (x, y)-plane, as shown in Fig.2. The corresponding time-loop in the (y, t)-plane (supposing that the Burgers vector is directed along x, like in Fig.1), is necessarily deformed, the deformation being proportional to the magnitude of the Burgers vector, ∆t ∝ b/v F . In Fig.3, we depict two possibilities, (I) and (II), both giving the deformed time-loop in the (y, t)-plane (III), but only (II) truly includes an holonomy. The shaded area could be seen as buckling out of the plane, so that to the antiparticle, traveling through it, will take longer than the particle to travel the corresponding portion lying into the undeformed crystal. If this is the only effect, then particle and antiparticle meet again, although not at 2t * , but at a later time 2t * + ∆t. This effect seems difficult to spot. The second possibility, though, is that the deformation induced by the nonzero Burgers is such that particles and holes do not meet. This gives rise, geometrically, to an holonomy in the time-loop, and physically to a net flux of particles and antiparticles, giving meaning to the vertex ψγ 5 φψ, hence directly related to the dislocations present in the material.
We conclude that, when time is duly included in the emergent relativistic-like picture of Dirac materials, there is room for torsion into the (2+1)-dimensional field theoretical description of their π-electrons dynamics. Although problems remain to be addressed (e.g., a unique assignment of torsion for the given distribution of Burgers vectors), our suggestion opens the doors to the use of these materials as analogs of many important theoretical scenarios where torsion plays a role. Our proposal, of a meaningful time-loop that could spot the presence of edge dislocations, routinely produced in Dirac materials, could be tested by realizing our Gedankenexperiment, based on the interaction with an external magnetic field. To this latter end, we notice that laser-graphene interaction, controlling electron dynamics on an unprecedented precision scale, is the focus of intense studies, both theoretical and experimental, see, e.g., [36,37]. Here we will recall the well-known argument, according to which, spinors are only coupled with the totally antisymmetric part of the torsion, in the minimal coupled prescription [28].
Suppose we have the following Hermitian and local Lorentz invariant action (here we used natural contain the contorsion part inside the spin connection, i.e., ω ab =ω ab + κ ab (de a +ω a b e b = 0 and T a = κ a b e b ). In order to obtain the field equations for ψ, we should variate the action under ψ. Therefore, we must integrate by parts the second term of (A1).
where BT is a boundary term, which could have some role in defining conserved charges, but we shall not take it into account here. Let us manipulate the last term in the first integral in (A2), where in the first equality we used the property [γ a , J bc ] = −i (γ c δ a b − γ b δ a c ). Now, We observe here that the term where in the last equality we used the antisymmetry of ω ab . Therefore, we can add safely the term (A3) to the action. So far, we have Now, we move to the second integral in (A2). First of all, remember that [26] √ −g = |e|, where for |e| we understand the determinant of the vielbein, i.e., |e| µνρ = abc e a µ e b ν e c ρ . So, Observe that the vielbein determinant fulfils the relation abc e a µ e b ν = |e| µνρ E ρ c . Then, It is important the property, Finally, we can compute the second integrand in (A2), as The action can be regrouped as which is the result given in [28] where− → D µ is the covariant derivative containing only the torsionless part of the connection. The second term of (A4) is of course the totally antisymmetric component of the torsion.