NNegative-curvature spacetime solutions for graphene
Antonio Gallerati
Politecnico di Torino, Dipartimento di Scienza Applicata e Tecnologia, corso Duca degli Abruzzi 24, 10129 Torino, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Torino, via Pietro Giuria 1, 10125 Torino, Italy [email protected]
Abstract
We provide a detailed analysis of the electronic properties of graphene-like materials withcharge carriers living on a curved substrate, focusing in particular on constant negative-curvature spacetime. An explicit parametrization is also worked out in the remarkable caseof Beltrami geometry, with an analytic solution for the pseudoparticles modes living onthe curved bidimensional surface. We will then exploit the correspondent massless Diracdescription, to determine how it affects the sample local density of states.
Contents a r X i v : . [ h e p - t h ] J a n Introduction
The recent developments in material science provide a new connection between condensed matterand quantum electrodynamics models. In particular, the study of the physics of carbon-basedmaterials like graphene opens a window on the possibility of a direct observation of quantumbehaviour in the curved background of a solid state system [1–3]. A graphene sheet is a bidimen-sional system of carbon atoms arranged in a honeycomb lattice of one-atom thickness, one of theclosest possible real two-dimensional objects. In 1984 Semenoff formulated the hypothesis thatgraphene could realize the physics of two dimensional massless Dirac fermions [4], this propertydiscriminating graphene from other 2D system. Graphene crystals were then produced in 2004as single carbon atom layers [5, 6].As we will discuss in detail, graphene and other 2D materials realize the physics of spinorialfields, whose Dirac properties emerge due to the structure of the space (lattice) with which thecharge carriers interact. The peculiar sheet structure then determines a natural description of itselectronic properties in terms of massless pseudoparticles, giving the possibility to study quasi-relativistic particle behaviour at sub-light speed regime [3, 7, 8]. A natural suggestion turns outto be that the geometric curvature of the two-dimensional sample, combined with the mentionedspecial relativistic-like behaviour, naturally leads to a general relativistic-like description forour pseudoparticles, which will then be regarded as Dirac fields in a 1+2 dimensional curvedspacetime background [9–14]. This gives us a real framework to study what is believed to be(as close as possible) a quantum field in a curved spacetime, with measurable effects pertainingto the electronic structure of the sample itself [12, 13, 15–17], so that the understanding of 2DDirac materials properties is important in condensed matter as well as in theoretical high energyphysics [1, 2, 17].The massless formulation is in general robust, since it emerges at the level of non-interactingsystem, the vanishing quasiparticle mass (gapless spectrum) protected by the combination ofparity and time-reversal symmetries. In general, interactions are not very efficient in introducinga gap and/or modifying the quasiparticle behavior [3, 18].In the context of high energy physics, the emergence of intrinsic and extrinsic curvaturein graphene-like materials can be used to investigate the fundamental physics of the quantumDirac dynamics in curved spacetimes, as well as to probe certain quantum gravity scenarios [19,20]. This formulation follows a bottom-up approach, where suitable condensed matter systemsprovide analogues of gravitational effects so that the propagation of quantum fields is dictatedby an effective metric, taking then advantage of mathematical tools from Einstein gravity (orextensions of the latter). The underlying idea is that suitable variants of this analogue modelscan be used as frameworks for different analysis and formulations of quantum gravity theories,gaining new insights into the corresponding problems.There are also some theoretical results that conjecture the use of graphene to have alter-native (unconventional) realizations of Supersymmetry [21, 22], the latter being instrumentalin describing the properties of graphene-like materials at the Dirac points, exploiting an holo-graphic top-down approach, the substrate description coming from a well-defined geometricformulation of a suitable gravitational model [23, 24]. Continuum limit and spacetime geometry.
The detailed study of suitable curved config-urations can highlight the peculiar properties of the charge carriers, derived from the discussedmassless Dirac description in a curved background [25–28]: the choice of the geometry, the our framework turns out to be the analog of a relativistic system, with characteristic limiting velocity givenby the Fermi velocity v f rather than the speed of light c (for graphene v f ∼ c ) E (cid:96) ∼ v f /(cid:96) ∼ . (cid:96) ∼ .
142 nm, so that thesecharge carriers see the graphene sheet as a continuum, justifying the quantum description in1+2 spacetime. Moreover, quasiparticles with large wavelength are sensitive to sheet curvatureeffects, claiming for a quantum field formulation in curved spacetime. In particular, this meansthat, in the continuum field approximation, we have to demand the charge carrier wavelengthsto be bigger than the lattice typical dimension, λ > πv f /E (cid:96) ∼ π(cid:96) .With the above prescriptions in mind, the challenge is now to find a suitable curved space-time where it can be easier to probe and study the relativistic-like quasiparticles quantumbehaviour. As we will see in Section 3, the Beltrami pseudosphere [29] is a promising candi-date where Dirac’s equation in curved space can be solved analytically, providing an explicitexpression for the Dirac spectrum and its effects on the electronic local density of states (LDOS).From an historical point of view, the Beltrami surface has been conjectured to provide apromising spacetime framework where to observe an Hawking–Unruh effect [30, 31], one of themost interesting predicted phenomena of a quantum field theory in curved background [32, 33].The possible formation of a Rindler-type horizon in a Beltrami geometry could then lead to acharacteristic thermal behaviour, related to the specific nature of quantum vacua and relativisticprocess of measurement [34–36]. In Section 4 we will provide an analytic expression for the Diracmodes of the charge carriers living in a Beltrami spacetime, so that experimental predictionsrelated to the electronic structure of the corresponding graphene-like sample can be explicitlyworked out.Finally, we point out that, although we have primarily graphene in mind, many of thefollowing considerations can be extended to other two-dimensional Dirac materials, includ-ing silicene, germanene, graphynes, several boron and carbon sheets, transition-metal oxides(TiO /VO ), organic and organometallic crystals (MoS ), artificial lattices (electron gases andultracold atoms) [37–45]. The quantum Dirac formulation introduced above emerges from the graphene lattice structure,where a unit cell is made of two adjacent atoms belonging to the two inequivalent, interpene-trating triangular sublattices. This means that we have two inequivalent sites per unit cell, thedistinction related to their topological inequivalence. The single-electron wave function can bethen conveniently arranged in a two-component Dirac spinor, so that the description of its elec-tronic properties can be given in terms of massless Dirac pseudoparticles [3], the characterizationbeing resistent to changes of the lattice preserving the topological structure.In the reciprocal lattice space, the first Brillouin zone (FBZ) results in a structure with thesame hexagonal form of the honeycomb lattice, rotated by a π/ K , K’ ( Dirac points ).3 ubstrate deformations and energy scales.
If we consider a graphene layer with hexagonallattice, every carbon atom has four electrons available for covalent bonds. Three of them formthe σ -bonds with three different nearest neighbors (merging of atomic 2 s orbitals); these bondsdefine the elastic properties of the sheet. The fourth electron forms a covalent π -bond with oneof the three neighbors (merging of atomic 2 p orbitals): being the latter π -bond much weakerthan the former σ , the involved π -electrons become charge carriers that are much more free tohop, determining then the electronic properties of the sample.If we want to construct the action that captures the physics of the π -electrons in the curvedsheet, we need to study the possible deformations that can be encoded in the Dirac description.In the large wavelength regime [12, 13, 46], we can find three kinds of deformation at work:extrinsic curvature, intrinsic curvature and strain [3]. The first deformation is an elastic effectthat can be expressed, at first order, using derivatives of the strain tensor [3, 47]; the second isan inelastic effect coming from the formation of disclination-type defects [48, 49]; the third isagain an elastic deformation that takes into account effects that are proportional to the straintensor (not to its derivatives), that turn out to work as potentials for a pseudo-magnetic field B µ and a scalar potential Φ [3, 47, 50–53].Since we are primarily interested in investigating the effects of curvature on the substratequantum (electronic) properties, we shall focus on inelastic deformations, since elastic deforma-tions cannot induce intrinsic sheet curvature [3]. This means that we will have to introduce alsothe energy scale E R ∼ v f /R < E (cid:96) , with R > (cid:96) and where 1 /R is a measure of the intrinsiccurvature. In fact, we want the curvature to be small if compared to the limiting value 1 /(cid:96) :this, in turn, means that we can formulate our theory using a smooth metric, ruling also out thedifficult bending of the strong σ -bonds. The previously introduced E (cid:96) energy now correspondsto the high-energy regime for our formulation and, when we are within the E R energy range,the charge carriers are still sensitive to the global effects of curvature.The above considerations suggest we should focus on electrons with longer wavelengths thanthe corresponding ones for the simple continuum approximation, λ > λ R > λ (cid:96) (with λ (cid:96) ∼ π(cid:96) , λ R ∼ πR ), so that our energies range is valid up to E R : in this situation, the elastic propertiesof the sample (involving much larger energies, of the order of tens of eV) are decoupled fromthe π -electrons dynamics, governed by inelastic effects, and in our mathematical formulation wewill neglect the contributions from B µ and Φ [54, 55]. Topological defects.
The formation of topological defects in 2D materials is the naturalway in which the sample layer heals vacancies and other analogous lattice damages. Amongthose, disclinations, dislocations and Stone–Wales defects (special dislocation dipoles) werefound to have the least formation energy and activation barrier, so that they result energet-ically favourable phenomenons [56, 57]. If we consider graphene-like materials, disclinations anddislocations are the most important topological sample defects , related, in the continuum limit,to curvature and torsion, respectively [48, 58].A disclination is a crystallographic defect associated with the violation of the (discrete)rotational symmetry. Positive (negative) disclinations are topological defects obtained by re-moving (adding) a semi-infinite wedge of material to an otherwise perfect lattice. If we considera bidimensional hexagonal lattice, a disclination consists in the substitution of an hexagon withother polygons: it therefore manifests itself by the presence of an n -sided polygon, with n (cid:54) = 6. we do not consider here impurities, Coulomb and resonant scattering or other issues mixing the Fermi points,with a corresponding chiral term in the action: we can assume that charge carriers mobility is not affected by thementioned effects at these energy scales [46]. The local lattice aspects can be then disregarded and the inelasticeffects will dominate, so that only intrinsic curvature must be taken into account (contributions from B µ and Φcan be neglected) [54, 55]
4f 3 ≤ n <
6, the associated singularity carries a positive intrinsic curvature, while, in the n >
Graphene-like flat substrates can be considered as 2D analogs of pseudorelativistic systems withcharacteristic velocity v f . The dynamics of the charge carriers ψ in the 1+2 dimensional flatspacetime can be then described, in the long wavelength continuum limit, by a massless Diracaction of the form S = i (cid:126) v f (cid:90) d x ¯ ψ ˇ γ a ∂ a ψ , (1)where a = 1 , , γ -matrices can be written interms of the Pauli matrices as ˇ γ a = (cid:0) i σ , σ , σ (cid:1) . (2)One can easily verify that the ˇ γ a = η ab ˇ γ b matrices satisfy the standard Clifford algebra (cid:8) ˇ γ a , ˇ γ b (cid:9) = 2 η ab , where η ab is the inverse of the flat 1+2 dimensional Minkowski metricin the mostly plus convention, η ab = diag( − , ,
1) .
Curved space.
Since we now want to include non-trivial intrinsic curvature effects, we arenaturally led to the customary generalization in a curved spacetime of the action for masslessDirac spinors in 1+2 dimensions [9, 11] S = i (cid:126) v f (cid:90) d x √ g ¯ ψ γ µ D µ ψ , (3)where µ = 0 , , g µν , andthe factor √ g ≡ (cid:112) − det( g µν ) comes from the request of a diffeomorphic-covariant form of theaction in the presence of curvature. The curved γ µ matrices are obtained from the constant ˇ γ a matrices of the flat frame by the action of the vielbein e µa (see App. A): γ µ = e µa ˇ γ a , (4)while the inverse vielbein e aµ performs the transformation in the other direction. The gammamatrices with upper indices γ µ = g µν γ ν satisfy the correspondent Clifford algebra relation incurved background, { γ µ , γ ν } = 2 g µν .The diffeomorphic covariant derivative is written as D µ = ∂ µ + Ω µ = ∂ µ + 14 ω µab M ab , (5)where M ab = [ˇ γ a , ˇ γ b ] are the Lorentz generators and ω µab defines the spin connection , thatcan be seen as the gauge field of the local Lorentz group. Since we are working in a torsionless5ramework, ω µab and e µa are not independent [61, 62] and the former can be expressed in termsof the latter as ω µab = e νa ∂ µ e νb + e νa Γ µλν e λb , (6)where Γ µνλ is the affine connectionΓ µνλ = 12 g σλ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) . (7)If we consider the above (5), Ω µ acts as a gauge field able to take into account all deformationsof the geometric kind.The equations of motion for the pseudorelativistic Dirac spinors coming from action (3)read [9, 10, 12–14] i γ µ D µ ψ = 0 , (8)that is, the generalized form of the massless Dirac equation in flat Minkowski spacetime obtainedthrough the substitutions ˇ γ a ! γ µ , ∂ a ! D µ . (9)Summarizing, we have constructed the large wavelength continuum description of the Diracquasiparticles living on a graphene-like sheet, modelling curvature effects through the couplingof the Dirac fields to a curved spatial metric, thus obtaining the physical description of thecharge carriers dynamics [12–14, 16, 17, 55]. If we consider a topologically trivial, purely strainedconfiguration (elastic membrane deformations), there are no relevant physical effects comingfrom the spin-connection. If instead the sample features a non-trivial intrinsic curvature, thespin connection dictates most of the physics for the Dirac fields. In particular, in our largewavelength continuum limit for the charge carriers, the spin connection can be associated withdisclination-type defects inducing curvature, encoding the physics imposed by the geometricsheet deformation.In the following sections we will consider an explicit parametrization of the curved mem-brane, working out an analytic solution of the Dirac equation in the corresponding curvedbackground. Among the class of negative-curvature surfaces, an important role is played by the subset ofsurfaces having constant negative Gaussian curvature. When embedded into R , those surfacesfeature essential singularities [63]: as a consequence of this, we find that it is impossible torepresent the whole Lobachevskian geometry on a real bidimensional surface, so that we areforced to restrict to mapping only a suitable stripe of the hyperbolic space [29].Another relevant point will concern explicit parametrization, so that hyperbolic (abstract)geometry can be expressed, together with the above singular boundaries, in terms of well definedcoordinates. In this regard, we notice that the line element of any surface of constant negativeGaussian curvature can be reduced to the one of the Beltrami or the hyperbolic or the ellipticpseudospheres [64]. The advantages of the Beltrami surface are that an embedding parametriza-tion can be given in terms of smooth, well-behaving single-valued functions and that it has onlyone well-defined singular boundary, corresponding to the maximal circle. from now on we work in (pseudo)natural units: (cid:126) = v f = 1 The Beltrami pseudosphere is a bidimensional surface that we here choose to parameterize as x = L exp (cid:0) uR (cid:1) cos ϕ ; y = L exp (cid:0) uR (cid:1) sin ϕ ; z = R (cid:0) arctanh f ( u ) − f ( u ) (cid:1) , f ( u ) = (cid:114) − (cid:16) LR exp (cid:0) uR (cid:1) (cid:17) . (10)As we can see by direct inspection, the parameterized Beltrami trumpet exists, in general, for u ∈ (cid:2) −∞ , R log (cid:0) RL (cid:1)(cid:3) . The surface can be suitably embedded in R and it is well-defined overthe whole non-singular part of the surface, the singular boundary being the maximal circle ofradius R corresponding to the limit value u (cid:63) = R log (cid:0) RL (cid:1) . Figure 1 : Beltrami trumpet parametric plot, with u ∈ [0 , u (cid:63) ] and v ∈ [0 , π ]. We can also notice that the above equations for the embedding are expressed in terms ofthe analogs of polar (or maybe better cylindrical) coordinates u, ϕ , so that we can easily movealong the “meridian” and the “parallel” of the trumpet. Every coordinate is in fact expressedas a smooth, well-behaving, single-valued function (taking a full turn on a parallel has no effectson the values of x, y ). Graphene-like substrate.
Even if the u –coordinate is unbounded from below ( u ! −∞ corresponding to z ! ∞ ), when we consider a graphene-like membrane we have to take into7ccount the considerations we made in Section 1 about the range of validity of the proposedcharge carriers dynamics (not holding for too small radii), the continuum limit approximationand the difficulties in bending the surface beyond certain limits. All these aspects lead to thephysical assumption of posing a limiting value for the surface (parallel) radius r = L e uR . Then,we should decide to make the natural assumption of minimum radius r ≥ L , being L much largerthan the lattice length (cid:96) ( r ≥ L (cid:29) (cid:96) ). This, in turn, means that the u –coordinate is restrictedto the interval u ∈ [0 , u (cid:63) ], with u (cid:63) = R log (cid:0) RL (cid:1) .The metric on the pseudosphere (10) has the form g µν = − L e uR , (11)so that the 1+2 spacetime line element reads ds = − dt + du + L e uR dϕ , (12)the resulting Ricci scalar R = − R being twice the Gaussian curvature K .The vielbein can be easily found to be e µa = L e uR , (13)and it correctly satisfies the relation g µν = e µa e νb η ab . The curved gamma matrices explicitlyread: γ µ = e µa ˇ γ a = (cid:0) i σ , σ , L e uR σ (cid:1) , (14)and the reader can verify that upper-indexed γ µ satisfy the Clifford algebra { γ µ , γ ν } = 2 g µν .Finally, the spin connection ω µab has non vanishing components ω = − ω = LR e uR , (15)so that now we are able to explicitly express the Dirac equation (8) in terms of the covariantderivative (5). Our challenge now is to obtain explicit ψ –solutions for the curved space Dirac fields satisfying(8), the corresponding relativistic pseudoparticles moving on a Beltrami surface.The analytic solution of Dirac equation (8) for charge carriers living on the Beltrami space-time described in previous Subsect. 3.1 has the explicit form ψ = e − i λ E t (cid:32) φ a φ b (cid:33) , (16)8ith φ a = C e i (cid:82) du ξ ( u ) e i k ϕ uR e − uR ,φ b = C e i (cid:82) du ξ ( u ) e i k ϕ i uR e − uR B ( u ) , (17)where λ = ± E , and C is a normalization constantthat can be determined from the condition (cid:90) d Σ √ g | ψ | = 1 , (18)being √ g = (cid:112) − det( g µν ) . The functions ξ ( u ) and B ( u ) are defined as ξ ( u ) = (cid:18) u − kL e − uR − R (cid:19) − λ E B ( u ) , B ( u ) = I ( − ) + I (+) I ( − ) − I (+) , (19)expressed in terms of the modified Bessel function of the first kind I ( ± ) = I (cid:18) ±
12 + i λ E R , kRL e − uR (cid:19) . (20) We are now going to consider a simple experimental application for the above solution forpseudorelativistic charge carriers living in a 1+2 dimensional curved background.Since we have obtained the explicit Dirac solution ψ , we can consider the probability density P = √ g | ψ | , in terms of which the normalization condition (18) in Beltrami geometry can beexplicitly written as (cid:90) d Σ P ( u, ϕ ) = 2 π (cid:90) du L e uR | ψ ( u, ϕ ) | = 1 . (21)Using now our new solutions (16), (17) for the Beltrami surface, it is possible to obtain the prop-erly normalized probability density, as shown in Figures 2 to 4 . We should use the appropriateprecautions discussed in Sections 1 and 3.1, that is considering the proper interval of variationfor the u –coordinate in order to satisfy the correct continuum limit for the membrane. the modified Bessel function of first kind I ( n, Z ) = Y is the function that satisfies the differential equation Z Y (cid:48)(cid:48) + Z Y (cid:48) − ( Z + n ) Y = 0 ; for certain arguments it has an explicit analytic expression, while it can alwaysbe evaluated to arbitrary numerical precision P ( u ) u Figure 2 : Normalized probability density as a function of u for a Beltrami-shaped sample surface withparameters E = 4 · − , λ = ± R = 5, L = 0 . C = 1 .
415 . P ( u ) u Figure 3 : Normalized probability density as a function of u for a Beltrami-shaped sample surface withparameters E = 0 . λ = ± R = 1, L = 0 . C = 3 .
174 .
20 40 60 800.0050.0100.0150.020 P ( u ) u Figure 4 : Normalized probability density as a function of u for a Beltrami-shaped sample surface withparameters E = 5 · − , λ = ± R = 10, L = 0 . C = 0 .
455 .
As we can appreciate, for the chosen parameters, the peak is localized in correspondence ofsmall values for u , the charge carriers being then mainly localized on the throat of the trumpet.For a given a location X on the layer surface and energy value E , the local density of states(LDOS) of the sample can be defined as [65] ρ s ( E, X ,
0) = 1 ε E (cid:88) E − ε P ( X ) , (22)for sufficiently small values of ε , where the 0–coordinate means that we are considering pseu-doparticles on the sample surface (i.e. zero distance from the substrate surface). The physicalmeaning of the LDOS for our bidimensional sample is the number of charge carriers per unitsurface and unit energy range of size ε , at a given surface location X and energy E . The sampleLDOS is not only an interesting direct observable for the predicted quantum behaviour, but alsoa substrate feature of immense importance for electronic applications, being the availability ofempty valence and conduction states (states below and above the Fermi level) crucial for thetransition rates. Measurements.
The sample LDOS can be detected using a scanning tunneling microscope(STM). The latter is an experimental device based on quantum mechanical tunneling, in whichthe wave-like properties of charge carries allow them to penetrate through a potential barrier,into regions that are forbidden to them in the classical picture. STM spectroscopy provides in-sight into the surface electronic properties of the substrate, being the tunneling current stronglyaffected by the local density of states ρ s . The latter is in turn related to the probability density P through definition (22).A typical STM device consists of a very sharp conductive tip which is brought within tun-neling distance ( < nm) from a sample surface, using a three-dimensional piezoelectric scanner.Let us imagine that electrons fill energy levels up to the Fermi level – which defines an upperboundary similar to the sea level – above which we find activated charge carriers. The Fermilevel of a material can be raised/lowered with respect to a second material by applying an ap-11ropriate voltage. Thus, to obtain a tunneling current through the gap between the sample andthe tip, a suitable bias voltage V can be applied, causing charge carriers to tunnel across thegap . In particular, when a negative bias is applied to the sample, its Fermi level is raised andthe charge carriers of the filled occupied states can tunnel into the unoccupied state of the tip,while the opposite occurs for a positive bias (filling of the sample empty states).We are interested in the STM–map setup, where the density of states at some fixed energy,is mapped as a function of the position ( u, ϕ ) on the sample surface. Let us assume ε = e V to be very small with respect to the work function Φ w (minimum energy required to extract anelectron from the surface), so that the sample states with energy lying between E f − ε and E f are very close to the Fermi level and have non-zero probability of tunneling into the tip. Theresulting tunneling current I is directly proportional to the number of states on the substratewithin our energy range of width ε , this number depending on the local properties of the surface.Including all the sample states in the chosen energy range, the measured tunneling current canbe modelled, in first approximation, as [65] I ∝ E f (cid:88) E f − e V P e − κ d , (23)where κ is some decay constant in the barrier separation depending on Φ w . The exponentialfunction gives the suppression for charge carriers tunneling in the classically forbidden region ofwidth d (sample-tip separation). The tunneling current can be then measured, for constant sep-aration d , at different X positions and, for sufficiently small V , it can be conveniently expressedin terms of the LDOS of the sample as [65]: I ( X ) ∝ ρ s ( E f , X , e − κ d e V . (24)In summary, tunneling current measured by STM mapping, at small bias voltage V and fixed tip-sample separation d , is proportional to the local density of states of the sample. In particular,we could scan our Beltrami surface at different positions varying the u –coordinate, mappinginhomogeneities in the local density of states ρ s ( E f , u, P ( u ).STM can operate in ambient atmosphere as well as in high vacuum; when a high-vacuumconfiguration is employed, its purpose is not to improve the performance of the STM but ratherto ensure the cleanliness of the sample surface. We also remark that the LDOS obtained with aSTM is not limited by the position of the Fermi energy, since both occupied and empty states areaccessible [66]. For finite bias voltage and different Fermi levels for the sample and tip material,the functional form of the tunneling current and its relation with the sample local density ofstates can be easily obtained from Bardeen time-dependent perturbation approach [65]. A deeper intertwining of different scientific areas can really provide an important step forwardin our understanding of various, fundamental physical aspects of our world. In particular, it hasbeen shown that the considerable gap between high energy physics and condensed matter – dueto mutually independent mathematical formulation and developments – can be reduced usinga multidisciplinary approach (see e.g. [19, 20, 23, 24, 67–79]). Following this spirit, we have usedalong the paper different techniques from high energy physics, differential geometry and general to simplify our discussion, we are assuming that both materials have the same Fermi level Acknowledgments
I would like to thank professor G. A. Ummarino for extremely helpful discussions during thepreparation of this report. I would also like to thank prof. M. Trigiante, prof. F. Laviano andprof. A. Carbone that supported these studies with their funds.13
Conventions
Dirac equation.
The Dirac equation in Minkowski spacetime is the result of the constructionof a relativistic field equation, whose squared wave function modulus could be consistentlyinterpreted as a probability density. To satisfy these conditions, the equation is of first orderin time-derivative, while relativistic invariance requires the equation to be first order in space-derivatives too. The final explicit form must fulfill the requests of Lorentz covariance and satisfythe Klein-Gordon equation, and reads( i ˇ γ a ∂ a − m ) ψ ( x ) = 0 , (A.1)together with the condition (cid:8) ˇ γ a , ˇ γ b (cid:9) = 2 η ab , (A.2)that is usually referred to as Clifford algebra, the matrix η ab being the inverse of the Minkowskiflat metric η ab .For the sake of notational simplicity, in eq. (A.1) we have omitted the spinorial indices of ψ ≡ ψ β and ˇ γ a = (ˇ γ a ) αβ . Spinors are objects that transform as scalars under general space-time coordinate transformations, while they trasform in a spinor representation R under thelocal Lorentz group: ψ (cid:48) α ( x ) = R (cid:2) Λ( x ) (cid:3) αβ ψ β ( x ) . (A.3)Using the explicit form of the Lorentz generators to construct the Pauli-Lubanski operator, itcan be easily shown that the particle has spin s = . A.1 Curved spaces
Einstein’s theory of gravitation relies on the symmetry principle of invariance under generalcoordinate transformations, that, in turn, can be viewed as local spacetime transformationsgenerated by the local translation generators. The gravitational force can be then geometricallymodelled in terms of the spacetime curvature.In order to conveniently describe general relativity scenarios together with spinorial fields,one should introduce some tools to describe transformation rules generalized to curved back-grounds, leading to the so-called vielbein formalism.
Vielbein formalism.
Let us consider a set of coordinates that is locally inertial, so that onecan apply the usual Lorentz spinor behaviour, and imagine to find a way to translate back to theoriginal coordinate frame. More precisely, let y a ( x ) denote a coordinate frame that is inertialat the space-time point x : we shall call these the “Lorentz” coordinates. Then, e µa ( x ) = ∂y a ( x ) ∂x µ (A.4)gives the so-called vielbein : it defines a local set of tangent frames of the spacetime manifoldand, under general coordinate transformations, it transforms covariantly as e (cid:48) µa ( x (cid:48) ) = ∂x ν ∂x (cid:48) µ e νa ( x ) , (A.5)while a Lorentz transformation leads to e (cid:48) µa ( x ) = Λ ab e µb ( x ) . (A.6)14he space-time metric, in particular, can be expressed as g µν ( x ) = e µa ( x ) e νb ( x ) η ab , (A.7)in terms of the Minkowski flat metric η ab . The original constant ˇ γ a matrices of the inertialframe can be converted into the new γ µ matrices of the curved background by the action of thevielbein: γ µ ( x ) = e µa ( x ) ˇ γ a , (A.8)while the inverse vielbein e aµ performs the transformation in the other direction. The vielbeinthus takes Lorentz (flat) latin indices to coordinate basis (curved) greek indices. The gammamatrices with upper indices γ µ = g µν γ ν , (A.9)satisfy the relation: { γ µ , γ ν } = 2 g µν , (A.10)that holds in curved backgrounds and is the equivalent form of the previous, flat Clifford algebra(A.2). Covariant derivative, spin connection.
The choice of the locally inertial frame y a is definedup to Lorentz transformations given by the Lorentz generators M ab . In order to couple fields,we define the covariant derivatives: D µ = ∂ µ + 14 ω µab M ab , (A.11)where M ab = 12 [ˇ γ a , ˇ γ b ] . (A.12)The ω µab object defines the spin connection , that can be seen as the gauge field of the localLorentz group, the corresponding field strength given by the Riemann curvature tensor, and isdetermined through the vielbein postulate (tetrad covariantly constant) [48, 49]: D µ e νa − Γ µνλ e λa = 0 . (A.13)The latter is written in terms of the affine connection Γ µνλ Γ µνλ = 12 g σλ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) . (A.14)while the explicitly expression for the spin connection is found to be ω µab = e νa ∂ µ e νb + e νa Γ µλν e λb . (A.15)Finally, the Dirac equation in curved spacetime can be written as:( i γ µ D µ − m ) ψ = 0 . (A.16)15 eferences [1] M.I. Katsnelson and K.S. Novoselov, “Graphene: New bridge between condensed matterphysics and quantum electrodynamics” , Solid State Communications (2007), n. 1, 3–13. [ p. ] [2] Andre K. Geim and Konstantin S. Novoselov, “The rise of graphene” , Nature materials (2007), n. 3, 183. [ p. ] [3] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim, “The electronicproperties of graphene” , Rev. Mod. Phys. (2009) 109–162, [ arXiv:0709.1163 ]. [ p.
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