Vol. 124 (2013) ACTA PHYSICA POLONICA A No. 4

Quantum Flatland and Monolayer Graphene from a Viewpoint of A. Dargys∗ Center for Physical Sciences and Technology, Semiconductor Physics Institute A. Go²tauto 11, LT-01108 Vilnius, Lithuania (Received May 11, 2013) Quantum mechanical properties of the graphene are, as a rule, treated within the Hilbert space formalism. However a dierent approach is possible using the geometric algebra, where quantum mechanics is done in a real space rather than in the abstract Hilbert space. In this article the geometric algebra is applied to a simple quantum system, a single valley of monolayer graphene, to show the advantages and drawbacks of geometric algebra over the

Hilbert space approach. In particular, 3D and 2D algebras Cl3,0 and Cl2,0 are applied to analyze relativistic properties of the graphene. It is shown that only three-dimensional Cl3,0 rather than two-dimensional Cl2,0 algebra is compatible with a relativistic atland. DOI: 10.12693/APhysPolA.124.732 PACS: 73.43.Cd, 81.05.U−, 03.65.Pm

1. Introduction mension has been reduced, one is automatically forced to change the type of Cliord algebra and its irreducible Graphene is a two-dimensional crystal made up of car- representation. bon atoms. The intense interest in graphene is stimulated On the other hand, if the problem was formulated in by its potential application in the construction of novel the Hilbert space terms then, generally, the neglect of nanodevices based on relativistic physics, or at least with one of space dimensions does not spoil the Hilbert space. a large ingredient of relativity [14]. From theoretical as- Of course, one has to reconsider the problem with re- pects the most interesting is the Dirac-like behavior of spect to its symmetry properties from the point of view electrons in 2D lattice, including a linear dispersion law, of group theory, since as a rule a new problem becomes the Klein tunnelling through potential barriers, conne- reducible to a simpler one. However, the situations may ment and integer Hall eect. The standard mathematical arise when the mechanical reduction of dimension of the language for this purpose is based on the Hilbert space Hilbert space makes the problem ill-posed. For example, with such mathematical objects as complex state vectors, this happens with the massless Dirac equation, the operators etc. A pedagogical overview of the sin- of which consists of 4×4 complex Dirac matrices, γ0 and gle valley graphene within the Hilbert space formalism γi with i = 1, 2, 3. If one assumes that the electron mass has recently appeared in [5], which can serve as an intro- m = 0 and γ3 = 0, then one gets the relativistic massless duction to the subject. In this paper a dierent mathe- Dirac equation for graphene. However, it appears that in matical language known as geometric or the Cliord al- this case one cannot construct the generators of the new gebra [610], which is formulated in real spaces, will be algebra [16]. used in the analysis of quantum properties of graphene. On the other hand, if the Cliord algebra is applied to Examples of applications of geometric algebra (GA) to the problem it will automatically insure correct symme- nanostructures can be found, for example, in [1115]. try of lower dimensional spaces and correct form of gen- As known, the Hilbert space as a such does not carry erators, because of strict group subordination of lower any attributes of a physical space. The space properties dimensional subalgebras. Accordingly, the respective are brought into Lagrangian or Hamiltonian to satisfy lower-dimensional massless Dirac equations, if they ex- symmetry requirements of the space or governing laws for ist, will have a correct form. physical objects. In contrast, the selection of a particu- In this article we shall rely on the GA hierarchy lar GA to solve the physical problem automatically takes Cl1,3 → Cl3,0 → Cl2,0 in transforming the equations into account the dimension, signature and symmetry rela- from the complex matrix representation in the Hilbert tions of the space or spacetime. This is best visible from space to GA . The simplest possible case the 8-periodicity table of GA [8], which predetermines of a single Dirac valley that is described by 2 × 2 com- types of irreducible matrices, or the Cliord groups that plex matrix is considered. In Sect. 2, the properties of should be used in solving the problem. So, if space di- graphene in terms of 3D space algebra Cl3,0 are consid- ered. In Sect. 3 the same problem is analyzed in 2D with

Cl2,0 algebra.

∗e-mail: [email protected]

(732) Quantum Flatland and Monolayer Graphene . . . 733

2. Graphene with Cl3,0 to GA space [9, 17]: 2.1. Graphene Hamiltonian in Cl3,0 |ψi ←→ ψ = a0 + a1Iσ1 + a2Iσ2 + a3Iσ3, Cl3,0 algebra describes Euclidean 3D space and simul- λ|ψi ←→ λψ, taneously is an even subalgebra of relativistic spacetime σˆ |ψi ←→ σ ψσ , (3) algebra , i.e., the isomorphism + is i i 3 Cl1,3 Cl1,3 ≈ Cl3,0 where is the and is one of the Pauli matrices. satised. The basis vectors of have positive sig- λ σˆi Cl3,0 Then, ˆ , where ˆ is given by (1), maps to the fol- nature: 2 2 2 . The vectors play Hg|ψi Hg σ1 = σ2 = σ3 = 1 σi lowing GA function for spinor : the role of space coordinates. The notation reects ψ H (ψ) = kψσ , (4) the fact that σi's are isomorphic to the Pauli matrices. g 3 Therefore, the basis vectors with dierent indices anti- where the wave vector k = kxσ1 + kyσ2 lies in the graphene , Fig. 1. Let us note that in the Hamilto- commute, σiσj + σjσi = 0. The orthogonal planes nian function (4) the vector represents the quantiza- in 3D space are described by Iσ1 = σ2σ3, σ3 tion axis, which is perpendicular to graphene plane. Iσ2 = σ3σ1, and Iσ3 = σ1σ2. With scalar and pseu- doscalar I = σ1σ2σ3 included, there are 8 basis ele- ments in Cl3,0. Such algebra is irreducible and closed under geometric multiplication (denoted by spaces be- tween the symbols). A general element of the algebra, usually called a , is a sum of all eight elements

M = a0+a1σ1+a2σ2+a3σ3+b1Iσ1+b2Iσ2+b3Iσ3+b4I, where the coecients ai's and bi's are real numbers. The product of any two multivectors gives new multivector that belongs to Cl3,0, i.e. Cl3,0 is closed under geomet- ric product. If one deletes a basis vector, for example equating σ3 = 0, then the remaining set of elements {1, σ1, σ2,Iσ1,Iσ2} will not make up the closed subal- Fig. 1. Cl3,0 basis vectors σk and wave vector k with gebra. Thus, the GA requires that either all its basis respect to graphene plane. elements, or a closed subalgebra, should be used to get a meaningful result. The Hamiltonian (4) is a linear function of 2D wave In the Hilbert space formalism (in this paper indicated vector k. The Rashba Hamiltonian and Dresselhaus by a hat over the symbol), the massless Dirac Hamilto- Hamiltonian for a quantum well that include relativistic nian for low energy electrons in the vicinity of critical K corrections are also proportional to 2D wave vector k. In point of graphene reduces to [5]: GA they have the following forms [17]: " # HR(ψ) = αRσ1σ2kψσ3, (5) ˆ 0 kx − iky (1) Hg = ~vF , (6) kx + iky 0 HD(ψ) = αDσ1kσ1ψσ3, 6 where αR and αD are, respective, coupling constants. Let where vF ≈ 10 m/s is the Fermi velocity. The wave us note that the basis vector σ3 in all Hamiltonians is per- vector k = (kx, ky) is measured with respect to valley pendicular to the plane k −k and represents the quan- minimum. In the following the natural units ~ = c = 1 x y tization axis, Fig. 1. Only the factors before ψ, namely along with vF = 1 are used, so that after multiplication of by one has the energy. The eigenenergies of the k, σ1σ2k, and σ1kσ2, determine the relativistic char- k ~vF Hamiltonian (1) are acter of 2D Hamiltonians. Keeping this in mind, simple GA calculations show that there are no new 2D relativis- q 2 2 (2) E = ± kx + ky = ±|k|, tic Hamiltonians. For example, the factor Iσikσi before where plus/minus sign corresponds to conduction/va- ψ leads to non-Hermitian Hamiltonian which must be discarded. The factors σ2kσ2, σ3kσ3, and σ2kσ1 give lence band. Since only two Pauli matrices, σˆx and σˆy ap- pear in the Hamiltonian (1), one gets a false impression already written Hamiltonian functions but with opposite that the problem is two-dimensional and therefore the signs or interchanged axes. So, it seems that proportional problem is limited to graphene plane only. The multipli- to k Hamiltonians (4), (5) and (6) are the only ones 2D Hamiltonians that include relativistic eects. cation table of Cl shows that the third space direction 3,0 Using GA projectors the Hamiltonian (4) can be σ3 which is perpendicular to the graphene plane as well as related GA elements cannot be deleted automatically. rewritten in physically more transparent way. In Cl3,0 As mentioned, similar situation appears in a relativistic algebra the elementary projectors that represent the mul- case, where 4D massless Dirac equation in momentum tivector ψ have the form [18]: representation µ , will remain in 1 γ pµ|Ψi = 0 µ = 0, 1, 2, 3 p (ψ) = (ψ ± σ ψσ ), 4D spacetime even if one limits the equation to two space ± 2 3 3 coordinates (µ = 1, 2) [16]. ψ = p+(ψ) + p−(ψ). (7) Now we shall transform the Hamiltonian (1) to Cl3,0 al- For example, if the projector (7) is applied to basis vec- gebra using the following mapping rules from the Hilbert tors ψ = σ1 or ψ = σ3 then one nds p+(±σ1) = 0 734 A. Dargys

and p+(±σ3) = ±σ3, i.e. p+ selects elements that be- The eigenmultivectors can be deduced from the same long to σ3 axis. The projector p− behaves in an op- Eq. (10) after insertion back of the eigenenergies E±. posite manner, p−(±σ1) = ±σ1 and p−(±σ3) = 0. Then, Eq. (10) reduces to The action of the projector on the that Iσ3 kˆψ σ = ±ψ . (15) lies in the graphene plane gives , but ± 3 ± p+(±Iσ3) = ±Iσ3 As known, in GA the eigenspinor can also be inter- . The projectors (7), as should be, are ψ± p−(±Iσ3) = 0 preted as a rotor [9] which is very helpful if rotations idempotent:  . p± p±(ψ) = p±(ψ) of vectors, planes etc. in Euclidean or Minkovsky type In the terms of projectors the graphene Hamilto- spaces are needed. Since the rotor satises ˜ ˜ , nian (4) can be rewritten as a sum of two projectors ψψ = ψψ = 1 Eq. (15) can be rewritten in a form of rotation of basis P+(ψ) and P−(ψ) in the following way: vector σ3, 1 1  H (ψ) = k ψ + kˆψσ  − ψ − kˆψσ  ψ σ ψ˜ = ±kˆ, (16) g 2 3 2 3 ± 3 ± where the tilde denotes the reversion involution. Thus, (8) = k[h+P+(ψ) + h−P−(ψ)], the left hand side represents the equation of rotation of q where 2 2, ˆ is the unit the perpendicular to graphene plane to k = |k| = kx + ky k = k/k σ3 other unit vector ±kˆ that lies in the graphene plane, wave vector, and h± = ±1, which as we shall see rep- resents the helicity quantum number. The projectors Fig. 1. From all this follows that ψ± in (16) can be rep- ˆ 1 ˆ select the parts of the multivec- resented as a rotor in the bivector k ∧ σ3 plane, P±(ψ) = 2 (ψ ± kψσ3) tor that are parallel (plus sign) or antiparallel (minus ˆ 1   ψ ±k∧σ3π/4 ˆ ψ± = e = √ 1 ± k ∧ σ3 , (17) sign) to k. If ϕ measured from the basis vector σ1 2 to k is introduced, Fig. 1, then in polar coordinates the ˆ ˆ where the wedge (or outer) product k ∧ σ3 = (kσ3 − projectors can be written ˆ σ3k)/2 represents the bivector. In expansion of the ex- 1 ponential function we have taken into account that ˆ P±(ψ) = (ψ ± cos ϕσ1ψσ3 ± sin ϕσ2ψσ3) . (9) k∧σ3 2 behaves as an imaginary unit, i.e. (kˆ ∧ σ )2 = −1. Since The projectors P (ψ) and P (ψ) are idempotent, i.e. 3 + − ˆ and are mutually perpendicular, the outer prod- they satisfy P P (ψ) = P (ψ). Thus, we nd that k σ3 ± ± ± ˆ ˆ the simplest graphene Hamiltonian is a sum of two pro- uct may be replaced by geometric, k ∧ σ3 = kσ3, or ˆ ˆ jectors of a multivector spinor onto wave vector k: the equivalently kσ3 = −σ3k. It should be noted that the rst is related with positive and the second is related eigenspinors (17) are written in a coordinate free way. ˜ ˜ with negative helicity. Below we shall see that positive/ The states (23) are normalized, ψ±ψ± = ψ±ψ± = 1, and negative helicity is related with the conduction/valence orthogonal in a sense that the scalar part of the product ˜ ˜ band states. is equal to zero, hψ+ψ−i = hψ+ψ−i = 0. If the obtained 2.2. Eigenenergies and eigenmultivectors spinors are inserted back into Hamiltonian function (8), the conduction and valence band eigenenergies are found From the projective representation (8) follows that ψ once more can be written as a sum of two parts having opposite he- ψ˜ H (ψ ) = h k = ±k. (18) licities: ψ = ψ+ +ψ−. Then, the respective eigenenergies ± g ± ± Finally, we shall note that the Hilbert space eigenstates E+ and E− can be found from two independent multi- of the Hamiltonian (1) (the concrete expressions can be vector eigenequations, H(ψ±) = E±ψ±, or in a full form found for example in [5]) have no physical meaning and kh ± ψ ± kˆψ σ  = E ψ . (10) if transformed to the Cliord algebra using the map- 2 ± ± 3 ± ± The equation for positive helicity can be solved in the ping (3) are found to be dierent from respective GA eigenstates (17). In contrast, the states (17) in GA rep- ˆ2 following way. Remembering that k = 1, after multi- resent the rotors in 3D physical space [9] and at the same plication from left and right of (10), respectively, by kˆ time are related with graphene band eigenenergies. and σ3 one has ˆ 2.3. Helicity and velocity kh+ψ+ = (2E+ − kh+)kψ+σ3. (11) ˆ Expressing kψ±σ3 from Eq. (10) and inserting back In graphene the electron can belong either to A or B into (11) one gets the scalar equation for eigenenergy sublattice. This property is characterized by pseudo-spin 2 2 degree of freedom described by operator , (kh+) = (2E+ − kh+) . (12) σˆ = {σˆx, σˆy} After squaring of the right hand side and cancellation where σˆx and σˆy are the Pauli matrices. The helicity of similar terms this equation yields conduction band operator is dened as a projection of the pseudospin op- eigenenergy erator on the wave vector direction [5]: ˆ ˆ (19) E+ = h+k. (13) C = k · σˆ. Similarly, for the negative helicity one nds valence band In geometric algebra the helicity operator (19) trans- eigenenergy formed by rules (3) becomes ˆ E− = h−k. (14) C(ψ) = kψσ3, (20) Quantum Flatland and Monolayer Graphene . . . 735 which can be expressed via Hamiltonian (4) bols ∇ and ψ˜. In GA terminology the brackets are called C(ψ) = H(ψ)/k, (21) grade projectors [9]. The bracket with zero (one) sub- from which follows that the eigenequation for helicity op- script indicates that only the scalar (vector) part of the erator is resulting multivector should be taken. For states (23), the formula (28) gives the average electron velocity C(ψ ) = h ψ (22) ± ± ± ˆ (29) and the respective eigenvalues are h = ±1. hvi1 = hvFk. ± This formula shows that the velocity is independent of In conclusion, the same result can be obtained in both wave vector magnitude and its direction in the conduc- the Hilbert space and GA formulations. However, in GA tion ( ) and valence ( ) bands has opposite the physics is more transparent since the full system of h = +1 h = −1 sense at xed k = kkˆ. basis vectors σ = {σ1, σ2, σ3}, which is isomorphic to the Pauli matrices, represents real 3D space. It should be It is interesting to calculate the average Cartesian com- ponents of the velocity, the projection operators of which noted that Cl algebra also takes into account the spin 3,0 are dened by equations quantization axis σ3, which can be seen in the Hamil- tonian (4) and helicity function (20), although the true vˆx|ψi = vFσˆx|ψi, spin was not included in the problem. The isospin reects vˆy|ψi = vFσˆy|ψi, (30) the property that there are two eigenvalues (h = ±1) or after mapping to GA, by multivector equations with respect to ˆ axis that is perpendicular to true spin k v (ψ) = v σ ψσ , quantization axis σ . Thus, the electron states in a x F 1 3 3 (31) graphene are characterized by two quantum numbers, k vy(ψ) = vFσ2ψσ3. and h = ±1, and can be written as Simple calculations give the following average projec- tions: 1  k  ψh(k) = √ 1 + h ∧ σ3 . (23) ˜ 2 |k| hvxi = ψvx(ψ) = vFkx/k, ˜ So, we see that it is preferable to use the term helicity hvyi = ψvy(ψ) = vFky/k. (32) which better reects the physical meaning rather than Since commutators of orthogonal components are equal the misnomer terms pseudospin, or isospin. to zero,  Since the helicity is a good quantum number one can vx vy(ψ) = 0, make a superposition of two states having opposite he-  vy vx(ψ) = 0, (33) licities we nd that for a monolayer graphene we have the same ˜ ψβ = cos βψ+ + sin βψ−, ψβψβ = 1, (24) formula as for a classical particle, where β is a mixing angle. Then the energy of this state 2 2 2 hvi = hvxi + hvyi . (34) is found to be However, if one includes the spinorbit interaction the ex- ˜ (25)  ψβH(ψβ) = k cos 2β. pression (33) and as a result the commutator vx vy(ψ) − This formula shows that the electron energy can be varied  vy vx(ψ) do not become equal to zero [11]. Then, the in two ways, either by changing the wave vector magni- square of the average velocity (34) will not be equal to tude, or by making an appropriate superposition with the sum of squares of average velocity components. opposite helicities. At β = π/4 the energy corresponds to the band crossing point, while at β = 0 and β = π/2 2.4. Time-dependence one returns to pure conduction and valence band states. The time-dependent SchrödingerPauli equation in This property is unique to monolayer graphene having a reads [17, 19]: linear dispersion. Cl3,0 ∂ψ In the Hilbert space the electron velocity operator is Iσ = H(ψ), (35) ∂t 3 ∂ where the Hamiltonian function is given by (4). vˆ = Hˆ = vFσˆ. (26) H(ψ) ∂k Multiplying both sides of (35) by bivector Iσ and not- The corresponding formula in GA is 3 ing that the pseudoscalar I commutes with all basis ele- vˆ|ψi ←→ ∇kH(ψ), ments, we can rewrite where the vectorial dierential operator was introduced ∂ψ(t) = −kIkˆψ(t), (36) ∂t ∂ ∂ ∂ (27) ∇k = σ1 + σ2 + σ3 . where Ikˆ is the unit bivector, the plane of which is per- ∂kx ∂ky ∂kz Then, in GA the average electron velocity is found to be pendicular to k. It satises (Ikˆ)2 = −1. The solution of ˜ the dierential multivector Eq. (36) is hvi1 = ∇˙ ψH˙ (ψ) , (28) 0 −kIkˆt ˆ where the overdot indicates the term which the dieren- ψ(t) = e ψ0 = (cos kt − Ik sin kt)ψ0, (37) tiation vector operator should act upon. We shall remind where ψ0 is the initial multivector at t = 0. The solution that such notation comes from a noncommutativity of the shows that the electron in the monolayer graphene can GA algebra. Therefore, it is forbidden to swap the sym- oscillate only with a single frequency 2 2 1/2, 2k = 2(kx + ky) 736 A. Dargys

which is equal to interband transition energy. The multi- tic graphene atland we shall address to Cl2,0 algebra, plier 2 is due to the fact that the rotor ψ(t) should be ap- where time is assumed to be the bivector [20]. Cl2,0 is plied twice in calculating the averages [compare Eq. (18)]. isomorphic to subalgebra {1, σ1, σ2,Iσ3} of Cl3,0 with To see how the rotor ψ(t) works, let us apply ψ(t) equivalence e1 = σ1, e2 = σ2, e12 ≡ e1e2 = Iσ3. A gen- to an arbitrary unit vector aˆ that lies in the graphene eral multivector of Cl2,0 is A = a0 + a2e1 + a1e2 + a3e12, plane, , the direction of which is where . Before considering the relativistic at- aˆ = cos γσ1 + sin γσ2 ai ∈ R determined by γ. Assuming that ψ0 = 1, the new vector land, however, at rst it is instructive to do calculations will be where time, as in the previous section, is treated as a 0 ˜ 0 0 0 (38) parameter. a (t) = ψ(t)aψ(t) = a1(t)σ1 + a2(t)σ2 + a3(t)σ3, where 3.1. Real matrix representation 0 2 2 a1(t) = cos (kt) cos γ + sin (kt) cos(γ − 2ϕ), The irreducible matrix representation of algebra a0 (t) = cos2(kt) sin γ − sin2(kt) sin(γ − 2ϕ), Cl2,0 2 consists of the following real matrices [21]: 0 (39) a3(t) = sin(2kt) sin(γ − ϕ), " # " # where angle is dened in Fig. 1. The module of the 1 0 1 0 ϕ 1ˆ = , eˆ = , (40) vector remains normalized at all times, (a0(t))2 = aˆ2 = 1. 0 1 1 0 −1 " # " # 0 1 0 1 eˆ = , eˆ = , (41) 2 1 0 12 −1 0

which satisfy Cl2,0 multiplication table. The following mapping rules between the Hilbert space and Cl2,0 alge- bras can be constructed " # a1 |ψi = ←→ ψ = a1 + a2e12, a2

eˆ1|ψi ←→ e1ψe1,

eˆ2|ψi ←→ −e2ψe1,

eˆ12|ψi ←→ −e12ψ. (42) However, it should be noted that real matrices cannot map the complex graphene Hamiltonian (1) onto GA. The mapping becomes possible only after Hamiltonian

(1) is brought to diagonal form Hˆg = diag(k, −k). Then the diagonal Hamiltonian in GA assumes the form q 2 2 (43) Hg(ψ) = ke1ψe1, k = kx + ky. Fig. 2. Precession trajectories of an arbitrary vector a. Taking into account that ψ consists of scalar and pseu- Thick shows the wave vectors ±k lying in the doscalar, and using the normalization condition ψψ˜ = 1, graphene plane. If the vector a becomes parallel or we can express ψ through a single parameter ϕ, antiparallel to electron wave vector the precession tra- e12ϕ/2 jectories shrink to a point. ψ = e = cos(ϕ/2) + e12 sin(ϕ/2). (44) If the rotor (44) is applied to a vector, for example to e1, Figure 2 illustrates the solution (38), where dier- the new vector will remain in the graphene plane, ent curves correspond to dierent γ. The char- ψe ψ˜ = cos ϕe − sin ϕe . (45) acteristic feature is that an arbitrary will leave the 1 1 2 a The average energy of the state (44) is graphene plane at later moments. However, when γ = ϕ q or , i.e. when is parallel or antiparallel to , ˜ 2 2 (46) γ = ϕ ± π aˆ k hψHg(ψ)i0 = kx + ky cos ϕ. there is no precession and 0 ˆ. Thus, the in-plane a (t) = a Thus, at ϕ = 0 and ϕ = π we have conduction and vectors a that are not aligned with k at later moments valence band energies which, as follows from (44), corre- will stick out from the graphene plane. For such vectors spond to eigenstates ψ+ = 1 and ψ− = e12, respectively. the precession axis coincides with k. As in Sect. 2 the energy bands can be related with the

helicity quantum numbers h±, if the Hamiltonian (43) is 3. Graphene with Cl2,0 rewritten in a form 1 1  The Cl3,0 algebra is connected with 3D Euclidean Hg(ψ) = k (ψ + e1ψe1) − (ψ − e1ψe1) space where time is a parameter rather than space-time 2 2  0 0  (47) coordinate. So, apart from a linear dispersion law, no = k h+P+(ψ) + h−P−(ψ) , other eects related with relativistic space-time transfor- where and 0 is the projection operator h± = ±1 P±(ψ) mations are to be expected. To have a true relativis- onto conduction/valence band states, Quantum Flatland and Monolayer Graphene . . . 737

1 ˆ 0 (48) Hg|ψi ←→ Hg(ψ) = kxe1ψ + kye2ψ = kψ. (54) P±(ψ) = (ψ ± e1ψe1). 2 However, the product ˜ apart from the scalar also con- Application of (48) to the eigenstates yields 0 , ψψ P+(1) = 1 tains the vector, P 0 (1) = 0 and P 0 (e ) = 0, P 0 (e ) = e . Since P 0 is − + 12 − 12 12 ± ˜ 2 2 2 2 directly related with the Hamiltonian, one can also intro- ψψ = a0 + a1 + a2 + a3 + 2(a0a2 − a1a3)e1 duce the helicity operator similar to (21), although now + 2(a0a1 + a2a3)e2, (55) the concept of the Pauli matrix-vector projection (20) and therefore cannot be normalized to unity. The nor- onto k is not needed altogether. Thus, in contrast to malization to scalar can be obtained if the following Cl3,0 case, the mapping by real matrices yields the Hamil- dagged multivector is introduced: tonian that has no vectors perpendicular to the graphene † ˜ (56) plane. ψ = −e12ψe12 = a0 − a2e1 − a1e2 − a3e12, then In the Hilbert space the dynamics of ψ is governed by † † 2 2 2 2 (57) equation ψψ = ψ ψ = (a0 + a3) − (a1 + a2). The formulae become more transparent if they are rewrit- ∂ i |ψi = Hˆ |ψi. (49) ten in a form of trigonometric functions ∂t g Since the imaginary i cannot be mapped onto Cl2,0, we ψ = r(cos α + e12 sin α) + s(sin βe1 + cos βe2). (58) are unable to transform the left hand side of this equa- Then, we nd that tion to and therefore to construct the equation of Cl2,0 ψ†ψ = r2 − s2. (59) motion. In fact, the deadlock here is more serious. Since the GA Hamiltonian (43) is proportional to the module The ψ-multivector (52) can be decomposed into a product of relativistic boost and rotation in a graphene of wave vector k the transformation from momentum to coordinate representation will result in the square root plane √ q φaˆ αe12 (60) for a coordinate operator: 2 2 2. Thus, ψ = ρe e , k = k → ∂x + ∂y √ where we require 2 2, and we nd that the mapping (42) should be rejected from ρ = ± r − s φ = arctanh(s/r) p 2 2 , and the beginning. aˆ = (a2e1 + a1e2)/ a1 + a2 = e1 sin β + e2 cos β where a1 = cos(β − α) and a2 = sin(β − α). The sec- ond exponent generates rotation in the graphene plane 3.2. Complex matrix representation by angle 2α and is related with trigonometric functions: αe12 e = cos α + e12 sin α. The rst exponent generates To circumvent the diculty the mapping by complex the boost in direction aˆ by magnitude φ and is related φaˆ matrices will be addressed. Since Cl2,0 is isomorphic with the hyperbolic functions: e = cosh φ + aˆ sinh φ. to the subalgebra {1, σ1, σ2,Iσ3} of Cl3,0, therefore, its The scalar ρ denes the magnitude of dilation. complex matrix representation may be related to the In (3 + 1) relativistic Cl1,3 algebra the event is dened Pauli matrices σˆx and σˆy and their product, as a sum of four vectors that span the space-time [9, 10]. " # " # The metric of this algebra ensures correct invariant inter- 1 0 0 1 1ˆ = , eˆ = , (50) val (x2 − t2) between space x and time t coordinates. In 0 1 1 1 0 Cl2,0 algebra, the both basis vectors e1 and e2 were used " # " # to construct the spatial vector x = x1e1 + x2e2, there- 0 −i i 0 fore, in order to have a correct scalar invariant interval eˆ2 = , eˆ12 = . (51) i 0 0 −i the time will be treated as a bivector. The relativistic

Now the state vector |ψi is complex. We shall assume event in Cl2,0 then can be described by the multivector that the following correspondence exists between the X = (t + x)e12, (61) complex Hilbert space vector and the general multivector the square of which gives the needed invariant interval of : Cl2,0 X2 = x2 − t2. " # The spectrum of the Hamiltonian (54) follows from the a0 + ia3 |ψi = ←→ ψ eigenenergy equation . Taking the re- a + ia Hg(ψ) = kψ = Eψ 2 1 verse of this equation ˜ ˜ and multiplying both (52) ψk = Eψ = a0 + a2e1 + a1e2 + a3e12, sides by e we obtain the dagged form of eigenmultivec- where . The mapping (52) between the wave func- 12 ai ∈ R tor equation tion in the Hilbert and GA space allows to construct the † † (62) following replacement rules for the Pauli matrices and ψ k = Eψ . their product Once more multiplying the initial and dagged eigenequa- tions from left and right by e12 and remembering that σˆx|ψi ←→ e1ψ, ψψ† is the scalar, we can write E2ψψ† = kψψ†k = σˆ |ψi ←→ e ψ, 2 †, which gives the relativistic spectrum y 2 k √ψψ E = 2 σˆxσˆy|ψi ←→ e12ψ, (53) ± k ≡ ±k. with the help of which the complex graphene Hamilto- Nonetheless, a note of caution is appropriate here. One nian mapped onto Cl2,0 becomes can check that formally the following mapping holds: 738 A. Dargys

i|ψi ≡ i1ˆ|ψi ←→ ψe12, (63) Since (±k + m) is the scalar, the spectrum by analogy which if used with the replacement rules (53) will gen- with Eq. (43) can be written automatically erate new rules, for example and p iσ ˆx|ψi ↔ e1ψe12 E = ± k2 + m. (72) . However, the matrix ˆ is not a mem- iσ ˆy|ψi ↔ e2ψe12 i1 If instead of real matrix representation one uses complex ber of and therefore it is illegitimate to map it onto Cl2,0 representation then one nds exactly the same dispersion algebra. Thus we are facing the same dilemma: Cl2,0 (72) for positive as well as for negative states. Thus we the term with time derivative in the Schrödinger i∂t|ψi nd that the geometric algebra which represents Eq. (49) cannot be mapped onto algebra. Cl2,0 Cl2,0 the true atland gives unacceptable relativistic disper- In the following we shall postulate that can be i∂t|ψi sion. In addition, we have seen that we cannot map the replaced by GA expression , where 2 plays ∂tψe12 e12 = −1 imaginary onto the atland. These problems do not the role of the imaginary unit. Then the time-dependent i arise in . Schrödinger equation in GA will be Cl3,0 ∂ψ e = kψ. (64) 4. Conclusion ∂t 12 This multivector equation can be solved after division Two subalgebras of relativistic Cliord algebra of ψ into even (scalar plus bivector) and odd (vector) Cl1,3 were considered as possible candidates for a description of parts, ψ = ψ+ + ψ−. This dierential equation splits into two equations, graphene quantum properties in Euclidean spaces with- out using the Hilbert space concepts. It is shown that ∂tψ+ = ke12ψ−, in case of a single valley graphene the three-dimensional ∂ ψ = −ke ψ , (65) t − 12 + Euclidean Cl3,0 algebra allows to construct Schrödinger- solutions of which are exponents with positive and nega- -type equation for electron with the relativistic Dirac tive energies. The general solution will be a sum spectrum. However, it was found impossible to construct

e12kt −e12kt such equation with two-dimensional algebra , which ψ(t) = e ψ+0 + e ψ−0. (66) Cl2,0 represents the true Euclidean atland. The preliminary The initial wave function ψ(0) = ψ+0 + ψ−0 can be rep- resented as rotation and boost in the graphene plane, investigations show that the full relativistic Cl3,1 algebra Eq. (60). It can be shown that at later moments the rel- is suitable for description of graphene with two inequiv- alent K and K0 valleys, when electron spin is included. ative contribution of even ψ+(t) = a0(t) + e12a3(t) and odd ψ−(t) = a1(t)e1 + e2a2(t) parts into the total ψ(t) will not change with time. Acknowledgments

3.3. Inclusion of mass term This work in part was supported by Agency in Science, √ Innovation and Technology (grant No. 31V-32) in the To obtain full relativistic spectrum E = ± k2 + m2 framework of High Technology Development Programme one should add the diagonal mass term diag(m, −m) to for 20112013. The author is thankful to A. Acus for massless relativistic Hamiltonian (1). In Cl such term 3,0 critical reading of the manuscript. corresponds to multivector mσ3ψσ3, so that GA Hamil- tonian (4) becomes 2 References Hg(ψ) = (k + mσ3)ψσ3. (67) The simplest way to nd the spectrum of this rela- tivistic Hamiltonian is to insert the stationary solution [1] D.S.L. Abergel, V. Apalkov, J. Barashevich, K. Ziegler, T. Chakraborty, Adv. Phys. 59, 261 ψ(k,E) = ψ(k)e−Iσ3Et into (35), which gives (2010). (68) Eψ(k) = (k + mσ3)ψ(k)σ3. [2] N.M.R. Peres, Rev. Mod. Phys. 82, 2673 (2010). Multiplying this algebraic multivector equation by its re- [3] S.D. Sarma, S. Adam, E.H. Hwang, E. Rossi, Rev. verse ˜ ˜ , and remembering that Eψ(k) = σ3ψ(k)(k+mσ3) Mod. Phys. 83, 407 (2011). and ˜ are scalars, one nds E ψ(k)ψ(k) [4] D.R. Cooper, B. D'Anjou, N. Ghattamaneni, Eψ(k)Eψ˜(k) B. Harack, M. Hilke, A. Horth, N. Majlis, M. Mas- sicotte, L. Vandsburger, E. Whiteway, V. Yu, e-print 2 2 ˜ = (k + m + mkσ3 + mσ3k)ψ(k)ψ(k). (69) arXiv:cond-mat/1110.6557v1 (2011). Since σ3k = −kσ3, we nd the correct relativistic dis- [5] P.E. Allain, J.N. Fuchs, Eur. Phys. J. B 83, 301 persion (2011). E2 = k2 + m2. (70) [6] D. Hestenes, Space-Time Algebra, Gordon and Breach, New York 1966. In Cl2,0, using real matrix representation (40) and (41), one nds the following mapping of the mass [7] D. Hestenes, G. Sobczyk, Cliord Algebra to Geomet- term: rical Calculus (A Unied Language for Mathematics and Physics), Reidel, Boston 1984. (71) mσˆ3|ψi ←→ me1ψe1. [8] P. Lounesto, Cliord Algebra and Spinors, Cam- Then, the Hamiltonian becomes Hg = (±k + m)e1ψe1. bridge University Press, Cambridge 1997. Quantum Flatland and Monolayer Graphene . . . 739

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