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Quantum Flatland and Monolayer Graphene from a Viewpoint of Geometric Algebra A Vol. 124 (2013) ACTA PHYSICA POLONICA A No. 4 Quantum Flatland and Monolayer Graphene from a Viewpoint of Geometric Algebra 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 Euclidean space 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 basis matrix 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 multivectors. 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 j i ! = a0 + a1Iσ1 + a2Iσ2 + a3Iσ3; Cl3;0 algebra describes Euclidean 3D space and simul- λj i ! λψ; taneously is an even subalgebra of relativistic spacetime σ^ j i ! σ σ ; (3) algebra , i.e., the isomorphism + is i i 3 Cl1;3 Cl1;3 ≈ Cl3;0 where is the scalar and is one of the Pauli matrices. satised. The basis vectors of have positive sig- λ σ^i Cl3;0 Then, ^ , where ^ is given by (1), maps to the fol- nature: 2 2 2 . The vectors play Hgj i Hg σ1 = σ2 = σ3 = 1 σi lowing GA function for spinor : the role of space coordinates. The notation reects H ( ) = k σ ; (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 plane, 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 bivectors 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 multivector, 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 f1; σ1; σ2;Iσ1;Iσ2g 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( ) = αRσ1σ2k σ3; (5) ^ 0 kx − iky (1) Hg = ~vF ; (6) kx + iky 0 HD( ) = αDσ1kσ1 σ3; 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 , 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 = ±|kj; tic Hamiltonians. For example, the factor Iσikσi before where plus/minus sign corresponds to conduction/va- leads to non-Hermitian Hamiltonian which must be discarded. The factors σ2kσ2, σ3kσ3, and σ2kσ1 give lence band. Since only two Pauli matrices, σ^x and σ^y 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 have the form [18]: representation µ , will remain in 1 γ pµjΨi = 0 µ = 0; 1; 2; 3 p ( ) = ( ± σ σ ); 4D spacetime even if one limits the equation to two space ± 2 3 3 coordinates (µ = 1; 2) [16].
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