Hall Effect on Non-commutative Plane with Space-Space Non-commutativity and Momentum-Momentum Non-commutativity

Transcription

1 Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 8, HIKARI Ltd, Hall Effect on Non-commutative Plane with Space-Space Non-commutativity and Momentum-Momentum Non-commutativity Won Sang Chung Department of Physics and Research Institute of Natural Science College of Natural Science, Gyeongsang National University Jinju , Korea Copyright c 2017 Won Sang Chung. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we consider the non-commutative plane with both space-space non-commutativity and momentum-momentum non-commutativity. We study the hamiltonian for an an electron moving on the non-commutative plane in the uniform external electric field along the x-axis and the uniform external magnetic field which is perpendicular to the plane. We solve the Schrödinger equation for this hamiltonian by using the standard factorization method and compute the Hall conductivity. 1 Introduction Ideas of non-commutative geometry were first proposed by Connes [1]. The major applications have been taken into account in the quantum field theory so as to understand the way how to remove UV singularities, and to construct the proper model for the quantum gravity. In Einstein gravity, the quantum vacuum fluctuations were shown to create mini black holes in which space-time points are not localized any more. It means that the space-time coordinate operators do not commute with each other. Instead, they obey the Heisenberg- Moyal commutation relation.

2 358 Won Sang Chung Physics in noncommutative space was considered in the course of studying the low energy effective theory of D-brane with a non-zero NS-NS B field background. The effects of non-commutative space arises at the string scale. The noncommutative quantum field theories are used in order to test the space non-commutativity. The noncommutative quantum field theories is related to the quantum mechanics in non-commutative space when we consider the low energy limit. Some progress has been accomplished in this direction [2-8]. In this paper we consider the non-commutative plane with both space-space non-commutativity and momentum-momentum non-commutativity. We study the hamiltonian for an an electron moving on the non-commutative plane in the uniform external electric field along the x-axis and the uniform external magnetic field which is perpendicular to the plane. We solve the Schrödinger equation for this hamiltonian by using the standard factorization method and compute the Hall conductivity. 2 Electron moving on the non-commutative plane The non-commutative quantum mechanics (NCQM) with space-space noncommutativity and momentum-momentum non-commutativity is shown to take the following form: [ˆx i, ˆx j ] = iθɛ ij, [ˆp i, ˆp j ] = i θɛ ij, [ˆx i, ˆp j ] = i hδ ij, (i, j = 1, 2), (1) where θ, θ are the constant, frame-dependent parameters; ɛ ij is an Levi-Civita symbol. For the algebra (1), we can find the following realization ˆx i = x i 1 2 h θɛ ijp j ˆp i = p i h θɛ ij x j, (2) where [x i, p j ] = i hδ ij, [x i, x j ] = [p i, p j ] = 0. From now on we use the following notations for the commutation relation (1): [ˆx, ŷ] = iθ [ˆp x, ˆp y ] = i θ [ˆx, ˆp x ] = i h [ŷ, ˆp y ] = i h [ˆx, ˆp y ] = [ŷ, ˆp x ] = 0 (3)

3 Hall effect on non-commutative plane with Then, the eq.(2) is rewritten as ˆx = x 1 2 h θp y ŷ = y h θp x ˆp x = p x h θy ˆp y = p y 1 2 h θx (4) Now consider an electron moving on the commutative plane in the uniform external electric field E 0 along the x-axis and the uniform external magnetic field B which is perpendicular to the plane. Then, the hamiltonian reads H = 1 2µ where x, y, p x, p y are operators obeying [ ( p x + e ) 2 ( c A x + p y + e ) ] 2 c A y + ee 0 x, (5) [x, p x ] = [y, p y ] = i h, [x, p y ] = [y, p x ] = 0 (6) and A x, A y are components of the vector potential and µ, c, e denote mass, light speed and charge, respectively. The corresponding Schrödinger equation reads [ ( 1 p x + e ) 2 ( 2µ c A x + p y + e ) ] 2 c A y ψ(x, y) + ee 0 xψ(x, y) = Eψ(x, y), (7) In this case the wave function ψ(x, y) is not an operator but a function. However, in the non-commutative plane (NC plane), the situation becomes complicated because the NC wave function ψ(ˆx, ŷ) should be regarded as an operator. Thus, we should define the space on which the NC wave function acts. Let H be the Hilbert space where the Hamiltonian acts. Then, the NC wave function acts in the auxiliary Hilbert space ( F ). In the space F, the scalar product is defined using the Moyal star product as f g = f g(x, y)dxdy, (8) where the star product is defined as 0 h θ 0 i = exp 2 ( h 0 0 θ x x px y py ) px θ 0 0 h y 0 θ h 0 py (9)

4 360 Won Sang Chung The commutation relations are then expressed in terms of the Moyal star product as follows: [x, y] = iθ, [x, p x ] = i h, [y, p y ] = i h [x, p y ] = 0, [y, p x ] = 0, [p x, p y ] = i θ (10) where [A, B] = A B B A. The Schrödinger equation is then given by where the hamiltonian is given by H ψ = Eψ (11) H = 1 [( p x + e ) ( 2µ c A x p x + e ) ( c A x + p y + e ) ( c A y p y + e )] c A y + ee 0 x (12) and the corresponding Schrödinger equation reads 1 [( p x + e ) ( 2µ c A x p x + e ) ( c A x + p y + e ) ( c A y p y + e ) ] c A y ψ(x, y) + ee 0 x ψ(x, y) (13) = Eψ(x, y) The eq.(13) is not gauge invariant in an ordinary ( commutative ) sense. But, if we redefine the gauge transformation via the star product as δψ = ie Λ(x, y) ψ(x, y) hc δa x = i[λ, A x ] i[x, Λ] δa y = i[λ, A y ] i[y, Λ], (14) we find that the eq.(13) is NC gauge invariant, which indeed holds because D i = p i + e c A i, (i = x, y) is NC gauge covariant, so D i D i is NC gauge invariant. If we adopt the symmetric gauge and use the realization (4), we can rewrite the Schrödinger equation as Hψ = Eψ (15) where hamiltonian H is given by H = [ ( 1 (1 κ)p x e ( 2 ( y) + (1 κ)p y + e ( ) 2 ( x ]+ee 0 x θ ) 2µ 2 h p y (16)

5 Hall effect on non-commutative plane with and p x = h i x, p y = h i y and κ = eθb. The solutions of the above equation 4c h can be obtained through the standard factorization method. Let us introduce two sets of step operators as follows: b = i(1 κ)p z + e ( z + λ b = i(1 κ)p z + e ( z + λ d = i(1 κ)p z + e ( z d = i(1 κ)p z + e ( z, (17) where z = x + iy, p z = p x ip y. They satisfy two independent boson algebras: where [b, b ] = 2µ hw, [d, d] = 2µ hw, [b, d] = [b, d ] = 0 (18) w = e µc Then, the hamiltonian becomes where ( 1 eθb 4c h ) ( H = 1 4µ (b b + bb ) λ + 2µ (d + d) λ2 2µ, (19) λ + = µee 0 2 λ = µee 0 2 2c e ( ) + B c θ 2c e ( ) B c θ θ 2 h(1 κ) θ 2 h(1 κ) The wave function and the energy eigenvalue are then given by ψ = n, α = 1 (2µ hw) n n! ei(αy+νxy) (b ) n 0 (20) and E n,α = hw 2 (2n + 1) + h(1 κ)λ + α λ2 µ 2µ where α R and n = 0, 1, 2, and ν = e ( ) B c θ 2c h(1 κ) (21)

6 362 Won Sang Chung Now we can explicitly obtain the coordinate representation of the ground wave function. To do so we set xy n, α = ψ n,α (x, y), (22) where ψ n,α (x, y) = N n (2µ hw) n n! ei(αy+νxy) ψ n (x, y) (23) From the relation bψ 0 = 0, we have ( ( e ψ 0 = N 0 exp z z λ ) 2(1 κ)c h(1 κ) z, (24) where N 0 = e ( ) B c θ 2πc(1 κ) exp λ 2 c 2 h 2 e(1 κ) ( ) (25) B c θ 3 Hall Conductivity on Noncommutative Plane Now let us find the Hall conductivity for the hamiltonian (19). The components of the current operator ĵ are defined as ĵ x = ieρ [H, ˆx], h ĵ y = ieρ [H, ŷ], (26) h where ρ denotes electron density. Using the commutation relation (3), we can rewrite the eq.(26) as ĵ x = eρ [ 1 ebθ ] (ˆp x + e µ 2c h câx) ĵ y = eρ [ 1 ebθ ] (ˆp y + e e2 E 0 ρθ (27) µ 2c h cây) h Using the eq. (4), we have ĵ x = eρ [ 1 ebθ ] [( 1 ebθ ) p x + µ 2c h 4c h ĵ x = eρ [ 1 ebθ ] [( 1 ebθ ) p y µ 2c h 4c h ( θ 2 h eb 2c ) ( θ 2 h eb ) ] x e2 E 0 ρθ 2c h y ] (28) Now, the expectation value of two components of the current operator can be calculated with respect to the eigenstates n, α leading to ĵ x = 0, ĵ y = e2 E 0 ρθ h (29)

7 Hall effect on non-commutative plane with Therefore, the Hall conductivity on non-commutative plane is given by σ NC H = e2 ρθ h (30) Comparing σh NC with the Hall conductivity σh C = ec ρ in the commutative B quantum mechanics, we know that when θ = hc eb. σ NC H = σ C H, (31) 4 Conclusion In this paper we considered the non-commutative plane with both space-space non-commutativity and momentum-momentum non-commutativity. We studied the hamiltonian for an an electron moving on the non-commutative plane in the uniform external electric field along the x-axis and the uniform external magnetic field which is perpendicular to the plane. We solved the Schrödinger equation for this hamiltonian by using the standard factorization method and computed the Hall conductivity. Finally we would like to mention the special case of θ = B. In this case, c the commutation relation (3) reduces to and the hamiltonian (16) becomes [ˆx, ŷ] = iθ [ˆp x, ˆp y ] = i c B [ˆx, ˆp x ] = i h [ŷ, ˆp y ] = i h [ˆx, ˆp y ] = [ŷ, ˆp x ] = 0 (32) H = 1 2µ (1 κ)2 (p 2 x + p 2 y) + ee(x θ 2 h p y) (33) Thus, in this special choice of θ, the hamiltonian of the form (16) in the non-commutative plane is equivalent to the one without magnetic field in the commutative plane. Acknowledgements. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF- 2015R1D1A1A ) and by the Gyeongsang National University Fund for Professors on Sabbatical Leave, 2016.

8 364 Won Sang Chung References [1] A. Connes, Non-commutative differential geometry, Publ. Mathematiques de IHES, 62 (1985), [2] N. Seiberg and E. Witten, String theory and noncommutative geometry, Journal of High Energy Physics, 1999 (1999), [3] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED, Phys. Rev. Lett., 86 (2001), [4] J. Gamboa, M. Loewe and J. C. Rojas, Noncommutative quantum mechanics, Phys. Rev. D, 64 (2001), [5] V. P. Nair and A. P. Polychronakos, Quantum mechanics on the noncommutative plane and sphere, Phys. Lett. B, 505 (2001), [6] B. Morariu and A. P. Polychronakos, Quantum mechanics on the noncommutative torus, Nucl. Phys. B, 610 (2001), [7] A. Hatzinikitas and I. Smyrnakis, The noncommutative harmonic oscillator in more than one dimension, J. Math. Phys., 43 (2002), [8] J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Noncommutative quantum mechanics: The two-dimensional cental field, Int. J. Mod. Phys. A, 17 (1999), Received: February 4, 2016; Published: June 16, 2017