Spin noncommutativity and the three dimensional harmonic oscillator

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(1)PHYSICAL REVIEW D 85, 025009 (2012). Spin noncommutativity and the three-dimensional harmonic oscillator H. Falomir* IFLP - CONICET and Departamento de Fı́sica, Facultad de Ciencias Exactas de la UNLP, C. C. 67, (1900) La Plata, Argentina. J. Gamboa† Departamento de Fı́sica, Universidad de Santiago de Chile, Casilla 307, Santiago, Chile Facultad de Fı́sica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile. M. Loewe‡ Facultad de Fı́sica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile Centre for Theoretical and Mathematical Physics, University of Cape Town, Rondebosch 7700, South Africa. F. Méndez§ Departamento de Fı́sica, Universidad de Santiago de Chile, Casilla 307, Santiago, Chile. J. C. Rojask Departamento de Fı́sica, Universidad Católica del Norte, Antofagasta, Chile (Received 8 November 2011; revised manuscript received 14 December 2011; published 18 January 2012) A three-dimensional harmonic oscillator with spin noncommutativity in the phase space is considered. The system has a regular symplectic structure and by using supersymmetric quantum mechanics techniques, the ground state is calculated exactly. We find that this state is infinitely degenerate and it has explicit spontaneous broken symmetry. Analyzing the Heisenberg equations, we show that the total angular momentum is conserved. DOI: 10.1103/PhysRevD.85.025009. PACS numbers: 03.65.w. In the last ten years the implications of noncommutative geometry [1] have been intensively investigated as a way, for example, to develop computational techniques for understanding problems that go beyond perturbation theory. A particularly interesting example is the quantum Hall effect where—due to the fact that the magnetic field is strong—nonperturbative techniques are required. Besides, the study of motion of charged particles in strong magnetic fields [2,3] is an interesting mathematical problem because, as it is known, the commutator of the momenta (more precisely the covariant derivatives), is different from zero. In other words, this is a genuine example of a noncommutative geometry problem. Some years ago, Nair and Polychronakos (NP) [4] studied the noncommutative harmonic oscillator, i.e. a system described by the Hamiltonian Hðx; pÞ ¼ 12ðp2 þ x2 Þ;. (1). and the deformed commutators [5] given by ½x^ i ; x^ j ¼ iij ;. ½p^ i ; p^ j ¼ iBij ;. ½xi ;pj ¼ iij ;. (2). with i; j 2 f1; 2; 3g. This problem is exactly soluble and presents a fixed point. *falomir@fisica.unlp.edu.ar † jgamboa55@gmail.com ‡ mloewe@fis.puc.cl § fernando.mendez.f@gmail.com k jurojas@ucn.cl. 1550-7998= 2012=85(2)=025009(5). In three dimensions we can write ij ¼ ijk k and Bij ¼ ijk Bk , with i ; Bj constant vectors. Then, the determinant of the symplectic matrix ab ¼ ½a ; b —with fa g ðx; pÞ—turn out to be detðÞ ¼ ð1 BÞ2 which, therefore, vanishes for B ¼ 1 [4]. Alternatively, it is possible to perform a coordinate transformation in the phase space in order to get a regular symplectic matrix but, in this case, the coordinate transformation turns out to be singular in the parameter space fi ; Bk g, again for B ¼ 1. In Refs. [6–8] a new model of noncommutative quantum mechanics was proposed where the spin degrees of freedom are explicitly considered in the commutation relations. Basically in this construction the algebra of dynamical variables is modified so that ij and Bij are chosen as Bij ¼ 2 ijk s^k ; (3) ij ¼ 2 ijk s^k ; where and are length and momentum scales, respectively, (in natural units, with ℏ ¼ 1) and s^k are spin matrices. The deformation of the second commutator in (2) with Bij specified in (3) is a kind of nonrelativistic version [9] of the Snyder-Yang algebra [10,11]. Indeed, the choice (3) corresponds to the reduction of M ¼ 2i ½ ; to just iijk k . In this article we propose a generalization of the NP oscillator [4] where spin degrees of freedom are introduced through this modification of the algebra of dynamical. 025009-1. Ó 2012 American Physical Society.

(2) FALOMIR et al.. PHYSICAL REVIEW D 85, 025009 (2012). variables and explore the physical structure of the resulting model. We show below that, contrarily to the case of the NP oscillator [4], the ground state of the system is infinitely degenerate as a consequence of a continuos symmetry breaking. This is an unexpected feature of our model that makes it worthwhile to be considered. Notice that the spontaneous breaking of a continuos symmetry is not a common fact in quantum mechanics. In general, the ground state exhibits the same symmetry as the Hamiltonian, even though there are exceptions. A particularly interesting system with three ground states is superconductivity at high temperatures where a triplet instead of a singlet state minimizes the energy, which requires the extension of the Cooper-pairing mechanism (see e.g. [7,12]). Another example is the breaking of SUSY in supersymmetric quantum mechanics in the real line [13] as a consequence of the potential behavior at infinity, or the breaking of SUSY and scale invariance in problems on the half line with a singular potential, as consequences of subtle definition-domain effects [14]. In order to explain our results, we write the complete set of consistent commutation relations ½x^ i ; x^ j ¼ i2 ijk s^k ;. ½p^ i ; p^ j ¼ i2 ijk s^k ;. ½x^ i ; p^ j ¼ iðij þ ijk s^k Þ; ½x^ i ; s^j ¼ iijk s^k ;. ½sî ; s^j ¼ iijk s^k ;. (4). ½p^ i ; s^j ¼ iijk s^k ;. which can be explicitly realized in terms of the usual canonical variables through the shift x^ i ! x^ i ¼ xi þ si ;. p^ i ! p^ i ¼ pi þ si ;. sî ! sî ¼ si ¼. 1 2. i;. ½sî ; p^ j . ½sî ; s^j . which is regular, with a constant nonvanishing determinant (independent of and ).1 This is a very interesting peculiarity of the oscillator with noncommutativity of spin, which justifies a more careful analysis of their properties. Thus, following the prescription in Eq. (5), the Schrödinger equation is H^ xi þ i ; pi þ i j c ðtÞ> ¼ i@t j c ðtÞ > : (7) 2 2 1. H^ ¼ 12ðp^ 2 þ x^ 2 Þ ¼ 12ðp^ i þ ix^ i Þðp^ i ix^ i Þ þ 32 ¼ Ayi Ai þ E0 ;. The same is also true for, at least, the model with spin 1.. (8). where E0 is the energy of the ground state and 1 Ai ¼ pffiffiffi ðp^ i ix^ i Þ; 2. 1 Ayi ¼ pffiffiffi ðp^ i þ ix^ i Þ 2. (9). are the analogous of the creation and destruction operators of the usual commutative case. These operators satisfy the algebra i ½Ai ; Aj ¼ ð iÞ2 ijk sk ; 2 i y y ½Ai ; Aj ¼ ð þ iÞ2 ijk sk ; 2 ½Ai ; Ayj ¼ ij þ ið2 þ 2 Þijk sk ;. (10). which is smooth in the commutative limit, i.e. when ; ! 0. To further discuss the physical content of this generalization of the harmonic oscillator with such a nonconventional algebraic structure, it is useful to consider the supersymmetric version associated with (8) and study, for example, the ground state of the model. In principle this is a direct calculation: First we redefine the zero energy level by subtracting the constant E0 to the Hamiltonian (H^ ! H^ þ E0 ), so that (9) changes into H^ ¼ Ayi Ai ;. (5). where ðxi ; pj Þ obey the standard Heisenberg algebra and the identification si ¼ 12 i corresponds to spin-1=2, the case we shall consider in this paper. These noncommutative variables give rise to the symplectic matrix 1 0 ½x^ i ; x^ j ½x^ i ; p^ j ½x^ i ; s^j C B B (6) A; @ ½p^ i ; x^ j ½p^ i ; p^ j ½p^ i ; s^j C ½sî ; x^ j . The model based on the harmonic oscillator we are considering is then defined by the Hamiltonian [15]. (11). which is a positive semidefinite operator. Following the supersymmetrization procedure proposed in [16,17] (see also [6]), the supercharges are defined as follows: Ai ! Q ¼ Ai c i ¼ Ai Ayi ! Qy ¼ Ayi c yi ¼ Ayi. i. i. . . A þ. ;. Ay . þ;. (12). where ¼ 12 ð 1 i 2 Þ fulfil 2 ¼ 0 and, therefore, c 2i ¼ 0 ¼ c y2 i . The supersymmetric Hamiltonian is then defined as 1 H ¼ fQy ; Qg 2 1 1 þ 3 1 y 122 3 ¼ Ay A 22 þ AA (13) 2 2 2 2 1 2 9 2 2 2 ¼ p þ x þ 3 x þ 3 p þ ð þ Þ 2 4 3 122 þ L 3 ; (14) 2 which is a positive semidefinite operator and commutes with the supercharges. 025009-2.

(3) SPIN NONCOMMUTATIVITY AND THE THREE- . . .. PHYSICAL REVIEW D 85, 025009 (2012). ½H; Q ¼ 0 ¼ ½Qy ; H: This is the standard supersymmetric algebra. The last term of (14) is the spin-orbit coupling that emerges from the usual three-dimensional supersymmetry, while x and p correspond to magnetic dipolar and Dresselhaus [18] interactions, respectively. In order to find the ground state 0 one can look for a vector which is annihilated by the supercharge Q, namely, Q0 ¼ 0: This equation reads as Ai i 0 ¼ 0 or, more explicitly, ! ! I0 0 0 ¼ 0 ) Ai Ai i 0 II 0. (15) (16). I i 0. ¼ 0;. (17). with I0 ; II 0 two-components spinors. In matrix form, Eq. (17) reads as ½ ðp ixÞ þ MI0 ¼ 0;. (18). 3 2 ð. iÞ is a complex number. where M ¼ Equation (18) is solved by functions of the form I0 ¼. ð3=4Þ ððxþkI Þ2 =2ÞþikR x. e. uðkÞ;. (19). where k ¼ kR þ ikI 2 C3 is a complex vector and uðkÞ is a normalized constant two-components spinor satisfying ð:k þ MÞuðkÞ ¼ 0. (20). uðkÞy uðkÞ. and ¼ 1. The condition (20) implies that the complex vector characterizing each ground state satisfies k2 ¼ M2 , and then 9 9 kR :kI ¼ ; (21) k 2R k2I ¼ ð2 2 Þ; 4 4 3 where kR ; kI 2 R . On the other hand, the condition Qy 0 ¼ 0 gives rise to no normalizable solution. From (19) we can compute the mean values in the vacuum of the relevant operators, namely, x; p and L :¼ x p. We get hxi0;k ¼ kI ;. hpi0;k ¼ kR ;. hLi0;k ¼ kR kI :. The noncommutative version of the orbital angular momentum should be defined as the (Hermitian) operator ^ :¼ 1fx^ p^ p^ xg ^ ¼ x^ p^ {s ¼ c L 2. ¼ L þ ðx pÞ s; whose vacuum mean value is given by ^ 0;k ¼ kR kI 1ðkR þ kI Þ ðuðkÞy uðkÞÞ: hLi 2. ^ 0;k ¼kR þ uðkÞy uðkÞ: ^ 0;k ¼kI þ uðkÞy uðkÞ; hpi hxi 2 2 (23). (25). ^ does not generate an SOð3Þ group. Contrarily to L, L From the algebraic point of view, notice that the conditions on k ¼ kR þ ikI determine only the square k2 . Then, there is a continuous of degenerate minima that can be obtained from one of them by a (complex) orthogonal transformation given by 3 3 complex matrices satisfying Ut U ¼ 13 . Near the identity, U ¼ eiM ’ 13 þ iM, the previous condition requires Mt ¼ M. Therefore, the Lie algebra of this group is the space of complex antisymmetric 3 3 matrices. This corresponds to the Lie algebra of the complexification of SOð3Þ, SOð3Þ, which has the covering group SLð2; CÞ. This Lie algebra is generated by 3 3 matrices fXi ; X i ; i ¼ 1; 2; 3g that satisfy the commutation relations ½Xi ; Xj ¼ iijk Xk , ½Xi ; X j ¼ iijk X k , and ½X i ; X j ¼ iijk Xk . Thus, for a given k 2 C3 with kR kkI , the solution for the ground state in Eq. (19) has the symmetry corresponding to rotations around this given direction and, therefore, So, its little group is SOð2Þ, generated by k X and k X. in this case the symmetry SOð3Þ of the Hamiltonian is broken down to SOð3Þ=SOð2Þ, and the transformations that move from this ground state to another one are elements of this quotient group. On the other hand, for nonparallel vectors kR ; kI the Hamiltonian symmetry is completely broken and each SOð3Þ transformation gives rise to a different ground state. Let us now discuss the problem concerning the dynami^ p^ and s. ^ The equations of cal evolution of the operators x; motion can be written in two equivalent ways, namely, using the Heisenberg equations for the noncommutative variables, i.e. 1 x^_ i ¼ ½x^ i ;H; i. (22). Therefore, the degenerate vacua are characterized by different mean values of the position, momentum, and orbital angular momentum, which vary continuously with k. In particular, the mean angular momentum is not quantized. This is consistent with a rotational SOð3Þ symmetry breaking. Anyway, as we will see below, the total angular momentum J ¼ L þ 12 is a conserved quantity. ^ p^ we For the noncommutative dynamical variables x; have. (24). 1 p^_ i ¼ ½p^ i ;H; i. 1 s_ i ¼ ½si ;H; (26) i. or, equivalently, by using the shift (5) and employing the conventional dynamical variables of the commutative case. Following the second point of view the equations of motion –after use of the Hamiltonian (14)–become d ðx 122 Þ ¼ ðp þ 3sÞ 122 þ 2ðx sÞ 3 ; dt d ðp 122 Þ ¼ ðx þ 3sÞ 122 þ 2ðp sÞ 3 ; dt d ðs 122 Þ ¼ 3ðp þ xÞ s 122 3ðL sÞ 3 : dt (27). 025009-3.

(4) FALOMIR et al.. PHYSICAL REVIEW D 85, 025009 (2012). These equations imply the conservation of the (conventional) total angular momentum J ¼ ðL þ sÞ 122 . Indeed, from the first two equations in (27) one gets d ðL 122 Þ ¼ 3ðp þ xÞ s 122 þ 3ðL sÞ dt. 3;. and using the third equation in (27) one finds d dJ ¼ ½ðL þ sÞ 122 ¼ 0; dt dt. (28). the aforementioned result. Therefore, J commutes with the Hamiltonian and generates the symmetry that is broken in ^ þ sÞ does not commute the ground state [notice that ðL with the Hamiltonian]. On the other hand, the different terms appearing in the Heisenberg equations of this extension of the threedimensional supersymmetric harmonic oscillator are relevant in different contexts. To see this, note first that the Hamiltonian in Eq. (14) is block-diagonal, acting on each block on the corresponding two-component spinor, I and II . But, as we have seen, only the upper component has a normalizable ground state at zero energy. The equations of motion restricted to this sector reduce to r_ ¼ ðp þ 3sÞ þ 2ðx sÞ;. of spin. The couplings r s and p s correspond to magnetic dipolar and Rashba forces, respectively, while L s is a spin-orbit coupling. The terms proportional to or are due to noncommutativity. In conclusion, we have studied an extension of the NP oscillator, which includes spin noncommutativity, and have discussed the differences with the standard version. The regular nature of the symplectic form and the infinite degeneracy of the ground state due to symmetry breaking suggest an underlying nonperturbative structure that, to our knowledge, is new in quantum mechanics. The degeneracy of the ground state in standard quantum mechanics is not a very common phenomenon. Usually, the Hamiltonian eigenvalue problem reduces to elliptic operators that have a unique ground state. We have shown that the noncommutative extension of the NP oscillator obtained by the modification of the algebra of dynamical variables considered in this paper circumvents this possibility, providing an example of a quantum-mechanical system with infinite vacuum degeneracy. It is interesting to notice that this kind of modification of the Heisenberg algebra could be of interest for the description of the low-energy excitations of graphene, as discussed in [19].. Here, the last three terms and on the right side of these equations are simply corrections due to supersymmetry and they are present regardless of the noncommutativity. We thank J. C. Retamal for useful discussions. This work was supported by grants from CONICET (PIP 01787), ANPCyT (PICT 00909), and UNLP (Proy. 11/X492), Argentina, and from FONDECYT-Chile Grant No. 1095106, 1095217, 1100777, and Proyecto Anillos ACT119.. [1] See e.g M. R. Douglas and N. Nekrasov, Rev. Mod. Phys. 73, 977 (2001). [2] See L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1962), Vol. III. [3] F. Wilczek, Fractional Statistics and Anyon Superconductivity (World Scientific, Singapore, 1990); J. Bellisard, in Solid State Physics, Proceedings on Localization and Disordered Systems, Bad Schandau, 1986 (Teuber, Lepzig, 1988; J. Bellisard, in Statistical Mechanics and Field Theory, Mathematical Aspects, Lectures Notes in Physics (Springer, Berlin 1986), pp. 99–156; see also A. Connés, Noncommutative Geometry (Pergamon Press, New York, 1991). [4] V. N. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001). [5] The literature is very extensive, some papers are G. V. Dunne, J. Jackiw, and C. Trugenberger, Phys. Rev. D 41, 661 (1990); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev. D 64, 067901 (2001); J. Gamboa, M. Loewe, J. C. Rojas, and F. Méndez, Int. J. Mod. Phys. A 17, 2555 (2002); J. Gamboa, M. Loewe, J. C. Rojas, and F.. Méndez, Mod. Phys. Lett. A 16, 2075 (2001); H. Falomir, J. Gamboa, M. Loewe, F. Méndez, and J. C. Rojas, Phys. Rev. D 66, 045018 (2002); K. Bolonek and P. Kosinski, arXiv:0704.2538; A. Kijanka and P. Kosinski, Phys. Rev. D 70, 127702 (2004); L. Mezincescu, arXiv:hep-th/0007046; C. Acatrinei, J. High Energy Phys. 09 (2001) 007; M. Gomes and V. G. Kupriyanov, Phys. Rev. D 79, 125011 (2009); O. Bertolami and C. Zarro, Phys. Rev. D 81, 025005 (2010); C. Bastos, O. Bertolami, N. Dias, and J. Prata, Phys. Rev. D 78, 023516 (2008); O. Bertolami, J. G. Rosa, C. M. L. de Aragao, P. Castorina, and D. Zappala, Phys. Rev. D 72, 025010 (2005); M. Gomes, V. G. Kupriyanov, and A. J. da Silva, Phys. Rev. D 81, 085024 (2010). [6] H. Falomir, J. Gamboa, J. Lopez-Sarrión, F. Méndez, and P. A. G. Pisani, Phys. Lett. B 680, 384 (2009). [7] A. Das, J. Gamboa, F. Méndez, and F. Torres, Phys. Lett. A 375, 1756 (2011). [8] A. Das, H. Falomir, M. Nieto, J. Gamboa, and F. Méndez, Phys. Rev. D 84, 045002 (2011).. p_ ¼ ðx þ 3sÞ þ 2ðp sÞ;. (29). s_ ¼ 3ðp þ xÞ s 3ðL sÞ:. 025009-4.

(5) SPIN NONCOMMUTATIVITY AND THE THREE- . . .. PHYSICAL REVIEW D 85, 025009 (2012). [9] For anyons see for instance S. Ghosh, Phys. Lett. B 338, 235 (1994); S. Ghosh, Phys. Rev. D 51, 5827 (1995). [10] H. Snyder, Phys. Rev. 71, 38 (1947). [11] C. N. Yang, Phys. Rev. 72, 874 (1947). [12] K. H. Bennemann and J. B. Ketterson, in Superconductivity; Conventional and Unconventional Superconductors, edited by K. H. Bennemann and J. B. Ketterson (Springer, New York, 2008), p. 3, Vol. I. [13] E. Witten, Nucl. Phys. B188, 513 (1981); 202, 253 (1982). [14] H. Falomir and P. A. G. Pisani, J. Phys. A 38, 4665 (2005).. [15] In a different context, the harmonic oscillator in a similar kind of noncommutative space has been discussed in A. Parmeggiani and M. Wakayama, Proc. Natl. Acad. Sci. U.S.A., 98, 26 (2000); A. Parmeggiani, Commun. Math. Phys. 279, 285 (2008). [16] J. Gamboa and J. Zanelli, Phys. Lett. B 165, 91 (1985). [17] E. Gozzi, Phys. Lett. B 129, 432 (1983); Phys. Rev. D 33, 3665 (1986). [18] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [19] H. Falomir, J. Gamboa, M. Loewe, and M. Nieto, arXiv:1109.6666.. 025009-5.

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