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The Nonlinear Dirac Equation in Bose-Einstein Condensates

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The Nonlinear Dirac Equation in Bose-Einstein Condensates The Nonlinear Dirac Equation in Bose-Einstein Condensates: Foundation and Symmetries L. H. Haddad and L. D. Carr Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA Abstract We show that Bose-Einstein condensates in a honeycomb optical lattice are described by a nonlinear Dirac equation in the long wavelength, mean field limit. Unlike nonlinear Dirac equations posited by particle theorists, which are designed to preserve the principle of relativity, i.e., Poincar´ecovariance, the nonlinear Dirac equation for Bose-Einstein condensates breaks this symmetry. We present a rigorous derivation of the nonlinear Dirac equation from first principles. We provide a thorough discussion of all symmetries broken and maintained. Key words: Nonlinear Dirac equation, Nonlinear Schrodinger equation, Bose-Einstein condensates, Ultra-cold atoms, Optical lattices, Graphene PACS: 05.45.-a, 03.75.-b, 67.85.Hj 1. Introduction perature and interaction sign, magnitude, and symmetry can be controlled externally. Moreover, 2D physics has Recently the first truly two-dimensional (2D) solid state recently been of great interest in this context, in the form material, graphene, was created in the laboratory [1,2]. One of the Berzinskii-Kosterlitz-Thouless crossover [5], and 2D of the most exciting aspects of this novel material is that systems underpinned by lattices are immediately available long wavelength excitations are described by a Dirac equa- in experiments. Instead of considering ultra-cold fermions, tion for massless particles, with a “speed of light” equal to which could be used to produce an near-exact analog of the Fermi velocity vF c/300 [3]. Thus one can study rela- graphene [6], we consider ultra-cold bosons. Bosonic statis- tivistic phenomena at≃ very low velocities in an experiment tics and interactions lead to a new feature in the massless far more accessible than a particle accelerator. The only real Dirac equation known in graphene – a naturally occurring requirement to obtain this equation is the simple hexago- nonlinear term, giving rise to a nonlinear Dirac equation. nal, or honeycomb lattice structure of the graphene [4]. One The study of nonlinear phenomena in ultra-cold atoms, arXiv:0803.3039v1 [cond-mat.other] 20 Mar 2008 can therefore consider any solid state system constructed especially in Bose-Einstein condensates (BECs) [7,8], on a honeycomb lattice, including artificial systems, in or- has been enormously fruitful. The recent text edited by der to study relativistic phenomena in novel materials ac- Kevrekidis, Frantzeskakis, and Carretero-Gonz´alez pro- cessible in tabletop experiments. vides an excellent summary of this field [9]. The nonlinear The most precise, cleanest, most controllable artificial mean field description given by the nonlinear Schrodinger solid state system is ultra-cold atoms in optical lattices. equation (NLSE) has been very accurate in the majority Such systems have no impurities and no disorder unless of experiments on BECs. Vector and non-local generaliza- specifically added in by hand. They are very versatile: tions of the NLSE have also proven useful [9]. In optical they can be constructed with an arbitrary lattice structure lattices, the mean field description remains accurate pro- in one, two or three dimensions; they can contain bosons vided the lasers creating the standing wave which is the and/or fermions, atoms and/or diatomic molecules; and optical lattice are not too intense, and the dimensionality they can even have a pseudo-spin structure. Their tem- of the system is greater than one [10,11]. In this article, we present a completely new class of non- linear phenomena in BECs, based on the nonlinear Dirac Email addresses: [email protected], [email protected] (L. D. equation (NLDE). Nonlinear Dirac equations have a long Carr). Preprint submitted to Elsevier 15 November 2018 history in the literature, particularly in the context of par- ey ticle and nuclear theory [12,13,14,15], but also in applied mathematics and nonlinear dynamics [16,17,18,19,20]. As nonlinearity is a ubiquitous aspect of Nature, it is natural to ask how nonlinearity might appear in a relativistic set- ting. However, this line of questioning has been strongly constrained by modeling, rather than first principles. That is, there is no standard first principle of quantum electro- B n dynamics (QED) which is nonlinear. So, the approach has 1 been to require symmetry constraints in nonlinear mod- els. One of these constraints is the principle of relativity, i.e., Poincar´ecovariance. Poincar´esymmetry includes ro- A B e tations, translations, and Lorentz boosts. x In contrast, our NLDE is not a model: it is derived from first principles for a weakly interacting bosonic gas in the presence of a honeycomb optical lattice. We show that n Poincar´esymmetry is naturally broken by the nonlinearity B 2 inherent in this system. Given that this form of nonlinear- ity, which depends only on the local condensate density, is one of the most common throughout nature, it is important (a) Hexagonal lattice structure to recognize that the principle of relativity may be broken by small nonlinearities even at a fundamental level, for ex- k ample of QED [21,22]. Thus, we suggest a new direction y B of investigation in particle physics of possible nonlineari- δ ties as well as providing a natural context in artificial solid K 1 state systems for the NLDE. δ3 This article is outlined as follows. In Sec. 2 we provide kx B a rigorous derivation of the NLDE from first principles. In A Sec. 3 we discuss both discrete and continuous symmetries common to relativistic systems, providing a clear physical −K δ interpretation in the present context. Finally, in Sec. 4 we B 2 conclude. (b) Reciprocal lattice (c) Nearest neighbor displace- ment vectors 2. The nonlinear Dirac equation Fig. 1. Characterization of a honeycomb lattice. 2.1. Two-Component Spinor Form of the NLDE A and B. Expanding in terms of Bloch states in the lowest band belonging to A or B sites of the honeycomb lattice, as The second quantized Hamiltonian for a weakly interact- shown in Fig. 1, we can break up the bosonic field operator ing bosonic gas in two spatial dimensions is into a sum over the two sublattices: 2 g 2 ψˆ = ψˆA + ψˆB , (3) Hˆ = d r ψˆ†H0ψˆ + d r ψˆ†ψˆ†ψˆψˆ , (1) Z 2 Z i~k (~r ~rA) ψˆ aeˆ · − u(~r ~r ) , (4) ~2 A ≡ − A 2 A H0 + V (~r) . (2) X ≡−2m∇ i~k (~r ~rB ) ψˆ ˆbe · − u(~r ~r ) , (5) B ≡ − B The bosonic field operators ψˆ = ψˆ(~r, t) obey bosonic com- XB mutation relations in the Heisenberg picture. In Eq. (1), 2 wherea ˆ and ˆb are the time-dependent destruction operators g 4π~ as/m is the coupling strength for binary contact ≡ at A and B sites and ~rA and ~rB are the positions of A interactions with as the s-wave scattering length and m the atomic mass. The external potential V (~r) is a honeycomb and B sites, respectively. The spatial dependence is then lattice formed by standing waves of three sets of counter- encapsulated outside the operator in the exponential and propagating laser beams [6]. The atoms experience this po- the functions u. The summation indices indicate sums over tential via the AC Stark effect. We assume that the third A or B sites. spatial dimension is frozen out by a tightly confining po- Inserting Eq. (3) into Eq. (1), the Hamiltonian can be tential which is locally harmonic, as in Ref. [5]. rewritten The honeycomb lattice has two sites in the lattice unit ˆ 2 ˆ ˆ ˆ ˆ cell. We refer to the resulting two degenerate sublattices as H = d r (ψA† + ψB† )H0(ψA + ψB) Z h 2 g ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ U U + (ψA† + ψB† )(ψA† + ψB† )(ψA + ψB )(ψA + ψB) . (6) + aˆ†aˆ†aˆaˆ + ˆb†ˆb†ˆbˆb . (10) 2 2 2 i XA XB In the integral over H0, imposing the restriction of nearest- neighbor interactions in the tight-binding, lowest band ap- Equation (10) is the Hubbard Hamiltonian divided into proximation eliminates all A-A and B-B transitions except two degenerate sublattices A and B, appropriate to the for on-site kinetic and potential terms; the latter can be ne- honeycomb optical lattice. In order to work towards the nonlinear Dirac equation, glected as an overall self-energy. Then only integrals involv- ˆ ing neighboring A-B sites remain in the sum. Similarly, in we calculate the time evolution ofa ˆ and b according to the the interaction term only on-site terms are non-negligible, standard Heisenberg picture prescription. This is similar i.e., overlap of functions u belonging to the same site. Thus to the approach taken by Pitaevskii in his landmark paper in the tight-binding, lowest band approximation, Eqs. (4)- which first obtained the NLSE, or Gross-Pitaevskii equa- (5) are substituted into Eq. (6) to yield: tion [25]. The Heisenberg equation of motion is i~ ∂taˆk = [ˆak, Hˆ ] . (11) 2 i~k ~χA i~k ~χA Hˆ = d r aˆ† e− · u(~χA)H0aeˆ · u(~χA) Z <A,B>X h The operatora ˆk, which destroys a boson at site k on sub- lattice A, satisfies the bosonic commutation relation i~k ~χA i~k ~χB +ˆa† e− · u(~χA)H0ˆbe · u(~χB) [ˆak, aˆ† ′ ]= δkk′ . (12) i~k ~χB i~k ~χA k +ˆb† e− · u(~χB)H0aeˆ · u(~χA) ~ ~ Then the commutator with the on-site interaction terms ˆ ik ~χB ˆ ik ~χB +b† e− · u(~χB)H0be · u(~χB) reduces to i g 2 4 + d r aˆ†aˆ†aˆaˆ [u(~χA)] [ˆak, aˆk† aˆk† aˆkaˆk]=ˆakaˆk† aˆk† aˆkaˆk aˆk† aˆk† aˆkaˆkaˆk.
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