PACS: Graphene words: Key equa Dirac nonlinear the maintained. of and broken derivation symmetries rigorous a present n equati the We Poincar´e Dirac covariance, i.e., nonlinear relativity, of Unlike principle limit. field mean wavelength, eso htBs-isencnestsi oecm optic honeycomb a in condensates Bose-Einstein that show We Abstract infrmsls atce,wt sedo ih”eulto equal light” of “speed velocity Fermi a the with that particles, equa- is Dirac massless material a for novel by tion described this are of excitations wavelength aspects long exciting most One the [1,2]. laboratory of the in created was graphene, material, Introduction 1. Carr). hycnee aeaped-pnsrcue hi tem- Their structure. pseudo- a and have molecules; even bosons diatomic contain can and/or can they they atoms dimensions; versatile: , three very and/or or structure are two lattice arbitrary one, They an in with unless hand. constructed disorder be by can no they in lattices. and optical added impurities in no specifically atoms have ultra-cold systems is Such system state solid ac- materials experiments. novel tabletop in in phenomena or- cessible relativistic in study systems, to artificial constructed including der system lattice, honeycomb state a solid on any consider One [4]. therefore hexago- graphene the can simple arXiv:0803.3039v1 of the structure lattice is honeycomb or equation nal, this obtain real to only The requirement accelerator. experiment particle an a than in accessible velocities more far low very at phenomena tivistic [cond-mat.other] 20 Mar 2008 rpitsbitdt Elsevier to submitted Preprint mi addresses: Email eetytefis rl w-iesoa 2)sldstate solid (2D) two-dimensional truly first the Recently h otpeie laet otcnrlal artificial controllable most cleanest, precise, most The 54.a 37.b 67.85.Hj 03.75.-b, 05.45.-a, olna ia qain olna crdne equation Schrodinger Nonlinear equation, Dirac Nonlinear h olna ia qaini oeEnti Condensates Bose-Einstein in Equation Dirac Nonlinear The [email protected] v F ≃ c/ eateto hsc,Clrd colo ie,Gle,Co Golden, Mines, of School Colorado Physics, of Department 0 3.Tu n a td rela- study can one Thus [3]. 300 , [email protected] onainadSymmetries and Foundation .H addadL .Carr D. L. and Haddad H. L. nierDrceuto o oeEnti odnae bre condensates Bose-Einstein for equation Dirac onlinear L D. (L. n oie ypril hoit,wihaedsge opre to designed are which theorists, particle by posited ons info rtpicpe.W rvd hruhdiscussion thorough a provide We principles. first from tion lltieaedsrbdb olna ia qaini the in equation Dirac nonlinear a by described are lattice al oeEnti odnae,Utacl tm,Otclla Optical atoms, Ultra-cold condensates, Bose-Einstein , olna em iigrs oa to occurring rise naturally massless giving a the term, – in nonlinear graphene feature in new known a equation of to Dirac analog lead interactions near-exact statis- and Bosonic an bosons. tics ultra-cold produce consider we to [6], used graphene be fermions, could ultra-cold which considering of Instead available experiments. immediately in are lattices form by 2D the underpinned and in systems [5], has context, crossover this physics Berzinskii-Kosterlitz-Thouless in the interest 2D of great Moreover, of externally. been recently controlled and be magnitude, can sign, interaction and perature qain(LE.NnierDrceutoshv long a have Dirac equations nonlinear Dirac the Nonlinear on (NLDE). based equation BECs, in phenomena linear [10,11]. one dimensionality than the greater the is and is system intense, which the too wave of not standing are the lattice optical pro- creating optical In accurate lasers [9]. remains the description useful vided proven field also mean have the NLSE lattices, generaliza- the majority non-local of the and Vector tions in BECs. accurate on very experiments of been Schrodinger has nonlinear the (NLSE) nonlinear by The equation given [9]. description field field this of mean by summary edited excellent pro- Carretero-Gonz´alez an text [7,8], vides recent and The Frantzeskakis, (BECs) fruitful. Kevrekidis, condensates enormously been Bose-Einstein has in especially nti ril,w rsn opeeynwcaso non- of class new completely a present we article, this In atoms, ultra-cold in phenomena nonlinear of study The oao841 USA 80401, lorado olna ia equation Dirac nonlinear k hssymmetry. this aks 5Nvme 2018 November 15 : ttices, ev the serve fall of long . 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 Fig. 1. Characterization of a honeycomb lattice. 2.1. Two-Component 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 . 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 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. (13) 2 Z − XA Taking the first product on the right and commuting the g 2 4 + d r ˆb†ˆb†ˆbˆb[u(~χB)] , (7) furthermost lefta ˆ through according to Eq. (12), one finds: 2 Z XB aˆkaˆk† aˆk† aˆkaˆk = 2ˆak† aˆkaˆk +ˆak† aˆk† aˆkaˆkaˆk. (14) ~χA ~r ~rA , ~χB ~r ~rB , ~χAB = ~rA ~rB . (8) ≡ − ≡ − − Substituting Eq. (14) into Eq. (13), one obtains Here the bracketed A and B summation index signifies a sum over nearest-neighbor A and B sites. Isolating the in- [ˆak, aˆk† aˆk† aˆkaˆk]=2ˆak† aˆkaˆk . (15) tegrals by pulling out all sums and terms not dependent on Substituting Eq. (15) into Eq. (11) and the Hubbard Hamil- ~r, Eq. (7) becomes tonian Eq. (10), one finds

~ ~ ~ ~ ˆ ˆ ik ~χAB 2 ik ~r ik ~r i~k (~r ~r ) ik (~rA ~rB − ) H = aˆ†be · d re− · u(~χA)H0e · u(~χB) i~ ∂ aˆ = t ˆb e Ak Bk + ˆb e · k − k n1  Z t k h k · − k n1 X − h − i~k (~rA ~rB ) ˆ k k−n2 ~ ~ ~ +bk n2 e · − + Uaˆ† aˆkaˆk (16) ˆ ik ~χAB 2 ik ~r ik ~r k + b†aeˆ − · d re− · u(~χB)H0e · u(~χA) − i Z  where the first three terms on the right hand side represent 2 i~k ~r i~k ~r th + aˆ†aˆ d re− · u(~χA)H0e · u(~χA) transitions from the three B-sites nearest the k site of the Z XA A sublattice and ~n1 and ~n2 are primitive cell translation vectors for the reciprocal lattice, as shown in Fig. 1. 2 i~k ~r i~k ~r + ˆb†ˆb d re− · u(~χB)H0e · u(~χB) Z In a similar fashion as Eqs. (11)-(16), we arrive at an XB expression of the same form for the B sublattice: g 2 4 + aˆ†aˆ†aˆaˆ d r[u(~χ )] ~ A i~k (~rA ~rB ) ik (~rA ~rB ) 2 Z i~ ∂ ˆb = t aˆ e− · k − k +ˆa e− · k+n1 − k XA t k h k k+n1 − h g i~k (~rA ~rB ) ˆ ˆ ˆˆ 2 4 k+n2 k ˆ ˆ ˆ + b†b†bb d r[u(~χB )] , (9) +ˆak+n2 e− · − + Ub† bkbk. (17) 2 Z k XB i Continuing to follow Pitaevskii’s method, we next take the Finally, we redefine the spatial integrals in Eq. (9) as hop- expectation value of Eqs. (16) and (17) with respect to on- ping energy th and interaction energy U, respectively, as is site coherent states. A tensor product over sites of such standard for the Hubbard Hamiltonian [23,24] (note that coherent states is also assumed [10]. A more formal, care- we reserve t for time). The terms in Eq. (9) proportional ful treatment of finite number states, rather than coherent ˆ ˆ toa ˆ†aˆ and b†b just count the total number of atoms in the states, has been worked out in the literature (see [26] and system, and can been neglected as an overall constant. This references therein). Either way, we obtain coupled equa- leads to the Hamiltonian tions of motion for discrete, on-site, complex-valued ampli- tudes. For simplicity of notation we take i~k (~rA ~rB ) i~k (~rA ~rB ) Hˆ = th aˆ†ˆbe · − + ˆb†aeˆ − · − − ˆ X h i ak aˆk , bk bk . (18) ≡ h i ≡ h i 3 Inserting the nearest-neighbor vectors ~δ1, ~δ2, and ~δ3 in the Up to now the “hat” symbol (circumflex accent) has been exponentials in Eqs. (16) and (17), as shown in Fig. 1, we reserved for operators alone. However, in Eqs. (26)-(27) we obtain use this symbol to indicate a unit vector in the x and y

~ ~ ~ ~ ~ ~ directions. Derivatives with respect to the unit-cell vectors ~ ik δ3 ik δ1 ik δ2 i a˙ k = th(bk e · + bk n1 e · + bk n2 e · ) take the form − − − +Ua∗a a , (19) k k k ~ √ ~ ~ ~ ~ ~ ~ ∂n1 = ~n1 = ( 3/2)∂x (1/2)∂y , (28) i~ b˙ = t (a e ik δ3 + a e ik δ1 + a e ik δ2 ) · ∇ − k h k − · k+n1 − · k+n2 − · ~ − ∂n2 = ~n2 = (√3/2)∂x + (1/2)∂y . (29) +Ubk∗bkbk , (20) · ∇ Substituting Eqs. (28)-(29) into Eq. (25), where n1 and n2 in the indices label the lattice sites in the two directions of the primitive-cell translation vectors ~n1 i~ ψ˙A = th [(√3/2)∂x (1/2)∂y]ψB [(1/2)+ i(√3/2)] and ~n2. − n − The NLDE is derived around the linear band crossings +[(√3/2)∂x + (1/2)∂y]ψB [(1/2) i(√3/2)] between the A and B sublattices at the Brillouin zone cor- − o ners [27], called Dirac cones in the graphene literature [3]. +UψA∗ ψAψA . (30) To this end, we insert particular values for the nearest- Further simplification of Eq. (30) leads to neighbor displacement vectors ~δ and evaluate ~k at the Bril- ~ ~ ~ louin zone corner, defined by k = K = (0, 4π/3) , δ1 = √ ~ ˙ th 3 ( 1 , 1 ), ~δ = ( 1 , 1 ), ~δ = ( 1 , 0), all in the x y i ψA = (∂xψB i∂yψB)+ UψA∗ ψAψA . (31) 2√3 − 2 2 2√3 2 3 − √3 − − 2 − plane. Then Eq. (19) becomes Similarly, for the continuum limit of b = b (t) ψ = k k → B ~ 0 i2π/3 i2π/3 ψB(~r, t), i a˙ k = th(bk e + bk n1 e− + bk n2 e ) − − − +Uak∗akak . (21) √ ~ ˙ th 3 i ψB = ( ∂xψA i∂yψA)+ UψB∗ ψBψB . (32) Reducing the exponentials, − 2 − − Equations (31)-(32) are in fact massless Dirac equations ~ i a˙ k = th[bk + bk n1 ( 1/2 i√3/2) − − − − with an added nonlinear term. To put this in a more fa- √ +bk n2 ( 1/2+ i 3/2)] + Uak∗akak . (22) miliar form, we use Pauli matrix notation. To this end, one − − must rename the coordinate axes so that x y, and, in In anticipation of taking the long wavelength, continuum order to preserve the handedness of the coordinate→ system, limit, as is necessary to obtain the NLDE, we group terms y x. We also reinsert the lattice constant a; note that a appropriately in Eq. (22) in order to construct discrete ver- →− is unrelated to the s-wave scattering length as briefly men- sions of derivatives: tioned in the definition of g following Eq. (1). Thus ∂ x → ~ √ ∂y and ∂y ∂x. Then Eqs. (31)-(32) become i a˙ k = th[bk + (bk bk n1 )(1/2+ i 3/2) →− − − − √ √ bk(1/2+ i 3/2)+(bk bk n2 )(1/2 i 3/2) tha√3 − ~ ˙ − − − i ψA = (∂yψB + i∂xψB)+ UψA∗ ψAψA , (33) bk(1/2 i√3/2)] + Uak∗akak , (23) − 2 − − √ ~ ˙ tha 3 which reduces to i ψB = ( ∂yψA + i∂xψA)+ UψB∗ ψBψB , (34) − 2 − ~ i a˙ k = th[(bk bk n1 )(1/2+ i√3/2) or in matrix form, − − − √ +(bk bk n2 )(1/2 i 3/2)] + Uak∗akak . (24) − − − ˙ ψA itha√3 0 ∂x i∂y ψA i~   = −  −    Taking the continuum limit and replacing the discrete 2 ψ˙B ∂x + i∂y 0 ψB quantities ak = ak(t) and bk = bk(t) by the continuous       functions ψA = ψA(~r, t) and ψB = ψB(~r, t), we arrive at ψA∗ ψAψA +U   . (35) ~ ˙ i ψA = th ∂n1 ψB(1/2+ i√3/2)+ ∂n2 (1/2 i√3/2) ψB∗ ψBψB − h − i   +UψA∗ ψAψA . (25) We can write Eq. (35) more compactly in terms of Pauli where the partial derivatives are in the directions of the matrices (σx, σy)= ~σ, unit-cell vectors ~n1 and ~n2. With a little trigonometry we ˙ find that the unit-cell vectors are ψA itha√3 ψA ψA∗ ψAψA i~   = − ~σ ~   + U  . (36) ψ˙ 2 · ∇ ψ ψ ψ ψ ~n = cos(π/6)ˆe sin(π/6)ˆe = √3/2ˆe 1/2ˆe , (26) B B B∗ B B 1 x − y x − y       ~n2 = cos(π/6)ˆex + sin(π/6)ˆey = √3/2ˆex +1/2ˆey . (27) Equation (36) is the NLDE in (2+1) dimensions.

4 However, we can make one further step by expressing To this end, we seek to put the NLDE into a form con- Eq. (36) in a more covariant-looking form as follows: sistent with the standard compact four-component spinor notation for the linear Dirac equation ψA ψA∗ ψAψA iγµ∂ Ψ=0 , (42) (iσ0∂t + ics~σ ~ )   U   =0 , (37) µ · ∇ − ψB ψB∗ ψBψB µ     where the γ are the 4 4 Dirac matrices [31]. We point out that this notation is not× only standard but more appropri- where ~σ and ~ are restricted to the x y plane. In Eq. (37) ∇ − ate if the quasi-particles, i.e., the long-wavelength excita- tions, develop a non-zero effective mass, as can be caused t a√3 h by lattice distortion [6]. cs ~ (38) ≡ 2 We obtained the NLDE by evaluating the exponentials ~ is an effective speed of sound of the condensate in the at the Brillouin Zone corner K+ = (0, 4π/3). There is lattice[28,29] (in graphene it would be replaced with the another inequivalent corner, as shown in Fig. 1, K~ − ≡ Fermi velocity [3]). This velocity is an effective speed of (0, 4π/3), near which perturbations in momentum, i.e., − light for excitations of the NLDE in our QED2+1 theory. long-wavelength quasiparticle excitations, are governed by Experimental values of cs in BECs areon the orderof cm/s, a similar first-order wave equation. By considering equa- ten orders of magnitude slower than the speed of light in a tions of motions derived around both Brillouin corners, we vacuum. Note also that in Eq. (37) U has now absorbed a can obtain the four-vector notation. The coupled equations factor of 1/~. evaluated at K~ are − Finally, a few additional definitions lead to a nicely com- tha√3 pact form for the NLDE. Let i~ψ˙ = − (∂ + i∂ )ψ + Uψ∗ ψ ψ , (43) A 2 x y B A A A √ 1 0 0 0 ψ ~ ˙ tha 3 A ¯ i ψB = − ( ∂x + i∂y)ψA + UψB∗ ψBψB . (44) A  ,B  , ψ  , ψ ψ∗ , ψ∗ . (39) 2 − ≡ ≡ ≡ ≡ A B 0 0 0 1 ψB h i       Following the same steps as before we obtain With the choice of metric which raises and lowers space- ˙ ψB itha√3 ψB ψB∗ ψBψB time indices restricted to (2+1) dimensions, i~   = ~σ ~   + U  . (45) ˙ 2 · ∇ ψA ψA ψA∗ ψAψA       10 0   We combine Eqs. (45) and (36) into one equation involving gµν = 0 1 0 , (40) a single 4-component object and attach subscripts to  −  ±   the wave functions Ψ to specify the corner of the Brillouin  0 0 1   −  Zone. The resulting expression is Eq. (37) becomes Ψ+ ~σ ~ 0 Ψ+ i∂t   + ics  · ∇    µ ~ (iσ ∂µ UψAψA¯ UψBψB¯ )ψ =0 , (41) Ψ 0 ~σ Ψ − −  −   − · ∇   −  where the standard Einstein summation rule is in effect, N+ µ 0, 1, 2 in keeping with (2+1) dimensions, and the U   =0 . (46) ∈ { } − N units are chosen such that cs = 1.  −  where the nonlinear terms are grouped into the 2-vectors N+ and N defined by 2.2. Maximally Compact Form of the NLDE −

(ψA∗ ψAψA)+ (ψB∗ ψB ψB) In Sec. 2.1 we developed ψ, a two-dimensional complex N+ =   , N =  −  (47) − object which brings to mind one member of a pair of Weyl- (ψB∗ ψBψB)+ (ψA∗ ψAψA) − in the (1/2,1/2) chiral representation of the Dirac     and the four-spinors are given by algebra used to describe massless neutrinos in the stan- dard QED3+1 theory. Such a treatment is appropriate for ψA+ any neutral Dirac viewed in the extreme relativis-   tic frame. In order to make the connection clear we must Ψ+ ψB+ Ψ     , (48) find the second member of the pair of Weyl-spinors and ver- ≡ ≡   Ψ  ψB  ify that the mapping is true. A thorough treatment of the  −   −    mapping of QED3+1 into the QED2+1 theory of graphene  ψA  is contained in [30] and references therein. We will continue  −  to restrict ourselves to (2+1) dimensions in the following. Ψ† Ψ+∗ Ψ∗ ψA∗ + ψB∗ + ψB∗ ψA∗ . (49) ≡ h − i ≡ h − − i 5 Here again the subscripts refer to the specific corner of In what follows we follow a similar route as in Ref. [15]. the Brillouin zone.± First we check the linear version with the delta interactions We can reduce Eq. (46) to a more compact form by in- turned off to ensure that Eq. (41) is indeed the massless troducing the following 4 4 matrices: Dirac equation in (2+1) dimensions with all the necessary × symmetries. For each symmetry we then check the nonlin-

1 0 σx 0 σy 0 ear interaction terms to determine whether they preserve Σ0 =   , Σ1 =   , Σ2 =   , (50) or break the symmetry. 0 1 0 σx 0 σy    −   −  A 0 0 0 3.1. Locality A+ =   , A =   , (51) 0 0 − 0 A     We do not necessarily require the evolution of the wave- B 0 0 0 function as described by an NLDE to be governed by a lo- B+ =   , B =   . (52) cal theory. A local theory is one in which the terms in the 0 0 − 0 B linear equations of motion involve only factors of the wave-     function and its derivatives evaluated at the same space- The boldface notation A and B denote the 2 2 matrices × time coordinate. Nonlocality arises in low energy limits of defined in Eq. (39). Also, the boldface entries 1 and 0 in some quantum field theories. However, the nonlinearity in the matrices in Eq. (47) refer to the 2 2 unit matrix and × our NLDE is manifestly local. Thus our NLDE is closer to zero matrix, respectively [15]. Then Eq. (46) becomes the standard Dirac equation on the classical level (no quan-

µ tum effects), modified by the on-site interaction term. In (iΣ ∂ UΨ†A ΨA UΨ†B ΨB µ − + + − + + Sec. 4 we discuss the possibility of non-local nonlinearities, UΨ†A ΨA UΨ†B ΨB )Ψ=0 . (53) including for graphene. − − − − − − We substitute into Eq. (53) the Dirac matrices in the Chiral 3.2. Poincar´eSymmetry representation:

0 1 0 0 A Poincar´e transformation takes the spatiotemporal 0 1 σx 2 σy γ   ,γ  −  ,γ =  −  (54) point defined by the the 4-vector rν into the point r′ ≡ ≡ µ 1 0 σx 0 σy 0 according to       0 ν and multiply on the left by γ . Note that µ 0, 1, 2 in rµ′ =Λµrν + dµ (57) keeping with (2+1) dimensions. Finally, for the∈ { nonlinear} where Λν is the coordinate matrix representation of the term we introduce µ Lorentz group and dµ is a space-time translation. The wave N UΨ† QΨQ. (55) function Ψ transforms as ≡− Q A+,AX−,B+,B− ∈ Ψ′(r ′)= M(Λ)Ψ(r) Implementing Dirac matrix notation as described, the NLDE of Eq. (53) becomes where the matrices M(Λ) form a representation of the sub- group of the Lorentz group consisting of spatial rotations µ 0 (iγ ∂µ + γ N)Ψ=0 . (56) and boosts. Boosts can be thought of as rotations in imag- inary time by imaginary angles mixing space and time co- Equation (56) is the most compact form of the massless ordinates. We restrict these transformations to (2+1) di- NLDE in 4-component spinor notation. mensions. The proof of of the standard massless 3. Symmetries and Constraints Dirac equation, Eq. (42), is arrived at with the aid of the transformations for the wave function and partial deriva- The NLDE as expressed by Eq. (37) or Eq. (41) looks tives: like the Dirac equation for a two-spinor, as stated in the 1 Ψ(r)= M − (Λ)Ψ′(r′) , (58) graphene problem [3], with the addition of two nonlinear j ∂ =Λ ∂ ′ . (59) on-site interaction terms, one for each A and B sublattice; xi i xj similarly, the more compact form developed in Sec. 2.2 also This yields the conditions for the Dirac matrices: appears to be a Dirac equation for a four-component spinor, with an additional term. However, one should not be too 1 γj =ΛjiMγiM − . (60) hasty in assigning characteristics based on appearances. We therefore make a careful and thorough exploration of the The standard form of the Dirac matrices obtained this way symmetries and other important mathematical properties can be found in the literature [31] and are identical to the of the NLDE. results of our theory.

6 ~ ~ Thus imposing Lorentz covariance on the NLDE as iK− δ3 +ak 2n1 e− · )+ Ubk∗ 2n1 bk 2n1 bk 2n1 . (72) expressed in Eq. (56) requires Ψ to transform under − − − − the irreducible representation of a subgroup of SL(2,C), Now we redefine the index k: in Eq. (71), k k +n1, while in Eq. (72), k k +2n . Then → the 2 2 complex matrices of unit determinant. The → 1 × (1/2,1/2) four-dimensional representation of SL(2,C), D , is ~ ~ ~ ~ ~ iK− δ1 iK− δ2 i a˙ k = th(bk n1 e · + bk n2 e · formed by taking the direct product of the two-dimensional − − − (1/2,0) (0,1/2) iK~ − ~δ3 representations D and D : +bk e · )+ Uak∗akak (73) ~ ~ ~ ~ (1/2,1/2) (1/2,0) (0,1/2) ~ ˙ iK− δ1 iK− δ2 D = D D . (61) i bk = th(ak+n2 e− · + ak+n2 e− · ⊗ − iK~ − ~δ3 This subgroup of SL(2,C) is isomorphic to a subgroup of +ak e− · )+ Ubk∗bkbk (74) the Lorentz group, the one obtained by restricting Lorentz To summarize, a rotation by π/3 which takes K~ + K~ ′ transformations to the plane of the honeycomb lattice. So, → + is identical to the unrotated theory but with K~ + K~ . the upper two components of Ψ transform as a spinor under − Note that the redefinition of the index k is different→ for the D(1/2,0), the lower two as a spinor under D(0,1/2), and Ψ two equations because old and new primitive cells are re- itself as a four-component spinor or bispinor [32]. lated by a rotation. Rotating by 2π/3 exchanges A and B The next task is to examine the behavior of Ψ, as defined sites once more and returns Ψ to± its original configuration. in Eq. (49) and governed by Eq. (56), under rotations in Since Ψ has four components, the effect of making one full the x y plane. In order to obtain Poincar´ecovariance we rotation of 2π is that the components acquire a net phase must show− that it is the same as that of a four-component so that Ψ Ψ. This Berry phase [2] endows Ψ with the spinor in the standard Dirac theory restricted to the 2D 3+1 characteristic→− double-valuedness of a genuine 4-spinor. plane. The honeycomb lattice is invariant under rotations However, we must be cautious when discussing chirality by 2π/3 but the four components of Ψ are also defined and helicity, since we treat (2+1) dimensions and can use by the± particular corner of the Brillouin Zone. Since we’re only the first two Pauli matrices. As in the (3+1) theory, one considering a discrete lattice it is natural to discuss discrete can define a pseudo-chirality operator in (2+1) dimensions, rotations which realign lattice points and in the continuum γ5, as the product of the other four γ matrices. In the Weyl, limit map the continuous rotations of QED onto our 2+1 or Chiral representation we have theory. Rotations by π/3 exchange A and B sites, and ± ~ ~ take the theory to that of the opposite K point: K+ does 1 0 not go to K~ but the result after calculating the relative γ5 iγ0γ1γ2γ3 =   , (75) − ≡ phase exponentials gives back the same theory. To see this 0 1  −  we chose a different primitive unit cell, the one obtained where again the boldface indicates a 2 2 submatrix. This by a rotation of 2π/3 about the ~n , ~n origin in Fig 1. This 1 2 is the natural representation for Ψ in× that the NLDE of is because the direction of K~ (defined as K~ rotated by +′ + Eq. (56) maps into this representation in a natural way: π/3) differs from that of K~ by 2π/3. − states of well defined chirality correspond to the upper and Thus under this discrete rotation lower 2-spinors. Thus the upper and lower spinors

bk bk 2n1 , (62) → − ψ+ 0 Ψ+   , Ψ   (76) bk n1 bk n1 n2 , (63) − → − − ≡ 0 − ≡ ψ − bk n2 bk n1 , (64)     − → − are eigenfunctions of γ5, ak ak n1 , (65) → − 5 5 ak+n1 ak 2n1 +n2 , (66) γ Ψ+ =Ψ+ , γ Ψ = Ψ . (77) → − − − − ak+n2 ak 2n1 . (67) → − The question of Poincar´ecovariance of the nonlinear Also we observe that Dirac equation remains. First, we check coordinate trans- lations. The wave function is required to transform as ~ ~ ~ ~ K+′ δ3 = K δ1 (68) · − · Ψ′(r + d )=Ψ(r ) . (78) ~ ~ ~ ~ µ µ µ K+′ δ1 = K δ2 (69) · − · ~ ~ ~ ~ Thus the nonlinear terms which have two factors of Ψ are K+′ δ2 = K δ3 (70) · − · invariant under translations. Under spatial rotations we ob- Substituting Eqs. (62)-(70) into Eqs. (19)-(20) we obtain serve that the interaction terms in the NLDE remain un- changed within the context of the full theory. We include ~ ~ ~ ~ ~ iK− δ1 iK− δ2 ~ ~ i a˙ k n1 = th(bk 2n1 e · + bk n1 n2 e · both K+ and K points in the full theory. As for the case − − − − − − iK~ − ~δ3 of boosts, we note that for the linear equation the compo- +bk n1 e · )+ Uak∗ n1 ak n1 ak n1 , (71) − − − − nents of the wave function transform in accordance with ~ ~ ~ ~ ~ ˙ iK− δ1 iK− δ2 i bk 2n1 = th(ak n1 e− · + ak 2n1+n2 e− · the transformation of the space-time coordinates in their − − − − 7 arguments. This is exactly canceled by the reciprocal trans- where µ 0, 1, 2 and we have used the chain rule and the formations of the partial derivatives. This is not the case anti-commuting∈{ properties} of the Dirac matrices. Taking for the nonlinear terms. One has some matrix product with the adjoint of the NLDE in four-component form, Eq. (56), two factors of the wave function. Thus the nonlinear Dirac yields equation is not invariant under Lorentz boosts. µ 0 For any truly fundamental theory of nature it is de- (iγ ∂µΨ)† = ( γ NΨ)† , (82) − manded that the governing equations be invariant with re- which implies spect to the Poincar´egroup. Sometimes this requirement µ 0 is loosened as in the of gauge theories, when ∂µΨ†γ = iΨ†N †γ . (83) fixing the gauge causes relativistic invariance to be non- − − manifest. Yet the theory itself certainly remains invariant. Then the (2+1)-divergence of the current is

Other times the breaking of Poincar´ecovariance implies µ 0 0 0 0 ∂ j = iΨ†N †γ γ Ψ+Ψ†γ (iγ NΨ) (84) that we are dealing with an effective theory in which deeper µ − physical processes are at work. Our case is an example of = iΨ†(N † N)Ψ = 0 (85) − − the latter. By including on-site interactions we break rela- tivistic covariance of the linear equation of motion by in- Thus current is conserved. This is in fact the statement troducing self-interactions which can be viewed as evidence that hermiticity implies current conservation. of “deeper physics” in our two-dimensional universe. 3.5. Chiral Current 3.3. Hermiticity The conserved chiral current for the linear Dirac equation is We require the Hamiltonian be Hermitian in order to guarantee that physically measurable quantities (observ- µ ¯ µ j5 Ψγ γ5Ψ . (86) ables) are real. Thus each term must be independently Her- ≡ mitian. In particular, we must show that N † = N. The Then the (2+1)-divergence of the chiral current is proof follows: µ ¯ µ ∂µj5 = ∂µ(Ψγ γ5Ψ) . (87) N † = (UΨ†A+ΨA+)† Following a similar route as in Sec. 3.4, we find

= UA+† Ψ†A+† (Ψ†)† µ ∂µj5 = iΨ†(Nγ5 + γ5N)Ψ (88) = UA+Ψ†A+Ψ − where we have used the hermiticity of N. This means that = UΨ†A ΨA + + in order for chiral current to be conserved we must have = N (79) N,γ5 =0 (89) where the last two steps work because A+ is real and sym- { } metric. The nonlinear terms, and therefore the NLDE, are where the curly braces signify the anti-commutator. The indeed Hermitian. anti-commutator is

N ,γ = UΨ†A ΨA γ + γ UΨ†A ΨA . (90) 3.4. Current Conservation { + 5} + + 5 5 + + Writing this out in explicit matrix form we find that Current conservation is expected in ordinary QED for a N ,γ =+2N, (91) closed system, i.e., an isolated volume of space. However, { + 5} most theories of do introduce Lorentz vio- N ,γ5 = 2N. (92) lating terms which bring in charge non-conservation effects. { − } − The conserved current for the linear Dirac equation is Therefore chiral current is not conserved. This fact is remi- niscent of the anomalous non-conservation of chiral current jµ = Ψ¯ γµΨ (80) in the case of some field theories upon quantization, which is mediated by instantons. It is interesting to consider that ¯ 0 where Ψ Ψ†γ . our nonlinear terms might be treated as quantum-induced ≡ We check that current is also conserved for the NLDE. nonlinearities. The 3-divergence of the current in (2+1) spatial dimensions is 3.6. Universality µ µ ∂µj = ∂µ(Ψ¯γ Ψ) 0 µ Universality refers to the invariance of a theory under = ∂µ(Ψ†γ γ Ψ) rescaling of the solution. For the case of the linear Dirac µ 0 0 µ = (∂ Ψ†)γ γ Ψ+Ψ†γ γ ∂ Ψ (81) equation, Ψ λΨ leaves the theory unchanged. Because − µ µ → 8 the nonlinear term in the NLDE contains one factor of Ψ 3.7.2. Charge Conjugation and one factor of its adjoint, it scales as λ2, thus breaking In order to maintain symmetry under charge conjugation the invariance of the nonlinear theory. For a treatment of one requires the nonlinear term to transform as NLDEs which are universal in this sense, but not relevant 0 ˆ 0 ˆ 1 to the present solid state system, see Ref. [15]. (γ N)′ = C γ N ∗C − (98) where in the Chiral representation the charge conjugation 3.7. Discrete Symmetries operator is

Cˆ 1 3.7.1. = γ In the case of the standard massless Dirac equation, in- and the wavefunction transforms in the standard way: variance under the parity operation Pˆ requires Ψ to trans- 1 0 T form as Ψ γ γ Ψ† . (99) → 0 Ψ Ψ′ = Pˆ Ψ= γ Ψ , (93) We determine if the NLDE maintains this symmetry. Let → F be one of the nonlinear terms. Then where, in contrast to the Chiral representation presented 0 0 1 0 T † 1 0 T in Eq. (54), we take γ in this section to be in the Dirac (γ N)′ = (γ γ Ψ† ) A+(γ γ Ψ† )A+ representation, T 0 1 1 0 =Ψ γ γ A+γ γ Ψ∗A+ (100) 1 0 Also, we have γ0 =   (94) 1 0 1 Cˆ 0 Cˆ 1 0 ∗ 1  −  γ N ∗ − = γ γ (Ψ†A+ΨA+) γ 1 0 T 1 We proceed to determine if the NLDE remains invariant = γ γ Ψ A+Ψ∗A+γ (101) under the transformation of Eq. (93). The parity operator Thus the NLDE breaks charge conjugation symmetry. acting on the honeycomb lattice inverts both coordinate axes, ~ ~ , and thus exchanges A and B sites while also exchanging∇→−∇ K~ -point indices. The transformed linear 3.7.3. Time Reversal equations (U = 0) are The usual time reversal, t t, requires that Ψ trans- form as → − ˙ ~ 0 ˆ 1 3 0 Ψ ~σ 0 Ψ Ψ Ψ′ = ΘΨ = iγ γ Ψ , (102) iγ  −  + ics  − · ∇  γ  −  =0 . (95) → ˙ ~ Ψ+ 0 ~σ Ψ+ where Θˆ is the time-reversal operator and    · ∇    Interchanging upper and lower spinors we obtain the equiv- σy 0 alent form iγ1γ3 =  −  , (103) 0 σy  −  Ψ˙ + ~σ ~ 0 Ψ+ i   + ics  · ∇    =0 . (96) In our theory the intrinsic effect of time reversal is to change Ψ˙ 0 ~σ ~ Ψ the direction of momentum so that K~ points are switched − − · ∇ −       without exchanging A and B indices. Combining these ef- Thus the linear equations are invariant under parity. fects, we determine whether or not the linear part of the Next we check the nonlinear term in the NLDE. With the NLDE, i.e., the linear Dirac equation, remains invariant: transformed nonlinear term added to Eq. (95) we obtain

∂ 1 3 Ψ ˙ ~ i iγ γ  −  Ψ ~σ 0 Ψ − ∂t i  −  +ics  − · ∇   −  Ψ+   Ψ˙ + 0 ~σ ~ Ψ+ − · ∇ − ~       ~σ 0 1 3 Ψ +ics  · ∇  iγ γ  −  =0 (104) N ~ U  −  =0 , (97) 0 ~σ Ψ+ −  − · ∇    N+  −  Then which after similar steps as above gives us back the NLDE. ~ Therefore parity is a symmetry of the nonlinear Dirac equa- ∂ Ψ+ ~σ 0 Ψ+ i   + ics  · ∇    =0 (105) tion. We note that parity is conserved in standard QED − ∂t Ψ 0 ~σ ~ Ψ and also in the theory of the strong interactions, but vio-  −   − · ∇   −  lated by weak interactions. Therefore, in this aspect, the The appearance of the negative sign in front of the time NLDE is consistent with QED. derivative indicates that these are the equations satisfied

9 by the negative energy solutions, or holes. Thus time rever- be looked for in under the proper sal takes the equations describing into holes and circumstances [21]. those for holes into electrons but keeps the overall theory All generalizations of the scalar nonlinear Schrodinger invariant. equation relevant to BECs are candidates for generalized We proceed to consider the nonlinear term. With the nonlinear Dirac equations. For example, pseudo-spin struc- transformed nonlinear term we obtain ture leads to a vector NLSE. One can therefore anticipate a vector NLDE as well. ∂ Ψ+ ~σ ~ 0 Ψ+ We have thoroughly explored both continuous and dis- i   + ics  · ∇    crete symmetries of the NLDE. In particular, we showed − ∂t Ψ 0 ~σ ~ Ψ  −   − · ∇   −  that pseudo-chiral current is not conserved, the NLDE is not covariant under Lorentz boosts, and it breaks charge N+ iU   =0 . (106) conjugation as well as time reversal symmetry. On the other − N hand, the NLDE is hermitian, local, conserves current, and −   is symmetric under parity. The NLDE maintains CPT sym- In Eq. (106) the interaction term has acquired a factor of i. metry, just as one finds in the . Therefore the NLDE is not invariant under time reversal. In a future work we will treat soliton and vortex solutions Thus, as one also observesin the Standard Model of particle of the NLDE, as a first step towards a complete classifica- physics, CP and T are not conserved independently but tion of nonlinear relativistic phenomena in BECs. CPT is conserved. We acknowledge useful discussions with Alex Flournoy and Mark Lusk. This work was supported by the National Science Foundation under Grant PHY-0547845 as part of 4. Discussion and Conclusions the NSF CAREER program.

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