Ultracold Fermions in a Graphene-Type Optical Lattice
Kean Loon Lee, Benoit Gremaud, Rui Han, Berthold-Georg Englert, Christian Miniatura
aa r X i v : . [ qu a n t - ph ] J un Ultracold Fermions in a Graphene-Type Optical Lattice
Kean Loon Lee,
1, 2, 3, ∗ Benoˆıt Gr´emaud,
2, 1, 4
Rui Han, Berthold-Georg Englert,
1, 4 and Christian Miniatura
5, 1, 4 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore Laboratoire Kastler Brossel, Ecole Normale Sup´erieure,CNRS, UPMC; 4 Place Jussieu, 75005 Paris, France NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore Institut Non Lin´eaire de Nice, UMR 6618, UNS,CNRS; 1361 route des Lucioles, 06560 Valbonne, France (Dated:
22 OCT 18 at 5:13 )Some important features of the graphene physics can be reproduced by loading ultracold fermionicatoms in a two-dimensional optical lattice with honeycomb symmetry and we address here its ex-perimental feasibility. We analyze in great details the optical lattice generated by the coherentsuperposition of three coplanar running laser waves with respective angles 2 π/
3. The correspondingband structure displays Dirac cones located at the corners of the Brillouin zone and close to half-filling this system is well described by massless Dirac fermions. We characterize their properties byaccurately deriving the nearest-neighbor hopping parameter t as a function of the optical latticeparameters. Our semi-classical instanton method proves in excellent agreement with an exact nu-merical diagonalization of the full Hamilton operator in the tight-binding regime. We conclude thatthe temperature range needed to access the Dirac fermions regime is within experimental reach. Wealso analyze imperfections in the laser configuration as they lead to optical lattice distortions whichaffect the Dirac fermions. We show that the Dirac cones do survive up to some critical intensity orangle mismatches which are easily controlled in actual experiments. In the tight-binding regime, wepredict, and numerically confirm, that these critical mismatches are inversely proportional to thesquare-root of the optical potential strength. We also briefly discuss the interesting possibility offine-tuning the mass of the Dirac fermions by controlling the laser phase in an optical lattice gener-ated by the incoherent superposition of three coplanar independent standing waves with respectiveangles 2 π/ PACS numbers: 03.75.Lm, 03.75.Ss, 37.10.Jk, 71.10.Fd
I. INTRODUCTION
In 2004, researchers in Manchester isolated one-atomthick sheets of carbon atoms, with the atoms organizedin a planar honeycomb structure [1]. Such a mate-rial is referred to as graphene and is of utmost im-portance in condensed-matter physics since by stackingit one gets the graphite structure, and by wrapping itone gets carbon nanotubes and fullerenes [2]. Grapheneis also of great theoretical interest because it providesa physical realization of two-dimensional field theorieswith quantum anomalies [3]. Indeed, the effective the-ory that describes the low-energy electronic excitations ingraphene is that of two-dimensional massless Weyl-Diracfermions. In graphene these massless fermions propagatewith about one 300th of the speed of light. Triggered bythe Manchester discovery, an intense activity has flour-ished in the field, and continues to flourish, as witnessedby Refs. [4, 5, 6, 7, 8, 9], for example. The reportedand predicted phenomena include the Klein paradox (theperfect transmission of relativistic particles through highand wide potential barriers) [7], the anomalous quantumHall effect induced by Berry phases [10, 11], and its cor- ∗ Electronic address: [email protected] responding modified Landau levels [12].It is now well established that some condensed-matterphenomena can be reproduced by loading ultracoldatoms into optical lattices [13, 14]. The great advan-tage is that the relevant parameters are accessible foraccurate control (shape and strength of the light poten-tial, atom-atom interaction strength via Feshbach res-onances [15], etc.) while spurious effects that destroythe quantum coherence are absent, such as the analog ofthe electron-phonon interaction. Our present objectiveis to analyze a scheme capable of reproducing in atomicphysics the unique situation found in graphene [16]. Itconsists of creating a two-dimensional honeycomb opti-cal lattice and loading it with ultracold fermions like theneutral Lithium-6 or Potassium-40 atoms.Parts of this paper recall known results. In addi-tion to the need of setting the stage and introducingthe notational conventions, there is also the intention tobridge the solid-state community and the atomic physicscommunity on the particular subject of massless Diracfermions as observed in graphene sheets and its counter-part in atomic physics. We also present extensions ofprevious solid-state works in the atomic physics contextand report a number of new results.We analyze the various experimental parameters thatneed to be controlled in order to reproduce, with coldatoms trapped in an optical lattice, the physics at workin graphene. After briefly introducing optical lattices,we first explain how to create an optical lattice with thehoneycomb symmetry and analyze its crystallographicfeatures. We then calculate the band structure in thetight-binding approximation and by exact diagonaliza-tion, thereby providing evidence for the occurrence of theso-called Dirac points. Next, we evaluate the nearest-neighbors hopping amplitude by using a semi-classicalinstanton method. For the benefit of possible experi-ments we give the necessary requirements for reachingthe massless Dirac fermions regime. Finally, we examinehow massless Dirac fermions survive lattice distortionsthat could result from intensity-unbalanced or misalignedlaser beams. These distortions open the way to newphysics related to the quantum Hall effect [17]. We willclose by briefly mentioning possible experiments to tar-get for noninteracting and interacting ultracold fermions[18, 19].
II. THE HONEYCOMB OPTICAL LATTICEA. Radiative forces and optical lattices
A two-level atom (with angular frequency separation ω at and excited-state angular frequency width Γ) thatinteracts with a monochromatic laser field with complexamplitude E ( r , t ) = E ( r ) e − i ω L t gets polarized and ex-periences radiative forces due to photon absorption andemission cycles [20, 21]. When the light frequency istuned far away from the atomic resonance, i.e., whenthe light detuning δ = ω L − ω at is much larger thanΓ, the field-induced saturation effects are negligible andthe atom essentially keeps staying in its ground state.In this situation, the atom-field interaction is dominatedby stimulated emission processes where the atomic dipoleabsorbs a photon from one Fourier component of the fieldand radiates it back into the same or another one of theseFourier modes. In each such stimulated cycle, there is amomentum transfer to the atom and, as a net result, theatom experiences an average force in the course of time.This dipole force exerted by the field onto the atom inits ground state is conservative. It derives from the po-larization energy shift of the atomic levels (AC Stark orlight shifts) [22] and the dipole potential V ( r ) is givenby V ( r ) = ~ Γ8 Γ δ I ( r ) I s , (1)where I ( r ) = ǫ c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / r of the atom and I s is the saturation in-tensity of the atom under consideration.For multi-level atoms, the situation is more compli-cated as the dipole potential now depends on the par-ticular atomic ground state sub-level under considera-tion. However, if the laser detuning δ is much larger than the fine and hyperfine structure splittings of theatomic electronic transition, then all ground state atomicsub-levels will essentially experience the same dipole po-tential. This common potential turns out to be given by(1) as well. Hence, by conveniently tailoring the spaceand time dependence of the laser field, one can producea great variety of dipole potentials and thus manipulatethe ground state atomic motion.Optical lattices are periodic intensity patterns of lightobtained through the interference of several monochro-matic laser beams [23]. By loading ultracold atoms intosuch artificial crystals of light one obtains periodic arraysof atoms. Indeed, as seen from (1), when the light fieldis blue-detuned from the atomic resonance ( δ > δ <
0) they can be trappedat the field intensity maxima. Such arrays of ultracoldatoms trapped in optical lattices have been used in awide variety of experiments. As recently evidenced bythe observation of the Mott-Hubbard transition with de-generate gases [24], they have proven to be a unique toolto mimic, test and go beyond phenomena observed un-til now in the condensed-matter realm [14, 25]. Theyalso have a promising potential for the implementationof quantum simulators and for quantum information pro-cessing purposes [13, 26, 27].
B. Optical lattice with honeycomb structure
1. Field configuration and associated dipole potential
The simplest possible optical lattice with honeycombstructure is generated by superposing three coplanartraveling plane waves that have the same angular fre-quency ω L = ck L , the same field strength E >
0, thesame polarization and the three wave vectors k a form atrine: their sum vanishes and the angle between any twoof them is 2 π/ k + k + k = 0 , k a · k b = k L (cid:16) δ ab − (cid:17) (2)with a, b = 1 , , δ ab is the Kronecker symbol [23].As is illustrated in Fig. 1, we choose the x, y -plane as thecommon plane of propagation and, to be specific, use k = k L e y , k k (cid:27) = k L ∓√ e x − e y E a ( r , t ) = E e i( k a · r − φ a ) e − i ω L t e z (4)where φ a is the phase of the a th field for t = 0 at r = 0.We note that a joint shift of the reference points in time xyz k k k π π π FIG. 1: The coplanar three-beam configuration used to gen-erate the honeycomb lattice. All beams have the same fre-quency, strength and linear polarization orthogonal to theircommon propagation plane. The honeycomb lattice underconsideration is obtained for blue-detuned beams with respec-tive angles 2 π/
3. For these symmetric laser beams, the time-averaged radiation pressure — albeit small at large detuning— vanishes in this configuration. By reversing the propaga-tion direction of one of the lasers, such that k = k + k , say,a triangular lattice of a different geometry is formed. We will,however, exclusively deal with the k + k + k = 0 case. and space, t → t − ω L X a φ a , r → r + 23 k L X a φ a k a , (5)removes the phases φ a from (4), so that the simple choice φ = φ = φ = 0 is permissible, and we adopt this con-vention. In an experimental implementation, one wouldneed to stabilize the phase differences φ a − φ b to preventa rapid jitter of the lattice that could perturb the atomstrapped in the potential minima.The dipole potential (1) generated by the electric field E = P a E a is of the form V ( r ) = V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = V v ( r ) with V = ~ Γ8 Γ δ I I s , (6)where I is the intensity associated with the field strength E . The total dimensionless field amplitude f ( r ) and thedimensionless optical potential v ( r ) are given by f ( r ) = 1 + exp( − i b · r ) + exp(i b · r ) (7)and v ( r ) = 3+2 cos( b · r )+2 cos( b · r )+2 cos (( b + b ) · r ) , (8)where b = k − k and b = k − k feature the recip-rocal primitive vectors. For the parameterization (3), wehave b b (cid:27) = κ e x ∓ √ e y κ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = √ k L . One may further notice that theperiodic patterns associated to each of the cosine termsin (8) have the same spatial period of (2 π/k L ) / √
3, about58% of the laser wavelength λ L = 2 π/k L . ...................................................................................................................... ........................................................................................................................ s ss ............................................................................................................................................................................................................................................................................................................................................................... ................. ............... ........ ........................................................................................... ........................................................................................... s s ........................................................................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................. . ............................................................................................................................................. s s Γ b b k k k K K ′ K K ′ K K ′ FIG. 2: The triangular reciprocal lattice ˜ B associated withthe triangular Bravais lattice of Fig. 4. It is spanned by thereciprocal primitive vectors b and b of (9), and is also atriangular lattice (as indicated by the full dots). The shadedregion identifies the first Brillouin zone Ω which is here a reg-ular hexagon. Its center is conventionally named Γ in thesolid-state literature. Opposite edges are in fact identical asthey only differ by a translation in the reciprocal lattice. Thisfeature is emphasized by drawing the identical edges with thesame (solid, dashed or dash-dotted) line. For the same rea-son, the three corners K a ( a = 1 , ,
3) are to be identified witheach other, and likewise the three corners K ′ a are really onlyone point in Ω. Thus only two of the six corners, collectivelylabeled as K and K ′ and known as the Dirac points, are dif-ferent. Also shown are the wave vectors of the three coplanarplane waves (dashed arrows). Linear combinations of the Brillouin vectors with inte-ger coefficients define the reciprocal lattice ˜ B , a regularpattern in k -space,˜ B = (cid:8) n b + n b (cid:12)(cid:12) n , n = 0 , ± , ± , . . . (cid:9) . (10)The reciprocal lattice is central to all studies of the dy-namics of particles that move under the influence of thegiven periodic potential [28].In particular, one domain in reciprocal space of utmostimportance is the first Brillouin zone Ω, defined as theso-called primitive Wigner-Seitz cell [28] of ˜ B , see Fig. 2.It is a regular hexagon but with the subtle feature thatopposite edges are to be identified with each other sincethey can be related by a displacement vector in ˜ B . Bythe same token the three corners K a (respectively K ′ a )have to be identified with one another and we collectivelydenote them by K (respectively K ′ ). These two differentcorners K and K ′ are known in the graphene literatureas the Dirac points for a reason that will become clear inthe next section. Upon denoting K ≡ K and K ′ ≡ K ′ ,their positions in Ω are given by the wave vector of thelasers that generate the optical honeycomb potential, K = − K ′ = 13 ( b − b ) = k (11)and K = k = K − b , K = k = K + b , as well as K a = − K ′ a . O QPR a a a a b aabb FIG. 3: The underlying Bravais lattice B of a two-dimensionalhoneycomb is the two-dimensional triangular Bravais latticewith a two-point basis A and B . The grey-shaded area is theprimitive cell Σ. The honeycomb lattice constant a is definedas the distance between nearest-neighbor sites.
2. Triangular Bravais lattice
The dimensionless potential (8) consists of a periodictwo-dimensional array of maxima, minima, and saddlepoints, generated by repeated translations of a primitiveunit tile called the basis . The underlying lattice geometryitself is encapsulated in the associated Bravais lattice B ,that is B = (cid:8) m a + m a (cid:12)(cid:12) m , m = 0 , ± , ± , . . . (cid:9) , (12)such that the value of the potential is not affected by anydisplacement R ∈ B , v ( r + R ) = v ( r ).The Bravais primitive vectors a a are constructed basedon the relation a a · b b = 2 πδ ab . (13)In other words, the Bravais lattice B and the Brillouinlattice ˜ B constitute dual spaces. Supplementing (9), wehave the explicit parameterization a a (cid:27) = Λ √ e x ∓ e y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 4 π/ (3 k L ) = 2 λ L / triangular one.We opt here for the diamond-shaped primitive cell Σ de-lineated by the two Bravais lattice vectors as a tilingfor the optical potential (8); see Fig. 3. Another possi-ble choice would have been the hexagonal Wigner-Seitzcell [28]. This cell is useful when discussing the symmetrygroup of the lattice.To proceed further one now needs to analyze the struc-ture of the optical potential (8) inside the primitive cell.In passing, we mention here that red detuned ( δ < V <
3. The honeycomb structure
When the optical lattice is instead blue-detuned ( δ > V is positive and atoms are “weak-field seekers”. Thepotential minima coincide with the minima of the elec-tric field strength, and the maxima coincide as well. Bychoice of coordinate system, the maxima locate at theBravais sites and the dimensionless potential (8) has itsmaximal value of v ( ) = 9 at the corners O, P, Q, R ofthe diamond-shaped primitive cell Σ, see Fig. 4.Two different potential minima, given by the zeros ofthe total dimensionless field amplitude f ( r ), are foundin Σ at r a = ( a + a ) = Λ √ e x and r b = 2 r a , (15)respectively. From a crystallographic point of view, Σis a primitive cell with a two-point basis. By applyingrepeated Bravais translations on Σ, one generates twodifferent sublattices of potential minima, one made upof a -type sites and the other made of b -type sites, seeFig. 3 and Fig. 4. Altogether the potential minima areorganized in a honeycomb structure reminiscent of thepositions of the carbon atoms in graphene sheets.The three displacements that move an a site to a neigh-boring b site — they translate the a sublattice to the b sublattice — are parameterized by c = 13 ( a + a ) = a e x , c = 13 ( a − a ) = a − e x + √ e y , c = 13 ( a − a ) = a − e x − √ e y , (16)where a = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Λ / √ π/ (3 κ ) = 2 λ L / √
27 is thehoneycomb lattice constant. It is the distance from an a site to a neighboring b site, or the distance from thecenter of the hexagon of minima to one of its corners.Halfway between two neighboring minima, the poten-tial has saddle points where v ( r ) = 1. They are lo-cated at the center and at the middle of the edges ofΣ, see Fig. 4. As the saddle points on opposite sides of Σare connected by Bravais displacements, there are there-fore three nonequivalent triangular sublattices of saddlepoints, and we thus count three saddle points per primi-tive cell.We also note that the potential is invariant under 120 ◦ rotations around the locations of the potential minimaand maxima and, therefore, that the potential is isotropicin the vicinity of these points. We anticipate that thelocal harmonic oscillator potential at a minimum will beisotropic; see (35) below. By contrast, the correspondinglocal potential at a saddle point is not isotropic.All these matters are illustrated in Fig. 4, where weclearly identify the various triangular sublattices. 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FIG. 4: [Color online]
Left:
The honeycomb pattern composed of the triangular lattices of minima at sites a and b , of maximaat sites c , as well as of the saddle points between neighboring a and b sites (marked by dots). The bottom plot shows thepotential along the x axis which is one of the . . . abcabc. . . lines with x = 0 at a c site. The saddle points s appear as localmaxima here, with a height that is one ninth of the global maxima at sites c . Cold atoms trapped in this optical potentialwould be found at the a and b sites. Right:
Equipotential lines for the optical honeycomb potential (6). Along the straightblack lines that connect the saddle points, we have V ( r ) = V . The red closed circular curves fill out a hexagonal area, centeredat the points of maximal potential; from inside out the respective values are V ( r ) = 8 V , 5 V , 2 V , and 1 . V . The closedcurves in blue and green fill out areas of the shape of equilateral triangles, their centers are the minima that constitute the a sublattice (blue) or the b sublattice (green); along the curves the potential has the values V ( r ) = 0 . V , 0 . V , 0 . V , and0 . V . One primitive diamond-shaped unit tile Σ spanned by a and a is traced out. It contains two different minima, oneof a -type (in blue, on the left inside) and one of b -type (in green, on the right inside). The trine of the a → b displacementvectors (16) is indicated as well. Finally, for completeness, we also trace out the Bravais Wigner-Seitz unit tile. It is a hexagoncentered at a potential maximum and with potential minima at its corners. be found at the a and b sites, similar to the binding ofelectrons in graphene to the carbon ions.As a side remark, it may be worth mentioning that thesaddle points affect the classical dynamics of a particleevolving in the honeycomb potential with a sufficientlylarge energy. Since the potential is nonseparable and an-gular momentum is not conserved here, the saddle-pointscould be the seed for instabilities in which case the mo-tion could turn out to be nonintegrable and chaotic. Ifso, this chaotic behavior should then be revealed, for ex-ample, in the statistical properties of the quantum spec-tra, whose level spacing fluctuations is expected to bedescribed by the gaussian orthogonal ensemble [29].
4. Optical honeycomb potential and graphene
In graphene sheets, the electrostatic potential that gov-erns the dynamics of electrons, the sum of the Coulombpotentials of the carbon ions, exhibits the symmetriesassociated to a honeycomb pattern. Of course, in thefiner details, the optical dipole potential of (6) and (8) differs markedly from the graphene potential. In particu-lar, the very strong forces that the electrons in grapheneexperience close to the ions have no counterpart in theoptical lattice, and the interaction between the atomsloaded into the optical potential is quite different fromthe electric repulsion between electrons. Nevertheless,the common symmetry group implies great similaritiesbetween the band structures of the two potentials, and inthe respective parameter regimes where the tight-bindingapproximation is valid, the effective Hamilton operatorsare virtually identical. In particular, experiments madewith atoms offers new knobs to play with and, with dueattention to the difference between the two physical sys-tems, these observations may deepen our understandingabout phenomena observed with graphene samples.In a very definite sense, the honeycomb potential (6)is the simplest of all graphene-type potentials [30]. Theirgeneral form is a Fourier sum over the Brillouin vectors, V ( r ) = X Q ∈ ˜ B e i Q · r v Q with v − Q = v ∗ Q . (17)The various symmetry properties of a honeycomb poten-tial ensure that the v Q s are grouped into sets of closelyrelated coefficients. If one coefficient in (17) is nonzero, awhole set of closely related coefficients have correspond-ing nonzero values as well.Other than the trivial constant solution V ( r ) = v , thesimplest case is obtained when all coefficients vanish ex-cept for the set associated with v b = V and, by conven-tion, v = 3 V . This yields the honeycomb potential (6)with v ( r ) of (8). III. MASSLESS DIRAC FERMIONSA. Band structure in the hopping picture
In the hopping picture, one envisions the particle ashopping from site to site with some quantum mechan-ical hopping (or tunneling) amplitude. In the simplestsituation, all sites have the same energy, only hops be-tween nearest-neighbors sites are considered and all hop-ping amplitudes take on the same complex value t . Theone-particle quantum dynamics is then conveniently de-scribed using second quantization. In the present situ-ation, as we have two different sub-lattices, one has tointroduce two sets of fermionic annihilation and creationoperators, one for the a sites, ( a i σ , a † i σ ), and one for the b sites ( b j σ , b † j σ ), where i and j label the sites in the two-dimensional lattices while σ stands for the spin index orany other pertinent quantum number of the particle. Thesecond-quantized Hamilton operator then reads H = X h i , j i ,σ (cid:0) t b † i σ a j σ + t ∗ a † i σ b j σ (cid:1) + ǫ X i σ (cid:0) a † i σ a i σ − b † i σ b i σ (cid:1) , (18)where h i , j i means that only nearest-neighbors are in-cluded in the sum. The model defined by this Hamiltonoperator accounts for hopping to neighboring lattice sitesbut does not permit a change of the internal quantumnumber σ during the hop. We have also included a pos-sible energy mismatch ǫ between the a and b sites [3].Using the Fourier transform in Ω of the fermionic oper-ators, the right-hand side of (18) can be recast into theform H = X k ∈ Ω ,σ ( a † k σ , b † k σ ) (cid:18) ǫ z k z ∗ k − ǫ (cid:19) (cid:18) a k σ b k σ (cid:19) (19)with z k = t X n e i k · c n , (20)from which we get the band spectrum ǫ ± ( k ) = ± q ǫ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (21)As expected from the fact that the honeycomb latticeconsists of two distinct sublattices, we find two bands: a conduction band (+) and a valence band ( − ). Thesebands are here independent of the spin index σ , meaningthat each k ∈ Ω accommodates 2 σ + 1 internal statesper subband. Without any real loss of generality, we willstick to spin- fermions in the sequel. As readily checked, z k vanishes when1 + e i k · a + e i k · a = 0 , (22)which is solved by the corners K and K ′ of Ω since K · a = K ′ · a = 2 π/
3. We thus see that the conduc-tion and the valence bands are gapped by ǫ , a situationtypical of a metal when the lattice is filled with particles.When there is exactly one particle per site (a situationknown as half-filling), all levels in the valence band arefilled at zero temperature, and the Fermi energy E F (theenergy of the highest filled level) precisely cuts the en-ergy surface at the K and K ′ points. In this case thelow-energy excitations of the system can be described bylinearizing the band spectrum in the neighborhood of K and K ′ . Denoting by q = p / ~ the small displacementfrom either K or K ′ , the linearization of z k gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ~ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v , (23)where the quantity v = 3 a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / (2 ~ ) is called the Fermivelocity in the solid-state community. We adopt this ter-minology although it is somewhat unfortunate, becauseit has nothing to do with the standard Fermi velocity p E F /m , which depends on the actual mass of the par-ticle.The dispersion relation now takes on the very sugges-tive form ǫ ± ( p ) ≈ ± q m ∗ v + p v (24)that is typical of a relativistic dispersion relation withparticle-hole symmetry. The effective mass m ∗ , definedthrough ǫ = m ∗ v , appears thus as the rest mass of theexcitations and relates to the energy imbalance of the twosub-lattices. The Fermi velocity v is the analog of thevelocity of light in relativity.The effective Hamilton operator that is derived fromthese considerations and describes the dynamics of theexcitations around K and K ′ , H = Z ( d r )(2 π ) ψ ( r ) (cid:18) i γ · ∇ + m ∗
00 i γ · ∇ − m ∗ (cid:19) ψ ( r ) , (25)where ψ ( r ) is a 4-component Dirac spinor encapsulatingthe excitations around K and K ′ while ψ = ψ † (cid:18) γ γ (cid:19) ,generates an equation of motion that resembles the Weyl-Dirac equation in two dimensions. This is why the name Dirac points is given to K and K ′ (see Refs. [3, 9] formore details). In this two-dimensional context, the Diracmatrices are γ µ = ( γ , γ ) = ( σ z , i σ x , i σ y ) in terms of thestandard Pauli matrices. √ √ ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k x /κ k y /κ FIG. 5: The tight-binding band structure of graphene (inunits of the tunneling strength ˛˛˛˛˛˛ t ˛˛˛˛˛˛ ) as a function of k ∈ Ωin units of κ = √ k L . The origin of energy has been chosenat the Dirac points and the axis ranges are ˛˛˛˛˛˛ k x /κ ˛˛˛˛˛˛ ≤ / ˛˛˛˛˛˛ k y /κ ˛˛˛˛˛˛ ≤ √ /
3. The bottom contour lines are lines ofconstant ˛˛˛˛˛˛ ǫ ˛˛˛˛˛˛ / ˛˛˛˛˛˛ t ˛˛˛˛˛˛ . When ǫ vanishes, as is the case in real graphene whereall lattice sites have the same energy, then ǫ ± ( k ) = ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and the two bands are degenerate at the corners ofΩ where they display circular conical intersections (seeFig. 5). In the literature, this situation is referred to asa semi-metal or a zero-gap semi-conductor and the cor-responding low-energy excitations are known as masslessDirac fermions. The total band width is W = 6 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and,at half-filling, the Fermi energy E F = 3 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (taking theenergy origin at the lower band minimum) precisely slicesthe energy bands at the Dirac points. Hence the Fermisurface reduces to these two points, so that the densityof states vanishes there [9], see Fig. 6. B. Tight-binding approximation
Mindful of possible experiments, the hopping param-eter t appears to be an important amplitude to eval-uate. We report three different methods for estimatingits strength (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We will start with the familiar tight-binding approximation using localised Wannier functions[31, 32] that are further approximated by Gaussians. Wewill then develop a more accurate semi-classical calcula-tion based on an instanton approach [33]. We will com-pare both results to a brute-force exact numerical com-putation.As a consequence of Bloch’s theorem [28, 34], the en-ergy spectrum of an atom of mass m moving in the hon- ρ ( E ) E FIG. 6: The noninteracting density of states per unit cell andper spin component ρ ( E ) as a function of the reduced energy E = E/ ˛˛˛˛˛˛ t ˛˛˛˛˛˛ . The origin of energy has been chosen at theDirac points. When E ≪
1, then ρ ( E ) ≈ ˛˛˛˛˛˛ E ˛˛˛˛˛˛ / ( √ π ) and thedensity of states vanishes at E = 0, a signature of the semi-metal behavior. Note the logarithmic Van Hove singularityat ˛˛˛˛˛˛ E ˛˛˛˛˛˛ = 1. eycomb lattice potential is obtained from H ψ n k ( r ) = (cid:20) − ~ ∇ m + V ( r ) (cid:21) ψ n k ( r ) = ǫ n ( k ) ψ n k ( r ) , (26)where we dropped the spin index σ which is not essentialhere. The Bloch waves ψ n k are given by ψ n k ( r ) = e i k · r u n k ( r ) (27)with k ∈ Ω, n the band index, and u n k ( r ) is a B -periodicfunction. The latter can be conveniently expanded usingWannier functions [28, 34, 35] in accordance with u n k ( r ) = X R ∈B e − i k · ( r − R ) w n ( r − R ) . (28)Wannier functions are very useful in describing modelswhere particles are localized in space, such as the Hub-bard model [36]. They form an orthonormal basis set offunctions centered at different Bravais lattice sites whichare copies of the same “seed” functions defined in a givenprimitive cell. The localization properties of the Wannierfunctions crucially depend on the analyticity propertiesof u n k as a function of k and decay exponentially in thesimple cases [37, 38, 39, 40].In the tight-binding approximation, the atoms are as-sumed to be sufficiently deeply-localized in the differentpotential wells where they only populate the lowest vi-brational levels. Vibrational states in different wells arealso assumed to have small overlap: the atomic motionis thus “frozen” except for the small tunneling amplitudebetween neighboring wells and are then effectively con-fined to move in the lowest bands of the lattice. Sincethe Wannier functions display the same symmetry as thelocal potential structure [41, 42], the natural idea here isthus to construct tight-binding Wannier functions fromlinear combinations of wave functions deeply-localized inthe two potential wells of the primitive cell (the so-calledatomic orbitals) [32, 43]. This trial wave function ex-ploits at best the sub-lattice structure of the honeycomblattice and should give good results at least for the firsttwo bands.After dropping the band index n , this approach, rem-iniscent of the LCAO method (linear combination ofatomic orbitals) [28, 34], leads to the ansatz ψ k ( r ) = α k ψ ( a ) k ( r ) + β k ψ ( b ) k ( r ) , (29)where the quasi-Bloch wavefunctions ψ ( a ) k ( r ) = X a e i k · r a w a ( r − r a ) ,ψ ( b ) k ( r ) = X b e i k · r b w b ( r − r b ) (30)essentially live on the type- a sublattice and the type- b sublattice, respectively. The sublattice Wannier func-tions w a ( r ) and w b ( r ) are normalized to unity. In thepresent case, we even have w b ( r ) = w a ( − r ) due to thereflection symmetry of the potential, V ( − r ) = V ( r ) [41].We define the on-site energies as E a = h w a |H| w a i ( a = a , b ) and use the parametrization E a = E + ∆ and E b = E − ∆ in the following, with E the mean on-siteenergy and ∆ half the on-site energy difference. Most im-portantly, the sublattice Wannier functions are orthogo-nal. However, obtaining their exact expressions is a diffi-cult task and one often resorts to simple approximationsthat do not have this property. This is why, in view ofthis very common practical situation, we will considerin the following that the Wannier functions w a ( r ) and w b ( r ) can overlap.Plugging now the ansatz (29)-(30) into (26), and onlyconsidering coupling between nearest-neighbor latticesites, we get the 2 × (cid:18) ∆ − E Z k − ER k Z ∗ k − ER ∗ k − (∆ + E ) (cid:19) (cid:18) α k β k (cid:19) = 0 , (31)where E = ǫ ( k ) − E and with the matrix entries Z k = X a t a e i k · c a ,t a = h w a | ( H − E ) | w b a i ,R k = X a h w a | w b a i e i k · c a . (32)Here b a = a + c a is a short-hand notation for the three b sites next to the a site.Several remarks are in order. First one notes that theoff-diagonal matrix entries depend on the energy as soonas the sublattice Wannier functions overlap. Second, asreadily checked, the hopping amplitudes t a and E areindependent of any energy shift in the Hamilton operatorand are thus independent of any particular choice for the energy origin as one expects. Note also that the valuesof E a and of E b do not depend on the particular choicefor point a or point b since H is B -translation invariant.By the same token, the values of t a and of h w a | w b a i donot depend on the particular choice of a , but b must beone of its three nearest neighbors.To have a nonzero solution, the 2 × h w a | w b a i ≪
1, the band structure isvery well approximated by ǫ ± ( k ) ≈ E ± q ∆ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (33)a form reminiscent of (21). For the honeycomb lattice,for which H is B -periodic and invariant under reflections,we further have E a = E b = E and ∆ = 0, which impliesthat the effective mass m ∗ of the Dirac fermions is in-deed zero. As a consequence, we get the two first bandsas ǫ ± ( k ) = E ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Furthermore, since V ( r ) is alsoinvariant under 2 π/ a ,all three tunneling amplitudes t a from a to b a acquirethe same value and Z k of (32) turns into z k of (20) with t = Z ( d r ) w ∗ a ( r )( H − E ) w a ( r − c ) , (34)where H is the differential operator of (26) and c is eitherone of the three displacement vectors in (16). C. Harmonic approximation
To proceed further one needs an approximation for theWannier functions w a and w b . One possibility is to relyon the harmonic approximation of the potential wellsaround sites a and b , that is to approximate w a and w b by the corresponding harmonic ground state wave func-tions. We find V ( r a + r ) ≈ V κ r = mω r for a = a , b with ~ ω = 3 p V E R , (35)where E R = ~ k L / (2 m ) is the recoil energy of the atom.In terms of ℓ = p ~ / ( mω ), the familiar length unit ofthe harmonic oscillator, the ground state wave functionis w a ( r a + r ) = w b ( r b + r ) ≈ √ πℓ e − r /ℓ . (36)From this we get E a = E b = E ≈ ~ ω and the overlapintegrals are simply h w a | w b a i = exp − π r V E R ! . (37)Keeping in mind that V ≫ E ≫ E R in the tight-binding regime, h w a | w b a i ≪ t ≈ − (cid:18) π − (cid:19) V exp − π r V E R ! , (38)at leading order. However, since the hopping amplitudeis given by the overlap integral of the localized wave func-tions w a and w b of two neighboring sites, we see that thevalue of t crucially depends on the tails of these wavefunctions. Wannier functions often decay exponentiallyand, therefore, they cannot be realistically approximatedby Gaussian wave functions. Hence (38) can, at best,serve as a rough underestimate [44]. In the next sectionwe will derive a reliable and accurate estimate of the tun-neling amplitudes in the tight-binding regime by use ofthe instanton method. D. Semiclassical estimate
Using k − L , p V /m , V , and p m/ ( k L V ) as length,velocity, energy, and time units, respectively, theSchr¨odinger equation can be conveniently recast into adimensionless form that features an effective Planck’sconstant ~ e (we keep the same symbols for the rescaledvariables for simplicity),i ~ e ∂ t ψ = − ~ e ∇ ψ + v ( r ) ψ , ~ e = r E R V , (39)with v ( r ) given by (8), here expressed in rescaled units.In the tight-binding approximation it is assumed that V ≫ E R , and thus ~ e ≪
1. In this situation, semiclassi-cal methods provide very efficient and very accurate waysfor evaluating dynamical and spectral quantities of inter-est. They generally amount to evaluating integrals withthe aid of semiclassical expressions for the quantum prop-agator, derived from its Feynman-path integral formu-lation through stationary-phase approximations aroundthe classical trajectories [45].For example, it is well-known that the energy splittingbetween the two lowest energy levels of an atom mov-ing in a one-dimensional symmetric double well can beaccurately calculated using the WKB method [45]. ThisWKB method can be extended to several dimensions andin the sequel we will derive a semiclassical estimate of t for the honeycomb lattice using the method proposed in[33]. It amounts to evaluating t using the classical com-plex trajectory (in rescaled units) that connects a and b through the classically forbidden region — the so-calledinstanton trajectory.Using ~ ω as an order of magnitude for the vibrationallevel inside a potential well, we see that in the rescaledunits, this energy is ~ ω /V = 3 ~ e / √ ≪
1. So we cansimply look for the instanton trajectory at zero energy. Inrescaled units, the hopping amplitude is then expressed as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V = α p ~ e e − S / ~ e , (40)where S is the (rescaled) classical action along the zero-energy instanton trajectory, and the numerical factor α is obtained from integrating out the fluctuations aroundthe zero-energy instanton trajectory (see below).As the zero-energy instanton fully runs in the classi-cally forbidden region, the variables take on complex val-ues. For our particular case, the good parameterizationturns out to keep r real while taking t = i τ and p = − i ˜ p purely imaginary with τ and ˜ p real. Hamilton’s classicalequations of motion in the new variables are just obtainedfrom the original ones by flipping V ( r ) to − V ( r ). Thesymmetry of the potential dictates that the zero-energyinstanton trajectory is simply the straight line connect-ing site a to b (see Fig. 4). In the following we calculatethe instanton between a and a + c . Integrating the equa-tion of motions, one gets the instanton trajectory in therescaled form r ( τ ) = k L ax ( τ ) e x withtan[ πx ( τ ) /
3] = −√ √ τ / . (41)The boundary conditions are x = 1, ˙ x = 0 when τ → −∞ and x = 2, ˙ x = 0 when τ → ∞ , meaning thatthe instanton starts at a with zero velocity and ends at b with zero velocity, the whole process requiring an infiniteamount of time. This is indeed what is expected as bothendpoints of the instanton are instable in the reversedpotential picture. Since the energy associated with thisinstanton trajectory is zero, the classical action is simply S = ka Z ka dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x, y = 0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 4 √ (cid:18) − π √ (cid:19) ≈ . , (42)where f ( x, y ) is given by (7).The computation of α proves technically more demand-ing. Following [33], it is given by the product α α with α = r S π s det[ − ∂ τ + ω ]det ′ [ − ∂ τ + ω x ( τ )] ,α = s det[ − ∂ τ + ω ]det[ − ∂ τ + ω y ( τ )] . (43)Here ω a ( τ ) = ( ∂ a v )( r ) ( a = x, y ) is the curvature of therescaled potential along the zero-energy instanton tra-jectory r ( τ ) while ω is the curvature of the rescaledharmonic potential approximation around a ; see (35). Inrescaled units, we have ω = 3 / √
2. The prime in the for-mula for α means that the determinant is calculated byexcluding the eigenspace of the operator − ∂ τ + ω x withthe smallest eigenvalue.The determinants of the differential operators involvedin the computation of α stem from the linear stabilityanalysis of the dynamical flow in the neighborhood of the0zero-energy instanton trajectory as encapsulated in themonodromy matrix. They can be straightforwardly com-puted from solutions of the linear Jacobi-Hill equationsof degree 2 associated with these differential operators[46]. For example, α is solved as α = lim T →∞ s J ( T ) J ( T ) (44)where the Jacobi fields J ( τ ) and J ( τ ) satisfy the differ-ential equations d J ( τ ) dτ − ω y ( τ ) J ( τ ) = 0 ,d J ( τ ) dτ − ω J ( τ ) = 0 , (45)with initial conditions J ( − T ) = J ( − T ) = 0 , ˙ J ( − T ) = ˙ J ( − T ) = 1 . (46)The interested reader is referred to [33, 46] for details.We simply give here the final result for the honeycomblattice: α = s √ π ≈ . , α ≈ . , α ≈ . . (47)Recasting the semiclassical calculation of the tunnelingamplitude in units of the recoil energy finally yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E R ≈ . (cid:18) V E R (cid:19) / exp " − . r V E R . (48)The same type of scaling laws has been obtained in thecase of the two-dimensional square optical lattice [44, 47].In the square-lattice geometry, however, the potential isseparable and the semiclassical calculation proves muchsimpler as it reduces to using the well-known Mathieuequation. E. Numerical computation of the band structure
Using Bloch’s theorem and plugging (27) into (26), weget a family of partial differential equations for the u n k slabeled by the Bloch vector k ∈ Ω. After scaling variableswith the same units as in the previous paragraph, theband structure is then extracted by numerically solving H k u n k ( r ) = ǫ n k u n k ( r ) , H k = − ~ e − i ∇ + k ) + v ( r ) (49)for each k ∈ Ω (expressed now in units of k L ). √ √ ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k x /κ k y /κ FIG. 7: Numerically calculated band structure of the two low-est energy bands for ~ e = 0 .
25 at discrete points in the Bril-louin zone Ω. The same conventions as in Fig. 5 are adopted.The value of ˛˛˛˛˛˛ t ˛˛˛˛˛˛ is determined by requiring that ǫ ± = ± ˛˛˛˛˛˛ t ˛˛˛˛˛˛ at the center Γ of the Brillouin zone. The similarity withFig. 5 shows that at V = 32 E R the tight-binding regime hasalready been reached. The u n k s being B -periodic, they are convenientlyFourier expanded in the reciprocal lattice ˜ B accordingto u n k ( r ) = X Q ∈ ˜ B C n Q e i Q · r . (50)The matrix representation of H k is sparse and banded.It is then appropriately truncated and diagonalized suchthat only a small number of coefficients C n Q are actuallysignificant for the corresponding energy bands. The en-ergy bands obtained in this way are exact and one caninvestigate their dependance on ~ e as done in Fig. 7 andFig. 8.The essential feature is to realize that the band de-generacies at points K and K ′ are generic and do notdepend on the actual value of the effective Planck’s con-stant. Indeed the existence of two degeneracy points inthe first Brillouin zone for the honeycomb lattice is a gen-eral consequence of the lattice symmetries [48, 49]. Thelattice symmetries are encapsulated in the point group ofthe lattice which is the set of operations that leave fixedone particular point of the lattice. The correspondingelements are rotations, reflections, inversions and theircombinations. Combined with B -translations, one getsthe space-group of the lattice. The graphene point grouphas been analyzed by Lomer [48] and contains twelve ele-ments. In terms of Bloch wave functions ψ n k , the latticespace-group operations translate into point group opera-tions on k ∈ Ω, possibly followed by a reciprocal latticetranslation to bring back the resulting new wave vectorin Ω. The key point is that degeneracies can only occurat Bloch wave vectors which are invariant (up to recipro-cal lattice translations) under the action of a nonabelian1 -0.4 -0.2 0 0.2 0.40.60.70.80.91 E n e r g y K K ′ k y /k L FIG. 8: Band structure for nearly-free particles moving in aweak honeycomb optical potential in units of V . The first 2levels are plotted as a function of k y /k L at k x /k L = √ / K to K ′ , see Fig. 2.The solid curves are obtained for ~ e = p E R /V = √ ~ e = √
5. As one can see the bandstructure is rather flat in the band centre but the levels curva-ture increases when ~ e is decreased. The Dirac degeneraciesin the ground state obtained at the Brillouin zone cornersare generic and can be inferred from group-theoretic consid-erations. Note however that the conical intersections do notextend much over the first Brillouin zone when the potentialis weak but start to spread when ~ e is decreased. subgroup G of the point group. For the graphene thishappens at the Dirac points. For example, at corner K ,beside unity, G is made of two rotations of angles ± π/ K . This group of order six admits an irreducibletwo-dimensional representation which explains the bandspectrum degeneracy at the Dirac points.This can be nicely illustrated in the weak V limit (orequivalently when ~ e is large). In this case, the particlesare quasi-free and the band spectrum can be understoodin two steps. First, one folds the parabolic dispersionrelation of the free particle into the first Brillouin zone(repeated-zone scheme [28]) and then one couples cross-ing levels at Bragg planes by the weak potential. At K , three plane waves fold with the same kinetic en-ergy, namely K = k , K = k and K = k (seeFig. 2). The weak periodic potential then couples thesethree plane wave states and the coupling matrix elementsare all identical. The eigenstates of this 3 × V is negative, the sin-glet is the ground state which is consistent with the trian-gular Bravais lattice obtained in this case ( δ < V is positive ( δ > K , see Fig. 8.From the exact numerical calculation, one can extract (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E R / ~ e FIG. 9: The hopping parameter ˛˛˛˛˛˛ t ˛˛˛˛˛˛ in units of the recoilenergy E R (crosses) as a function of the inverse of the effec-tive Planck’s constant ~ e = p E R /V as obtained from theexact numerical computation. The harmonic approximation(dashed curve) and the semiclassical calculation (solid curve)of the hopping parameter have been added for comparisoneven if their range of validity is restricted to the tight-bindingregime ~ e ≪ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E R / ~ e FIG. 10: The hopping parameter ˛˛˛˛˛˛ t ˛˛˛˛˛˛ (in units of the recoilenergy E R ) as a function of the inverse effective Planck’sconstant ~ e = p E R /V in the tight-binding regime where ~ e ≪
1. As one can see, the harmonic approximation (dashedcurve) is completely off. For example at V = 32 E R (or ~ e = 0 . ˛˛˛˛˛˛ t ˛˛˛˛˛˛ is underestimated by a factor 10 and the dis-crepancy gets worse as V increases. On the other hand, theagreement between the semiclassical calculation (solid curve)and the exact numerical computation (crosses) just provesexcellent. the slope of the dispersion relation at the Dirac pointsand then the corresponding tunneling strength (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) as afunction of ~ − e , see Fig. 9. Figure 10 gives the com-parison between the exact calculation, the harmonic andthe semiclassical calculations as a function of ~ − e in thetight-binding regime where ~ e ≪
1. As one can see, theharmonic approximation is way off whereas the semiclas-2 ǫ ( k ) K ′ K Γ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k y FIG. 11: Cut of the linear dispersion approximation along Oy at k x = 0 in the first Brillouin zone Ω as compared to theactual band spectrum in the tight-binding regime. At half-filling, the Fermi energy cuts the band spectrum at the Diracpoints K and K ′ . Doping the system away from half-fillingmoves the Fermi energy up or down but the system can stillbe described in terms of massless Dirac fermions provided a ˛˛˛˛˛˛ q ˛˛˛˛˛˛ ≪
2, i.e. provided the change in the Fermi energy ismuch less than the band-width W = 6 ˛˛˛˛˛˛ t ˛˛˛˛˛˛ itself. By the sametoken, thermal excitations of the system can still be describedas thermal massless Dirac fermions provided k B T ≪ W . sical estimate proves excellent. F. Reaching the massless Dirac fermions regime
To access the massless Dirac fermions regime one firstneeds to completely fill the ground state band alone, asituation known as half-filling. This is achieved by havingexactly one fermion with spin state σ = ± / ρ = 1 in thetight-binding picture. When this is achieved, the Fermienergy slices the band structure at the Dirac points. Forexperiments that study transport phenomena, one wouldalso need to subsequently dope the sample away fromhalf-filling such that the Fermi energy of the system isvaried in the linear part of the band structure.In a usual experiment, atoms are generally held in anexternal harmonic potential and the optical lattice po-tential is superimposed. Reaching half-filling could thenbe done in two steps, first by significantly increasing re-pulsive interactions U between fermions through a Fes-hbach resonance and then by driving the system intothe Mott-Hubbard phase with one fermion per site asdone in [50, 51]. Then setting U to zero again shouldmaintain the system at number density ρ = 1. Obvi-ous candidates for such experiments are Potassium-40as well as Lithium-6 atoms [50, 51, 52]. In the exter-nal trap, the Mott insulator appears first where the lo-cal filling is approximately one atom per site and one needs to ensure that adding more atoms (or increasingthe chemical potential µ ) does not favor the appearanceof the doubly-occupied Mott phase. This will be thecase for very strong repulsion U ≫ µ, t , k B T in whichcase one expects the entire centre of the trap to con-tain a Mott insulating phase with single occupancy andnegligible thermally-activated doubly-occupied sites. Inthe case of the honeycomb lattice, starting from a spin-unpolarized sample, it is known that half-filling is reachedfor U c ∼ t , the atoms displaying at the same time ananti-ferromagnetic order [18]. Note that U c ∼ W , where W = 2 E F = 6 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is the band-width.Doping the system could be done in the followingway. The external harmonic confinement (with angu-lar trap frequency Ω t ) defines a characteristic length ζ = q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / ( m Ω t ) over which the energy is shifted byprecisely the tunneling energy t [53, 54, 55]. This lengthdefines the distance over which one given lattice site iscoupled by tunneling to its neighbors. In turn, havingloaded N F fermions into the trap, one can define a char-acteristic filling factor through ˜ ρ = N F ( a/ζ ) . Varying (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) by changing the lattice potential height V or tighten-ing/loosening the trap by changing Ω t would thus allowto tune ˜ ρ in a controlled way and hence to dope the sys-tem.For the conical intersection at the Dirac points to sig-nificantly spread over the Brillouin zone Ω, one needs toreach the tight-binding regime where V is large enough(typically V > E R will do). Inspection of the Tay-lor expansion of (20) then shows that it is sufficient tohave (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ≪ q being the small displacement from aDirac point) for the band structure to be well approxi-mated by a linear dispersion relation around the Diracpoints. The available energy range ∆ E is thus set bythe band-width itself, namely ∆ E ≪ W . So tuning thefilling factor away from half-filling and residual thermalfluctuations will keep the system in the massless Diracfermions regime provided µ, k B T ≪ W (Fig. 11). Forexample, at V = 32 E R , the temperature constraint, asderived from (48), is T < T R /
50 whereas it is
T < T R / V = 10 E R . There is thus room left for reaching themassless Dirac fermions regime within the current state-of-art cooling technology. IV. ROBUSTNESS OF THE MASSLESS DIRACFERMIONS
As the very existence of the massless Dirac fermionsregime rests on the two conical degeneracies in the bandstructure, one may wonder if this regime would resistimperfections of the system. Indeed the argument wegave to explain the conical degeneracies relied on group-theoretic arguments which were specific to the hexagonalsymmetry of the honeycomb structure. In practice, it isimpossible to control the laser configuration to the pointwhere all intensities and alignment angles would all be3exactly equal. Such imperfections in the system wouldobviously break the hexagonal symmetry and one couldthink that the Dirac fermions would just be destroyed.In fact, as we will see shortly, massless Dirac fermionsare quite robust and survive small imperfections that areeasily within experimental reach.
A. Imbalanced hopping amplitudes
To understand why massless Dirac fermions are robust,we will start by analyzing the case of imbalanced hop-ping amplitudes as done in [56]. For real graphene, thiswould correspond to stretching the graphene sheet. Inthis case, the tight-binding band structure is given by ǫ ± ( k ) = ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where Z k is defined in (20). The de-generacies are found at points k D ∈ Ω canceling Z k = 0.This condition boils down to sum up three vectors to zeroin the two-dimensional plane, see Fig. 12. As such, a so-lution is only possible provided the hopping amplitudessatisfy one of the norm inequalities given by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (51)and cyclic permutations. If this is the case, defining theangles ϕ , = arg t , − arg t , the Dirac points solvecos( k D · a − ϕ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , cos( k D · a − ϕ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (52)subject to the condition (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( k D · a − ϕ ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( k D · a − ϕ ) = 0 . (53)We find the important result that the system self-adaptsto changes in the hopping amplitudes by shifting theDirac points away from the corners of the Brillouin zoneuntil the norm inequalities (51) break and degeneraciesdisappear. Thus, provided the hopping imbalance is nottoo strong, the massless Dirac fermions do survive im-perfections in the system and the hexagonal symmetrybreaking.We illustrate this important feature in the simple caseof only one imbalanced hopping amplitude, namely t = γt , t = t = t . We further choose γ real and 0 < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D γ and D ′ γ given by k D = − k ′ D = ϕ ( b − b ) where ϕ ∈ [0 , /
2] solves cos(2 πϕ ) = − γ/
2. Thismeans that the two Dirac points D γ and D ′ γ move alongopposite paths in the Brillouin zone Ω. The fact thatDirac points always come in by pairs of opposite locationin Ω is generic [57]. When γ is increased from 0 to 2, D γ starts at k = (3 k L / e y for γ = 0, then moves alongaxis Oy and reach corner K at γ = 1. Note that when γ →
0, the physical situation is that of weakly coupled“zig-zag” linear chains. For γ > D γ leaves Ω but u α α α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) FIG. 12: The condition Z k = 0 is equivalent to cancel theresultant vector u of three vectors, each with length ˛˛˛˛˛˛ t n ˛˛˛˛˛˛ andpolar angle α n = k · c n + arg( t n ). There will always be asolution provided one of the norm inequalities ˛˛˛˛˛˛˛˛˛˛˛˛ t ˛˛˛˛˛˛ − ˛˛˛˛˛˛ t ˛˛˛˛˛˛˛˛˛˛˛˛ ≤ ˛˛˛˛˛˛ t ˛˛˛˛˛˛ ≤ ˛˛˛˛˛˛ t ˛˛˛˛˛˛ + ˛˛˛˛˛˛ t ˛˛˛˛˛˛ (and cyclic permutations) is satisfied. a translation in reciprocal lattice brings it back on thevertical edges of Ω (technically we get two copies of thesame point). D γ reaches the middle of the vertical edgeat γ = 2 where it merges with D ′ γ into a single Diracpoint, see Fig. 13. Interesting physics occurs at γ = 2in connection with the quantum Hall effect [17, 58]. Assoon as γ >
2, the degeneracy is lifted and the masslessDirac fermions do not exist anymore. For negative γ , D γ and D ′ γ move back from ± (3 k L / e y to the centre Γ ofthe Brillouin zone where they merge and disappear, seeFig. 13. The fact that Dirac points can only merge atthe centre and mid-edge points of Ω is also generic [57].Hence, far from being a nuisance, we see that con-trolling the hopping amplitude imbalance proves an in-teresting way of exploring the massless Dirac fermionsphysics under different circumstances by moving aroundthe Dirac points in the Brillouin zone. B. Optical lattice distortions
The previous discussion concentrated on the impactof imbalanced hopping amplitudes irrespective of thechange of symmetry of the lattice potential. We willnow analyze these lattice distortions in more detail andgive quantitative estimates about the experimental de-gree of control which is required to target the masslessDirac fermions regime. We will consider in-plane laserbeams with different (positive) strengths E n = s n E andwith respective angles away from 2 π/
3, see Fig. 14. It isimportant to note that we will always stick to imperfec-tions which are compatible with a two-point Bravais cell.They will only induce distortions of the hexagonal spa-tial structure of field minima but without breaking thispattern.The new optical lattice potential is now given by V ′ ( r ) = V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ′ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) with the new total dimensionless field4 D ′ = D D = D ′ D D ′ Γ K K ′ K K ′ K ′ K FIG. 13: When the three hopping amplitudes t n are unbal-anced, the Dirac points are shifted in the Brillouin zone Ωand disappear when the norm inequality ˛˛˛˛˛˛˛˛˛˛˛˛ t ˛˛˛˛˛˛ − ˛˛˛˛˛˛ t ˛˛˛˛˛˛˛˛˛˛˛˛ ≤ ˛˛˛˛˛˛ t ˛˛˛˛˛˛ ≤ ˛˛˛˛˛˛ t ˛˛˛˛˛˛ + ˛˛˛˛˛˛ t ˛˛˛˛˛˛ is no longer satisfied. We depict here how the Diracpoints D γ and D ′ γ move in Ω when only one hopping ampli-tude is imbalanced, namely t = γt and t = t = t . Points D γ (thick path) and D ′ γ (thin path) move along opposite paths.Increasing γ from 0, point D γ starts at D and moves upward.It reaches point K at γ = 1 (balanced amplitudes case) thenmoves along the vertical edge of Ω where it reaches its middlepoint D at γ = 2. The Dirac points cease to exist when γ >
2. For negative γ , D γ moves downward from D (dottedthick path), reaches the zone center Γ for γ = − γ < − amplitude f ′ ( r ) = s + s exp( − i b ′ · r ) + s exp(i b ′ · r ) . (54)Here the b ′ n (n=1,2) feature the new reciprocal latticebasis vectors. They define in turn a new set of Bravaislattice basis vectors a ′ n giving rise to a new primitivediamond-shaped cell Σ ′ . Unless the angle mismatchesvanish, the new Bravais and reciprocal lattices are nolonger hexagonal but oblique with no special symmetryexcept for inversion. As a consequence, the new firstBrillouin zone Ω ′ is still a hexagon but no longer a regularone .Since we assume a two-point primitive cell, the minimaof the new optical potential still identify with zeros of f ′ ( r ). Similarly with the case of imbalanced hoppingamplitudes, we find two solutions if and only if the fieldstrengths s n satisfy one of the norm inequalities (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s ≤ s + s (and cyclic ones). In this case theminima are given bycos( b ′ · r ) = s − s − s s s , cos( b ′ · r ) = s − s − s s s , (55)subject to the condition s sin( b ′ · r ) = s sin( b ′ · r ).In the following we will examine separately the effectof strength imbalance and angle mismatch. xyz k k k π π θ θ (a) O P QR a a (b)FIG. 14: (a) [Color online] (a) The asymmetric in-plane3-beam configuration. Three monochromatic and linearly-polarized laser beams with wave vectors k n interfere with dif-ferent strengths E n = s n E ( n = 1 , , π/
3. (b) Distorted optical lattice obtainedwith ϑ = ϑ = 5 × − and s = 1, s = 1 . s = 0 .
97. Forweak enough distortions, the primitive diamond-shape cell Σstill contains two field minima as evidenced in the plot.
1. Critical field strength imbalance
To give an estimate of the critical field strength im-balance beyond which the Dirac points cannot survive,we consider the simple case of only one imbalanced laserbeam and no angle mismatch, namely θ = θ = 0, s = 1 + η and s = s = 1. In this case the Bravaislattice, the reciprocal lattice, the primitive cell Σ andthe Brillouin zone Ω are not modified. The new opticalpotential V ′ ( r ) = V v ′ ( r ) reads v ′ ( r ) = v ( r ) + 2 η δv ( r ) + η ( η + 2) ,δv ( r ) = cos( b · r ) + cos( b · r ) , (56)where v ( r ) is given by (8). Note that when only one fieldstrength is imbalanced, the corresponding potential stilldisplays a reflection symmetry. In the present case, itis the Ox -reflection symmetry because V ′ ( r ) is invariantunder the exchange b ↔ b . Requiring now that the5 O P QR a a FIG. 15: [Color online] Slightly distorted lattice obtained withvanishing mismatch angles and one imbalanced field strength,namely s = 10 / s = s = 1. In this particular case thehexagon of field minima is slightly squeezed along the horizon-tal axis Ox and the vectors c ′ n connecting a given minimumto its three nearest-neghbors have now different lengths. Inthe situation depicted ˛˛˛˛˛˛ c ′ ˛˛˛˛˛˛ = ˛˛˛˛˛˛ c ′ ˛˛˛˛˛˛ = ˛˛˛˛˛˛ c ′ ˛˛˛˛˛˛ . In turn, due to thereflection symmetry about Ox , the tight-binding hopping am-plitudes satisfy ˛˛˛˛˛˛ t ˛˛˛˛˛˛ = ˛˛˛˛˛˛ t ˛˛˛˛˛˛ = ˛˛˛˛˛˛ t ˛˛˛˛˛˛ . primitive cell Σ exhibits two field minima imposes − ≤ η ≤
1. Their positions in Σ are given by r ′ a , b = ϕ a , b ( a + a ) with cos(2 πϕ a , b ) = − (1 + η ) /
2. Their mid-point r ′ s = ( r ′ a + r ′ b ) / a + a ) / V ′ s to cross to go from a and b in Σ. One finds V ′ s = ( η − V .As a whole the field minima organize in a hexagonwhich is stretched ( η negative) or compressed ( η positive)along Ox , see Fig. 15. As a consequence two of the threenew vectors c ′ n joining one minimum to its three nearest-neighbors will have equal length. In the present situationwe get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The potential barrier height V ′′ s to cross to go from a to b along c ′ and c ′ is given bythe corresponding saddle points located at the middle ofthe edges of Σ. One finds V ′′ s = ( η + 1) V .Now, when η is increased from 0, the minima movecloser along c ′ and move away along c ′ and c ′ . At thesame time, the potential barrier V ′ s along c ′ is loweredand the the potential barrier V ′′ s along c ′ and c ′ is in-creased. As a net effect, in the tight-binding picture, weexpect the tunneling amplitude (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) to increase while (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) decrease. We get the opposite conclusion when η is lowered from 0. Since the potential is invariant through b ↔ b , we further have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and we recover thecase of one imbalanced hopping amplitude analyzed inthe previous section.One could try to derive a semiclassical expression ofthe t n as a function of η using the instanton method but,actually, such a tedious calculation proves unnecessary,at least when η is small. Indeed, by inspection of thesemiclassical expression (40), we expect the ratio (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t /t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) to scale as exp(∆ S ( η ) / ~ e ) at leading order, where ∆ S ( η ) is the action difference between the two instanton trajec-tories linking sites a and b along c ′ and c ′ respectively.For small enough η we expect ∆ S ( η ) to grow linearly with η , the slope being positive since the ratio (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t /t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) shouldincrease with η . The Dirac degeneracies disappear whenthis ratio is 2 (see previous section), thus we get the semi-classical prediction that this will happen when η ∝ ~ e .This result can also be inferred by saying that the Diracpoints will disappear as soon as the perturbing potential2 ηδV ( r ), see (56), strongly mixes the unperturbed states.This will happen when the corresponding coupling energyequals the mean level spacing of the unperturbed systemand we get back to the prediction η ∝ ~ e .To check our semiclassical prediction we have com-puted, for each value of the effective Planck’s constant ~ e , the ground state and first excited-state levels for dif-ferent values of η and we have extracted the correspond-ing critical value η c for which the Dirac degeneracies arelifted. Figure 16 gives an example of the band structureobtained at ~ e = 1 / √ ≈ .
16 for η ranging from 0 to0 . η c as a function of ~ e , seeFig. 17. We have fitted the data with the quadratic fitfunction α ~ e + β ~ e and found α ≈ . β ≈ . ~ e ≪ η from 0 is not symmetrical. When η is decreased from 0, the Dirac degeneracies are pre-dicted to disappear when (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t /t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) →
0. However the bestthat we can do is to let η → −
1. This unfortunatelymeans that one laser beam is almost extinguished andthe situation is more that of very weakly coupled one-dimensional chains, a situation we postpone to futurestudy as it proves interesting for high- T c superconduc-tivity [59]. We thus see that decreasing slightly η from 0does not harm the Dirac degeneracies. They move insideΩ but do survive. By contrast, increasing slightly η from0 does destroy the Dirac degeneracies as soon as η ∼ ~ e .As one can see from the plots, the tolerance aboutthe intensity mismatch of the laser beams increases with ~ e , or equivalently when the optical lattice depth V de-creases. On the other hand, as we already saw, the Diraccones do not extend much over the Brillouin zone if V istoo small. So there is a trade-off to make. The situationis however really favorable since the intensity mismatchtolerance is already in the 10% range for V ∼ E R .This means that the massless Dirac fermions prove quiterobust and should be easily accessed experimentally.
2. Critical in-plane angle mismatch
We now estimate the critical angle mismatch when alllaser beams have the same intensities ( s = s = s = 1).We see from (54) that the new optical potential still dis-plays the exchange symmetry b ′ ↔ b ′ and thus a re-6 E n e r g y η K K ′ FIG. 16: The band diagram for the two lowest levels as afunction of η for V = 80 E R ( ~ e ≈ . K and K ′ of the balanced situation, see Fig. 2. The originof energy is fixed at the Fermi energy for a half-filled bandand all bands have been shifted such that the upper and lowerbands intersect at zero energy difference. Dirac points disappearDirac points exist η c ~ e FIG. 17: The critical laser strength imbalance η c at whichthe Dirac degeneracies are lifted as a function of the effectivePlanck’s constant ~ e = p E R /V . The solid line correspondsto a quadratic fit of the numerical data. The linear coefficientis α ≈ . β ≈ . η c ∝ ~ e . The degree of control of theintensity imbalance of the laser fields gets more stringent asthe optical lattice depth V is increased. Nevertheless, at al-ready V = 20 E R ( ~ e ≈ . flection invariance with respect to their bisectrix. In thefollowing we stick to the simple case where θ = − θ = θ and θ is small. In this case both the Bravais lattice, thereciprocal lattice, the Brillouin zone Ω and the diamond-shaped primitive cell Σ get modified. The new reciprocalbasis vectors turn out to be b ′ = b + δb , b ′ = b + δb where δb = ( θ/ √ b and δb = ( θ/ √ b . Since theexchange symmetry b ↔ b is again preserved, the newpotential continues to display the Ox -reflection invari-ance. Figure 18 gives a plot of the new potential struc- O PQ R a a FIG. 18: [Color online] Distorted lattice obtained with bal-anced field strengths s n = 1 and angle mismatch θ = − θ = − π/
10. In this particular case the hexagon of field min-ima is stretched along the horizontal axis Ox and the vec-tors c ′ n connecting a given minimum to its three nearest-neghbors have now different lengths. In the situation depicted ˛˛˛˛˛˛ c ′ ˛˛˛˛˛˛ = ˛˛˛˛˛˛ c ′ ˛˛˛˛˛˛ = ˛˛˛˛˛˛ c ′ ˛˛˛˛˛˛ . In turn, due to the reflection symme-try about Ox , the tight-binding hopping amplitudes satisfy ˛˛˛˛˛˛ t ˛˛˛˛˛˛ = ˛˛˛˛˛˛ t ˛˛˛˛˛˛ = ˛˛˛˛˛˛ t ˛˛˛˛˛˛ . ture for θ = − π/ b ′ and b ′ decreases when θ is increased from 0. Inturn the angle between the corresponding a ′ n increasesand the hexagon structure made by the a and b minimaget compressed along Ox . The opposite conclusion holdswhen θ is decreased from 0. We get again the situationwhere (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t /t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ θ ≥ θ > θ < closing theangle between the b ′ n , so for θ = − θ = θ > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t /t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) then increases and the threshold (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t /t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 ismore rapidly hit. This is the situation we explore.Applying the same reasoning as before, we thus predictthe critical angle mismatch beyond which the masslessDirac fermions are destroyed to scale as θ c ∝ ~ e . Again,to get θ c as a function of ~ e , we numerically compute theband structure at a given ~ e for different in-plane mis-match angles θ and then extract the value θ c for whichthe Dirac degeneracy is lifted. We then repeat the pro-cedure for different ~ e . As one can see, our predictionis in very good agreement with the numerical calcula-tions, see Fig. 19, and well supported by a quadraticfit. As θ c increases with ~ e , there is a trade-off to makebetween reaching the tight-binding regime where V islarge and achieving an experimentally reasonable anglemismatch tolerance which requires V to be small. Thetrade-off turns out to be a favorable one since already for V = 20 E R ( ~ e ≈ . ◦ on the laser beams alignment. We expect the same typeof scaling for small out-of-plane angle mismatches. Fur-7 Dirac points existDirac points disappear θ c / π ~ e FIG. 19: The critical angle mismatch θ c (in units of π ) beyondwhich the Dirac degeneracies disappear as a function of theeffective Planck’s constant ~ e = p E R /V . The dashed linecorresponds to a quadratic fit of the numerical data. The lin-ear coefficient is 0 .
109 while the quadratic one is − . θ c ∝ ~ e . The degree of control ofthe angle mismatch gets more stringent as the optical latticedepth V is increased. Nevertheless, at already V = 20 E R ( ~ e ≈ . ◦ whichis not particularly demanding. thermore, when several small distortions combine, theireffects should add up and thus the critical imperfectionthreshold should still scale with ~ e .As an overall conclusion we see that massless Diracfermions are quite robust to moderate lattice distortions.Demonstrating them in an experiment should not be par-ticularly demanding in terms of the control of the laserconfiguration. C. Inequivalent potential wells
We finally briefly mention how to distort the opticallattice in a systematic manner as it allows for an experi-mental control of the mass of the Dirac fermions as wellas for a continuous switch from a honeycomb lattice to atriangular one.In Sec. II B 4, we observed that the honeycomb poten-tial (6) is the simplest of all graphene-type potentials,characterized by choosing v and v b real (in fact, pos-itive) while putting all unrelated coefficients in (17) tozero. Now, letting v b to acquire a phase ϕ , such thate − i ϕ v b is positive, will break the reflection symmetry ofthe honeycomb potential [30].In the r -dependent part of the dimensionless potential(8), this phase ϕ is introduced by the replacement X a =1 cos( b a · r ) → X a =1 cos( b a · r + ϕ ) , (57)where b = − b − b . This can be implemented by super- imposing three independent standing waves, of the samewavelength and with equal intensity, whose wave vectorsform the trine of Fig. 1 [60]. As a consequence of theincoherent superposition, the t replacement of (5) is notavailable, and the r replacement alone cannot remove allthree phases of the standing waves. One can, however,shift r such that the three phases are the same, and thenone has an intensity pattern proportional to the right-hand side of (57).Most of the hexagon structure of Fig. 4 remains un-changed by this modification: lattice sites a , b , c con-tinue to be the locations of local minima and maxima,whereas the saddle points s acquire new positions on the. . . abcabc . . . lines.Figure 20 confirms that, for small ϕ values, the minimaof the honeycomb dipole potential are still organized ina hexagonal pattern but we now have different potentialdepths at sites a and b . The potential energy mismatchis 2 ǫ ≈ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / √
3. In view of (21) and (24), this meansthat the Dirac fermions acquire a mass m ∗ ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) or, inother words, that the Dirac degeneracies are lifted. Thepossibility of fine-tuning the mass of the Dirac fermionsthrough the parameter ϕ is an interesting experimentalknob to play with.Increasing (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) further, one can also see that, for theparticular values (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = π/ π/
2, the three sublatticesof saddle points merge into a single triangular lattice,which coincides with the a , b , or c lattice, respectively;see Fig. 20. This merging of a potential minimum ormaximum with three saddle points, leads to a peculiarthird-order saddle point. For ϕ = π/
6, say, the s sitesmerge with the b sites and we have X a cos( b a · r + ϕ ) (cid:12)(cid:12)(cid:12) ϕ = π/ ≈ − X a (cid:2) b a · ( r − r b ) (cid:3) (58)for (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − r b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ κ − , hence a cubic saddle point ratherthan the usual quadratic saddle point.An unpolarized ultracold gas of spin- fermions loadedinto such a potential at half-filling would lead to twofermions per well. By driving the system through attrac-tive interactions, one could even get a Mott insulator offermion pairs. By switching off all interactions and set-ting ϕ = 0, one should be able to study oscillations ofatoms between the a and b sublattices. We will analyzethis situation in a follow-up paper. V. CONCLUSION
Motivated by the vivid field of graphene physics, wehave explained and analyzed how to reproduce masslessDirac fermions by loading ultracold fermions in an opti-cal lattice with honeycomb structure. We have describedthe two-dimensional laser configuration that gives riseto an optical potential where field minima are organizedin a honeycomb structure (with lattice constant a ) and8 ϕ = 0 . .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................... x ....................................................................................................................................................................................................................... V ϕ ( x e x ) ....... V ....... V ....... a ....... a ....... b ....... b ....... b ....... c ....... c ....... s ....... s ϕ = π/ . .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................... x ....................................................................................................................................................................................................................... V ϕ ( x e x ) ....... V ....... V ....... a ....... a ....... b ....... b ....... b ....... c ....... c ....... s ....... s ϕ = π/ . .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................... x ....................................................................................................................................................................................................................... V ϕ ( x e x ) ....... V ....... V ....... a ....... a ....... bs ....... bs ....... bs ....... c ....... c ϕ = π/ . .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...................... x ....................................................................................................................................................................................................................... V ϕ ( x e x ) ....... V ....... V ....... a ....... a ....... b ....... b ....... b ....... c ....... c ....... s ....... s FIG. 20: For various values of the phase parameter ϕ of (57),the plot shows the potential energy along a . . . abcabc . . . linein Fig. 4. The top plot, for ϕ = 0, repeats the bottom-left plotof Fig. 4 for reference. The degeneracy between sites a and b is lifted for the small ϕ value of ϕ = π/
24, the saddle pointshave moved closer to the b sites, where we continue to havelocal minima. In this situation the Dirac fermions acquire amass m ∗ ∝ ˛˛˛˛˛˛ ϕ ˛˛˛˛˛˛ . When ϕ = π/
6, the saddle points s coincidewith the b sites, and we have cubic saddle points there. Fi-nally, in the bottom plot, we have ϕ = π/ b and c sites, with po-tential maxima at both of them. Except for a displacement,the potential in the bottom plot is the negative of the poten-tial in the top plot, and thus identical with the honeycombpotential (6) for red rather than blue detuning of the threerunning wave lasers. For ease of comparison, the potentialconstants are adjusted such that the maxima and minima areat V = 0 and V = 9 V , respectively, for all ϕ values. we have thoroughly detailed the corresponding crystallo-graphic features. The behavior of atoms propagating insuch an optical potential in the tight-binding regime isin one-to-one correspondence with the behavior of elec-trons propagating in a graphene sheet. The ground stateand first-excited levels of the band structure exhibit twoconical degeneracies located at the corners of the first Brillouin zone, as dictated by symmetry arguments. Inthe neighborhood of these degeneracies, the band spec-trum is linear.When the lattice is loaded with fermions at half-filling,the Fermi energy slices the band structure at these de-generacy points, known as the Dirac points. Aroundhalf-filling, the tight-binding Hamilton operator can thenbe recast in a form reminiscent of the relativistic Weyl-Dirac Hamilton operator and featuring so-called masslessDirac fermions. The important parameter driving thedynamics turns out to be the hopping amplitude t be-tween nearest-neighbors sites as it gives the band width W = 6 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and the “Fermi velocity” v = 3 a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / (2 ~ ). Wehave derived a semiclassical expression for (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) in termsof the effective Planck’s constant of the problem, namely ~ e = p E R /V (with V the optical potential strengthand E R the recoil energy) and have compared it to anexact numerical calculation of the band spectrum. Fromthis we have derived quantitative experimental criteria(such as the required initial temperature of the atomicgas) to reach the massless Dirac fermions regime.We have also examined the robustness of the masslessDirac fermions to imperfections of the laser configura-tion (field strengths imbalances and angle mismatches).Massless Dirac fermions turn out to be quite robust asthe equality of the beam intensities should be controlledwithin the few percent range while the respective beamangles should equal 2 π/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is strong enough. For repulsive interac-tions, quantum Monte-Carlo calculations predict anti-ferromagnetic order to occur at half-filling [18]. For at-tractive interactions, mean-field calculations have startedto analyze the BEC-BCS crossover and predict a semi-metal/superconductor transition [19]. Recent Monte-carlo studies have even started to analyze this BEC-BCScrossover [64] and one can expect an increase of suchstudies in the near future. Very recently, implementa-tions of massless Dirac fermions in square lattices havebeen proposed [65, 66]. The situation seems thus ma-ture for an experimental effort towards loading ultracoldfermions in a honeycomb optical lattice. Acknowledgments
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