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Negative Refraction in Mo¨bius Molecules Y. N. Fang,1,2,3 Yao Shen,4 Qing Ai,5,∗ and C. P. Sun1,3,† 1Beijing Computational Science Research Center, Beijing 100084, China 2State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, and University of the Chinese Academy of Sciences, Beijing 100190, China 3Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 4School of Criminal Science and Technology, People’s Public Security University of China, Beijing 100038, China 5Department of Physics, Beijing Normal University, Beijing 100875, China We theoretically show the negative refraction existing in Möbius molecules. The negative refractive index is induced by the non-trivial topology of the molecules. With the Möbius boundary 6 condition,theeffectiveelectromagnetic fieldsfelt bytheelectron in aMöbius ringisspatially inho- 1 0 mogeneous. Inthisregard,theDN symmetryisbrokeninMöbiusmoleculesandthusthemagnetic response is induced through the effective magnetic field. Our findings open up a new architecture 2 for negative refractive index materials based on the non-trivialtopology of Möbius molecules. p e PACSnumbers: 81.05.Xj,03.65.Vf S 9 I. INTRODUCTION Instead of fabricating those delicately designed struc- ] turesfromtheconventional“top-down”approach,anat- h tractive alternative is to generate the dielectric media p It is recognized that materials with simultaneously with self-assembled functional atoms or molecules [13– - negative permittivity and permeability also support the t 15]. This “bottom-up” approach is particularly inter- n coherentpropagationofelectromagneticfields[1],likethe esting due to the small size of molecules and their as- a usual case where both permittivity and permeability are u sociated quantum effects. For example, by utilizing the positive. Due to a reversed phase velocity with respect q quantuminterferenceamongmulti-levelatoms,it is pos- to the group velocity [2], the boundary condition makes [ sible to suppress the absorption meanwhile keep reason- the light to bend in an unexpected direction as if the re- able optical response [16]. We notice that the molecular 2 fractive index appears negative according to the Snell’s v law. Hence, the effect is termed as negative refraction. ringcannoteffectivelyrespondtothemagneticfield,be- 9 causetheredoesnotexistanysplittomakethesymmetry Negative refractive index materials are very promis- 2 breaking as shown in Refs. [15, 17] and Appendix A. 7 ing, e.g., in achieving the electromagnetic field cloaking In this paper, instead of relying on the split-ring con- 5 [3, 4], facilitating the sub-wavelength imaging [5], and figuration,wedescribeapotentialnegative-refractingdi- 0 crime scene investigation [6]. However, such materials electricmediumwiththeMo¨biusmolecules[18,19]. Such 1. do not naturally exist for the absence of magnetic re- molecules were theoretically proposed [18] and had been 0 sponse at the same frequency as electric response. To fabricated [20–22] in several experiments. Compared 5 overcome this difficulty, delicately designed metamateri- with other molecular-ring-like annulenes, the symmetry 1 als have been proposed and tested. Metamaterials rely : ofMo¨biusmoleculeisloweredbyitsboundarycondition, v on structural designing of the sub-wavelength unit cells i.e., the usual D symmetry is broken here to C [19]. i totunetheelectromagneticresonantcharacteristic[4]. A N 2 X Similartoasuperconductingringwithmagneticfluxgo- key element commonly involved in such artificial archi- ing through [23], this particular boundary condition for r tecturesistheconfigurationofsplit-ringresonator[7–10]. a Mo¨biusmoleculecouldbecanceledbysomelocalunitary ThisstructureisanalogoustoanLCresonatorwithchar- transformations. Afterthetransformation,themotionof acteristic frequency ω 1/√LC with L and C being 0 ∼ an electron in the Mo¨bius ring is subject to an effective self-inductance and capacitance of the resonator, which spatially-inhomogeneousmagneticfield, eventhoughthe is tunable by engineering. A metamaterial with split- original system is exposed to a physically uniform elec- ring resonators can negatively respond to the magnetic tromagnetic field [19]. Both electric and magnetic re- field when ω & ω [9]. Although the negative refrac- 0 sponses can be induced at the same transitions and thus tion has been demonstrated at the visible wavelength, both negative permittivity and permeability can be ob- manufacturing of microstructureat the size of 30-100nm servedatthesamefrequencies. Inthissense,thenegative is required [11]. Such technically requesting fabrication refraction in Mo¨bius molecules is intrinsically caused by makes it still an open challenge to scale the materials to the non-trivial topology of the molecules. Motivated by 3D bulk. this discovery, we propose realizing a new kind of metamaterials based on the Mo¨bius molecules. This paper is organized as follows. In the subsequent ∗Electronicaddress: [email protected] section,theMo¨biusmoleculeisbrieflyintroducedfollow- †Electronicaddress: [email protected] ing Ref.[19]. InSec. III,the allowedtransitionsbetween 2 ThedifferencebetweentheMo¨bius molecularringand thecommonchemicalannulenesliesintheboundarycon- dition[19]. Here,theNthnucleusintheAringisexactly the 0th nucleus of the B ring. Therefore, a = b , (4) 0 N b = a (5) 0 N imply that the operators do not obey the periodical boundarycondition. FurthercalculationsinAppendix A demonstrate that the vanishing magnetic dipole for the perfect ring results in the absence of negative refraction. FIG.1: (color online) (a) Schematicofa molecular ringwith With a local unitary transformation U (see Ref. [19] j Möbius topology: carbon atoms are shown by black circles, and Appendix B), the Hamiltonian becomes withthebondingatomslinkedtogether. (b)Energyspectrum Ekσ ofthemolecular ring. Two energybandsaredenotedby N−1 their different pseudo spin labels σ = and . Allowed inter- H = [B†Vσ B ξ(B†QB +h.c.)], (6) band transitions from the ground sta↑te are↓indicated by the j z j − j j+1 j=0 colored arrows, where dashed (x and y) and solid (x, y, and X z) arrows indicate thepossible field polarizations. where c B = U A j↑ , (7) differenteigenstatesofaMo¨biusmoleculeinducedbyap- j j j ≡ cj↓ (cid:20) (cid:21) plied electromagnetic fields are discussed under dipole eiδ/2 0 approximation. Then,inSec.IVweinvestigatethepossi- Q = , (8) 0 1 bilityofhavingsimultaneouslynegativepermittivityand (cid:20) (cid:21) permeability in a medium consisting of non-interacting δ = 2π/N, (9) Mo¨bius molecules. In Sec. V, the negative refraction is the Pauli matrix σ is written in the pseudo spin space, analyzed at a planar medium interface for two different z i.e., B†σ B = c† c c† c . We emphasize that the incident conditions. And discussions regarding loss and j z j j↑ j↑ − j↓ j↓ new operators c ’s satisfy the periodical boundary con- bandwidth of negative refraction are also given. Finally, jσ dition. Equation (6) represents a pseudo spin in a fic- the main results are summarized in Sec. VI. titious ring [19]. For the spin-up state, the ring is fur- ther penetrated by a perpendicular homogeneous mag- II. THE MO¨BIUS MOLECULAR RING netic field, as indicated by the phase factor exp(iδ/2) in the hopping term. More insight on the non-trivial boundaryconditionstemsfromthecorrespondingenergy TheMo¨biusmolecularringhasseveralphysicalrealiza- spectrum, cf. Fig. 1(b) and Appendix B: tions,e.g.,throughgraphene[24]andthenon-conjugated molecules[25]. Forageneraldouble-ringsystemwith2N δ atoms,asshowninFig.1(a),theHu¨ckelHamiltonianfor Ek↑ = V 2ξcos(k ), (10) − − 2 a single electron reads [19, 26] E = V 2ξcosk, (11) k↓ − − N−1 H = A†MA ξ A†A +h.c. , (1) which are obtained by performing the Fourier transform j j − j j+1 on B with the resulting transformed operator denoted Xj=0 h (cid:16) (cid:17)i asCjandk =lδ(l =0,1, N 1). Comparedwiththe k ··· − where spectrumofatopologicallytrivialannulene,theeffective magneticfluxresultsinashiftoftheE bandsuchthat a k↑ A = j , (2) the two lowest states become degenerate. Similar shift- j b (cid:20) j (cid:21) ing behaviorhas alsobeen suggestedfora cyclic polyene ǫ V under a real magnetic field [26]. M = − . (3) V ǫ (cid:20) − − (cid:21) Here, the fermionic operators a†(b†) create a localized III. TRANSITION SELECTION RULES j j atomic-orbital φ (φ )atthejthnuclearsiteofthe j+ j− | i | i A (B) ring respectively; 2ǫ describes the on-site energy ThegeneralinteractionHamiltonianofamoleculewith difference between atoms of two rings; The inter (intra) externalelectromagneticfieldiscomplicated. Butdueto -ring resonance integral is denoted by V (ξ). Hereafter, the small size of a molecule (typically around a few nm) weconsideraspecialcasewhereallatomsareofthesame with respect to wavelength the dipole approximation is species, i.e., ǫ=0 [26], as indicated in Fig. 1(a). applicable. In the linear response regime, non-vanishing 3 electromagnetic response implies that the corresponding integral. For the Mo¨bius molecular ring, β is either ξ ij dipole operators have off-diagonal elements between dif- or V, depending on whether the atoms are on the same ferent eigenstates of H. Hence it is useful to analyze the ring or not. Consequently, the magnetic dipole operator transitionselectionrulesunderthedipoleapproximation. is explicitly given as We investigate the dipole transition selection rule of the Mo¨bius molecular ring under the perturbation of an ex- m~ = e [A†(VS~+−σ )A +ξ(iA†N~ A +h.c.)] ternaloscillatingelectricfieldwithamplitudeE~0 andfre- − Xj j j,j y j j j j+1 quency ω. The interaction Hamiltonian is written under (17) the dipole approximationas with the effective area HE′ =−d~·E~0cosωt, (12) S~iαjβ ≡ 21R~iα×R~jβ (18) where the electric dipole operator d~= e~r and ~r is the − (α,β = ) and the vectorial matrix position operator for the electron. ± The transition selection rule can be inferred from the wmhaitcrhixareeletmheenltinseoafr cHoEm′ bbinetawtieoennsothfetheeigaetnosmtaicteosrboiftaHls., N~j ="S~j+,0j++1 S~j−,0j−+1 #. (19) In the following we ignore the overlapintegrals from different atomic orbitals and assume [15, 26] Under the dipole approximation, the interaction Hamiltonian describing the coupling of the Mo¨bius φjs ~r φj′s′ =δjj′δss′R~js, (13) molecular ring to an external oscillating magnetic field h | | i withamplitudeB~ andfrequencyωissimilartoEq.(12): where R~ (R~ ) denotes the position vector of the jth 0 j+ j− nuclear site at the A(B) ring. For the Mo¨bius molecular H′ = m~ B~ cosωt. (20) ring with radius R and width 4W, the nuclear positions B − · 0 areexplicitlywritteninthemolecularcoordinatesystem, The explicit expressionsofthe matrix elements are com- cf. Fig. 1(a) and Ref. [19], as plicatedinthecaseofmagneticperturbationandthuswe ϕ ϕ list them in Appendix C. Through straightforward cal- R~j± =(R W sin j)cosϕjeˆx+(R W sin j)sinϕjeˆy culations, we find the following transition selection rules ± 2 ± 2 ϕ as illustrated in Fig. 1(b): j Wcos eˆ , (14) z ± 2 xyz xy k, ⇋ k, , k, ⇋ k+2δ, , where | ↓ixy | ↑i | ↓i xy|z ↑i (21) k, ⇋ k+δ, , k, ⇋ k+δ, , | ↑i | ↓i | ↓i | ↑i ϕ =jδ (15) j where only the inter-band transitions are shown since in is a rotating angle which locates atoms on the rings. the limit of large N, the spectra E ’s become continu- kσ Thetransitionselectionrulesfortheelectricdipoleop- ouswithrespecttok andthus the inter-bandtransitions erator are briefly summarized as, cf. Appendix C, are more relevant to the response at high frequency as compared to the intra-band transitions. A comparison x,y,z x,y k, ⇋ k, , k, ⇋ k+2δ, , between Eq. (21) and Eq. (16) indicates that the selec- | ↓ix,y | ↑i | ↓i z| ↑i (16) tion rules for the inter-band transitions are almost the k,σ ⇋ k δ,σ′ , k, ⇋ k+δ, . | i | ± i | ↓i | ↑i same for both electric and magnetic couplings. Here, k,σ denotes an eigenstate of H with energy E . kσ | i Thesuperscriptsx,y,zindicatetheelectricfieldpolariza- tions. Those allowedinter-bandtransitions are schemat- IV. PERMITTIVITY AND PERMEABILITY ically shown in Fig. 1(b). Since the negative refraction depends on the simulta- In order to realize the negative refraction, the simul- neously negative permittivity and permeability, we fur- taneously negative permittivity ←ε→r and permeability ←µ→r therinvestigatethetransitionselectionrulesforthemag- arerequired[1]. To evaluate those quantities,let us con- netic dipole m~. For molecular systems the current is re- sider a negative index medium realized by single crystal stricted to the chemical bond, thus m~ is written as [27] of Mo¨bius molecules, with the same orientation for each m~ = J S~ ,whereS~ isaneffectiveareaspecified molecule. Although the permittivity and permeability hi,ji ij ij ij below. The bond current J flows from the ith atom to are originally treated as scalars, for anisotropic medium ij P the jth atom, and by the tight-binding approximation theycouldgenerallybesecondordertensorsthatdepend the summation only runs over bonded-atoms pairs. The onthefrequencyωofexternalfieldsE~ andH~. ←ε→r and←µ→r operator corresponding to the bond current is given by are calculated by considering that the electric displace- Jˆ = ieβ a†a + h.c. [27], where β is the resonance ment D~ and magnetic induction B~ are related to the ij ij i j ij 4 polarization P~ and the magnetization M~ of the medium [15, 28]: D~ = ε0←ε→rE~ =ε0E~ +P~, (22) B~ = µ0←µ→rH~ =µ0H~ +µ0M~. (23) Here, P~ and M~ are quantum mechanical averaging of the electric and magnetic dipole operators in the per- turbed molecular ground state. In the linear response regime, applying the Green-Kubo formula yields [30], cf. Appendix D, FIG. 2: (color online) Relative permittivity ←ε→r and permeability ←µ→r as a function of the detuning ∆ω = ω ∆0,↑ − 1 d~ (d~ E~) between the frequencies of incident field ω and transition P~ = −~υ Re ′ ωg,kσ∆kσ,g+·iγ , (24) |0,↓i⇋|0,↑i. Onlynegativediagonalelementsof←ε→r and←µ→r 0 kσ in their corresponding principal-axis coordinate systems are k,σ − X shown. Here we adopt following parameters V = ξ = 3.6eV M~ = µ0 Re ′m~g,kσ(m~kσ,g ·H~). (25) [31], W =0.077nm [32], R=NW/π, γ−1 =4ns [39]. −~υ ω ∆ +iγ 0 kσ k,σ − X Here, ∆kσ = Ekσ Eg is the molecular resonant transi- long lifetime τ, ←ε→r is simplified from Eq. (24) by only tion frequency bet−ween the excited and ground states of including the summation term whose denominator con- HamiltonianH. Og,kσ = g Ok,σ isthematrixelement tains a transition frequency that is equal to ∆0↑. With ofoperatorObetweenthehu|np|ertuirbedmolecularground thisapproximation,therelativepermittivityissimplified state g and excited state k,σ . The ground state is in the molecular coordinate system as | i | i excluded in the summation, as indicated by the prime. 1 η′(ω) 0 0 Ttohbeelifiedteinmtiecsaτl. =υ01/isγtohfetvhoeluemxceitoecdcustpaiteedsbayreaaMssuo¨mbieuds ←ε→r(ω)= −00 1−2ηη′′((ωω)) 1−24ηη′(′ω(ω)), (26) molecule in the medium. In the above calculation, the − − inter-molecule interactions have been neglected, i.e., the   where polarization and magnetization are assumed to be the contribution by a collection of non-interacting particles. 1 e2W2 η(ω)= , (27) Thisapproximationisvalideitherinthelowdensitylimit 8~ε0υ0ω−2V −2ξ(1−cos2δ)+iγ or for medium with crystal structure, e.g., Zinc-Blende [28]. Also, we have verified that both the central fre- andη′(η′′)isreal(imaginary)partofη. Thepermittivity quencyandbandwidthofnegativerefractionarenotsub- tensorisnotdiagonalinthemolecularcoordinatesystem stantially modified when the inter-molecular interaction because of the non-zero off-diagonal term ε(yz). Since r isconsideredatthelevelofLorentzlocalfieldtheory[29] the tensor is real-symmetric, ←ε→r is diagonalized in the as proven in Appendix E. principal-axiscoordinatesystembyarotationaroundthe Because ←ε→r 1 O(P~ /E~ ) and ←µ→r 1 x axis. This yields ≃ − | | | | ≃ − O(M~ /H~ ), Eqs. (24,25) imply that the negative val- ues| of|←ε→|r |and ←µ→r occur necessarily around the molec- ε1 =1−5η′, (28) ular resonant transition frequencies ∆kσ. Furthermore, 1 η′ and1forrelativepermittivityalongthreeprincipal the responses exist only if the corresponding transitions ax−es, respectively. In Fig. 2(a), the relative permittivity are allowed by both electric and magnetic dipole inter- is numerically demonstrated with the following parame- actions with fields. As for the Mo¨bius molecular ring, ters, V =ξ =3.6eV[31], W =0.077nm[32], πR=NW, it is particularly interesting that the inter-band transi- γ−1 =4ns [39], whichareobtainedby fitting the spectra tion k, ⇋ k′, could be allowed both electrically in experiments. | ↑i | ↓i and magnetically by the dipole couplings. Besides, the On the other hand, the tensor of the relative perme- typical value for resonance integral is around the order ability ←µ→r is represented by of a few eV [31, 43], already in the region of visible frequency. Therefore, simultaneously negative permittivity 1 α2η′ 0 0 − and permeability in the visible frequency regime might ←µ→r(ω)= 0 1 α2η′ 2αβη′ , (29) be realized in a medium containing molecules with the  0 −2αβη′ 1− 4β2η′  − − Mo¨bius topology.   where To quantitatively investigate this possibility, ←ε→r is ex- plicitlycalculatedfortheMo¨biusringbyusingthematrix R δ elements d~kσ,g. For positive V and ξ, the ground state α = ~c[V +ξ(cosδ−cos2)], (30) oinfteHr-bisan|0d,↓triaannsidtiothnufsreEqgue=ncEy0∆↓.0↑Wahnednfoωr issuffinecairentthlye β = 2R~cξ sin2 2δ cos2δ. (31) 5 The permeability tensor is not diagonalin the molecular coordinate systemeither. Because α and β are generally different from 1, ←µ→r and ←ε→r cannot be diagonalized by the same rotating transformation. In other words, the principal axes in which ←µ→r is diagonal do not coincide with the principal axes of ←ε→r. This yields µ =1 α2+4β2 η′, (32) 1 − 1 α2η′,and1forrelative(cid:0)permeabil(cid:1)ityalongthreeprin- − cipal axes, cf. Fig. 2(b), respectively. FIG.3: (coloronline)Schematicplotsofthereflectionandthe refractionatthemediuminterfacez=0. (a)Theelectricfield V. NEGATIVE REFRACTION AT MEDIUM and (b) the magnetic field of the refracted light is polarized INTERFACE along thex-direction,respectively. TheMo¨biusmediumisanisotropicinthesensethatthe wavevectorandthezaxis. Weconsidertwoindependent relativepermittivity ←ε→r and permeability ←µ→r are tensors incident configurations respectively: rather than scalars. Therefore, apart from demonstrat- ingsimultaneously←µ→r <0and←ε→r <0insomefrequency region, the more concrete way to show the existence of E~ =E eˆ ei(kiyy+kizz−ωt), (E-polarized) (33) i 0 x negative refraction is to investigate the behavior of refracted electromagnetic waves at the medium interface. and Here, we investigate the reflection and refraction at a planarinterfacebetweentheMo¨biusmediumandtheair H~i =H0eˆxei(kiyy+kizz−ωt), (H-polarized) (34) fortwospecificincidentconfigurations. Theresultsshow that the medium is “left-handed” for E-polarized propa- where ~ki = kiyeˆy +kizeˆz is the wave vector of incident gating mode [33]. light. These twoconfigurationsareknownasE-polarized The behavior of a propagating wave inside a medium andH-polarized respectively,astheelectricandmagnetic is captured by its phase and group velocities. For ex- fields of the refracted light are perpendicular to the re- ample, when the negative refractionwas first introduced fracted wave vector [33], respectively. in Veselago’s seminal paper [1], a “left-handed” material wasdescribedas a medium inwhichthe electromagnetic wave propagates with the opposite phase velocity with A. E-polarized Incident Configuration respecttothegroupvelocity. Inmediawherethepermit- tivityandpermeabilitytensorsaresymmetric,the group For the specific incident configuration given by velocity is along the same direction as the Poynting vec- Eq. (33), the three criteria for a “left-handed” medium tor [33]. In such case, the reversal of the phase velocity could be checked one by one. It follows from the bound- tothePoyntingvectorhasbeenappliedasacriterionfor aryconditionsderivedfromtheMaxwell’sequationsthat “left-handed” materials: ~k S~ < 0 [35], where ~k and [28] t t t · S~ are respectively the wave vector and Poynting vector oftrefracted field. Besides this, if we consider the refrac- eˆz (E~i+E~r E~t) = 0, × − tion from the medium interface, the causality requires eˆ (H~ +H~ H~ ) = 0, (35) z i r t that the normal component of its Poynting vector is in × − k =k = k sinθ, iy ty i the same direction as that for the incident light [36]. In summary,weadoptthefollowingthreecriteriaascharac- where the subscript r denotes the reflected field back to teristics of a “left-handed” medium: (i) The wave vector the air. In accordance with above boundary conditions, ~kt in the medium is a real-valued vector; (ii) The wave the electric fields of the reflected and refracted lights vector ~k and Poynting vector S~ in the medium satisfy could be written as t t ~k S~ < 0; (iii) The normal component of the Poynting t· t E~ = rE eˆ ei(kiyy−kizz−ωt), (36) vector remains the same sign across the medium inter- r 0 x face,e.g.,ifinterfacenormaliseˆz thenSizStz >0,where E~t = tE0eˆxei(kiyy+ktzz−ωt), (37) S~ is the Poynting vector of the incident light. i Let us consider a linearly polarized monochromatic where we also use r and t to representthe reflectionand light which is incident from the air onto the interface refraction coefficients respectively, and 1 + r = t. To of Mo¨bius medium as illustrated in Fig. 3. Let z =0 be determinetherefractedwavevector,wecombinethetwo themediuminterfaceandsupposethattheincidentplane Maxwell’s equations~kt E~t = ωµ0←µ→rH~t and~kt H~t = × × is the y-z plane and θ is the angle between the incident ωε0←ε→rE~t: by multiplying both hand sides of the first − 6 equation with ←µ→r−1 and then inserting it to the second (ktx,kty,ktz) space (or equivalently in the (ntx,nty,ntz) equation, the following equation could be derived, i.e. space by ~nt ~kt/ω√µ0ε0). This geometry relation then ≡ ~kt×[(←µ→r)−1(~kt×E~t)]=−ω2µ0ε0←ε→rE~t. (38) aPsosyenrttsin:ginvethcteormfeodriuamgiwveitnh~ksyimsmeiethtreircp←ε→arraallnedl o←µ→rr,anthtie- parallel to the normal vector of the wave vector surface, Since the equation is homogeneous in E , E and E , tx ty tz as proven in Appendix F. the necessary condition for non-vanishing refracted light The shape of the wave vector surface, as shown in is that the determinant of its coefficient matrix is zero. Combined with E~ = E eˆ and k = 0, this necessary Fig. 5, strongly depends on the incident frequency. And t tx x tx those different shapes will in turn give rise to contrast- condition could be explicitly written as ing propagationproperties of the refracted fields. Let us first consider an incident field with frequency ω ∆ . 0↑ ωc22ε(rxx)− µ1 (ki2yµ(rxx)+2µ(ryz)kiyktz +µr(zz)kt2z)=0. iIstofofltlohwesofrrdoemr oEfqu.n(i2ty7,)wthhailteiαn this1caansde ηβ′ < 01.a≪Tndhe|rηe′-| 1 ≪ ≪ (39) fore, after ignoring the second order terms of α and β in Inserting Eqs. (26,29) into the above equation yields Eq.(39),thewavevectorsurfaceisnearlyofcircleshape, i.e., k = k µ (1 η′)(1 4β2η′) sin2θ tz i 1 {± − − − n2 +n2 1 η′. (47) µ(yqz)k (cid:2)/µ(zz). (cid:3) (40) ty tz ≈ − − r iy} r Obviously, a real solution of k is admitted for all inci- tz The time-averaged Poynting vector of refracted light dent angles, cf. Fig. 5. Since µ(yz) 1 and meanwhile S~ =Re E~ H~∗ /2 reads r ≪ t { t× t} µ1,µ(rxx),andµr(zz)areoforderonewhenω ∆0↑,itfol- ≪ S~t = Styeˆy+Stzeˆz, (41) lowsfromEqs.(41-43)thatthePoyntingvectorS~t isap- proximatelyalongthe directionof~k . Because~k S~ <0 t t t where is violated, the refracted field is “right-handed”. · E2t2 When the frequency of the incident field is increased Sty = 2ωµ0 µ ktzµ(ryz)+kiyµ(rxx) , (42) to just above the lowest inter-band transition frequency 0 1 ω &∆ , since η′ 1, µ in Eq. (32) could be negative. (cid:16) (cid:17) 0↑ 1 E2t2 ≫ S = 0 k µ(yz)+k µ(zz) . (43) By linearly combining nty and ntz to eliminate the cross tz 2ωµ0µ1 iy r tz r terminEq.(39),thewavevectorsurfaceisahyperboloid, (cid:16) (cid:17) i.e., According to Eqs. (29,40,41), for the E-polarized configuration, the three criteria for “left-handed” medium n˜2 n˜2 are specified as~k S~ <0, and tz ty =1, (48) t· t (1 η′)µ1 − η′ 1 − − (1 η′) 1 4β2η′ sin2θ, if µ >0, (44) − − ≥ 1 where n˜tz=ntzsinφ+ntycosφ, n˜ty=ntzcosφ ntysinφ, (1 η′)(cid:0)1 4β2η′(cid:1) sin2θ, if µ1 <0, (45) and the mixing angle φ = tan−1[ 4αβ/(β2− α2)]/2. − − ≤ − − Similar non-closed wave vector surface has been sug- and (cid:0) (cid:1) gested in uniaxial left-handed materials [34]. As µ <0, 1 E2t2 Eq. (41) implies that the Poynting vector is opposite to Stz = 0 [ 2αβη′kiy +(1 4β2η′)ktz]>0. (46) the normal vector of the wave vector surface and thus 2ωµ µ − − 0 1 ~k S~ < 0, showing that the wave propagating in the t t · Figure4(a)showsthe“phasediagram”intheθ-ωplane Mo¨bius medium is now “left-handed”. Furthermore, in as determined by the above three criteria. Obviously, ordertofulfilltherequirementforcausality,therefracted thereisafrequencywindowinwhichtheappliedelectro- wave vector should be on the lower branch, cf. Fig. 5. magneticfieldscanbenegativelyrefracted,whileinmost Equation (48) also indicates the frequency region regions of the θ-ω plane the refracted light are “right- where “left-handed” propagating wave is allowed in the handed”. Besides these, there is also a considerable re- Mo¨bius medium. Since α and β are small quantities, gionwherethelightwillbetotallyreflected. Theinsetof the sign of µ will change along with the increase of η′. 1 Fig.4(a)alsoshowsµ neartheinter-bandtransitionfre- This sign change will make the shape of the wave vector 1 quencyoftheindividualMo¨biusmolecule. Acomparison surface deform from a hyperboloid to a spheroid. How- shows that there is a correspondence between the “left- ever,because 1 η′ is still negativeand µ >0,Eq. (48) 1 − handed” phase boundary and the zeros of µ , which will willnothavesolutionforrealn . Animaginaryn indi- 1 tz tz be illustrated later. catesthatthewavecannotpropagateinsidethemedium Toquantitativelyunderstandthe“phasediagram”,we and the incident wave is totally reflected. Therefore, the rely on the geometry relation between the Poynting vec- frequency region for the propagating “left-handed” E- tor and the wave vector surface [33]. The wave vec- polarized wave in the Mo¨bius medium is determined by tor surface is defined by solutions to Eq. (38) in the the sign change of µ , cf. Fig. 4(a), and the resulting 1 7 FIG. 4: (color online) “Phase diagram” of left-handedness in the θ-ω plane for (a) E-polarized incident field, (b) H-polarized incident field. Red and yellow regions indicate respectively the parameter regions where the refracted wave is “right-handed” (RH)andtheincidentfieldistotallyreflected(TR)backtotheair. Therefractedfieldis“left-handed”(LH)intheblueregion. The insertsshow a magnified part in themain plot where ω is close to thelowest inter-band transition. Thered dashed curve shows µ1 as a function of ω near theinter-band transition. Here we use thesame parameters as in Fig. 2. | | Furthermore,thetotalreflectionoftheincidentlightwill happened as there is no real solution for n at large tz incident angle. B. H-polarized Incident Configuration The analysisforthe H-polarized case is similarto that fortheE-polarized case. InaccordancewithEqs.(35,34), thereflectedandrefractedmagneticfieldsarewrittenas, cf. Fig. 3(b) H~ = rH eˆ ei(kiyy−kizz−ωt), (49) r 0 x H~ = tH eˆ ei(kiyy+ktzz−ωt). (50) t 0 x The equation for determining the refracted wave vector component k is similar to Eq. (38), i.e., tz FIG. 5: (color online) Cross sections of the wave vector sur- faces of the refracted fields at different incident frequencies: ~kt (←ε→r)−1 ~kt H~t = ω2µ0ε0←µ→rH~t. (51) × × − ω ∆0↑ (thick red, magnified by 100), ω & ∆0↑ (thick h (cid:16) (cid:17)i blu≪e), ω ∆0↑ (thin red). The green circle, magnified by By solving this equation, we obtain the refracted wave 100, depic≫ts the wave vector surface of the incident field (E- vector as dpoirleacrtizioend)o.fIncoerarechspcoansdei,ntgheProeyfnraticntgedvwecatvoervS~etct(obrla~kctkaanrdrotwhe) ktz={±ki ε1[(1−η′α2)(1−4η′)−sin2θ] are explicitly shown, while the wave vector of the incident ε(yzq)k /ε(zz). (52) field~ki =ω√µ0ω0(sinθeˆy+cosθeˆz) is depicted as the green − r iy} r arrow. The black dashed line represents the constraint from The time-averaged Poynting vector for the refracted theboundarycondition. Hereweusethesameparametersas light is S~ =S eˆ +S eˆ , with t ty y tz z in Fig. 2. H2t2 S = 0 (k ε(yz)+k ε(xx)), (53) ty 2ωε ε tz r iy r 0 1 bandwidth, i.e., width of the frequency region support- and ing negative refraction, is = ω ω , where ω s are B | 1− 2| j H2t2 the two solutions to µ1(ωj)=0. Stz = 2ωε0 ε (kiyε(ryz)+ktzεr(zz)). (54) Ifwefurthertunethefrequencyωtobefarbiggerthan 0 1 ∆ , η′ approaches zero, i.e. η′ 0+. Equation (47) The three criteria for “left-handed” medium are then 0↑ is again valid in this case, but t→he radius of the wave summarized as~k S~ <0, and t t · vector surface for the refracted field is smaller than that of the incident field. However, ~kt S~t > 0 suggests that 1−η′α2 (1−4η′) ≥ sin2θ, if 5η′ <1, (55) the refracted light is also “right-h·anded” in this case. (cid:0)1 η′α2(cid:1)(1 4η′) sin2θ, if 5η′ >1, (56) − − ≤ (cid:0) (cid:1) 8 and 0.51ns is enough for observing negative refraction. Since the excited-state lifetime of Mo¨bius systems can reach (1 5η′)[ 2η′kiy +(1 4η′)ktz]>0. (57) as long as 350ns [39, 40], it is reasonable to expect a − − − medium with Mo¨bius molecules as a potential material The “phase diagram” in the θ-ω plane for the H- to show the negative refraction. polarized case is deduced according to above three criteria, as shown in Fig. 4(b). Similarly to the E-polarized case, there are regions in the θ-ω plane where the re- Although in our calculation only one electron is con- fracted propagating field is “right-handed” and regions sidered, our result is consistent with the more realistic where the incident field is totally reflected. However, in case when all π electrons from all atoms in the Mo¨bius contrast to the E-polarized case, there are no frequency molecule are taken into account. In that case, when the regionsintheH-polarized casesuchthatthepropagating spin degree of freedom is considered, two electrons with refracted field could be “left-handed”. different spin states can stay in the same energy eigenstate k,σ . For the ground state of the total system | i including all electrons, all the states of the lower energy C. Discussions band will be filled. Theoretically, there could be negative refractionaround4N possible transitionfrequencies Intheprevioussubsections,wehavediscussedtheneg- iftheexcited-statelifetime issufficientlylong,asimplied ative refraction regardless of loss. Generally speaking, by Eqs. (16,21,58). The inter-band transition in our cal- there will be loss in the medium due to couplings of the culationisjustthespecialcaseof4N possibletransitions. molecules to the bath. The condition for negative re- Inthis sense,oursimplifiedcalculationclearlyillustrates fraction in the presence of loss is slightly different from the key factors for demonstrating negative refraction in that without loss [37, 38]. Because of the loss effect, Mo¨bius molecules. the imaginary part η′′ should also be considered in the relative permittivity and permeability. This results in the replacement of η′ by η = η′ +iη′′ in Eqs. (26,29). Furthermore, the Hamiltonian describing a molecule Secondly, the refracted wave vector as determined from interactingwithelectromagneticfieldisapproximatedas the Maxwell’s equations is now generally complex, and dipole interaction in our calculation, cf. Eqs. (12,20). the phase velocity is along the real part of the refracted Generallyspeaking,therearemulti-polecontributionsto wave vector [37, 38]. Through considering the normal Coulomb interaction between molecule and electromag- incidence of an E-polarized field onto the medium inter- netic field. As long as the molecule is small, the dipole face, we can prove that the frequency region of negative approximation is valid and has been frequently used in refraction is not qualitatively changed when the loss is the investigationof metamaterials[14, 28]. Onthe other taken into consideration. hand,forthesakeofsimplicity,theinter-molecularinter- On the other hand, as depicted in Eq. (27), the nega- actionhasbeenneglectedinourcalculation. Asshownin tiverefractionwillapparentlydisappearifthedecayrate the Appendix E, based on the Lorentz local field theory, of excited stats is sufficiently large. A limitation on the boththecentralfrequencyandbandwidthofnegativere- lifetimeoftheexcitedstateforachievingnegativerefrac- fraction have not been substantially modified when the tioncouldbe deducedfromthe bandwidth = ω ω interactionbetweenmoleculesistakenintoaccount. Asa 1 2 B | − | according to zeros of µ (ω), i.e. result, by modeling Mo¨bius molecules as non-interacting 1 particles, the key factors influencing negative refraction e2(α2+4β2)W2 2 can be captured. = Re 4γ2. (58) B s(cid:20) 8ε0υ0~ (cid:21) − TheMo¨biusmoleculediscussedinthispaperistermed This indicates a restriction on the excited state lifetime an equilateral Mo¨bius strip as the twist density is 16ε υ ~ the same everywhere. In Ref. [25], it was reported 0 0 τ = , (59) c e2(α2+4β2)W2 that two conformations of tetrahydroxymethylethylene Mo¨bius molecule withchemicalformulaC H O have 42 72 18 above which the negative refraction from the Mo¨bius been synthesized. Although the equilateral Mo¨bius medium is expectable. Numerical verification has been molecule has not yet been synthesized, it was predicted performed and shows that the bandwidth is not qual- that it could be achievable [25]. On the other hand, we itatively changed with the inclusion of the loss effect. notice that the Hu¨ckel model with empirical parameters In experiments, it is possible to synthesize a Mo¨bius has been successfully applied to describing experimental ring of carbon atoms with N = 12 and radius 0.29nm data of more than 60 organic molecules with maximum [21, 22]. And it was theoretically predicted that Mo¨bius deviation no more than 15%[31, 32, 43]. Therefore, it is molecules with more than 60 atoms are as stable as reasonable to expect that the theoretical predictions for Hu¨ckel molecules [41]. By taking the value 3.6eV for Mo¨bius molecules could also be observed in the future V and ξ [31], one finds that an excited state lifetime of experiments. 9 VI. CONCLUSION the interaction does not mix different eigenstates of H. Consequently, the negative refraction is absent in this We have explored the Mo¨bius molecular ring as a po- situation. tential candidate for negative refraction. The previous Furthermore, we can show that for a common double investigationswiththefunctionalatomsormoleculesrely ring,i.e.,achemicalannulenewiththeperiodicalbound- on the conceptual analogy of the split-ring resonator for ary condition a0 =aN and b0 =bN, the magnetic-dipole magnetic response [15], while for Mo¨bius ring this is in- transition is not at the same frequency as the electric- duced byits non-trivialboundary condition. Our results dipole transition. As a result, there will not be the neg- demonstrate that engineering on the topology is benefi- ative refraction either. cial in realizing the high frequency magnetic response at the molecularlevel. This findingopens upanalternative Appendix B: Diagonalization of Möbius Hamiltonian approach to design molecular negative index materials, which is promising in achieving 3D bulk negative refraction at the visible wavelength. Beforeevaluatingmatrixelementsfordipoleoperators, Wefurtherremarkthatourproposaliscomplementary wesolveexplicitlytheenergyspectrumandthemolecular to the previous experimental investigation [10], where eigenstates. Consider a Mo¨bius molecular ring with 2N the classical metamaterial was fabricated with Mo¨bius sites,whichisdescribedbytheHu¨ckelHamiltonianH as topology in a “top-down” fashion. In order to induce illustrated in Eq. (1). The Hamiltonian H is in diagonal the magneticresponse,their elementis stillbasedonthe form when expressed in terms of C and C†, i.e., k k configuration of the split-ring resonator. Moreover, due to quantum effect, our architecture is two order smaller H = C†E C , (B1) k k k in size than theirs. k X This work was supported by the National Natu- with ral Science Foundation of China (Grant No. 11121403 and No. 11505007), the National 973-program (Grant d 1 N−1 No. 2012CB922104 and No. 2014CB921403), and the Ck = dkk↑↓ = √N eikjBj, (B2) Youth Scholars Program of Beijing Normal Univer- (cid:20) (cid:21) j=0 X sity (Grant No. 2014NT28), and the Open Research V 2ξcos(k δ) 0 Fund Program of the State Key Laboratory of Low- Ek = − 0 − 2 V 2ξcosk . (B3) Dimensional Quantum Physics, Tsinghua University (cid:20) − − (cid:21) Grant No. KF201502. Therefore,thesingle-electronmoleculareigenstates k,σ | i are k, =d† 0 , k, =d† 0 . (B4) Appendix A: Absence of Magnetic Dipole Transition | ↑i k↑| i | ↓i k↓| i in a Perfect Ring Here, 0 denotes the vacuum state and k = lδ with l = | i 0,1,...,N 1 and δ =2π/N. ConsideramolecularringformedbyN identicalatoms − It is useful to express k,σ in terms of the localized with the nearest neighbour hopping strength ξ and site | i atomic-orbitals for later use. To achieve this one notice energy ǫ. The single electron Hamiltonian is written as that,accordingtoRef.[19],B isrelatedtoA byalocal j j unitary transformation, i.e. N−1 H = ǫa†a ξa†a +h.c. , (A1) j j − j j+1 B = cj+ =U A (B5) Xj=0 h (cid:16) (cid:17)i j cj− j j (cid:20) (cid:21) where periodical boundary condition a0 = aN is as- with sumed. The operator a† creates an excitation at the jth j 1 e−iϕj/2 e−iϕj/2 atom, which is located at R~j = Rcosϕjeˆx +Rsinϕjeˆy Uj = √2 1 − 1 , (B6) andϕ isdefinedinEq.(15), Rdenotesthe radiusofthe (cid:20) (cid:21) j molecular ring. andj =0,1,...,N 1. Itfollows fromEqs.(B2,B5) that Based on the bond current formalism in Ref. [27], the the eigenstates k,−σ can be written in terms of A = j magnetic dipole operator m~ reads T | i a b as j j i mx =my =0, mz = 2eξR2sinδ a†jaj+1+h.c. (A2) (cid:2) |k(cid:3),↑i =N−1e−ikj c†j↑ 0 Xj (cid:20)|k,↓i(cid:21) j=0 √N "c†j↓ #| i X Equations (A1,A2) indicate that a perfect ring does not N−1e−ikj eiϕj/2 eiϕj/2 a† caonudpltehetointtheeramctaiognneHtiacmfiieltldon[i1a7n].isBpercoapuosreti[om~n,aHl t]o=m~0, =j=0 √2N(cid:20) 1 − 1 (cid:21)" b†jj #|0i.(B7) X 10 Furthermore, by noticing that the atomic orbitals φ where d~ = e~r is the electric dipole operator with e j± | i − − are created by acting a† or b† on the vacuum state, the being the electric charge and~r being the position vector j j molecular eigenstates are rewritten as of electron, σ± = (σx iσy)/2 with σα (α = x,y,z) ± being Pauli operators in the pseudo spin space spanned N−1 by k,σ and k′,σ , e.g., for Eq. (C1) σ is defined as 1 z k, = e−i(k−δ2)j(φj+ φj− ), (B8) | i | i | ↑i √2N | i−| i j=0 X N−1 1 k, = e−ikj(φ + φ ). (B9) j+ j− | ↓i √2N | i | i j=0 X Especially,iftheresonanceintegralsV andξarepositive, the molecular ground state is N−1 1 g = 0, = (φ + φ ). (B10) j+ j− | i | ↓i √2N | i | i σz = k, k, k, k, , (C4) j=0 | ↑ih ↑|−| ↓ih ↓| X Appendix C: Matrix Elements of Electric and Magnetic Dipoles The matrix elements of electric dipole operator are eˆ (α = x,y,z) is the unit vector in α direction, R and α W 0 C d~C† 0 = e [(eˆ +2eˆ )σ eˆ σ ](,C1) 4W are respectively the radius and width of the Mo¨bius k k − 4 y z x− x y molecule. D (cid:12) (cid:12) E 1 0 C(cid:12) d~C† (cid:12)0 = e [(eˆ ieˆ )(2R+Wσ ) (cid:12)k k±δ(cid:12) − 4 x∓ y y D (cid:12) (cid:12) E +2Weˆ σ ], (C2) (cid:12) (cid:12) z ∓ (cid:12) (cid:12) The matrix elements ofmagnetic dipole operatorm~ = W 0 C d~C† 0 = e ( ieˆ eˆ )σ , (C3) ie~r [H,~r]/2~ are summarized as follows: k k±2δ − 4 ∓ x− y ∓ − × D (cid:12) (cid:12) E (cid:12) (cid:12) (cid:12) (cid:12) For k,σ ⇋ k,σ′ transitions | i | i 0 C m~C† 0 − k k D (cid:12)(cid:12)(cid:12) −81(cid:12)(cid:12)(cid:12)ξ{Eco2sW(k2[co2sδ()k−cδo)s(−kcosδk)]+eˆyco+s([kW+2(δc)o)sk 41RW{− V +ξ(cos(k−δ)−cos(k+ 2δ)) (eˆx−ieˆy) = 2e~ −+4R2(−cos(k−+ 2δ)−c−os(k− 32δ))]eˆz} −2iξ(cid:2)cos4δ cos(k− 54δ)−cos(k+ 43δ(cid:3)) eˆz} ,  41RW−{2−iξ(cid:2)cVos+4δξ(ccooss((kk+−34δδ))−−ccooss((kk+−2δ45)δ(cid:3))(eêxˆz+}ieˆy) −21ξsink W2(cid:2)sinδ2eˆy−(2R2+W2cos2δ)(cid:3)sinδeˆz   (cid:2) ((cid:3)C5) (cid:2) (cid:3) for k,σ ⇋ k δ,σ′ transitions | i | ± i 0 C m~C† 0 − k k+δ e D (cid:12)(cid:12)(cid:12) 18W2ξ[co(cid:12)(cid:12)(cid:12)s(Ek−δ)−cos(k+δ)](ieˆx+eˆy−eˆz) −41−R2Wiξ{c(cid:2)oVsδ4+cξo(cid:0)sc(oks−k−45δc)o−s(cko+s(k2δ)+(cid:1)(cid:3)34(δeˆ)x−eˆzi}eˆy)  = 1RW V +ξ(cos(k+δ) cos(k δ)) (eˆ ieˆ ) , 2~ 4 { − − 2 x− y (cid:2) (cid:3)  iξ[cos(k δ) cos(k+δ) cos(k+ 3δ) 1W2ξ cos(k δ) cos(k+ 3δ) (ieˆ +eˆ eˆ )   − (cid:2) − − − (cid:3) 2 8 − 2 − 2 x y− z   +cos(k δ)]eˆ   − 2 z} (cid:2) (cid:3)    (C6)