Electrostatic analogy of the Jackiw-Rebbi zero energy state
Gabriel Gonzalez, Javier Mendez, Ramon Diaz de Leon-Zapata, Francisco Javier Gonzalez
EElectrostatic analogy of the Jackiw-Rebbi zero energy state
Gabriel Gonz´alez,
1, 2, ∗ Javier M´endez, Ram´on Diaz de Le´on-Zapata,
3, 2 and Francisco Javier Gonz´alez C´atedras CONACYT, Universidad Aut´onoma de San Luis Potos´ı, San Luis Potos´ı, 78000 MEXICO Coordinaci´on para la Innovaci´on y la Aplicaci´on de la Ciencia y la Tecnolog´ıa,Universidad Aut´onoma de San Luis Potos´ı,San Luis Potos´ı, 78000 MEXICO Instituto Tecnol´ogico de San Luis Potos´ı, Avenida Tecnol´ogico s/n, 78376 Soledad de Graciano S´anchez, SLP, MEXICO
We present an analogy between the one dimensional Poisson equation in inhomogeneous media and the Diracequation in one space dimension with a Lorentz scalar potential for zero energy. We illustrate how the zeroenergy state in the Jackiw-Rebbi model can be implemented in a simple one dimensional electrostatic settingby using an inhomogeneous electric permittivity and an infinite charged sheet. Our approach provides a novelinsight into the Jackiw-Rebbi zero energy state and provides a helpful view in teaching this important quantumfield theory model using basic electrostatics.
PACS numbers: 42.25.Bs, 42.82.Et, 42.50.Xa, O3.65.PmKeywords: Poisson equation, Dirac equation, Jackiw-Rebbi model
I. INTRODUCTION
The Dirac equation is one of the fundamental equationsin theoretical physics that accounts fully for special relativ-ity in the context of quantum mechanics for elementary spin-1/2 particles. The Dirac equation plays a key role to manyexotic physical phenomena such as graphene, topologicalinsulators and superconductors. These systems proved to beideal testing grounds for theories of the coexistence of quan-tum and relativistic effects in condensed matter physics.Recently, a significant number of studies has addressed theproblem of simulating relativistic quantum mechanics us-ing different physical platforms such as optical structures, metamaterials and ion traps. These studies are based on themathematical analogies found between different physical the-ories which provides a way to explore at a macroscopic levelmany quantum phenomena which are currently inaccessiblein microscopic quantum systems. Among the wide varietyof quantum-classical analogies investigated so far it appearsthat the most fruitful one is given by the analogy between op-tics with quantum phenomena due naturally to the duality be-tween matter and optical waves. The study of quantum-opticalanalogies is based on the formal similarity between the parax-ial optical wave equation in dielectric media and the singleparticle Schrodinger equation. Among the wide variety ofquantum-optical analogies we can mention the Bloch oscil-lations and Zener tunneling, dynamic localization, Andersonlocalization, quantum Zeno effect, Rabi flopping and coher-ent population trapping. All these progress has led to the areaof research of how relativistic quantum systems can be mimicby optical waves. More recently, optical systems governed bythe relativistic Dirac equation have been investigated exper-imentally such as Klein Tunneling, Zitterbewegung and theJackiw-Rebbi model.The purpose of this article is to demonstrate that electrostat-ics can provide a laboratory tool where physical phenomenadescribed by the Dirac equation can be explore. In particu-lar, we demonstrate that the Poisson equation in one dimen-sional inhomogeneous media can be mapped into the zeroenergy state of the Dirac equation in one dimension with aLorentz scalar potential. By tailoring the electric permittiv- ity we propose an electrostatic experiment that simulates ahistorically important relativistic model known as the Jackiw-Rebbi model. Since the derivation of this important modelmany useful variations of the Jackiw-Rebbi model have beeninvestigated such as the Ramajaran-Bell model, the massiveJackiw-Rebbi model, the coupled fermion-kink model andthe Jackiw-Rebbi model in distinct kinklike backgrounds. The article is organized as follows. First we will start witha brief review of the Jackiw-Rebbi model and how one canobtain the zero energy state of the JR model. Then we willshow how the Poisson equation can be mapped into a Dirac-like equation, and illustrate how the zero energy state in theJackiw-Rebbi model can be implemented in a simple one di-mensional electrostatic setting by using an infinite chargedsheet separating two different media. The conclusions aresummarized in the last section.
II. JACKIW-REBBI MODEL IN ONE DIMENSION
The Jackiw-Rebbi model describes a one dimensionalDirac field coupled to a static background soliton field and isknown as one of the earliest theoretical description of a topo-logical insulator where the zero energy mode can be under-stood as the edge state. In particular, the Jackiw-Rebbi modelhas been studied by Su, Shrieffer and Heeger in the continuumlimit of polyacetylene. The one dimensional Dirac equationin the presence of an external field ϕ ( x ) and with (cid:126) = c = 1 is given by ˆ H D Ψ( x ) = [ σ y ˆ p + σ x ϕ ( x )] Ψ( x ) = E Ψ( x ) (1)where σ y = (cid:18) − ii (cid:19) , σ x = (cid:18) (cid:19) and Ψ = (cid:18) ψ ψ (cid:19) . (2)We use the Pauli matrices σ x and σ y in order to have a real twocomponent spinor Ψ( x ) . From eq. (2) it follows that the DiracHamiltonian possesses a chiral symmetry defined by the oper-ator σ z , which anticommutes with the Dirac Hamiltonian, i.e. { ˆ H D , σ z } = 0 . The chiral symmetry implies that eigenstates a r X i v : . [ c ond - m a t . o t h e r] D ec - - - - φ ( x ) FIG. 1: The figure shows the external scalar potential ϕ ( x ) whichchanges sign at the interface x = 0 . come in pairs with positive and negative energy ±E , respec-tively. It is possible however for an eigenstate to be its ownpartner for E = 0 , if this is the case then the state is topologi-cally protected. The resulting zero energy state is protected bythe topology of the scalar field, whose existence is guaranteedby the index theorem, which is localised around the soliton. The Jackiw-Rebbi model uses ϕ ( x ) = m tanh ( λx ) for theexternal scalar field, with m > and λ > . For simplicitywe will consider a external scalar field given by ϕ ( x ) = m x | x | (3)forming a domain wall at x = 0 where ϕ ( x = 0) = 0 . Thescalar field given by eq. (3) is a simplification of the Jackiw-Rebbi model first proposed by Rajaraman-Bell. The preciseform of the external scalar potential is not important as longas it asymptotically approaches an opposite sign at x → ±∞ .The wave function may change corresponding to a particularform of the external scalar potential, but the existence of thezero energy state is determined solely by the fact that the massis positive on one side and negative on the other. Therefore,the solution is very robust against the external scalar potential.The solution of the Dirac equation at exactly zero energy forthe scalar field given by eq. (3) is obtained by solving thefollowing equation (cid:18) − ∂ x + ϕ ( x ) ∂ x + ϕ ( x ) 0 (cid:19) (cid:18) ψ ( x ) ψ ( x ) (cid:19) = 0 (4)which gives ψ i = C ∓ exp [ ∓ m | x | ] , for i = 1 , . (5)where C ∓ is a normalization constant and the double sign ineq.(5) is − (+) for i = 1(2) . Note that ψ , cannot be bothnormalized. If we impose that lim x →±∞ ψ i ( x ) → we needto make C + = 0 in order to have a properly normalized state.In Fig.(2) we show the wave function for the zero energy stateof the Jackiw-Rebbi model. - - ψ ( x ) FIG. 2: The figure shows the Jackiw-Rebbi zero energy mode givenby Eq.(5) for the external scalar field depicted in Fig.(1). Note howthe zero energy state is localized around the interface x = 0 . III. ELECTROSTATIC ANALOG OF THE JACKIW-REBBIMODEL
In this section we show that the zero energy Jackiw-Rebbistate can be generated at the interface of two dielectric mate-rials separated by a infinite charged sheet. The use of infinitecharged sheets for emulating physical systems has been usedextensively in the past for a wide range of applications such asa simple parallel plate capacitor or the study of the one di-mensional Coulomb gas. Since we will be working witha planar charge distribution we will consider only the one di-mensional Poisson equation with an inhomogeneous electricpermittivity, i.e. (cid:15) ( x ) , which is given by ddx (cid:18) (cid:15) ( x ) dVdx (cid:19) = − ρ ( x ) , (6)where V ( x ) is the electrostatic potential and ρ ( x ) is the vol-ume charge distribution. Expanding the left hand side of eq.(6) and multiplying by /(cid:15) we have d Vdx + (cid:15) (cid:48) (cid:15) dVdx = − ρ(cid:15) , (7)where (cid:15) (cid:48) represents the total derivative with respect to thespace coordinate x . Let us now make the following trans-formation V ( x ) = V ln ( ψ ( x ) /A ) , (8)where V and A are constants to ensure dimensional consis-tensy, and ψ ( x ) is an arbitrary function. Substituting eq. (8)into eq. (7) we have V ψ (cid:48)(cid:48) ψ − (cid:15) (cid:48) (cid:15) E x − V E x = − ρ(cid:15) (9)where we have used the identity E x = − dV /dx . If we useeq. (6) in the right hand side of eq. (9) we end up with thefollowing equation V ψ (cid:48)(cid:48) ψ − dE x dx − V E x = 0 , (10)note that eq. (10) does not depend on the electric permittivityanymore. Multiplying eq. (10) by ψ /V and adding and sub-tracting the term E x ψ (cid:48) /V to the left hand side of eq. (10) wehave lim E→ (cid:40)(cid:20) ψ (cid:48) + E x V ψ (cid:21) (cid:48) − E x V ψ − E x V ψ (cid:48) + E ψ (cid:41) = 0 (11)where E is an auxiliary constant that we will set to zero at theend of our calculations. If we make the following substitution ψ (cid:48) + E x ψ /V = E ψ into eq. (11), we end up with the fol-lowing equation − ψ (cid:48) + E x ψ /V = E ψ . These two coupleddifferential equations can be written in the same mathematicalform as the Dirac equation with c = (cid:126) = 1 , i.e. ˆ H D Ψ = (cid:20) σ y ˆ p + σ x (cid:18) E x V (cid:19)(cid:21) Ψ = E Ψ . (12)Equations (12) can be reduced to two uncoupled Schr¨odingerequations ˆ H i ψ i = 0 , for i = 1 , , given by ˆ H i ψ i = (cid:18) ∂ ∂x + U i ( x ) (cid:19) ψ i ( x ) = 0 (13)where U , ( x ) = (cid:34) − (cid:18) E x V (cid:19) + E ± V dE x dx (cid:35) . (14)Clearly, ˆ H , are supersymmetric partner Hamiltonians whichcan be factorized as ˆ H = ˆ A † ˆ A − E and ˆ H = ˆ A ˆ A † − E where ˆ A = ( ∂ x + E x /V ) and ˆ A † = ( − ∂ x + E x /V ) . Therelation between Poisson’s equation and Schr¨odinger equationin one dimension has been pointed out before by one of theauthors (GG). We can easily construct the zero energy mode by setting E = 0 in eq.(12) and solving for the uncoupled first order differentialequations for ψ , , i.e. ψ i = C ∓ exp (cid:20) ∓ (cid:90) (cid:18) E x V (cid:19) dx (cid:21) (15)where C ∓ is a normalization constant and the double sign ineq.(15) is − (+) for i = 1(2) . The existence of a zero energystate then depends on the asymptotic behavior of E x .We know from basic electrostatics that the electric field due toan infinite charged sheet with volume charge density ρ ( x ) = σδ ( x ) with σ > separating two dielectric materials withelectric permittivity (cid:15) and (cid:15) is given by E x ( x ) = (cid:26) σ (cid:15) , for x > − σ (cid:15) , for x < . (16)Interestingly, the elestrostatic field given by eq.(16) has thesame form as the external scalar field given by eq.(3) that al-lows the existence of the zero energy state in the JR model.The electrostatic potential for the electric field given by eq.(16) is V ( x ) = − (cid:90) x E x ( x ) dx = (cid:26) − σ (cid:15) x, for x > σ (cid:15) x, for x < . (17) - - - - ψ ( x ) FIG. 3: The figure shows the electrostatic Jackiw-Rebbi zero energymode given by Eq.(18) for the following values σ = V = 1 , (cid:15) = 1 and (cid:15) = 2 . Using eq. (15) we see that we need to set C + = 0 in order tomake the two-component spinor normalizable. Therefore, thenormalized wave function for the zero mode is given by Ψ( x ) = (cid:114) σV ( (cid:15) + (cid:15) ) (cid:18) e V ( x ) /V (cid:19) . (18)In Fig.(3) we show the electrostatic zero energy wave func-tion for the Jackiw-Rebbi model, the wave function domi-nantly distributes near the interface x = 0 and decays ex-ponentially away. The solution given in eq. (18) for (cid:15) = (cid:15) is the same as the Jackiw-Rebbi zero energy state. IV. CONCLUSIONS
In conclusion we have shown that the Poisson equation inone dimensional inhomogeneous media can be used to simu-late the Jackiw-Rebbi model in one space dimension for thezero energy state. In particular, we demonstrate how the zeroenergy state of the Jackiw-Rebbi model can be implementedin an electrostatic set up with an infinite charged sheet thatseparates two different media. Based on these findings, wehave introduced an electrostatic platform for realizing the zeroenergy state of the Jackiw-Rebbi model which allows one toprobe in the laboratory.
V. ACKNOWLEDGMENTS
This work was supported by the program “C´atedras CONA-CYT”. FJG would like to acknowledge support from project32 of “Centro Mexicano de Innovaci´on en Energ´ıa Solar”and by the National Laboratory program from CONACYTthrough the Terahertz Science and Technology National Lab(LANCYTT). ∗ Electronic address: [email protected] Dirac, P.A.M., “The quantum theory of the electron”, Proc. R.Soc. A Novoselov, K.S. et al ., “Two dimensional gas of massless Diracfermions in graphene”, Nature , 197-200 (2005) Hasan, M.Z. and Kane, C.L., “Topological insulators”, Rev. Mod.Phys. , 3045-3067 (2010) Qi, X.L. and Zhang, S.C., “Topological insulators and supercon-ductors”, Rev. Mod. Phys. , 1057-1110 (2011) Mohammad-Ali Miri, Mathias Heinrich, Ramy El-Ganainy andDemetrios N. Christodoulides, “Supersymmetric Optical Struc-tures”, Phys. Rev. Lett. , 233902 (2013) G. Gonz´alez, “Dirac equation and optical wave propagation in onedimension”, accepted for publication in PSS (RRL) Wei Tan, Yong Sun, Hong Chen and Shun-Qing Shen, “Photonicsimulation of topological excitations in metamaterials”, Sci. Rep. , 3842 (2014) Lamata, L., Le´on, J., Schatz, T. and Solano, E., “Dirac equationand quantum relativistic effects in single trapped ion”, Phys. Rev.Lett. , 253005 (2007) S. Longhi, “Classical simulation of relativistic quantum mechan-ics in periodic optical structures”, Appl. Phys. B Jackiw, R. and Rebbi, C., “Solitons with fermion number”, Phys.Rev. D , 3398 (1976) R. Rajaraman and J.S. Bell, “On solitons with half integral charge”, Phys. Lett. B , 115 (1982) F. Charmchi and S.S. Gousheh, “Massive Jackiw-Rebbi model”,Nucl. Phys. B , 256-266 (2014) Amado, A. and Mohammadi, A., “Coupled fermion-kink systemin Jackiw-Rebbi model”, Eur. Phys. J. C , 465 (2017) Bazeia, D. and Mohammadi, A., “Fermionic bound states in dis-tinct kinklike backgrounds”, Eur. Phys. J. C , 203 (2017) Su, W.P., Shrieffer, J.R. and Heeger, A.J., “Soliton excitations inpolyacetylene”, Phys. Rev. B , 2099 (1980) David J. Griffiths,
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