PT phase transition in a (2+1)-d relativistic system
Bhabani Prasad Mandal, Brijesh K. Mourya, Kawsar Ali, Ananya Ghatak
aa r X i v : . [ qu a n t - ph ] S e p PT phase transition in a (2+1)-d relativistic system
B. P. Mandal a , B. K. Mourya a , K. Ali b , A. Ghatak aa Department of Physics, Banaras Hindu University, Varanasi -221005, India b Condensed Matter Division, Bhabha Atomic Research Centre, Mumbai 400085, India
Abstract
We study a massless Dirac particle with PT symmetric non-Hermitian Rashba inter-action in the background of Dirac oscillator potential to show the PT phase transitionin a (2+1) dimensional relativistic system analytically. PT phase transition occurs whenstrength of the (i) imaginary Rashba interaction or (ii) transverse magnetic field exceedtheir respective critical values. Small mass gap in the spectrum, consistent with other ap-proaches is generated as long as the system is in the unbroken phase. Relativistic Landaulevels are constructed explicitly for such a system. e-mails address: [email protected] address: [email protected] address: [email protected] address: [email protected] Introduction
Consistent quantum theory with real energy eigenvalues, unitary time evolution andprobabilistic interpretation for combined Parity (P) and Time reversal (T) symmetricnon-Hermitian theories in different Hilbert space equipped with positive definite innerproduct has been the subject of intrinsic research in frontier physics over the last oneand half decades [1]-[3]. The huge success of the complex quantum theory has lead toextension to many other branches of physics and has found many applications [4]-[28].Such non-Hermitian PT symmetric systems generally exhibit a phase transition [6]-[9] (or more specifically a PT breaking transition ) that separates two parametric regions (i)region of the unbroken PT symmetry in which the entire spectrum is real and eigenstatesof the systems respect PT symmetry and (ii) a region of the broken PT symmetry inwhich the whole spectrum (or a part of it) appears as complex conjugate pairs and theeigenstates of the systems do not respect PT symmetry. The study of PT phase transi-tion has become extremely important due to the fact that such phase transition and itsrich consequences are really observed in variety of physical systems [10]-[16]. Howevermost of the analytical studies on PT phase transition are restricted to one dimensionalnon relativistic systems. Few groups [17]-[18] have studied PT phase transition in higherdimensions for restricted class of non-Hermitian potentials, in non-relativistic situations.In the present work we consider a relativistic PT symmetric system in (2+1) dimensionsto study the PT phase transition and its consequences.As a model we consider a relativistic system of a massless Dirac particle in (2+1)dimension [29]. Such systems have become extremely important due to the discovery ofgraphene, a single layer of the carbon atoms arranged in a honeycomb lattice. The studyof graphene has created an avalanche in the frontier research over the past decade due toits exceptional electronic properties and potentially significant applications [30]-[35]. Thelow energy excitations of the atoms in the graphene obey massless Dirac equation in (2+1)dimensions. The two dimensional graphene sheet is essentially a zero gap semiconductorwith Fermi level E f precisely at E = 0 with a linear dispersion relation given by E = ~ v f k where v f is the Fermi velocity and k is the carrier wave vector. However a small gap hasbeen found experimentally in the graphene spectrum [36]. Such a mass gap is generatedby breaking the inversion symmetry either (i) by considering spin-orbit interaction (e.g.Rashba interaction) or (ii) by the coupling the system with quantized circularly polarizedfields. In this present work we consider a massless Dirac particle with PT symmetricimaginary Rashba interaction [37] in the background of Dirac oscillator (DO) potential[38]-[45] to demonstrate the PT phase transition in (2+1) dimensional relativistic system.We analytically show that PT phase transition occurs in such a system when the strengthof (i) the imaginary Rashba interaction or (ii) the transverse magnetic field exceeds acritical value. However it is challenging to make a realistic experimental setup to observethe PT-phase transition for the system of graphene with imaginary Rashba interaction.Small mass gap which is consistent with other approaches are also generated as long asthe system is in PT unbroken phase. The mass gap vanishes at the phase transitionpoint and becomes imaginary in the broken phase of the system. We construct the2olutions for the other valley of the system and show the PT phase transition in thesimilar fashion. Relativistic lowest order Landau levels are constructed explicitly for thesystem. Connection of this system to well-known Jaynes-Cumming (JC) [46]-[47] modelin optics is also indicated.Now we present the plan of the paper. In section (2) we introduce the model and itssymmetries. PT phase transition and generation of mass gap in the spectrum are discussedin section (3). Section (4) is kept for conclusions. The massless Dirac particle with imaginary Rashba interaction in the transverse static magnetic field in the background of DO potential is described by the Hamiltonian H = v f [ ~σ. ( ~ Π − iK ~rβ ) + iλ ( ~σ × ~ Π) . ˆ z, (1)where ~ Π = ~p + e ~Ac , and K , λ are real constants and v f is Fermi velocity.We consider the magnetic field along z-direction, B = B ˆ k and choose the vector potentialas ~A = ( − B y , B x ,
0) in the symmetric gauge. The term linear in ~r is DO potential [38] asin the non-relativistic limit, it reduces to simple harmonic oscillator (SHO) potential withstrong spin-orbit coupling. Initially, DO was introduced in the context of many bodytheory, mainly in connection with quark confinement models in QCD [39]. Later thesubsequent studies have revealed several exciting properties connected to the symmetriesof the theory and DO has found many physical applications in various branches of physics[40]-[45]. Non-Hermitian version of Dirac equation is also considred earlier.In particular,(2 + 1) dimensional massless Dirac equation in the presence of complex vector potentialand its bound state solutions are discussed in [54]. Exact solutions for the bound statesof a graphene Dirac electron in various magnetic field with translational symmetry areobtained in [55].The Hamiltonian in component form is written as H = v f ( σ x Π x + σ y Π y ) − iK v f ( σ x x + σ y y ) β + iλ ( σ x Π y − σ y Π x ) (2)where, Π x = p x − eB y c and Π y = p y + eB x c .Now we discuss the symmetry properties of this model. Parity transformation, (cid:18) x ′ y ′ (cid:19) = A (cid:18) xy (cid:19) is an improper Lorentz transformation (i.e. det A = −
1) and is defined in two alternativeways in two dimension as, P : x −→ − x, y −→ y, p x −→ − p x , p y −→ p y . (3) Various solutions of Dirac equation in presence of inhomogeneous[48]-[50] as well as time dependent[51]-[53] magnetic field have also been obtained for hermitian model : x −→ x, y −→ − y, p x −→ p x , p y −→ − p y . (4)Under such parity transformations Dirac wavefunctions transform as P ψ ( x, y, t ) = σ y ψ ( − x, y, t ) P ψ ( x, y, t ) = σ x ψ ( x, − y, t ) (5)such that free Dirac equation remains invariant under P and P . The time reversaltransformation ( t −→ − t, i −→ − i, p x −→ − p x , p y −→ − p y ) in (2+1) dimensionDirac theory is defined as T = iσ y ˜ K , where ˜ K is complex conjugation operation suchthat, T ψ ( x, y, t ) = iσ y ˜ Kψ ( x, y, − t ) = iσ y ψ ⋆ ( x, y, − t ). It is straight forward to check that H † = v f ( σ x Π x + σ y Π y ) − iK v f ( σ x x + σ y y ) β − iλ ( σ x Π y − σ y Π x ) = H This system, however is invariant under both P T and P T as P T H ( P T ) − = HP T H ( P T ) − = H A massless relativistic system has two degenerate but inequivalent solutions known asvalleys of the system. For the present system in this work there exists another Diracequation corresponding to the other valley. The Hamiltonian corresponding to othervalley is obtained by taking a time reversal transformation of H [56, 57] as,˜ H = T HT − = v f ( σ x ˜Π x + σ y ˜Π y ) − iK v f ( σ x x + σ y y ) β − iλ ( σ x ˜Π y − σ y ˜Π x ) (6)where ˜Π x = p x + eB y c and ˜Π y = p y − eB x c . It is straight forward to show that ˜ H = ˜ H † and[ ˜ H, P T ] = [ ˜ H, P T ] = 0. Now in the next section we find the solution of Dirac equationfor both the valleys to show the PT phase transition and mass gap generation in suchsystems. In this section we solve the Dirac equation with Hamiltonian H and ˜ H corresponding tothe valleys for the massless particle explicitly to demonstrate PT phase transition in (2+1)dimensional relativistic system. In both cases we found that mass gap is generated dueto imaginary Rashba interaction in the background of DO potential as long as the systemis in the unbroken phase. We further show how the different solutions of Dirac equationfor both the valleys in this particular model are interrelated. We begin with writing theHamiltonians in Eqs.(2) and (6) in a compact form on a complex plane z = ( x + iy ) as, H = (cid:18) A Π z + iC ¯ zB Π ¯ z + iC z (cid:19) , ˜ H = B Π z + iC ¯ zA Π ¯ z + iC z (7)4here, A = 2( v f − λ ), B = 2( v f + λ ), C = K v f − ( v f − λ ) B e c and C = − K v f + ( v f + λ ) B e c are constants and canonical conjugate momenta in complexplane Π z and Π ¯ z are defined as Π z = − i ~ ddz = ( p x − ip y ), Π ¯ z = − i ~ dd ¯ z = ( p x + ip y )with the properties[¯ z, Π ¯ z ] = i ~ ; [ z, Π ¯ z ] = 0; [¯ z, Π z ] = 0; [Π ¯ z , Π z ] = 0; [ z, Π z ] = i ~ (8) Now we first present the solutions corresponding to the Hamiltonian H . To find the solu-tion of the Dirac equation corresponding to H we assume a solution of the correspondingDirac equation Hψ = Eψ in the two components form as, ψ = (cid:18) φiχ (cid:19) (9)Dirac equation is then written in components form as( A Π z + iC ¯ z )( B Π ¯ z + iC z ) φ = E φ (10)( B Π ¯ z + iC z )( A Π z + iC ¯ z ) χ = E χ (11)Now we look for a solution of the kind φ = ξ ( z, ¯ z ) e d z ¯ z χ = η ( z, ¯ z ) e d z ¯ z (12)Substituting these in Eq.(10) or in Eq.(11) and comparing the coefficient of ( z ¯ z ) in bothsides we obtain d = C A ~ or C B ~ . For d = C A ~ the Eqs.(10) and (11) reduce to, − AB ~ ∂ ξ∂z∂ ¯ z + Kz ∂ξ∂z − ( E − K ) ξ = 0 (13) − AB ~ ∂ η∂z∂ ¯ z + Kz ∂η∂z − E η = 0 (14)Similarly for d = C B ~ the Eqs.(10) and (11) reduce to, − AB ~ ∂ ξ∂z∂ ¯ z − K ¯ z ∂ξ∂ ¯ z − E ξ = 0 (15) − AB ~ ∂ η∂z∂ ¯ z − K ¯ z ∂η∂ ¯ z − ( E + K ) η = 0 (16)We find the solution of the Eqs.(13) and (14)(i.e. for d = C A ~ ) as ξ n = a In z n η n = a In z n +1 with , E n = ( n + 1)( AC − BC )5here a In are real constants. Thus the general solution for the n th level for the case d = C A ~ is ψ In ( z, ¯ z, t ) = (cid:18) ξ n iη n (cid:19) e C A ~ z ¯ z e − i ~ E n t = a In (cid:18) z n iz n +1 (cid:19) e C A ~ z ¯ z e − i ~ E n t (17)In exactly similar fashion we obtain the solution corresponding to d = C B ~ as, ψ IIn ( z, ¯ z, t ) = a IIn (cid:18) ¯ z n +1 i ¯ z n (cid:19) e C B ~ z ¯ z e − i ~ ˜ E n t , with , ˜ E n = ( n + 1)( BC − AC ) = − E n (18) To find the solution corresponding to the Dirac equation for ˜ H we observe that, H A ↔ B − − − −→ C ↔ C ˜ H Interchanging A ↔ B, C ↔ C is equivalent to λ → − λ, K → − K and B → − B .Thus it is straight forward to obtain the solutions for the Dirac equation correspondingto ˜ H which describes their valley, by using these changes of parameters in the solutionscorresponding to Dirac equation with Hamiltonian H. The solutions for the systems ˜ H are given as, For ˜ d = C A ~ , ˜ ψ In ( z, ¯ z, t ) = ˜ a In (cid:18) ¯ z n +1 i ¯ z n (cid:19) e C A ~ z ¯ z e − i ~ E n t (19)For ˜ d = C B ~ , ˜ ψ IIn ( z, ¯ z, t ) = ˜ a IIn (cid:18) z n iz n +1 (cid:19) e C B ~ z ¯ z e − i ~ ˜ E n t (20)where E n = ( n + 1)( AC − BC ) = − ˜ E n . We will be using these results to discuss PTphase transition and generation of mass gap in the next subsection. The energy eigenvalues for the solutions corresponding to d = ˜ d = C A ~ in both the valleyare, E n = ± s ( n + 1) (cid:20) v f − λ ) B e ~ c − K v f ~ (cid:21) (21)This indicates a mass gap ∆ = r v f − λ ) B e ~ c − K v f ~ (22)is generated between the positive and negative energy solutions due to the interactionspresent in this theory. Now we proceed to show that this system passes through a PT6hase transition depending on the different parametric values. The energy eigenvaluesare real for these solutions when(i) λ ≤ v f (1 − K cB e ) ≡ λ c OR(ii) B > K v f c ( v f − λ ) e ≡ B c (23)As long as E n is real ψ In and ˜ ψ In are an eigenstate of both P T and P TP T ψ In ( z, ¯ z, t ) = σ y iσ y (cid:18) ξ n ( − z, − ¯ z ) − iη n ( − z, − ¯ z ) (cid:19) e C A ~ z ¯ z e − i ~ E n t , as P T z = − z = i ( − n (cid:18) ξ n iη n (cid:19) e C A ~ z ¯ z e − i ~ E n t = i ( − n ψ In ( z, ¯ z, t ) (24)Similarly, P T ψ In ( z, ¯ z, t ) = σ x iσ y (cid:18) ξ n ( z, ¯ z ) − iη n ( z, ¯ z ) (cid:19) e C A ~ z ¯ z e − i ~ E n t , as P T z = z = − iσ z (cid:18) ξ n − iη n (cid:19) e C A ~ z ¯ z e − i ~ E n t = − (cid:18) ξ n iη n (cid:19) e C A ~ z ¯ z e − i ~ E n t = − ψ In ( z, ¯ z, t ) (25)Similarly we obtained for the other valley as, P T ˜ ψ In ( z, ¯ z, t ) = − i ( − n ˜ ψ In ( z, ¯ z, t ) P T ˜ ψ In ( z, ¯ z, t ) = − ˜ ψ In ( z, ¯ z, t ) (26)Thus for d = ˜ d = C A ~ system corresponding to both the valleys are in unbroken PTphase as long as the coupling for Rashba interaction is less or equal to λ c or strengthstrength of the magnetic field is greater or equal to a critical value B c . Now we observein this situation the other two solutions ( ψ IIn , ˜ ψ IIn ) (i.e for d = ˜ d = C B ~ ) correspond tothe broken PT phase as the eigenvalues for both the valleys are imaginary, ˜ E n = ± iE n with real E n . In this case ψ IIn and ˜ ψ IIn are not eigenstates of either P T or P T i.e., P i T ψ
IIn = a i ψ IIn , P i T ˜ ψ IIn = b i ˜ ψ IIn for i = 1 , E n is imaginary. Hence PT symmetry is broken spontaneously for d = ˜ d = C B ¯ h . Thus the system with d = ˜ d = C A ~ passes through P T and P T phasetransition where either (a) strength of the imaginary Rashba interaction is equal to acritical value ( λ c ) or (b) the external magnetic field B is equal to a critical value ( B c ).On the other hand if λ > λ c , for fixed B or B < B c for fixed λ , the solutions for boththe valleys for d = ˜ d = C B ~ , i.e. ψ IIn , ˜ ψ IIn are in the unbroken phase and the solutions7orresponding to d = ˜ d = C A ~ , i.e. ψ In , ˜ ψ In are in broken phase. PT phase transition for d = ˜ d = C A ~ is demonstrated graphically in Fig. 1 by plotting E n vs λ for fixed value of B in Fig. 1(a) and by plotting E n vs B for fixed λ in Fig 1(b). unbroken region broken region n = n = n = n = n = J Ψ n I , ΨŽ n I N J Ψ n II , ΨŽ n II N Λ c Λ E n broken region unbroken region n = n = n = n = n = J Ψ n I , ΨŽ n I NJ Ψ n II , ΨŽ n II N B c B E n (a) (b) Fig. 1: PT phase transition for the case for d = ˜ d = C A ~ . Fig (a) Real (Solidlines) and imaginary (broken lines) parts of E n have been plotted with Rashba couplingstrength for fixed values of B = 100 , K = 0 . , v f = 0 . c , e = 1 , ~ = 1 , c = 137 for n = 0 , , , , . Phase transition occurs at λ c = 1 . . Fig (b) Similarly E n is plottedwith strength of magnetic field ( B ) for fixed values of λ = 0 . , keeping other parameterssame as in Fig 1 (a) for n = 0 , , , , . Phase transition occurs at B c = 6 . . Now we would like to concentrate on mass gap ∆ when the system passes a PT phasetransition;(i) For d = ˜ d = C A ~ , when λ < λ c or B > B c we see from Eq.(22) mass gap isalways positive and becomes imaginary when λ > λ c or B < B c . Thus a mass gap isgenerated as long as system is unbroken PT phase and given by the expression in in Eq.(22). Further in unbroken phase ∆ varies as √ B for large enough magnetic field. Thisresult is consistent with other approaches.(ii) For d = ˜ d = C B ~ when λ > λ c or B < B c the system is in unbroken phase andagain real mass gap is generated which is consistent with other approaches. Thus massgap is always generated in this theory which is consist with other approaches. In Fig. 2variation of mass gap is shown with respected to the Rashba coupling strength as well asstrength of the magnetic field. Real mass gap for the massless systems is generated aslong as the system is in PT unbroken phase.8 nbroken region broken region0.0 0.5 1.0 1.5 2.0 2.5 Λ D brokenregion unbroken region B B D (a) (b) Fig. 2: Variation of mass gap with λ and B for the case d = ˜ d = C A ~ . Fig. (a)Real (Solid line) and imaginary (broken line) part of mass gap is plotted with strength ofthe Rashba interaction for fixed values of B = 100 , K = 0 . , v f = 0 . c , e=1, ~ = 1 , λ ≤ λ c = 1 . , mass gap is real and system is unbroken PT phase. Fig. (b) Real(Solid line) and imaginary (broken line) part of mass gap is plotted with the strength ofmagnetic field by keeping the strength of Rashba interaction fixed for a value of λ = 0 . , K = 0 . , v f = 0 . c , e=1, ~ = 1 . B ≥ B c = 6 . , system is in unbroken phase,and real mass gap varies as ∆ ∝ √ B . Special cases :(i) In absence of DO ( K = 0), system passes to broken phase when strength ofRashba interaction exceeds Fermi velocity λ > v f . The mass gap in that case (unbrokenPT phase) is given by ∆ = q v f − λ ) B e ~ c which is real as λ < v f for the unbrokenphase. Mass gap vanishes at the transition point λ = v f .(ii) In absence of Rashba interaction ( λ = 0), the system is Hermitian, but can havereal energy eigenvalues only when the magnetic field is sufficiently large i.e. B > K ce ,the mass gap generated in this case is ∆ = q v f ~ ( B ec − K ) which is always real for B > K ce . In both the cases for large enough magnetic field ∆ ∝ √ B , which is consis-tent with other approaches. Again in this case, mass gap vanishes at the transition point, B = K ce . This critical value of magnetic field is same as obtained in [44]. The factor of2 is due to the choice of vector potential in the symmetric gauge. This discussion in validfor both the valleys. 9 owest Landau Levels (LLL) :The LLLs are obtained by putting the condition Q χ = 0, which implies( A Π z + iC ¯ z ) χ = 0 . (28)We substitute χ = u ( z, ¯ z ) e C A ~ z ¯ z to solve Eq.(28) and obtain ∂u ( z, ¯ z ) ∂z = 0 , (29)as the defining rule for LLL in this system. We obtain the LLL in the coordinate repre-sentation, which is infinitely degenerate as χ ( z, ¯ z ) = ¯ z l e C A ~ z ¯ z , l = 0 , , , ..... ∞ (30)The monomials ¯ z l with l = 0 , , , ... ∞ serve as a linearly independent basis. The firstand higher excited states in the coordinate representation are then obtained by applying Q † given in Eq.(30) to the LLL repeatedly. The LLL corresponding to the other valleyis obtained by putting ( B Π z + iC ¯ z ) ˜ χ = 0. Following similar calculation we obtain LLL˜ χ ( z, ¯ z ) = ¯ z l e C B ~ z ¯ z , l = 0 , , , ... ∞ and other higher excited states for this valley. Connection with JC Model :We would like to point out the connection of this model with generalized version ofwell known JC model for massless particles in optics. The Hamiltonian corresponding tothe simple version of (JC) model is given as H JC = g ( σ + a + σ − a † ) + σ z mc , (31)where σ ± = √ ( σ x ± iσ y ) are usual spin raising and lowering operators. This model iswidely used in optics to study the atomic transition in two level systems.Now the Hamiltonian H in Eq.(7)can be expressed as, H = √ K (cid:18) Q Q † (cid:19) (32)where, Q = √ K ( A Π z + iC ¯ z ) and Q † = √ K ( B Π ¯ z + iC z ) with K = ( AC − BC ) ~ . Itis straight forward to check, h Q , Q † i = 1. Note Q = Q for real Rashba interaction.The Hamiltonian is not Hermitian but invariant under P T and P T both and is furtherwritten as, H = √ K ( σ + Q + σ − Q † ) (33)This is the extended version of JC model for particle with rest mass zero. It is veryexciting to note how two completely different theories are related in this fashion. This10rovides an alternative approach to study any massless Dirac particle (such as graphene)with Rashba interaction in the background of DO potential in the transverse magneticfield using JC model in quantum optics. For a real Rashba interaction, Q = Q andthe extended JC model in Eq.(33) exactly coincide with JC model for a massless particle.Similar discussion is also valid for the other valley. We have studied PT phase transition in a (2+1) dimensional relativistic system to enlightdifferent aspects of this important phenomenon. For this purpose we have considered themassless Dirac particle with imaginary Rashba interaction in the background of DO poten-tial in (2+1) dimension. Such a system is non-Hermitian due to the imaginary coupling.We have constructed two different Parity- Time reversal transformations ( P T and P T ) in (2+1) dimensional relativistic system such that the system is invariant under both P T and P T . We have constructed different possible solutions corresponding to both thevalleys for this system. The energy eigenvalues are real as long as the eigenstates respect P T and P T symmetry. We further investigate P T and P T phase transitions in thissystem to show that system passes from unbroken P T and P T phases to broken phasesdepending on the strength of the Rashba interaction or transverse magnetic field. Theinter-relation of different solutions for both the valleys with respect to PT-phase transi-tion is presented. Even though graphene is zero gap semiconductor with linear dispersionrelation, a small mass gap in its spectrum is observed experimentally. In our formulationsmall mass gap consistent with other approaches is generated for the system of masslessDirac particle when the system is in unbroken phase. The mass gap vanishes at the pointof PT phase transition. We would like to point out that results on PT-phase transitionpresented in this paper is valid for any system with massless Dirac particle in presenceof imaginary Rashba interaction in the background of Dirac oscillator. In particular ourmodel has some relevance to the possible physics in graphene when the solution for boththe valleys are combined. The lowest order Landau levels for this system have been cal-culated explicitly to show the infinite degeneracy in such systems. The connection ofsuch system with JC model has also been discussed. It will be exciting to correlate otherexciting properties of graphene with PT phase transition. Acknowledgments
BKM and BPM acknowledge the financial support from the De-partment of Science and Technology (DST), Govt. of India under SERC project sanctiongrant No. SR/S2/HEP-0009/2012. AG acknowledges the Council of Scientific & Indus-trial Research (CSIR), India for Senior Research Fellowship.
References [1] C. M. Bender and S. Boettcher
Phys. Rev. Lett. , 5243 (1998).112] A. Mostafazadeh, Int. J. Geom. Meth. Mod. Phys. , 1191 (2010) and referencestherein.[3] C. M. Bender, Rep. Prog. Phys. , 947 (2007) and references therein.[4] Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys.Rev. Lett. , 030402 (2008).[5] B. Basu-Mallick and B.P. Mandal,
Phys. Lett. A , 231 (2001); B. Basu-Mallick,T. Bhattacharyya and B. P. Mandal,
Mod. Phys. Lett. A , 543 (2004).[6] G. Levai, J. Phys.
A 41 (2008) 244015.[7] C. M. Bender, G. V. Dunne, P. N. Meisinger, M. Simsek
Phys. Lett. A
281 (2001)311-316.[8] C. M. Bender, David J. Weir
J. Phys.
A 45 (2012)425303.[9] B. P. Mandal, B. K. Mourya, and R. K. Yadav (BHU),
Phys. Lett. A , 1043(2013).[10] C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodulides, M. Segev, and D.Kip:
Nature (London) Phys. , 192 (2010).[11] A. Guo, G. J. Salamo: Phys. Rev. Lett. , 093902 (2009).[12] C. M. Bender, S. Boettcher and P. N. Meisinger :
J. Math. Phys. Phys. Rev. Lett. , 054102(2010).[14] A. Nanayakkara:
Phys. Lett.
A 304 , 67 (2002).[15] N. Hatano and D. R. Nelson:
Phys. Rev.
B 58 , 8384 (1998).[16] N. Hatano and D. R. Nelson:
Phys. Rev. Lett. , 570 (1996).[17] M. Znojil J. Phys.
A 36 (2003) 7825.[18] C. M. Bender, G. V. Dunne, P. N. Meisinger, M. Simsek
Phys. Lett. A
281 (2001)311-316.[19] M. V. Berry,
Czech. J. Phys. , 1039 (2004).[20] W. D. Heiss, Phys. Rep. , 443 (1994).[21] A. Ghatak, R. D. Ray Mandal, B. P. Mandal,
Ann. of Phys. , 540 (2013).[22] A. Ghatak, J. A. Nathan, B. P. Mandal, and Z. Ahmed,
J. Phys. A: Math. Theor. , 465305 (2012). 1223] A. Mostafazadeh, Phys. Rev. A , 012103 (2013).[24] L. Deak, T. Fulop Ann. of Phys. , 1050 (2012).[25] M. Hasan, A. Ghatak, B. P. Mandal,
Ann. of Phys. , 17 (2014).[26] A. Ghatak and B. P. Mandal
J. Phys. A: Math. Theor. , 355301 (2012).[27] B. P. Mandal and S S. Mahajan arXiv :1312.0757, (2013) [ Accepted in Coumnicationin Theoretical Physics].[28] B. P. Mandal and A. Ghatak, J. Phys. A: Math. Theor. , 444022 (2012).[29] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V.Grigorieva, S. V. Dubonos and A. A. Forsov Nature , 197 (2005).[30] Novoselov K. S. et al 2004 Science 306 666.[31] Castro Neto A H et al 2009 Rev. Mod. Phys. 81 109.[32] M. I. Katsnelson, K. S. Novoselov, A. K. Geim,
Nature Phys. , 620 (2006).[33] K. Shizuya, Phys. Rev. B , 075419 (2008).[34] J. Schliemann, Phys. Rev. B , 195426 (2008).[35] J. Schliemann, New J. Phys. , 043024 (2008).[36] Y. Araki J. Phys.: Conf. Ser. , 012022 (2011).[37] E. I. Rashba,
Phys. Rev. B (2003), 241315; ibid. B , (2004), 201309[38] M. Moshinsky, A. Szczepaniak. J. Phys. A , 1817 (1989).[39] D. Ito, K. Mori, E. Carrieri, Nuovo Cimento A , 1119, (1967).[40] B. P. Mandal, S. Verma Phys. Lett. A , 1021 (2010).[41] B. P. Mandal, S. K. Rai
Phys. Lett. A , 2467 (2012).[42] B. P. Mandal
Mod. Phys. Lett. A , 655 (2005).[43] B. P. Mandal and S. Gupta Mod. Phys. Lett. A , 1723 (2010).[44] O. Panella and P. Roy, Phy. Rev. A Ann. of Phys. (2014).[46] E. T. Jaynes, F. W. Cummings
Proc. IEEE , 89 (1963).[47] Jonas Larso, Phys. Scr. , 146 (2007).1348] V N Rodionov, arXiv: 1404.0503[49] P. Roy, T K Ghosh and K Bhattacharya, J. Of Phys: Cond. Matt , 055301, (2012).[50] A. Jellal, A. E. Mouhafid, J Phys A44, 015302 (2011).[51] S. Deffner and A Saxena, arXiv: 1506.09131.[52] B Khantoul and A Fring, arXiv: 1505.02087.[53] J Oertel and R Schutzhold, arXiv: 1503.06140.[54] O. Panella and P. Roy , Symmetry, , 103 (2014).[55] S. Kuru, J. Negro and L. M. Nieto, J. Of Phys: Cond. Matt , 455305 (2009).[56] A. F. Morpurgo, F. Guinea Phys. Rev. L , 196804 (2006).[57] Feng Zhai, Kai Chang Phys. Rev. B85