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J. Mod. Ph y s . doi:10.4236/jm p Copyright © 2 A M Abstract Some aspec t of the Hall c equation for m a single non l Keywords: A 1. Introdu c The quantu m now well kn o dimensional e p henomena w magnetic fiel d the integer q u and was sub s effect [3,4]. R tum Hall effe tally to exhibi where f is tion. Thus (1 ) integer case , 1,2, 3, , an d case appear i The two effe c differences i n electrons and for the forma t At the transit i teger quantu m observed. Th e to solve the defined by th e j e 1 2m H . , 2010, 1, 158 p .2010.13023 P 2 010 SciRes. M odel f D Receive d t s of anyon p c onductance. m ulated fro m l inear differe n A nyon, Hall c tion m Hall effect i o wn phenome n e lectron syste m w hen subject e d [1]. First en c u antum Hall e ff s equently foll o R eferring to t h ct, the Hall r e t plateaus at t h H R either an inte g ) incorporates , f takes d some promin e i n sequences c ts show rema r n origin. In bo quasiparticles t ion of the pla t i ons between s m Hall effect e oretically, th e many- b ody q e many body H jj e () e p Ar c -162 P ublished Onlin e f or the Q in th D epartment o f d January 25, 2 p hysics are r e A single p a m the anyon p n tial equatio n Resistance, C i s a recently n on which ap p m which exhi b e d to an int e c ountered exp ff ect has recei v owed by the h ese two effe c e sistance is fo u h e quantized v a 2 h f e g er or a simp l both of these on integer e nt fractions fo such as f r kable similar i th effects, th e s is believed t o t eaus in the H a s uccessive pl a , scaling beh a e aim in unde r q uantum mec h H amiltonian gi v 22 jk j k 1e 2r e August 2010 ( h Q uanti z e Qua n Pa u f Mathematics , E-mail: b r 2 010; revised F e viewed with a rticle Schrö d icture. The S n and solutio n C onductance , discovered a n p ears in a tw o b its spectacul a e nse transver s erimentally [ 2 v ed much stud y fractional H a c ts as the qua n u nd experime n a lues ( 1 l e rational fra c effects. For t h values f n fo r the fraction 21 n n, 4 1 n n i ties despite t h e localization o o be responsib a ll conductivit y a teaus in the i n a vior has be e r standing this h anical proble m v en by j j k U( r) ( 2 h ttp://www.scir p z ation o n tum H a u l Bracken , University o f acken@pana m F ebruary 26, 2 the intentio n d inger mode l S chrödinger e n s for the m o Composite P n d o - a r s e 2 ], y , a ll n - n - 1 ) c - h e al 1 . h e o f l e y . n - e n is m 2 ) The energy field; t h and th e form p to mov It is versio n tures o c are det e a har m model constr u tions c some s p hysic a lished f the Ha l up the introd u 2. Set Let Ψ though t numbe r an ope r The p .org/journal / j m o f the H a ll Eff e f Texas, Edinb u m .edu 2 010; acc e p te d n of establis h l is introduc e quation-con s o del can be p r P articles first term o n in the prese n h e second ter m e third term i s ositive backg r e in the two-d i the intention n of (2) subje c c cur often in t e rmined by s o m onic oscillato which emph a u cted for a H a c an be found. s pecific assu m a l properties t f rom the mod l l resistance, ( model some u ced. Let us pr o tin g up the Ψ ()x b e the e t of as a flux c r . A composi t r ator phase tra n ( ) x phase field Θ Θ() x m p) H all Re s e ct u rg, USA d March 20, 2 0 h ing a model e d and coupl s traint syste m r oduced. n the righ t -ha n n ce of a cons t m is the Coulo m s a one- b ody p r ound. The el e i mensional here to set u c t to a physic a t his area, for e o lving the Sch r r potential. T h a sizes geome t a ll system and A wavefunc t m ptions. It w i t hat are very r el; in particul a ( 1) can be ob t more physica o ceed to this [ Model-Co m e lectron fiel d c arrying a bos o t e- p article fiel n sformation Θ() ) im x e x Θ () x is defin e 2 ()dyx y s istanc e 0 10 for the qua n ed with a c o m can be con v n d side is th e t ant external m m b interactio n p otential due t e ctrons are co n -plane. u p and solve a a l constraint. S e xample Land a r ödinger equa t h us a simple t ry in the pr o it is shown t h t ion is obtain e i ll be seen t h r elevant can b a r, the quanti z t ained. To be g l concepts ne e 5,6]. m posite Pa r d . An anyon o n or fermion q d ()x is de ) x . e d by ()y , JMP e n tization o nstraint v erted to e kinetic m agnetic n energy; t o a uni- n strained a simple S uch pic- a u levels t ion with physical o blem is h at solu- e d under h at some b e estab- z ation of g in to set e d to be r ticles may be q uantum fined by (3) (4) P. BRACKEN Copyright © 2010 SciRes. JMP 159 where in (3) is an integer and () x y in (4) is the angle made between the vector x y and the -axis; represents anyon density. The effect of the operator phase transformation (3) is to attach m flux quanta to each electron. Composite particles experience the effec- tive magnetic field B() eff x described by the potential Α() j x , where Α() j x depends on the external vector potential () ext j A x and a field () k Cx , which is an aux- iliary field determined solely by the density () x , ()() () ext jj A xAxCx . (5) Therefore, from (5), it follows that () ()() effij ijD BxAx Bmx , (6) and so the effective magnetic flux is the sum of the real magnetic flux and a term which can be regarded as a Chern-Simons flux. Now suppose that () j A x in (4) satisfies the Coulomb gauge condition () 0 jj Ax . (7) It is possible to express () j A x in terms of a scalar field () A x as () A() jjkk A xx e . (8) This conclusion is only possible in a planar geometry. Substituting (8) into () eff Bx , the field A() x can be regarded as the scalar potential of the effective magnetic field, 2 () A() eff Bx x e . (9) This is basically the type of constraint we would like to apply in order to solve (2); that is, by taking a particu- lar reasonable form for () eff Bx . The state vector Ψ is assumed to fully or very nearly characterize the electronic state of the system. The total free charge is given by 22 s Qe dx . (10) The steady state time-independent wavefunction is given by /iEt e , where 0 Ψ is time-independent and will have to satisfy the time-independent Schrödinger equation HE . (11) Let us incorporate an additional assumption into the construction of this model here. Let us suppose that we can write () () eff Bx Bx in the following form 2 ()Bx k , (12) where k is related to the total magnetic flux through the surface; that is, the number of flux quanta of the magnetic field and other constants. The magnetic flux density affects the electronic states as it modifies the Hamiltonian. Of course, the Hamiltonian is modified by the vector potential, which in a simply-connected domain is given by the usual formula A Bx . For exam- ple, suppose we write and use (12) in the form 2 0 ()Bxa , (13) and a is a constant which satisfies 22 0 Φ = () ss Q Bxdxadx aaN e (14) In (14), N is the number of relevant current carrying charge quanta. Moreover, let M denote the number of magnetic flux quanta, which means the total flux can be written as Φ 2 h Me . (15) When the flux and charge are quantized, these results imply that a is a fraction which can be expressed in terms of the flux quantum 2 M h aNe . (16) On a simply connected region, the vector potential can be represented as a one-form given in terms of a single function , which stands for A here, as xy A dy dx . (17) Using (17), the magnetic field can be calculated and then (13) yields a constraint equation 2 0 Φ xxyy a . (18) 3. Solution of the Schrödinger Equation The main objective here is to solve the time-independent Schrödinger equation coupled with Equation (18) to ob- tain Ψ. Of course, vector potential (17) appears in the Schrödinger equation, as can be clearly seen from (2). This procedure will lead to a nonlinear equation; howev- er, it will be found that solutions with the correct physi- cal properties can be determined in closed form. Keeping the first term in (2), the left hand side without the overall multiplicative constant applying (17) leads to 2 222 2 00,x0,y 0 2 2i xyy xxy ee P. BRACKEN Copyright © 2010 SciRes. JMP 160 Therefore (11) written out in full takes the form, 2 222 22 00,x0,y 0 2 0 2i 2. xyy xxy ee mE (19) Now the problem takes the form of finding solutions to (19) subject to the condition (18) This will not be done in a completely general way, but with some assumptions which will lead to a physically relevant result. Suppose the electron system describes a rectangular geometry in the xy plane. Moreover, let 0 Ψ have a plane wave dependence in the x direction, so solutions which have the structure 0 Ψ,. ikx x ye y (20) is sought where y is a real function of y. Let us take the function in the vector potential to be indepen- dent of x , () y (21) The derivatives of 0 Ψ can be calculated based on (20) and then substituted into (19), 22 22 yy 2e 2. y kyykyyey mE y (22) This takes the form of a second order equation for y , but it is coupled to y in (17), 2 yy 2. y ykey mEy (23) If is assumed to have the form (21), then 0 xx and (18) assumes the simple form 2 yy .ay (24) Since the right-hand side of (24) depends only on y, (24) can be integrated once to obtainy , which appears in (23), in terms of y as 0 2 y. y y yad (25) Imposing 0 0 y y . Substituting (25) into (23), this coupled system is reduced to the following nonlinear eigenvalue problem 0 22 yy 2 2 () . y y emE ykad yy (26) Therefore, the dependent variable in (26) is y . In addition to (26), it is useful to write down a decoupled version which is obtained by introducing a new variable y given by 0 2. y y e yk ad (27) Equation (26) can be written in the form of a pair of equations as follows, 22 yyy 2 2 ,. emE yayy yyy (28) The Hall resistance for this two-dimensional system can be calculated based on (28), in fact it can be written in terms of y . The geometry is that of a rectangular plate with edges which are parallel to the x and -axes. To be consistent with (20), where the x -dependence in Φ is assumed to be a plane wave, only the y dimen- sion will be of significance here. The terminations for integration localized at fixed -coordinates, are termed the left (L) and right (R) edges of the geometry. The Hall potential is defined as the difference of potentials be- tween these two edges of the rectangle. In fact, the Hall potential can be obtained from (26), or better in terms of the solution for y by means of 2 22 H, 2 VRL me (29) where R and L refer to right and left. Only the longitu- dinal x or plane wave component of the current density contributes 0 22 x0 0 Re A. y y eee jiekady mm (30) The potential H V is transverse to the current. From (28), since 2 can be related to y , the current density can be represented entirely in terms of the variable as 22 22 xy . 2 y eh jmamam Integrating x j and using the definition of H V given in (29), x I can be related to H V as follows, 22 222 xx H. 22 LL y RR e IjdydyRL V am ama (31) By means of (16), the quantity a can be eliminated from (31) to produce the following remarkable formula, 2 xH 2. Ne I V Mh (32) The result in (32) immediately implies the Hall resis- tance is quantized according to, H H2 x . 2 V M h RIN e (33) P. BRACKEN Copyright © 2010 SciRes. JMP 161 Finally, it will be shown that a wavefunction can be determined based on the coupled system (28). In fact, the coupled equations in (28) can be combined into a single nonlinear differential equation for the function y , from which y can be determined. To begin to do this, differentiate the first equation in (28) and then divide this by y to obtain yy y y 2. (34) Differentiating both sides of this, there follows 22 yy yyy 22 22 . y yy y (35) Squaring both sides of (34), an additional expression for 22 / y is obtained. Substituting this into the right hand side of (35), 2 yy yyyyy yy 1. 24 (36) From the second equation in (28), upon dividing by , it follows that yy 2 2 2mE . (37) Substituting (37) into (36), a third order nonlinear eq- uation in terms of the independent variable results, 2 yyy yy 2 yy 122, 2E (38) where we put 2 2mE . E (39) A general solution to (38) may not be possible, how- ever, something can be done. Note that upon omitting 2 (22)E from (38), the equation can be integrated. Thus, we have yy y 1 lnln 0 2 yy , and inte- grating gives 0. This can be integrated as well to give 3 123 1 1 yy 3cc c c . A specific physically realistic solution to the general form of (38) can be approached as follows. The first equation in (28) implies that the sign of y is determined by a, therefore, when does not vanish, must be a mo- notonic function. Consequently, one way in which a class of solution can be obtained is to consider the case in which y is only a function of , y.w (40) In fact, g can be determined explicitly. Diffe- rentiating both sides of (40) with respect to , we get 2 yy yyyy w,w .ww www (41) Substituting (40) and (41) into (38) gives rise to the following equation for w, 22 1 w2, 2 wwE (42) Clearly (42) is nonlinear, however, there is a way to produce a solution which is physically reasonable. There exists a quadratic polynomial solution for w which can be expressed in terms of as 2 w( ) These constants can be specified upon substitution in (42), and it will constitute a solution provided that 0 and 2 1 w2. 2E (43) Taking (43) and replacing the result in (40), it is clear the resulting equation can be separated to give 2 2yc. 2 d E (44) The negative sign gives a tangent function solution which will be prone to have poles and can be written 2tan 2yc.yE However, the other choice of sign in (44) gives rise to the result, 1arctan()y c. 2EE This can be solved explicitly for the function y , 2y 2y 1C 2tanh yc2. 1C E E e yEE E e (45) By differentiating (45), an expression for 2y is obtained. The function y which we need to write the wavefunction (20) is found from the square root of this, namely 1 2 1y 4 1 2y 1 22 C. 1C E E Ee yea e (46) The wavefunction is then determined using (46) by means of, P. BRACKEN Copyright © 2010 SciRes. JMP 162 / Ψ. iEt ikx ee y This is a bounded function on any right half axis and square integrable over the rectangular area. Thus there exists a solution with the desired physical properties. Therefore, it has been seen how (1) emerges and that physical classes of solutions to (2) can be investigated. Most importantly, a link between the wavefunctions im- plied by the model and the calculation of a corresponding resistence for the model has been shown. 4. Conclusions An elementary model for the quantum Hall effect has been developed. It is known in this field that simple models based on Schrödinger equations can be very use- ful in studying the effect. For example, the equation is solved with the harmonic oscillator potential to describe and obtain the energies of Landau levels. The model emphasizes several aspects of the geometry of the system in obtaining the results (32,33). It is quite interesting that a single particle Schrödinger equation can be obtained and solved in closed form, and which incorporates a sig- nificant amount of the physics involved. 5. References [1] J. K. Jain, “Composite Fermions, Perspectives in Quan- tum Hall Effects,” In: S. D. Sarma and A. Pinczuk Eds., J. Wiley and Sons, New York, 1987. [2] K. von Klitzing, G. Dorda and M. Pepper, “New Method for High-Accuracy Determination of the Fine Structure Constant Based on Quantized Hall Resistance,” Physics Review Letters, Vol. 45, No. 6, 1980, pp. 494-497. [3] R. B. Laughlin, “Quantized Hall conductivity in Two dimensions,” Physical Review B, Vol. 23, No. 10, 1981, pp. 5632-5633. [4] R. E. Prange and S. M. Girvin, “The Quantum Hall Ef- fect,” Springer Verlag, New York, 1987. [5] Z. F. Ezawa, “The Quantum Hall Effects, Field Theoreti- cal Approach and Related Topics,” World Scientific, Singapore, 2000. [6] R. B. Laughlin, “Fractional Quantization,” Reviews of Modern Physics, Vol. 71, No. 4, 1999, 863-874. |