Time-dependent Magnetic Fields and the Quantum Hall Effect

T ime-dependent M agnetic F ields and the Quantum H all Effect T.R. Go vindarajan a and V.P. Nair b a The Institute of Mathematical Sciences, Chennai 600 113, India Krea Univ ersity , Sri Cit y , 517646, CCSP , SGT Univ ersit y , Gurugram 122505, India b Ph ysics Department, Cit y College of New Y ork, CUNY New Y ork, NY 10031 E-mail : vpnair@ccny.cuny.edu trg@imsc.res.in, govindarajan.thupil@krea.edu.in Abstract Ermako v has sho wn how the solution to the classical harmonic oscillator in one spa- tial dimension with general time-dependent fr equency can be reduced to the time- independent case and an associated nonlinear ordinary differ ential equation, an anal- ysis which has been applied to the Schr ¨ odinger equation as well. W e extend this anal- ysis to the Landau problem of a charged par ticle in a uniform magnetic ﬁeld in two dimensions and construct the gener alize d Laughlin wave functions for the case when the magnetic ﬁeld is time-dependent. W e also work out the dynamics of density ﬂuc- tuations (the Girvin,MacDonald,Platzman or GMP mode) and argue that it is possi- ble to tune the frequency of the magnetic ﬁeld to obtain a compr essible droplet of fermions. W e also analyze the dynamics of the edge modes of the droplet for the integer H all effect. 1 Intr oduction E ver since its disco very in the 1980s, quantum H all effect has been a fascinating phe- nomenon, generating an enor mous amount of r esearch both on the experimental and theoretical aspects of the problem[1]. An elaborate and successful theoretical structure based on wav e functions, ﬂux attachment procedur es, conformal ﬁeld the- ory , etc. has been built up b y now [2]. W e may also note that the Hall effect has been theoretically generalized to higher dimensions [3]-[5]. Some of the results or predic- tions obtained in this case can perhaps be accessible experimentally using the idea of synthetic dimensions [6]. The mathematical aspects of the quantum H all effect, in- cluding connections to noncommutative geometr y and von Neumann algebras, have also been of great resear ch interest [7]. F or the standard quantum H all effect in 2+1 dimensions, we may also note that a Chern-Simons action, which includes the elec- tromagnetic vector potential as well as a statistical gauge ﬁeld, has been shown to apply to the bulk dynamics of a dr oplet of electrons [8]. For such a droplet, which is a state of ﬁnite volume with a boundar y , the edge dynamics, which is given b y a chiral boson ﬁeld, is important as well. Among other features, it helps to cancel the gauge anomaly r esulting from the v ariation of the Chern-S imons bulk action. The r esponse function of the system to v ariations of the electromagnetic ﬁeld, for example, the H all conductivity , is also directly obtained from the Chern-Simons action. Ev en though a ﬂat background metric is what is r elevant for almost all experimental situations, con- siderations of the Hall effect on spaces with a nontrivial background metric and/or nontrivial topology are useful [1, 9]. They lead to the identiﬁcation of new transport coefﬁcients. F or instance, the term proportional to the time-derivative of the met- ric in the energy-momentum tensor of the system gives the Hall viscosity . F or this analysis, one is thus looking at a time-dependent (albeit slo wly var ying) background metric. A natural question is then: What is the dynamics of a quantum H all state if the magnetic ﬁeld, which is used to deﬁne the Landau levels and the H all state, is itself time-dependent? This is the subject we focus in the present paper . E ven though there is a large body of literatur e on the H all effect, this particular question seems not to have received any attention. W e do take note that adiabatic changes of the magnetic ﬁeld have been considered before, see the r eview [10]. Also the case of time-dependent electrostatic or conﬁning potentials has been analyzed in [11]. Our approach to the problem of time-dependent magnetic ﬁelds starts with a simple observation. F or the harmonic oscillator with a time-dependent frequency ω ( t ) , the general solution of the time-dependent Schr ¨ odinger equation can be con- 2 structed pro vided one can solve an auxiliary nonlinear equation kno wn as the Er - mako v equation [12]. This method is brieﬂy reviewed in the next section. Since the single-particle Landau problem (of a charged particle in a uniform magnetic ﬁeld) is reducible to a harmonic oscillator , we can expect that a similar strategy can be applied. But the Landau problem of inter est is in two spatial dimensions and the degeneracy due to magnetic translations must be taken into account. The holomor - phicity of the lo w est Landau level wave functions is a key featur e of the Laughlin-type many -particle wave functions . Therefor e, one would like to expr ess the solution to the time-dependent Schr ¨ odinger equation in terms of coherent states of the oscil- lator with time-independent frequency . This means that a simple generalization of the Ermakov method is needed, and this will be presented in section 3. W e will also construct the Laughlin-type states for ﬁling fraction ν = (1 / 2 p + 1) . The change in magnetic ﬂux due to the time-dependence of the magnetic ﬁeld B will cr eate electric ﬁelds (by F ar aday’ s law) and consequent currents . W e work out the expressions for the charge and curr ent densities as well. But even more signiﬁcantly , while for a constant magnetic ﬁeld a droplet of elec- trons behaves as an incompressible ﬂuid, the time-dependence of B allo ws for com- pression and dilation since the magnetic length of a single-particle state can change with time. ( W e consider keeping the ﬁlling fraction ﬁxed as we change B .) This will also drive the dynamics of the Girvin-MacDonald-Platzman (GMP) mode [13]. The relev ant equations are worked out in section 4. As an example, we also consider a magnetic ﬁeld of the for m B ( t ) = B 0 + B 1 sin Ω t , where B 0 is the constant (time- independent) magnetic ﬁeld and the second term is a time-dependent per turbation of it. S uch a magnetic ﬁeld will drive the GMP mode; it can then have frequencies ω k ± Ω wher e ω k is the unperturbed fr equency of the GMP mode . These should be ex- perimentally detectable, perhaps, in a suitable scattering process . Also , for a suitable Ω , one may even be able to eliminate the gap of the GMP mode and obtain a tran- sition to a compressible situation. ( This could be a compressible ﬂuid or a cr ystal depending on the wave numbers for the energy proﬁle of the GMP mode .) A natural extension of our analysis would be to a droplet of fer mions . F or a con- stant magnetic ﬁeld, these are descr ibed b y area-pr eserving diffeomorphisms of the droplet. The time-dependence of B will affect the edge dynamics since compression and dilation are now additional modes . F or the case of the integer Hall effect, we work out the generalization of the area-pr eserving diffeomorphsims and the action for the edge modes in section 5. The effect of allowing for compression and dilation or changes in the radius of a droplet as the magnetic ﬁeld changes is that the equation 3 go verning the edge modes becomes an integro-differential equation. The analysis of the edge modes for the fractional QHE will be signiﬁcantly mor e involved due to the fact that interpar ticle interactions are a key to obtaining the re- quired states. With a time-dependent magnetic ﬁeld, it is not clear at this point how the interactions r espond to ar ea-preserving diffeomorphisms. This issue, as well as the question of solving the equations for the edge ﬂuctuations, will be left to future work. W e conclude with a short r esume of the r esults in section 6. There ar e two appen- dices giving technical details of the simpliﬁcation of the actions for density ﬂuctua- tions and edge modes. 2 Ermako v method for the 1d-oscillator with time-dependent frequency W e start by brieﬂy recalling the salient featur es of the one-dimensional oscillator with time-dependent frequency . The Hamiltonian for this case is given b y H ( t ) = 1 2  p 2 + ω 2 ( t ) q 2  (1) The Schr ¨ odinger equation can thus be written out as i ∂ Ψ ∂ t = 1 2 h − ∂ 2 q + ω ( t ) 2 q 2 i Ψ (2) In the E rmakov analysis, the solution to this equation is giv en by Ψ( t ) = 1 √ b e iϕ Ψ 0 ( ξ , 0) ϕ = ˙ b q 2 2 b − E τ , τ = Z t 0 dt ′ 1 b 2 ( t ) (3) where ξ = q /b , with b as a time-dependent scale factor . H ere Ψ 0 ( ξ , 0) is the wave function obeying the equation 1 2  − ∂ 2 ∂ ξ 2 + ω (0) 2 ξ 2  Ψ 0 ( ξ , 0) = E Ψ 0 ( ξ , 0) (4) Thus Ψ 0 is the wave function for a ﬁxed frequency ω (0) with energy E . The allow ed values for E are ( n + 1 2 ) ω (0) as usual. By direct differentiation of Ψ( t ) as given in (3), one can easily check that it is a solution of the Schr ¨ odinger equation (2) if the scale factor b obeys the equation ¨ b + ω 2 ( t ) b = ω 2 (0) b 3 (5) 4 This is the Er mako v equation. If we have a solution to this equation, we have an exact solution to the Schr ¨ odinger equation. N otice also that, from (3) Z dq Ψ ∗ Ψ = Z dq b Ψ ∗ 0 ( ξ , 0)Ψ 0 ( ξ , 0) = Z dξ Ψ ∗ 0 ( ξ , 0)Ψ 0 ( ξ , 0) (6) sho wing that if Ψ 0 is normalized, then so is Ψ . The essence of the Ermako v method hinges on the existence of an invariant for the oscillator even when ω is time-dependent. T o wards this, we can start with the general operator I ( t ) = 1 2  α ( t ) q 2 + β ( t ) p 2 + γ ( t )( q p + pq )  (7) The time-derivative of I is given b y ˙ I = dI dt = ∂ I ∂ t − i [ I , H ] (8) where the commutator is evaluated by the standard r ule [ q , p ] = i . I mposing time- independence of I ( t ) , i.e., ˙ I = 0 , we get equations for the functions α ( t ) , β ( t ) , γ ( t ) [14]. They can be solved in terms of b ( t ) obeying (5). The result for I is then I ( t ) = 1 2  q 2 b 2 + ( b p − ˙ bq ) 2  (9) where , for br evity , we have set ω 2 (0) = 1 . I t is also possible to construct the eigenvalues of the new invariant I ( t ) . T owar ds this, one can deﬁne new ladder operators a , a † b y a = 1 √ 2  q b + i ( bp − ˙ bq )  a † = 1 √ 2  q b − i ( bp − ˙ bq )  (10) I t is easy to verify that these obey the expected commutation r ule  a , a †  = 1 . The invariant I is given in terms of these operators b y I =  a † a + 1 2  (11) But we should r emember our H amiltonian (1) is not diagonal in the diagonal basis of I ( t ) . The key point here is the variable ξ which is a scaled version of q , with the scale factor obeying the Ermako v equation. W e can generalize the method to the two- dimensional problem of a charged particle in a magnetic ﬁeld b y identifying the scaled version of the coor dinates. W e turn to this no w . 5 3 The Ermako v equations for Landau levels F or the Landau problem or the problem of a charged particle in a magnetic ﬁeld B ( t ) with time dependence in two spatial dimensions, the H amiltonian is given b y H 0 = ( p − eA ) 2 2 m − zero-p oint energy = − 2 m  ∂ ∂ z − λ ¯ z   ∂ ∂ ¯ z + λz  (12) where λ = eB / 4 . In the second line we hav e introduced the combinations z = x 1 + ix 2 , ¯ z = x 1 − ix 2 ∂ ∂ z = 1 2 ( ∂ 1 − i∂ 2 ) , ∂ ∂ ¯ z = 1 2 ( ∂ 1 + i∂ 2 ) (13) The solution to the time-de pendent Schr ¨ odinger equation with the H amiltonian (12) is then given b y Ψ( z , ¯ z , t ) = 1 √ ¯ bb e i Φ Ψ 0 ( ξ , ¯ ξ , 0) (14) where ξ = z /b , ¯ ξ = ¯ z / ¯ b and Ψ 0 ( ξ , ¯ ξ , 0) is an eigenstate of H at time t = 0 with energy eigenvalue E . N otice that we need a complex function b in the present case. I t is convenient to write b as b = √ ρ e iθ (15) In terms of ρ and θ , the Ermakov equations ar e ¨ ρ − ˙ ρ 2 2 ρ − 8 m 2  λ 2 0 ρ − λ 2  = 0 (16) θ = Z dt ′ 2 m  λ 0 ρ ( t ′ ) − λ ( t ′ )  (17) where λ 0 = eB 0 / 4 . The phase Φ in Ψ in (14) is given by Φ = m ˙ ρ 4 ρ ¯ z z − θ ( t ) − Z t dt ′ E ρ ( t ′ ) (18) V eriﬁcation that (14) does indeed satisfy the time-dependent Schr ¨ odinger equation is somewhat tedious but can be done in a straightforwar d way b y dir ect differ entiation. One can verify that (14) is the solution if Ψ 0 obeys the equation − 2 m  ∂ ∂ ξ − λ 0 ¯ ξ   ∂ ∂ ¯ ξ + λ 0 ξ  Ψ 0 ( ξ , ¯ ξ , 0) = E Ψ 0 ( ξ , ¯ ξ , 0) (19) 6 The normalization of the wave function in (14) is given b y Z d 2 z Ψ ∗ Ψ = Z d 2 z ρ Ψ ∗ 0 ( ξ , ¯ ξ , 0) Ψ 0 ( ξ , ¯ ξ , 0) = Z d 2 ξ Ψ ∗ 0 ( ξ , ¯ ξ , 0) Ψ 0 ( ξ , ¯ ξ , 0) (20) W e see that Ψ is nor malized if Ψ 0 is normalized. In order to work out the time- dependence of Ψ explicitly , we need to solve the Ermakov equations. In principle, one can solve (16) for ρ ( t ) and then use it to obtain θ , Φ and Ψ . Admittedly , solving the nonlinear equation (16) is difﬁcult in general. But the point for us is not the ex- plicit solution, but rather (14) gives the general form of the wave function and hence it can be used to construct multiparticle wave functions to analyze density ﬂuctua- tions and edge modes. W e will now focus on the lowest Landau level for which the solutions to (19) may be written as Ψ 0 n = e − λ 0 ¯ ξ ξ ξ n √ n ! = e − ¯ z z / 2 κ ξ n √ n ! (21) where κ = ( ρ/ 2 λ 0 ) . The Slater deter minant of Ψ from (14) with the choice (21) for Ψ 0 will give the wave function for the ν = 1 quantum Hall state . Similarly , for the ν = 1 2 p +1 state, w e can write down the Laughlin wave function as Ψ (2 p +1) ( x 1 , x 2 , · · · , x N ) = C 2 p +1 ρ N 2 exp X i i Φ i − ¯ z i z i 2 κ ! Y i

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