Modeling a Quantum Hall System via Elliptic Equations

Abstract

Quantum Hall systems are a suitable theme for a case study in the general area of nanotechnology. In particular, it is a good framework to search for universal principles relevant to nanosystem modeling and nanosystem-specific signal processing. Recently, we have been able to construct a partial differential equations-based model of a quantum Hall system, which consists of the Schrödinger equation supplemented with a special-type nonlinear feedback loop. This result stems from a novel theoretical approach, which in particular brings to the fore the notion of quantum information. Here we undertake to modify the original model by substituting the dynamics based on the Dirac operator. This leads to a model that consists of a system of three nonlinearly coupled first-order elliptic equations in the plane.

1. Quantum Entanglement of Composite Systems and Its Associated Hamiltonian Dynamics

In this section, we give a very brief overview of the principles behind the mesoscopic loop models as developed in [1–5]. The theme of quantum information is extremely relevant to our approach, but it has been brought to light in explicit terms only recently; an in-depth discussion will appear elsewhere [6, 7].

Since the measurement of Hall resistance in a quantum Hall system results in a classical signal, it is natural to ask if any quantum information is in the process transduced to the classical medium. Let us develop this point of view into a formal discussion. Suppose that the electronic solid-state system and the ambient magnetic field are represented by two Hilbert spaces; say 𝐇 and 𝐇, respectively. According to the principles of quantum mechanics [8], the combined system is then represented by the tensor product space 𝐇⊗𝐇. Moreover, the states of the composite system assume the form|Ψ_combined⟩=_𝑚,𝑛𝐴^∗_𝑚𝑛|𝜑_𝑚⟩⊗|𝜓_𝑛⟩∈𝐇⊗𝐇. (1.1)We will associate with every state as above an entanglement operator (with no intention to imply that the composite state (1.1) is necessarily entangled in the rigorous sense of the term [8]):𝐾=𝐴_𝑚𝑛|𝜑_𝑚⟩⟨𝜓_𝑛|. (1.2) Thus, if an electronic system is found in the state 𝜓∈𝐇, then the electromagnetic system will collapse into the state𝐾𝜓∈𝐇. (1.3)Let us define the density operator of system 𝐇 in the usual way:𝐇𝜌=Tr(|Ψ_combined⟩⟨Ψ_combined|). (1.4)One readily finds𝜌=𝐾^∗𝐾. (1.5)This lays out the vocabulary for a quantum-mechanical description of a composite system. Next, we postulate a model relevant specifically to the quantum Hall systems. Namely, we propose that, in the presence of a magnetic field, the dynamics of the electronic system are derived from the energy functional of the formTr(𝐻𝜌)+𝛽logdet(𝜌), (1.6)where 𝐻 is the “regular” Hamiltonian governing the microscopic dynamics of the electronic subsystem. For the purposes at hand, there is no need to discuss the constant 𝛽 beyond the simple statement that it is nonzero when the magnetic field is nonvanishing. We would like to point out that this energy functional is a special case of functionals given in the form, say, Tr(𝐻𝜌)+𝛽Tr(𝑓(𝜌)), where 𝑓 is a suitable analytic function. Indeed, logdet(𝜌)=Tr(log𝜌). Roughly speaking, function 𝑓 describes the interaction between two subsystems of a composite quantum system. Many different aspects of the general models of this kind are discussed in [6, 7]. As it is shown in [4], and again here, the functional with 𝑓=log captures the characteristic of a quantum Hall system, which justifies our special interest in it. It is also worthwhile mentioning that −Tr(log𝜌) is the von Neumann relative quantum entropy 𝑆(𝐼||𝜌) [8].

Henceforth we assume for simplicity that 𝐻 is diagonalizable in the basis consisting of its eigenvalues:𝐻𝜓_𝑘=𝐸_𝑘𝜓_𝑘. (1.7)Now, substituting the decomposition (1.5) into (1.6) yields the following Hamiltonian:Ξ(𝐾)=Tr(𝐾𝐻𝐾^∗)+𝛽logdet(𝐾^∗𝐾). (1.8)This in turn leads to the following system of Hamiltonian equations (see [1, 2]):−𝑖ℏ𝐾̇=𝐾𝐻+𝛽𝐾^∗−1. (1.9)(From this point on we will set ℏ=1.) The corresponding eigenvalue problem assumes the form𝐾𝐻+𝛽𝐾^∗−1=𝜈𝐾. (1.10)Equation (1.11) is a part of the proposed model of quantum Hall systems in homeostasis. Solutions of (1.10) are found to assume the form (see [1, 2]) 𝐾=_{𝐸𝑘<𝜈}(𝛽𝜈−𝐸_𝑘)^1/2|𝑈𝜓_𝑘⟩⟨𝜓_𝑘|, (1.11)where 𝑈∶𝐇→𝐇 is an arbitrary unitary transformation.

2. The Mesoscopic-Loop Model Based on the Dirac-Type Equations

The model we are about to present deals with two types of particles, which will be called electrons and holes. Both of them will have the same effective mass 𝑚^∗=1 and charge 𝑒=1. It seems appropriate to emphasize that, of course, the notion of effective mass depends on the model. All analyses will be performed in a two-dimensional plane. Recall that the Hodge star ∗ in two dimensions is a linear operation on differential forms determined by the relations ∗𝑑𝑥=𝑑𝑦,∗𝑑𝑦=−𝑑𝑥,∗1=𝑑𝑥⋀𝑑𝑦, and ∗(𝑑𝑥⋀𝑑𝑦)=1. The exterior derivative 𝑑 defines the coderivative 𝛿=−∗𝑑∗. Let the vector potential 𝐴 be given in local coordinates as 𝐴=𝐴₁𝑑𝑥+𝐴₂𝑑𝑦. In particular, 𝛿𝐴=−(𝐴_1,𝑥+𝐴_2,𝑦), while𝐵(𝑥,𝑦)=∗𝑑𝐴=𝐴_2,𝑥−𝐴_1,𝑦 (2.1)is the magnetic flux density. The corresponding Schrödinger operator 𝑆_𝐴 acts on a wave function 𝜓 as follows:2𝑆_𝐴𝜓=−(𝜕²_𝑥+𝜕²_𝑦)𝜓+2𝑖(𝐴₁𝜕_𝑥𝜓+𝐴₂𝜕_𝑦𝜓)+(𝐴²₁+𝐴²₂)𝜓−𝑖(𝛿𝐴)𝜓. (2.2)(The factor of 2 in front of 𝑆_𝐴 stems from the fact that in the adopted units the kinetic energy is multiplied by a factor of ℏ²/2𝑚^∗=1/2.) The Schrödinger operator can be successfully used in a construction of mesoscopic loop models [5]. In this paper, we will consider a modification based on a Dirac-type factorization of the Schrödinger operator. To this end, we use the two-dimensional Dirac operator [9] with its parameters set at 𝑚=0 and 𝑐=1. Namely, let𝐻_𝐴=0𝐷^∗𝐷0, (2.3)where𝐷=−𝑖𝜕_𝑥−𝐴₁+𝜕_𝑦−𝑖𝐴₂ (2.4)so that𝐷^∗=−𝑖𝜕_𝑥−𝐴₁−𝜕_𝑦+𝑖𝐴₂. (2.5)A direct calculation shows that𝐻²_𝐴=𝐷^∗𝐷00𝐷𝐷^∗=2𝑆_𝐴−𝐵002𝑆_𝐴+𝐵, (2.6)where 𝐵=𝐵(𝑥,𝑦) is determined by 𝐴 via (2.1). Observe the difference of signs at the magnetic flux density term. This necessitates the interpretation of the two wave functions defining the state vector|𝜓₁𝜓₂⟩ (2.7)as representing particles with opposite pseudospins. At this point, we venture to describe a quantum Hall system in a manner discussed in the previous section, which brings the notion of entanglement to the fore. Namely, we consider the following system of equations:𝐻_𝐴|𝜓₁𝜓₂𝜓⟩=𝐸|¹𝜓₂𝜓⟩,∗𝑑𝐴(𝑥,𝑦)=|𝐾|¹𝜓₂⟩(𝑥,𝑦)|², (2.8)which involves a coupling realized via the entanglement transform 𝐾. Note that this forces an appropriate normalization of 𝐾 so as to guarantee the correct charge to flux ratio.

3. Quantization of Hall Resistance

Note that if |^𝜓1_𝜓2⟩ is a solution of (2.8), then in particular it is an eigenfunction of 𝐻_𝐴. We now request that 𝐾 be a stationary solution as in (1.11) with 𝑈=𝐼, where|𝜓_𝑘𝜓⟩=|^𝑘,1𝜓_𝑘,2⟩ (3.1)are the eigenvectors of the Dirac Hamiltonian 𝐻_𝐴. In general, properties of the model depend on the choice of 𝑈. The case 𝑈=𝐼 is interpreted as corresponding to the phase-correlated regime; see [1] for a more detailed discussion of this point. Next, assume 𝜈>𝐸. Without loss of generality, we may also assume that |^𝜓1_𝜓2⟩ is on the list of mutually orthogonal eigenfunctions defining 𝐾 via (1.11). (This can always be arranged even if 𝐸 has multiplicity greater than 1.) Thus, we have𝜓𝐾|¹𝜓₂𝜓⟩=𝑐|¹𝜓₂⟩ (3.2)for a constant 𝑐. Consequently,𝜓|𝐾|¹𝜓₂⟩(𝑥,𝑦)|²∼|𝜓₁|²+|𝜓₂|². (3.3)This implies that the last equation of (2.8) is equivalent to the statement𝜕_𝑥𝐴₂−𝜕_𝑦𝐴₁=𝑅_𝐻(|𝜓₁|²+|𝜓₂|²). (3.4)Note that 𝑅_𝐻 is the ratio of the total flux to the total charge counted between the two types of particles. In particular, due to quantization of flux and charge, 𝑅_𝐻 is a rational number when expressed in appropriate units. We will demonstrate that in a symmetric realization of this model the Hall resistance is equal to 2𝑅_𝐻. We conduct the analysis in a rectangular-domain setting with the assumption that the wave functions are periodic in 𝑥, both with the same period; that is,𝜓₁=exp(𝑖𝑘𝑥)𝜒₁𝜓(𝑦),₂=exp(𝑖𝑘𝑥)𝜒₂(𝑦). (3.5)By substituting to (2.8), we readily obtain𝜒^′₁=−(𝑘−𝐴₁)𝜒₁+𝐸𝜒₂,𝜒^′₂=(𝑘−𝐴₁)𝜒₂−𝐸𝜒₁. (3.6)Furthermore, let us assume 𝐴₂=0,𝐴₁=𝐴₁(𝑦), and 𝐴₁(0)=0. Let us also introduce an auxiliary variable 𝑤=𝑘−𝐴₁. In such a case, (2.8) is equivalent to the system𝑤^′=𝑅_𝐻(𝜒²₁+𝜒²₂𝜒),𝑤(0)=𝑘,_′1=−𝑤𝜒₁+𝐸𝜒₂,𝜒^′₂=𝑤𝜒₂−𝐸𝜒₁. (3.7)By differentiation, we readily obtain𝜒₁^′′=(−𝐸²+𝑤²−𝑤^′)𝜒₁,𝜒₂^′′=(−𝐸²+𝑤²+𝑤^′)𝜒₂. (3.8)These are 1D stationary Schrödinger equations with energy 𝐸²/2. Note that 𝜒₁ is affected by potential (𝑤²−𝑤^′)/2, while 𝜒₂ is affected by potential (𝑤²+𝑤^′)/2. Both quantities result from the presence of the magnetic field, and thus they represent the Hall effect. There are two potential functions, differing by the sign of the correction term 𝑤^′, because two types of carriers are involved in the charge transport. Note that𝑤(𝑦)=𝑘+𝑅_𝐻^𝑦₀(𝜒²₁(𝑦^′)+𝜒²₂(𝑦^′))𝑑𝑦^′. (3.9)We calculate the current next. Recall that𝑗=Re{𝜓^∗₁(𝑖⋅𝑑𝜓₁+𝜓₁𝐴)}+Re{𝜓^∗₂(𝑖⋅𝑑𝜓₂+𝜓₂𝐴)}. (3.10)A direct calculation shows that the two 1 -forms amount toRe{𝜓^∗_𝑙(𝑖⋅𝑑𝜓_𝑙+𝜓_𝑙𝐴)}=−𝜒²_𝑙(𝑦)𝑤(𝑦)𝑑𝑥,𝑙=1,2. (3.11)In summary,1𝑗=−𝑅_𝐻𝑤^′1(𝑦)𝑤(𝑦)𝑑𝑥=𝑅_𝐻𝑤∗𝑑²2. (3.12)(We remind the reader that ∗ denotes the Hodge star.) Observe that the current flows along the 𝑥-axis. Since both types of carriers contribute to the current, the total Hall potential is the sum of the two separate potentials pointed out above, that is,𝑉_𝐻=𝑤²+𝑤^′2+𝑤²−𝑤^′2=𝑤². (3.13)It follows that1𝑗=2𝑅_𝐻∗𝑑𝑉_𝐻. (3.14)By integration, we then obtain that the Hall resistance is 2𝑅_𝐻. We remark that this is twice the value obtained in the Schrödinger-type model (cf. [4, 5]). Recall that 𝑅_𝐻 has been introduced as the ratio of the number of quanta of the magnetic field to the number of quanta of electric charge 𝑒=1 of both types of particles. If particles of type, say, 𝜓₂ were not accounted for in the calculation of the filling factor, then there would be a discrepancy between the inverse filling factor and the quantity 𝑅_𝐻.

It is interesting to ask when the electrons and holes are engaged as charge carriers in equal numbers. Let us position the edges of the plate symmetrically about the 𝑥-axis, that is, 𝑦∈[−𝑏,𝑏]. Note that the number of carriers of each type is proportional to^𝑏_−𝑏𝜒²_𝑙(𝑦^′)𝑑𝑦^′,𝑙=1,2. (3.15)Now assume 𝜒₂(0)=𝜒₁(0), and consider the last two equations of (3.7). In such a case, one can find a solution satisfying the condition 𝜒₂(𝑦)=𝜒₁(−𝑦). For this solution, particles of each type occur in equal numbers.

4. The Elliptic System in the Plane

In Section 3, we demonstrated that Hall resistance is quantized under certain symmetry assumptions which reduce the problem to a system of ordinary differential equations. We emphasize that Hall potentials and Hall resistance can be defined and discussed in the general two-dimensional setting [5]. In fact, the two-dimensional, that is, partial differential equations (PDEs), setting cannot be avoided if we want to study the effect of lattice potential and so forth. We will now briefly discuss the ramifications of the PDE problem. Consider (2.8) jointly with the assumptions about 𝐾 put forth in the previous section. In this case, it is equivalent to the following system of first-order partial differential equations:𝜕_𝑦𝜓₁=𝑖𝜕_𝑥𝜓₁+(𝐴₁+𝑖𝐴₂)𝜓₁+𝐸𝜓₂,𝜕_𝑦𝜓₂=−𝑖𝜕_𝑥𝜓₂−(𝐴₁−𝑖𝐴₂)𝜓₂−𝐸𝜓₁,𝜕_𝑦𝐴₁=𝜕_𝑥𝐴₂−𝑅_𝐻(|𝜓₁|²+|𝜓₂|²). (4.1)A direct calculation shows that if a triplet (𝜓₁,𝜓₂,𝐴) is a solution of the system, then so is (𝑒^𝑖𝜑𝜓₁,𝑒^𝑖𝜑𝜓₂,𝐴+𝑑𝜑) for an arbitrary real function 𝜑=𝜑(𝑥,𝑦). Function 𝜑 represents gauge freedom and therefore carries no physical information. In particular, only the relative phase of the pair (𝜓₁,𝜓₂) is physically meaningful. Next, observe that while system (4.1) is not elliptic by itself, it will become strictly elliptic when amended with the gauge-fixing equation𝜕_𝑦𝐴₂=−𝜕_𝑥𝐴₁. (4.2)Consider a system that consists of (4.1) and (4.2). Note that the nonlinear terms in (4.1) are (real) analytic functions of the dependent variables. Therefore, the classical Cauchy-Kovalevskaya theorem [10] guarantees local existence of solutions of the initial value problem when the boundary conditions are also analytic. The initial value problem may help gain some insight into the nature of the system; an example of a solution obtained numerically with a finite difference method is displayed in Figure 1. Since the boundary value problems spring to mind in this context, one should emphasize that in view of ellipticity, the natural setting is that of the Riemann-Hilbert-type problem [11]. However, we wish to signal briefly that the boundary value problem is not the only type of approach appropriate for the study of the mesoscopic-loop model; there is a promising complementary approach. Namely, setting 𝙰=𝐴₁+𝑖𝐴₂, we can represent (4.1)-(4.2) as a system of three nonlinearly coupled 𝜕-equations:𝜕𝜓₁=𝑖2(𝙰𝜓₁+𝐸𝜓₂),𝜕𝜓₂𝑖=−2(𝙰𝜓₂+𝐸𝜓₁),𝜕𝑖𝙰=−2𝑅_𝐻(|𝜓₁|²+|𝜓₂|²). (4.3)This suggests the relevance of a variety of complex methods [12], but this theme will not be discussed in the present paper.

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Figure 1 

A numerical solution of (4.1) with 𝐸=5 (in units adopted throughout the paper), 𝑅𝐻=3/5, and 𝐴2≡0. The initial values 𝜓2(𝑥,0)=0.3exp(𝑖𝑥)+0.2∗exp(2𝑖𝑥),𝜓1(𝑥,0)=−0.9𝜓2(𝑥,0), and 𝐴1(𝑥,0)=0 are prescribed on the edge facing front right.

5. Closing Comments

The Dirac equation is undergoing a renaissance in condensed matter physics due to its relevance to the unusual two-dimensional crystal called graphene (see, e.g., [13]). I emphasize that in this paper we have not addressed any of the issues related to that stream of research. Rather, we have used the Dirac equation as an analytic alternative to the Schrödiner equation. It helped us find an almost equivalent reformulation of the nonrelativistic model.

Acknowledgment

The author is grateful to the journal referees whose thoughtful questions and comments helped improve this paper.