# Modeling a Quantum Hall System via  Elliptic Equations

Authors: Artur Sowa
Journal: Advances in Mathematical Physics
 (2009)

---

## 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.


---

## Body

## 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 aquantum 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(1.1)|Ψcombined〉=∑m,nAmn*|φm〉⊗|ψn〉∈H^⊗H.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]):(1.2)K=∑Amn|φm〉〈ψn|. Thus, if an
electronic system is found in the state ψ∈H,
then the electromagnetic system will collapse into the state(1.3)Kψ∈H^.Let us define the density
operator of system H in the usual way:(1.4)ρ=TrH^(|Ψcombined〉〈Ψcombined|).One readily
finds(1.5)ρ=K*K.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 form(1.6)Tr(Hρ)+βlogdet(ρ),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 thatH is diagonalizable in the basis consisting of
its eigenvalues:(1.7)Hψk=Ekψk.Now, substituting  the
decomposition (1.5) into (1.6) yields the following
Hamiltonian:(1.8)Ξ(K)=Tr(KHK*)+βlogdet(K*K).This in turn leads to the
following system of Hamiltonian equations (see [1, 2]):(1.9)−iℏK˙=KH+βK*−1.(From this point on we will set ℏ=1.)
The corresponding eigenvalue problem assumes the form(1.10)KH+βK*−1=νK.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])
(1.11)K=∑Ek<ν(βν−Ek)1/2|Uψk〉〈ψk|,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 massm*=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=−(A1,x+A2,y),
while(2.1)B(x,y)=*dA=A2,x−A1,yis the magnetic flux density.
The corresponding Schrödinger operator SA acts on a wave function ψ as follows:(2.2)2SAψ=−(∂x2+∂y2)ψ+2i(A1∂xψ+A2∂yψ)+(A12+A22)ψ−i(δA)ψ.(The factor of 2 in front of SA 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, let(2.3)HA=[0D*D0],where(2.4)D=−i∂x−A1+∂y−iA2so that(2.5)D*=−i∂x−A1−∂y+iA2.A direct calculation shows that(2.6)HA2=[D*D00DD*]=[2SA−B002SA+B],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(2.7)|ψ1ψ2〉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:(2.8)HA|ψ1ψ2〉=E|ψ1ψ2〉,*dA(x,y)=|K|ψ1ψ2〉(x,y)|2,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|ψ1ψ2〉 is a solution of (2.8), then in particular it
is an eigenfunction of HA.
We now request that K be a stationary solution as in (1.11) with U=I,
where(3.1)|ψk〉=|ψk,1ψk,2〉are the eigenvectors of the
Dirac Hamiltonian HA.
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 |ψ1ψ2〉 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 have(3.2)K|ψ1ψ2〉=c|ψ1ψ2〉for a constant c.
Consequently,(3.3)|K|ψ1ψ2〉(x,y)|2∼|ψ1|2+|ψ2|2.This implies that the last equation
of (2.8) is equivalent to the
statement(3.4)∂xA2−∂yA1=RH(|ψ1|2+|ψ2|2).Note that RH 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, RH 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 2RH.
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,(3.5)ψ1=exp(ikx)χ1(y),ψ2=exp(ikx)χ2(y).By substituting to (2.8), we
readily obtain(3.6)χ1′=−(k−A1)χ1+Eχ2,χ2′=(k−A1)χ2−Eχ1.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 system(3.7)w′=RH(χ12+χ22),w(0)=k,χ1′=−wχ1+Eχ2,χ2′=wχ2−Eχ1.By differentiation, we readily
obtain(3.8)χ1′′=(−E2+w2−w′)χ1,χ2′′=(−E2+w2+w′)χ2.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
that(3.9)w(y)=k+RH∫0y(χ12(y′)+χ22(y′))dy′.We calculate the current next.
Recall that(3.10)j=Re{ψ1*(i⋅dψ1+ψ1A)}+Re{ψ2*(i⋅dψ2+ψ2A)}.A direct calculation shows that
the two 1 -forms amount to(3.11)Re{ψl*(i⋅dψl+ψlA)}=−χl2(y)w(y)dx,l=1,2.In summary,(3.12)j=−1RHw′(y)w(y)dx=1RH*dw22.(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,(3.13)VH=w2+w′2+w2−w′2=w2.It follows that(3.14)j=12RH*dVH.By integration, we then obtain
that the Hall resistance is 2RH.
We remark that this is twice the value obtained in the Schrödinger-type model
(cf. [4, 5]). Recall that RH 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 RH.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 thex-axis, that is, y∈[−b,b].
Note that the number of carriers of each type is proportional
to(3.15)∫−bbχl2(y′)dy′,l=1,2.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 Section3,
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:(4.1)∂yψ1=i∂xψ1+(A1+iA2)ψ1+Eψ2,∂yψ2=−i∂xψ2−(A1−iA2)ψ2−Eψ1,∂yA1=∂xA2−RH(|ψ1|2+|ψ2|2).A direct calculation shows that
if a triplet (ψ1,ψ2,A) is a solution of the system, then so is (eiφψ1,eiφψ2,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(4.2)∂yA2=−∂xA1.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:(4.3)∂¯ψ1=i2(Aψ1+Eψ2),∂¯ψ¯2=−i2(Aψ¯2+Eψ¯1),∂¯A¯=−i2RH(|ψ1|2+|ψ2|2).This suggests the relevance of a
variety of complex methods [12], but this theme will not be discussed in the present
paper.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.
(a)(b)(c)


## 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.



---

*Source: 514081-2008-11-06.xml*