New degeneracies and modification of Landau levels in the presence of a parallel linear electric field

We consider a three-dimensional system where an electron moves under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. The linear electric field leads to harmonic oscillations along the z-direction. There are therefore two frequencies characterizing the system: the usual cyclotron frequency $\omega_c$ corresponding to motion along the x-y plane and associated with Landau levels and a second frequency $\omega_z$ for motion along the z-direction. Most importantly, when the ratio $W=\omega_c/\omega_z$ is a rational number, the degeneracy of the energy levels does not remain always constant as the energy increases. At some energies, the degeneracy jumps i.e. it increases. In particular, when the two frequencies are equal, the degeneracy increases with each energy level. This is in stark contrast to the usual Landau levels where the degeneracy is independent of the energy. We derive compact analytical formulas for the degeneracy. We also obtain an analytical formula for the energy levels and plot them as a function of $W$. The increase in degeneracy can readily be seen in the plot at points where lines intersect. For concreteness, we consider the electric field produced by a uniformly charged ring. Besides a linear electric field in the z direction the ring produces an extra electric field in the xy plane which we treat via perturbation theory. The Landau degeneracy is now lifted and replaced by tightly spaced levels that come in"bands". The plot of the energy levels shows that there is still a degeneracy where the bands intersect.


Introduction
An electron moving in two dimensions (x-y plane) under a constant magnetic field (in the z-direction)has discrete energies known as Landau levels. The energies, neglecting spin, are simply those of the one-dimensional harmonic oscillator with cyclotron frequency ω c . In this paper we consider adding to this scenario a linear electric field which is parallel to the magnetic field above the z=0 plane and anti-parallel to the magnetic field below the z=0 plane. This leads the particle to oscillate harmonically about z=0 along the z-direction with frequency ω z . We therefore obtain a three-dimensional system with two characteristic frequencies: ω c and ω z . The linear electric field clearly modifies the energy levels of the electron. But the most crucial point is that the degeneracy of the system does not necessarily remain constant as the energy increases when the ratio of the two frequencies, W = ω c /ω z , is a rational number. At particular energies, the degeneracy will jump i.e. it will increase. This is most apparent when the two frequencies are equal. In that case, the degeneracy increases with each energy level. Recall that the Landau levels themselves have a significant degeneracy but it is constant for every energy level. Adding the linear electric field therefore alters the degeneracy of the system in a profound way. We calculate the degeneracy and obtain compact analytical formulas for them. We also obtain analytical formulas for the energy levels as a function of W and two quantum numbers n and n z . We plot the energy as a function of W and each line represents a pair (n, n z ). The degeneracies increase at points where lines for different (n, n z ) pairs intersect.
The increase in degeneracy is reminiscent of the two-dimensional Fock-Darwin system [1,2] where besides an applied constant magnetic field, an extra quadratic potential of the form 1 2 m ω 2 0 (x 2 + y 2 ) is often added by hand in order to confine the electrons in the x-y plane. Such a system is characterized by two frequencies, ω c and ω 0 , and the degeneracy is also altered significantly for particular ratios of the two frequencies. The Fock-Darwin system has been very popular and successful in studying quantum dots [3][4][5][6][7][8][9], a major area of research in nanotechnology.
2 A charged particle moving in a linear electric field parallel to a uniform magnetic field The Hamiltonian for an electron of charge −e, mass m e and spin s moving non-relativistically in a general electromagnetic field is given by where p is the canonical momentum, A(x, t) is the vector potential, φ(x, t) is the scalar potential, µ e is the magnetic moment of the electron and B(x, t) is the magnetic field. The magnetic and electric fields are given by B = ∇ × A and E = −∇φ − ∂A ∂t respectively. We now consider the case of a uniform magnetic field of magnitude B 0 acting in the +z direction, B = B 0ẑ and a linear electric field also in the z direction E = k zẑ where k is a positive constant. The electric field is zero on the plane z = 0 and points in the +z direction above the plane (z > 0) and in the -z direction below the plane (z < 0). The magnetic and electric field therefore point in the same direction above the plane but in opposite directions below the plane. We choose the vector potential to be A y = x B 0 , A x = A z = 0 whereas the scalar potential is given by φ(z) = −k z 2 /2 (chosen such that the scalar potential is zero at z=0). The choice of vector potential is of course not unique but all results are gauge invariant. Substituting the potentials into the Hamiltonian (1) yields The Hamiltonian H commutes with p y , s z and also H z = p 2 z 2m e + e 2 k z 2 and therefore these operators share a common eigenfunction ψ: Note that H z is the Hamiltonian of a harmonic oscillator in the z-direction with angular frequency and energies given by With H given by (2) and the results (3) and (5), the eigenvalue equation H ψ = E ψ takes the form, where and The operator on the left hand side of (6) corresponds to that of a harmonic oscillator moving in the x-direction with angular frequency ω c (referred to as the cyclotron frequency) with center located at x = x 0 . Its eigenvalues are therefore given by The above energy is degenerate since states with different values of k y have the same energy. If the magnetic field is applied to a rectangular area of sides L x and L y with periodic boundary conditions in the y direction, then k y takes on values 2 π n y /L y where n y is a positive or negative integer. The value of x 0 must lie between −L x /2 and L x /2 so that − . The degeneracy D, which is the number of possible integral values of n y , is then given by the integer part of where Φ = B 0 A is the magnetic flux through the rectangular area and Φ 0 = h e is a fundamental unit of quantum flux.
Substituting (9) for the eigenvalue of equation (6), one obtains The magnetic moment of the electron is given by where δ = 0.0011 is a very small radiative correction. Expressing B 0 in terms of ω c via (7) we obtain Substituting (13) into (11) we obtain the exact expression for the energies: The first term on the right hand side has a near two-fold degeneracy due to the contribution of the spin. Setting n c = n (with n ≥ 1) and choosing the negative sign one obtains (n − δ/2) ω c whereas setting n c = n−1 and choosing the positive sign yields (n+δ/2) ω c . The two energies are basically equal as the ratio of the latter to the former is to first order given by 1 + δ/n where δ/n is less than one in a thousand. For all practical purposes, we can therefore neglect δ and consider the two-fold degeneracy to be exact 1 . This allows one to express the above energy in the convenient form E = n ω c + n z + 1 2 ω z ; n, n z = 0, 1, 2, 3, ...
where there is a two-fold degeneracy due to the spin for every n except n = 0 which is nondegenerate. The energy (15) splits neatly into two terms. The first term depends on the cyclotron frequency ω c and is due to the uniform magnetic field only. One recognizes this term as the Landau levels with spin included. The second term depends on the angular frequency ω z and is due to the linear electric field alone. There are no cross terms that depend on both the magnetic and electric fields.
An important point is that if ω c /ω z is a rational number, different values of n and n z can yield the same energy. The degeneracy in this case will be studied in the next subsection. For all other cases -where ω c /ω z is an irrational number -the total degeneracy is obtained by multiplying the degeneracy due to the spin with the factor D given by (10)

Degeneracy when ω c ω z is a rational number
We can express the energy (15) in the following form We now show that different values of n and n z can yield the same energy only if the ratio ω c /ω z is a rational number. Let Q = n ωc ωz + n z . Let n change by an integer i and n z by an integer j. Then the new value of Q, which we label Q , is Q = (n + i) ωc ωz + (n z + j) = Q + i ωc ωz + j. The energy remains the same if Q = Q which implies that ωc ωz = − j i . This means that there can be a degeneracy due to different pairs (n, n z ) only if ωc ωz is a positive rational number with j and i having opposite signs (i.e. if n increases then n z decreases and vice versa).
Before we proceed with analyzing the general case where ω c /ω z is any rational number, it is instructive to look at the special case when the two angular frequencies are equal: ω c = ω z = ω. This occurs when the magnetic field strength B 0 reaches the value of m e k/e. Then (17) reduces to E ωc=ωz = n + n z + 1 2 where N = n + n z = 0, 1, 2, 3, .... There are many possible pairs (n, n z ) that yield a given N and hence a given energy. The different pairs are (N, 0), (N −1, 1), (N −2, 2), ..., (0, N ). There are N pairs with n = 0, each of which are two-fold degenerate due to the spin and one pair with n = 0 which is non-degenerate. The degeneracy due to the possible (n, n z ) pairs and the spin is therefore given by 2N + 1. The total degeneracy is obtained by multiplying this value with the factor D given by (10): When the two frequencies are equal, we see that the degeneracy depends on N and hence on the energy; higher energy levels are more degenerate. This is in sharp contrast to the usual Landau levels, where the degeneracy is constant for all energy levels 2 .
We now consider the general case where ωc ωz is a fraction I J where I and J are positive integers and J is the smallest possible denominator. Let P be defined as P = n I J + n z ; n, n z = 0, 1, 2, 3, ...
so that the energy (17) is given by E = (P + 1 2 ) ω z . The degeneracy of the energy E is therefore same as the degeneracy of P . The goal is to find the degeneracy for a given pair (n, n z ).
For a given value of P , there is a maximum value that n z can attain, which we label n zmax . At that point, n is at its minimum possible value n min . Starting with n z = n zmax and n = n min , decreasing n z by I and increasing n by J yields the same value of P . This procedure can be repeated [ nz max I ] times at which point n z reaches n z min , its minimum value (here [x] denotes the greatest integer less than or equal to x). The number of different pairs (n, n z ) that yield the same value of P is then [ nz max I ] + 1. We would now like to express [ nz max I ] in terms of the quantum numbers n and n z . First note that n min < J. We can therefore write [ nz max I ] = [ nz max I ]+[ n min J ] since [ n min J ] = 0. The possible values of n and n z that yield the same value of P are n z = n zmax − I and n = n min + J where = 0, 1, 2, 3, ..., [ nz max I ]. Substituting this into the above, we obtain that n zmax If P is an integer, then n zmax = P and n min = 0. Of the [ nz max I ] + 1 possible pairs that yield the same value of P , one pair has n = 0 and this has a spin degeneracy of unity while the remaining [ nz max I ] pairs have n = 0 and have a spin degeneracy of 2. So the degeneracy due to both the spin and the possible (n, n z ) pairs is 2 [ nz max I ] + 1. We must also include the degeneracy factor D given by (10). The total degeneracy when P is an integer is then given by Note that when I = 1, the above degeneracy is the same as that of (19).
When P is not an integer, n min = 0. In that case, all the [ nz max I ] + 1 possible pairs have a spin degeneracy of 2. So the degeneracy due to both the spin and the possible (n, n z ) pairs is 2([ nz max I ] + 1). Including the degeneracy factor D we obtain We plot below the energy (17) (in units of ω z ) as a function of the ratio W = ω c /ω z (you can think of ω z as being held fixed while ω c changes). Points of intersection occur when W is rational and represent degeneracies. The red horizontal lines represent n = 0 (the first one occurs at n z = 0, the second one at n z = 1, etc.  Figure 1: Energy levels (in units of ω z ) as a function of the ratio W = ω c /ω z . Each line corresponds to an (n, n z ) pair. Lines intersect where W is rational and represent degeneracies. These are quite evident at ratios of unity and 2 but can also be seen at other rational numbers.

Conclusion
In this work we showed that adding a linear electric field to the usual Landau level scenario has consequences not only for the energy levels but most importantly for the degeneracy of those levels. When the ratio of the two frequencies W = ω c /ω z is rational, at particular energies, the degeneracy increases in contrast to the degeneracy of the Landau levels which are independent of the energy. We obtained analytical formulas for the degeneracy: equation (16) when W is irrational, equation (22) when P , given by (20), is an integer and equation (23) when P is not an integer. We also obtained an analytical formula for the energy, equation (15). This energy formula is valid in empty space and is determined entirely by the magnetic and electric field only.
In the future, we would like to study the behavior of electrons in a given material subject to a constant magnetic field and linear electric field. In most materials there would be modifications to the energy formula; you would have to multiply the Zeeman term by a g-factor and replace the other energy terms by an effective energy since the effective mass of the electrons depends on the band structure. But the important finding here would persist in materials: the degeneracy would be altered at specific ratios of W . There may also be reason to add a confining quadratic potential as is done in work on the Fock-Darwin system as this might model the boundaries of the material in a better fashion. If that is the case, the energy and degeneracy landscape would be quite rich because the system would now be characterized by three frequencies.