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Derivation of a Hamiltonian for formation of
particles in a rotating system subjected to a
homogeneous magnetic field.
Sergio Manzetti 1,2and Alexander Trounev 3
1. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596,
SE-75124 Uppsala, Sweden.
2. Fjordforsk A/S, Bygdavegen 155, 6894 Vangsnes, Norway.
3. A & E Trounev IT Consulting, Toronto, Canada.
December 17, 2018
1
1 Abstract
Bose-einstein condensates have received wide attention in the last decades given
their unique properties. In this study, we apply supersymmetry rules on an exist-
ing Hamiltonian and derive a new model which represents an inverse scattering
problem with similarities to the circuit and the harmonic oscillator equations.
The new Hamiltonian is analysed and its numerical solutions are presented. The
1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) ”Derivation of a Hamilto-
nian for formation of particles in a rotating system subjected to a homogeneous magnetic field”
In: Modeling of quantum vorticity and turbulence with applications to quantum computations
and the quantum Hall effect. Report no. 142020. Copyright Fjordforsk A/S Publications.
Vangsnes, Norway. www.fjordforsk.no
1
results suggest that the SUSY Hamiltonian describes the formation of vortices
in a homogeneous magnetic field in a rotating system.
2 Introduction
Hamiltonian operators are probably the most fundamental part in quantum
physics and form the basis for calculating the physical properties for elementary
particles, their energy and observables. Operators possess properties, such as
self-adjointness and unboundness, which allows them to give sensible physical
answers to quantized systems, where operators such as the Hamiltonian in the
anyon-model of Laughlin [1] are of particular appeal. Supersymmetry is also
an appreciable aspect of matter and energy, and by its models developed in the
last decades, a concise foundation for its application in various fields of physics
shows potential towards problems in the areas of conductance, magnetism and
gravity fluctuations [2, 3, 4, 5]. A possible application of supersymmetry is
also in quantum computing algorithms and can be based on solving computa-
tional chains of commands using elementary particles, their spin identity and/or
charge and a supersymmetric rule in interpreting their signal. Supersymmetry
has previously been investigated in relation to particles with emphasis on the
spin-population in terms of non-relativistic supersymmetric terms by Valenzuela
et al [6]. Brink et al. [7] have also provided new insight to the field of research
on supersymmetry and particles and reported a system based on supersymmetry
on a particular operator to include fermions and anyons where they obtained a
solution of the supersymmetric Calogero model and a supersymmetric extension
of the deformed Heisenberg algebra. Hlousek and Spector [8] also contributed to
this field and developed a supersymmetric model for anyons where the interplay
between supersymmetry and physical properties of anyons (such as rotations
and confinement) was analyzed. Hlousek and Spector constructed an own su-
persymmetric generalization of the Hopf term, which generates the anyon model
as supersolitons. This led to the proposition that the transition from unbroken
gauge symmetry to the Higgs phase is simultaneously a transition from anyon
2
to canonical spin and statistics. In the present paper one rather focuses on an
approach where we simplify an existing Hamiltonian [1] into a supersymmetric
counter-part which we study its operator properties, and from which we develop
a master equation using the Gross-Pitaevskii model [9, 10].
Supersymmetry
A supersymmetric state evolves from factorizing the Schr¨odinger equation into
a super-pair of the Hamiltonian. This can be easily represented by splitting
the kinetic term into two first-order derivative operators, A and A†. This well-
known procedure [11] gives the superpotential
V(x) = W2(x)−(~/√2m)W0(x),(1)
with its solution:
W(x) = −~/√2mψ0
0(x)
ψ0(x),(2)
where ψ0(x) is the original wavefunction for the non-supersymmetric Schr¨odinger
equation. The charges of the particles are also represented by supersymmetric
terms, using the operators:
Q=
0 0
A0
Q†=
0A†
0 0
,
which commute by elementary superalgebra [9]. The supercharges have an
advantage over regular charge-representation for elementary particles in classi-
cal physics in that they can be considered as operators which change bosonic
degrees of freedom into fermionic degrees of freedom and vice-versa [11]. In or-
der to develop a supersymmetric analogue of the Laughlin Hamiltonian, we first
revisit the Laughlin state of anyons [1], to then devise the supersymmetric sign
duality (see above) on the factorized components of the Laughlin Hamiltonian.
3
Supersymmetric representation of the Laughlin state
Laughlin [1] defined the Hamiltonian operator:
H=|~/i∇ − (e/c)~
A|2,(3)
where ~
A=1
2H0[xˆy−yˆx], is the symmetric gauge vector potential, where
H0is the magnetic field intensity. However, presenting this treatise in a first
quantized formalism, we use the regular form:
H= [~/i∇ − (e/c)~
A]2,(4)
where the Hamiltonian is simply the normal square of the covariant deriva-
tive. Supersymmetry rules introduced above are here adapted to form a super-
symmetric model for the states of the quantum hall particles. We do this by
looking at the Hamiltonian in eqn. (4), which is composed of the two first-order
differential operators:
H= [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
A].(5)
Applying SUSY [11] rules on the two factorized components in eqn. (5), we in-
vert the sign of the second first-order differential operator, and get the superpair
of the Laughlin Hamiltonian:
HSU SY = [~/i∇ − (e/c)~
A]×[−~/i∇ − (e/c)~
A].(6)
The SUSY counter-part of the Hamiltonian from (5) becomes:
HSU SY =T T ∗,(7)
with
T= [~/i∇ − (e/c)~
A],(8)
4
and
T∗= [−~/i∇ − (e/c)~
A],(9)
where Tand T∗are the superpair in the SUSY Hamiltonian and one an-
other’s complex conjugate.
We consider then Tand T∗, with γ= (e/c)~
A, in one dimension:
T=h
i
d
dx −γ·˜
I
T∗=−h
i
d
dx −γ·˜
I.
which commute by the relation:
[T T ∗]−[T∗T] = 0 (10)
making Tand T∗two commutating complex operators in Hilbert space H.
Both Tand T∗are unbounded operators given the condition:
||T x|| c||x||,(11)
Tand T∗are also non-linear given the γconstant, being a constant trans-
lation from the origin. Tand T∗are non-self-adjoint in Has the following
condition is not satisfied:
hT φ, ψi=hφ, T∗ψi,(12)
From this, it follows that:
HSU SY :D(HS U SY )−→ H,
and
D(HSU SY )⊂H,
5
where HSU SY is an unbounded nonlinear and non-self-adjoint operator com-
plex on Hilbert space H=L2[−∞,+∞]. However, HSUS Y is self-adjoint in its
domain, D(HSU SY ) on L2[a, b].
From supersymmetry theory in quantum mechanics [11] it follows that
H†Ψ = HΨ†=EΨ = EΨ†, therefore we can assume that:
HSU SY Ψ = EΨ,(13)
where E is the energy of the system, which generates the boundaries of
D(HSU SY ) on L2[0, L], where the interval of quantization (L) is contained in
the zero-point energy term E= n2~2π2
2mL2.
We are primarily interested in studying the system in eqn (13) at singularity,
which yield the form:
HSU SY Ψ = 0,(14)
and follow by developing an analytical solution to eqn. (13). In the one-
dimensional case, we represent (13) in the form
−γψ(x) + i~ψ0(x) = ψ1(x), Eψ (x) = −γψ1(x)−i~ψ0
1(x) (15)
This system of equations has periodic solutions at E < γ2and exponential
for E > γ2. Consequently, if we consider special solutions for E= 0, then they
will be periodic everywhere, where the vector potential describing the magnetic
field is nonzero. We shall consider the behavior of a large number of particles
in a homogeneous magnetic field in a rotating system. The vector potential in
this case has the form ~
A= [ ~
B~r]/2. Magnetic field has one component Bzonly,
therefore A2=B2
z(x2+y2)/4. For normal charge particles these potential acts
like a trap. But in a case of SUSY Hamiltonian the potential change sign, as we
can see from eqn. (15), so we have opposite action and special kind of dynamics
lids to the vorticity formation.
6
3 Numerical solution to the SUSY-Hamiltonian
We use Gross-Pitaevskii model [9, 10] and Sobolev gradient method [12]. Put
ψ=ψ(x, y, t), then in nondimensional variables master equation has a form:
∂ψ
∂t =1
2∇2ψ+iΩ(x∂ψ
∂y −y∂ ψ
∂x )−β|ψ|2ψ+γ2(x2+y2)ψ(16)
Note that nonlinear Schr¨odinger equation (Gross-Pitaevskii model) follows
from (16) when replacing t→it, γ →iγ. Eqn. (16) describes evolution of the
wave function from some initial state ψ(x, y, 0) = ψ0(x, y) and up to stationary
state with E= 0. The energy approximates zero asymptotically with time,
with some relatively small positive or negative deviation (Fig 1). We can stop
calculation at E= 0, or at t=topt , where solution comes to stationary state
with E > 0 or E < 0, but close to E= 0.
.
Figure 1.The change in the energy of the system as a function of time (top)
and the wave function at different instants of time with the corresponding en-
ergy levels (bottom).
Since we investigate periodic solutions of equation (16), the initial data can
also be given in the form of a periodic function, where we select,
ψ(x, y, 0) = (cos(xπ/2L) cos(yπ/2L))n, n = 1,2,3,4,(17)
which is presented in fig. 2 where the simulation data is shown from equation
7
(16) with the initial conditions in eqn. (17) and zero boundary conditions. The
parameters of equation (16) were given the following: Ω = 20, β = 1000, γ2=
1/2, where Ω is the angular momentum and βand γ2are arbitrary constants.
As it follows from the data, shown in Fig. 2, vorticity is formed in all cases
with an equal number of vortices (twelve holes/vortices). Externally, the final
distribution of the amplitude of the wave function looks the same for n = 1,2,3,4,
although the initial data differ quite strongly. The vortices are formed around
a quantum void, depicted in blue in the figures.
.
Figure 2.The amplitude of the wave function at different instants of time for
the initial data (17) with n= 1,2,3,4 and Ω = 20, β = 1000, γ 2= 1/2 . Black:
holes/vortices; confined blue region: quantum void.
We also investigate the behaviour of the wavefunction at energy level n=4,
with angular momentum Ω=15 and 20 (Fig. 3).
8
.
Figure 3.The amplitude of the wave function at different instants of time for
the initial data (17) with n= 4 and Ω = 15,18,20,30; β= 1000, γ 2= 1/2 .
We note the essential difference between this problem and the analogous
problem for the Bose-Einstein condensate [13]. The vortices arise in a com-
pletely symmetric fashion during the decay of the symmetry of the periodic
boundary conditions. Also, we note that the value of Ω is an order of magni-
tude higher than in the analogous problem [13]. We found out what caused
such a big difference. For this, we took as the initial data the ground states
described in [13], namely
ψ(x, y, 0) = ψv(x, y) = aexp[−(x2+y2)/2], a = 1,1/√π
ψ(x, y, 0) = ψv1(x, y)=(x+iy)ψv(x, y).
(18)
The simulation data of the amplitude of this wave function with initial data
in the form of ψvwith a= 1 is hereby shown in figure 4. In this case, 4 vortices
arise even at Ω = 0.95, β = 100, while for Ω = 1.4 their number is 24. This
agrees with the results of [13], however there is also a difference in the geometry
of the vortex distribution. This can be seen in figure 5, which shows the 3D
9
distributions of the amplitude of the wave function corresponding to the data
in Fig. 4. Hence, figure 5 shows that individual vortices are represented as
holes. We furthermore show in fig. 6 shows the simulation data of the ampli-
tude of the wave function with initial data in the form of ψv1with a= 1 and
Ω=1.4, β = 100. Comparing the data shown in Fig. 4 and 6, we find a great
difference in the final states with equal values of the parameters, based on the
smoothness of the initial state. The difference of the two initial states in (16)
and (18) is therefore of paramount importance to study the SUSY Hamiltonian
further.
.
Figure 4.The amplitude of the wave function at different instants of time for
the initial data (18) in form ψvwith a= 1 and Ω = 0.95,1.1,1.2,1.4; β=
100, γ2= 1/2 .
10
.
Figure 5.3D distributions of the amplitude of the wave function at differ-
ent instants of time for the initial data (18) in form ψvwith a= 1 and
Ω=0.95,1.1,1.3,1.4; β= 100, γ 2= 1/2 .
.
Figure 6.2D and 3D distributions of the amplitude of the wave function at
different instants of time for the initial data (18) in form ψv1with a= 1 and
Ω=1.4; β= 100, γ 2= 1/2 .
In Fig. 7 shows 3D distribution of the amplitude of the wave function in
the final state in Fig. 6 in two angles, illustrating the depth of penetration of
holes. 2D and 3D distributions of the amplitude square of the wave function at
11
different instants of time for the initial data (18) in form ψvwith a= 1/√πand
Ω=1.4; β= 100, γ 2= 1/2 are shown in Fig. 8.
Thus, we have shown that the supersymmetric Hamiltonian has a wide field
of application including the simulation of vorticity in quantum systems in a
magnetic field and possibly for quantum rotation. The results obtained by
us convincingly testify to the existence of periodic solutions of equation (16)
containing vortices in a wide range of parameters and under different initial
conditions.
The numerical model for equation (16) is much simpler than in the case of a
similar problem formulated for the Bose-Einstein condensate. This is explained
by the fact that equation (16) has steady-state solutions that correspond to
states with zero energy for the initial model. To find these solutions, we used
standard package for PDE solving in the system Mathematica version 10.x-11.x.
.
Figure 7.3D distributions of the amplitude of the wave function at t= 2 for
the initial data (18) in form ψv1with a= 1 and Ω = 1.4; β= 100, γ 2= 1/2 .
12
.
Figure 8.2D and 3D distributions of the amplitude square of the wave function
at different instants of time for the initial data (18) in form ψvwith a= 1/√π
and Ω = 1.1,1.4; β= 100, γ 2= 1/2 .
Finally, we note that the SUSY Hamiltonian does not differ in its application
from the well-known Hamiltonian (4) defined by Laughlin [1]. Equation (16)
can be regarded as a natural extension of model [1] for the numerical study of
the dynamics of quasiparticles in magnetic fields.
4 Discussion
The SUSY Hamiltonian (6) describes a system composed of a damping part
and an angular motion part combined together, which is similar to the damped
harmonic oscillator [14]. The Hamiltonian can, in its given form (eqn. (6)
therefore describe a scattering phenomenon [15] where the oscillating particle
13
has a small imaginary part in its mass and experiences decay (with negative
value of γ)=(m)6=0, releasing energy, or the particle experiences gain and ab-
sorbs energy (with a positive value of γ). This system is also a retrospective
inverse scattering problem [15], which can describe physical parameters (i.e.
electromagnetic field intensity or conductance) from observations of the evolu-
tion of the wavefunction. In its form given in eqn. (6), it has a non-unitary
time-evolution, which requires the generation of a master equation in order to
explain a quantum-event with a fully developed time-description [16, 17]. We
have generated a master eqn. in eqn (16) which yields a supersymmetric model
and thus drives the general state in eqn. (6) into a pure quantum state [18].
This gives a description of a physical properties of a new equilibrium system
possibly occurring in quantum Halls and in superconductors and/or alkali met-
als. In its general form however (eqn. (6)), the SUSY Hamiltonian can be made
more general given its retrospective time-properties [16, 19], and because the
physical law described by it is currently unknown, it can be applied also to
other phenomena, for instance non-equilibrium phenomena in quantum electro-
dynamics and quantum field theory.
5 Conclusions
A supersymmetric version of the ”Laughlin Hamiltonian” for bose-einstein con-
densates has here been developed and its properties discussed. Given the emerg-
ing interest in anyons in the last decades and the putative application in quan-
tum computing and quantum algorithms [20, 21, 22, 23, 24, 25], the approach
of considering the supersymmetric variant of this Hamiltonian is expected to
be a resource to the quantum physics community. The SUSY Hamiltonian de-
scribes a problem which is similar to the harmonic oscillator equations and the
circuit equations and can, therefore, be of importance to interpret particles in
one-dimensional systems in the future. Further work will evolve on studying the
evolution of the SUSY Hamiltonian model and the difference between smooth
14
and less smooth initial states, as well as the evolution of angular momenta under
and formation of vortices.
15
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6 Acknowledgements and correspondence
A sincere gratitude is owed to A. Prof. Michael Rogers at the Oxford College of
Emory University for critical discussion on the manuscript. The author would
also like to thank Professor Ronald Friedman at Indiana University Purdue
University Fort Wayne and Prof. Thors Hans Hansson at Stockholm University
for useful comments, and Prof Anna Fino from the Mathematics Department,
University of Torino, for inspiring to write this paper.
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