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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

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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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