TOPOLOGICAL QUANTUM NUMBERS.docx

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TOPOLOGICAL QUANTUM NUMBERS

IN

CONDENSED MATTER PHYSICS

Course # PH421

B.Tech Project Report

V.NITHINRoll no:08012119

Project Instructor: GIRISH S SETLUR


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ACKNOWLEDGMENT

I take this opportunity to thank those who have helped me in one or the other way

during my project tenure. First of all, I express my sincere gratitude to my projectsupervisor Dr. Girish S. Setlur , for his sincere guidance, obliging suggestions and

interesting and valuable ideas, which helped me to carry out my project work infruitful manner. .

I also owe my thanks and affections to my parents and my friends, from whom I

got mental support and affection every time in maintaining the work properly. Atlast I thank the Department of Physics, Indian Institute of Technology Guwahati

for providing me with the facilities necessary to carry out my work and also all thefaculties, Department of Physics, Indian Institute of Technology, Guwahati.


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CERTIFICATE

This is to declare that the project entitled Topological Quantum Numbers inCondensed Matter Physics . was carried out by me under the guidance of Assosc.

Prof. Girish S Setlur, as part of B.Tech (Engineering Physics) programme of the

Department of Physics, IIT Guwahati.

Signature:

Date:16/04/12

Name of student: V Nithin

Roll No.: 08012119


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CONTENTS Page No:

Description of Topics

1) Abstract 5

2) Introduction 6

3)Two-Fluid Model 7

4) Relationship Between Superfluid Velocity and BEC wavefunction

A) Gross-Pitaevskii equation 10

B) Quantized Vortices 13

C) Vortex lines 16

D) Energy of the quantized vertex 17

5) Results and Discussion 19

6) Conclusion 20

7) References 21


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ABSTRACT :

Ever since the discovery of Integer Quantum Hall Effect, topologicalquantum numbers namely quantum numbers that are tied to properties thatare global in nature have become the topic of intense research. Topologicalproperties are distinct from most other physical properties that are local,such as density, conductivity, etc. Topological properties relate to thesystem as a whole and encode information such as the number of `holes' insome state space or the number of times a curve winds before it closes inon itself. These are likely to be of use in building a `topological quantumcomputer' The Project involves an in depth study of these issues and anexploration of potential practical applications.


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1 Introduction:

The liquid state of Helium(atomic number 4) exists in two phases: the high temperature phase

called Helium I and low temperature phase called Helium II. These two phases are marked by

lambda transition which occurs at the critical temperature Tc =2.172 at saturated vapor

pressure. This marks the onset of Bose-Einstein condensation and quantum order. Helium I is

called classical fluid and obeys classical hydrodynamics, whereas Helium II is called superfluid

and deviates from the former.

Onsager and Feynman proposed the quantization of superfluid circulation and presence of free

quantized vortices. Experimental confirmation came from the work of Hall and Vinen with the

discovery of mutual friction in rotating helium and with the direct observation in a macroscopic

experiment of the quantization of circulation. This work highlighted the importance of Londons

:quantum mechanism on a macroscopic scale, and of importance of Bose condensation in super

fluidity.

superfluid is a state of matter in which the matter behaves like a fluid without viscosity and

with extremely high thermal conductivity. The substance, which appears to be a normal liquid,

will flow without friction past any surface, which allows it to continue to circulate over

obstructions and through pores in containers which hold it, subject only to its own inertia.


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2) Two Fluid Model:

Helium II is considered as being composed of a penetrating mixture of two fluids: normal fluid

and superfluid. Each fluid component has its own density( ) and velocity field( ). Thesuperfluid component is irrotational and it does not have entropy or viscosity. The normal fluid

component is a gas of thermal excitations, depending on their wavelength they are called

phonons and rotons.The normal fluid carries the entire entropy and viscosity of Helium II. The

relative proportions of the two fluids depends on absolute temperature T. At T = 0,Helium II iscompletely superfluid[1]:

On increasing the temperature the superfluidity decreases and at ,it entirely vanishes:

The superfluid has potential flow and it can flow through boundaries without getting attached

to them. Super fluidity is closely related to BEC.

The ground state of a system of interacting bosons is node less-real and positive.

A) Criteria for Super fluidity:

Landaus criteria is based on Galilean transformation of energy and momentum. A quantum

system cannot change its energy continuously ,therefore the system creates an excitation toabsorb the dissipated energy.

E and P are the energy and momentum of the fluid in reference frame L. In order to express

these quantities of same fluid in other reference frame L, with a relative velocity -v w.r.t L

frame.

P = P + Mv;


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E =

| |

M, is the total mas of the fluid

Considering the fluid at zero temperature, that is all particles are in ground state and flows

through capillary with velocity v. Assuming the fluid is viscous, the motion will produce

dissipation of energy. Disiipation can occur through creation of elementary excitations,

Bogoliubov quasi-particle for the case of an interacting Bose gas .In the reference frame L the

fluid is at rest. Let p be the momentum of the single elementary excitations in the fluid. The

total energy of the fluid in L frame is + ,where

is the ground state energy and

is the elementary excitation energy.

In L frame, fluid is moving with velocity v, but capillary is at rest.

In the L frame the excitation energy is

Spontaneous creation of elementary excitations can only occur if the process is energetically

favorable.

i.e

Implies there is dissipation of energy.

i.e inequality is satisfied if ||

and when the momentum p of the elementary excitation is opposite to fluid velocity v,

The critical value of the fluid velocity,


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If ,no elementary excitations will be spontaneously formed. This is Landaus criteria forsuper fluidity.(At absolute zero).

For temperature T>0,thermally created excitations will be present,which on movement have

friction with the container walls on interaction. For flows with ,there is apart of fluid thatmoves without dissipation. The total mass current density of the fluid as the sum of two parts.

Index n indicates the excitations.


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3) Relationship Between Superfluid Velocity and BEC wavefunction (order

parameter):A) Gross-Pitaevskii equation:

Consider the second quantized many body Hamiltonian:

]

(1)

(r) is the trapping potential. is the interaction potential.where Hamiltonian is expressed in terms of Boson Field Operators and .According to Bose Einstein commutation relations

[ ] = [ [ For short-range interaction, can be approximated as: (2)

Where is a coupling constant, which is characterized by a (s- wave scattering length)In cold dilute gases, only low energy two particle collisions are significant and the collisions are

independent of the detail of two-body potential.

The Heisenberg equation of motion of field operator satisifies,

] (3)


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Substituting ,

| |] (4)

Where

is the macroscopic wavefunction.

The above equation is called Gross Pitaevskii(GP) equation.

This equation is valid when the s-wave scattering length is much smaller than the average

distance between atoms, and the number of atoms in the condensate is much larger than unity.

According to BEC,the local density of particles is related to the square of the amplitude.

| | (5)Therefore the total number of particles is given by,

| |The particle current density[2] is

(6)Multiplying the equation (4) by and subtracting the complex conjugate of the resultingequation, then using j(r,t) from above,

(7)Thus from the above equation we can see the conservation of the total particle number N.

By expressing the condensate wave function in amplitude and phase we get, (8)Putting back and using (6) and (7) equations, The particle density now can be written as

(9)Also we know Thus, the superfluid velocity of the condensate particle can be written as:

` (10)


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is related to the gradient of the phase S of the order parameter.The phase of the orderparameter acts as velocity potential and is called velocity field.By inserting equation (8) into (4),equation of S can be written as:

(11)

Quantum pressure: Because of the term containing gradient of particle density, Heisenberg

uncertainty relationship between particle number and phase can be seen. The states with

definite number of particles have completely uncertain phase.

The stationary solution of equation (4) has the form

(12)

Where = is the chemical potential.The total energy of the system E is

E = || || || (13)In order to get time-independent GP equation, inserting (12) in (4) :

| |] = 0 (14)

The solution of the above equation corresponding to lowest energy gives the order parameter

of the ground state.

The excited states are complex functions. Quantized vortex which is an excited state is

considered .


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B) Quantized Vortices:

In rigid systems, the tangential velocity corresponding to a rotation is given by

x rWhere w is the angular velocity, r is the displacement vector form the origin.

These systems have diffused vorticity,

X v =

X (w X r)

But, the super fluid velocity equation (10) satisfies

(15)

Cylinder vessel containing superfluid.


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S =0,for the flow to be incompressible.Thus,the superfluid is irrotational and deviates fromrigid rotator.

Consider a superfluid which is confined to a macroscopic cylinder of radius R and length L.

The solution of Gross-Pitaevskii equation for a rotation around the z-axis of the cylinderobtained using cylindrical co-ordinates (r,,z)[3] .

Due to the symmetry of the problem, the modulus depends only on radial part,

||

Substituting above amplitude and expanding Laplace operator in cyndrical coordinates,

employing separation of variables method. we get

|| (16)

For the order parameter to be single valued, the parameter k must be an integer.Using equations (10) and (16),the tangential velocity is given by,

*||

(17)

Here, the velocity decreases with increasing r, which is in contrast to rigid rotator. At large

distances from the z-axis, the tangential velocity approaches zero and thus irrotational.The loop of around a closed counter about z-axis is given by,

(18)

Thus the circulation is quantized in the units of h/m, which is independent of the radius of the

contour. For helium, the quantized unit is macroscopically large.It arises because Bosons are

indistinguishable, have a symmetrical single wavefunction. Quantisation of angular momentum


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and all particles must have the same angular momentum is the cause for this quantization. The

quantization of circulation is analogous to quantization of flux in superconductivity. The

topological excitations in super fluids appear in the form of vortex structures in which

circulation is quantized. The excitations decay only at the boundary of the system or during

collision. Hence they are topological in nature.

Substituting (16) into (14) gives the equation for the modulus of order parameter.

||

|| || || (19)

Let the solution for the above equation be of the form:

||

(20)

Where t =r/p and p=h/ .Then substituting (20) into (19),the function f(t) satisfies,

(

) (

) (21)

Since as t approaches infinity density must approach its unperturbed value n,

i.e |

|

implies f() = 1.

As t0,the function f(t) tends to zero as || ,that is superfluid density | | tends to zero on the axis of the vortex .The ground state solution ,corresponding to k =0 has uniform density f(t) =1.

The topological excitations in superfluids appear in the form of vortex structures in which

circulation is quantized. The excitations decay only at the boundary of the system or duringcollision. Hence they are topological in nature.


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C) Vortex lines:

Free quantized vortex line in the superfluid component is a quantum of circulation around a

tiny cylindrical hole in the helium(superfluid).Such a line has one quantum of circulation and

the size of the hole depends on the counterbalancing of Kinetic Energy of flow and the surface

energy of the hole.Vortex lines allows the superfluid component to rotate as if the Helium is

placed in a rotating vessel,otherwise due to equation (15) scuh rotations are not allowed. Any

rotation of fluid must be contained in the form of vertex lines featuring quantized circulation.

Vortex lines scatter the excitations in normal fluid,thus they are responsible for frictional force

between two fluids.

Vortex lines in uniformly rotating superfluid component.


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D) Energy of the quantized vertex:

Since the quantized vertex is an excited state,the energy of the excited state is the difference

between E(k and E(k=0) ground state .The total energy E is given by equation(13),expressing energy interms of function f(t).

[()

] (22)

Integrating the equation for ,the excitation energy is,

(23)

Where is the size of the vertex core.


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Comparing the tangential velocity field of rigid rotation and irrotational flow.

As seen from the figure the irrotaional velocity field diverges as r tends to zero.

The solutions of GP equation for vortices(k=1,solid line;k=2,dashed line) as a function of radial

component dependent t (t=r/p)


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4 ) Results and Discussion:

From the two fluid model ,the Superfluid acts as if it were a mixture of a normal component,

with all the properties associated with normal fluid, and a superfluid component. The superfluid

component has zero viscosity , zero entropy.

Landau criterion on Superfluid velocity states that if fluid velocity is less than critical velocity, no

spontaneous excitations are formed .Thus for superfluidity the flow velocity must exceed the

critical value.T he critical value depends on particle-particle interaction.

The macroscopic occupation of a single quantum state in the Bose-condensed Helium gives rise

to macroscopic quantum effects.

As superfluids, Bose-Einstein condensates support rotational flow only through quantized

vortices.

The velocity of the superfluid is proportional to gradient of phase of order parameter. The

superfluid is irrotational. This phase winds through an integer multiple of 2 around a vortex

line. The circulation of superfluid is quantized. The circulation around any closed loop in the

superfluid is zero, if the region enclosed is simply connected.

The tangential velocity of irrotational fluid decreases with the distance from the center,but for

rigid rotator it increases constantly with distance.

The energy of the exicted state in case of scylinder container ,is proportional to length of the

cylinder and to logarithm of radius


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5) Conclusion:

The superfluid phase of liquid Helium II shows two-fluid behaviour; a normal fluid coexisting

with a superfluid component. The superfluid component can exhibit frictionless flow at low

velocities and in narrow channels. Rotational motion in the superfluid component is severely

restricted by quantum effects, associated with the quantization of circulation.


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6) References:

[1] David Thouless Topological quantum numbers in non-relativistic physics World

Scientific Publishing (1998)

[2] B.H.Bransden,C.J.Joachain Quantum Mechanics Pearson Education.Ltd ,2000

[3] R. J. Donnelly, Quantized Vortices in Helium II(Cambridge University Press, 1991).[4] L D Landau. The theory of superfluidity of helium II J. Phys. USSR, 5:71, 1941.

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TOPOLOGICAL QUANTUM NUMBERS.docx