Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 27: Renormalization group approach
to the quantum rotor model: Finite temperature properties
In this lecture, we will develop the renormalization group equations that will
allow us to understand the properties of systems at small nonzero temperat
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 25: Renormalization group for the quantum rotor model: Details and the ow equation
In order to implement the renormalization group approach outlined in the
last lecture, it is useful to discuss some geometric preliminaries rst. Since
the calculati
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 24: Renormalization group approach
to the breakdown of spinwave theory: Basic
ideas and formalism
In the last couple of lectures, we have understood in a couple of very dierent
ways that the n eld in the O(3) quantum rotor model cannot develop lon
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 20: Eective eld theory for the BoseHubbard model
In the previous lecture, we have sketched the expected phase diagram of
the BoseHubbard model, and introduced a meaneld treatment that can in
principle calculate in a selfconsistent way the value
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 15: Manyparticle quantum mechanics: Algebraic preliminaries and wavefunction
description
In the last module, we have seen how a simple perturbative calculation of
the eective Hamiltonian for the oneband Hubbard model in the limit of
large U/t at
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 22: of Quantum rotor analysis of instability of Nel state to quantum and thermal
e
uctuations.
In the last lecture, we saw how one can expand about the fully ordered
state of the quantum rotor model to obtain a description of the lowenergy
spectr
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 18: Calculating with the Bosonic path
integral.
In the last lecture, we have developed the coherent state path integral representation for a system of many bosonic particles. As was clear from the
derivation, the path integral is dened as a formal
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 19: Phases and excitations of the BoseHubbard model
In this and the next few lectures, we will study a simple lattice model that
will serve as a concrete example within which we will explore the physics
of systems of bosons at very low temperature
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 17: Path integral description of manybody physics
In the last lecture, we have seen that the formalism of secondquantization
in which various terms in the Hamiltonian of manybody systems are described in terms of creation and annihilation operat
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 7: Path integral representation and
spin coherent states
In the rst ve lectures of this course, we have seen how Statistical Mechanics
provides a prescription for the calculation of macroscopic properties of manyparticle systems, and understood th
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 12: Expanding the Berry phase term
In this lecture, we will pick up the threads from where we left o last time,
and use the decomposition of N into an alternating Neel part n and a uniform
part L to rewrite the Berry phase term in a more useful an
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 29: Vortices and their interactions
In the last lecture on linearized superuid hydrodynamics, we saw that a
phase dierence once established remains static according to the equations
of motion of the linearized (harmonic) theory and drives a steady
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Selfassessment  Module 6
A standard duality transformation of the xy model interchanges the
roles of vortices with vorticity m and mfold anisotropies, i.e. terms of
the form
d2 r cos(m(r)
m
(1)
, and sends the stiness g to g 1 under this duality trans
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 28: Low energy rotor description of
the superuid state and transition to insulating behaviour
In Lectures 19 and 20, we have seen how to describe a system of bosons
in a lattice potential, with integer density per latticesite in terms of an
eecti
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 30: Statistical mechanics of vortices
consequences for superuid density
In the last lecture, we have seen that the contribution to the energy of the
xy model coming from vortices can be rewritten as the energy of a plasma
of charges interacting wi
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 23: The MerminWagner theorem
As we mentioned in the previous lecture, it is actually possible to rule out, in a
mathematically rigorous way, the existence of a longrange ordered state that
spontaneously breaks a continuous symmetry in low dimens
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 26: Renormalization group approach
to the quantum rotor model: Phases and phase
transitions
As we have already argued in the last but one lecture, to characterize the
g dependence of the large correlation length scale beyond which a d = 1
dimensio
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Self Assessment  Module 5
Consider two species of bosonic atoms trapped in a coldatom experiment and living in an optical lattice potential that forms a two dimensional square lattice. Label the two species by = 1, 2. The
Hamiltonian reads
(b bj + b bi
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Self Assessment  Module 2
1. Consider a modern atomtrap apparatus in which alkali atoms are
trapped in a harmonic potential well created by laser beams. Assume
that the total number of atoms is xed, equal to N . Assume that this
gas of N atoms has reach
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 10: Introduction to quantum antiferromagnets
In the last two lectures, we have developed a nice path integral representation
of the d dimensional quantum Heisenberg antiferromagnet, with Hamiltonian
S(ri ) S(rj )
H = J
(1)
ij
where ij denote neare
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 21: Spin wave theory for quantum rotor model
In the last three lectures, we have introduced a simple model for a system
of interacting bosons in a lattice potential, and developed caricatures for the
superuid and insulating states of this systems,
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Self Assessment  Module 4
Consider a system of fermionic atoms F that can be of two types which
we label as and (you may think of these as two possible polarizations
of an internal spin degree of freedom, although in real life examples it
is a bit more
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Self Assessment  Module 3
Consider a particle in a onedimensional harmonic potential with Hamiltonian
p2
1
H=
+ m2 x2
(1)
2m 2
Compute the linear response of the particle coordinate x due to an
oscillating electric eld that couples to the particle via
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 16: Manybody physics in secondquantized
language
In the last lecture, we went over two sets of algebraic preliminaries: One having to do with the abstract algebra of creation and annihilation operators for
fermions and bosons, with dening relati
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 14: Probes of quantum antiferromagnetism
In this lecture, our main goal is to understand the physics behind some
standard probes of antiferromagnetic Mott insulators. It is important to
discuss this since, insulators, unlike good or bad metals or
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
Lecture 2: Review and preview
Let us begin with a review of undergraduate statistical physics, looking back
at what you have already learnt in your rst course on statistical physics.
Both Newtonian mechanics and its quantum counterpart, the HeisenbergSchr
Korea Advanced Institute of Science and Technology
OPERATION RESEARCH 2
INDUSTRIAL IE332

Spring 2014
MODULE SIX
EMERGING ASPECTS OF ORGANIZATIONAL BEHAVIOUR
Chapter Twenty Four Organizational behaviour across cultures (One Hour)
Reference:
Organizational behaviourHuman behaviour at work by John W Newstrom, 12th edition,
McGrawHill
Prepared By
Dr Susmit