symmetry

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What is symmetry? Immunity (of aspects of a
system) to a possible change
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The natural language of Symmetry - Group Theory
We need a super mathematics in which the
operations are as unknown as the quantities they
operate on, and a super-mathematician who does
not know what he is doing when he performs these
operations. Such a super-mathematics is the
Theory of Groups. - Sir Arthur Stanley Eddington

• GROUP set of objects (denoted G) that can be
  combined by a binary operation (called group
  multiplication - denoted by ?)
• ELEMENTS the objects that form the group
  (generally denoted by g)
• GENERATORS Minimal set of elements that can be
  used to obtain (via group multiplication) all
  elements of the group
• RULES FOR GROUPS
• Must be closed under multiplication (?) - if a,b
  are in G then a?b is also in G
• Must contain identity (the do nothing element)
  - call it E
• Inverse of each element must also be part of
  group (g?g -1 E)
• Multiplication must be associative - a ? (b ? c)
  (a ? b) ? c not necessarily commutative

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Ex. Of continuous group (also Lie gp.) Group of
all Rotations in 2D space - SO(2) group
Det(U) 1
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Lie Groups

• Lie Group A group whose elements can be
  parameterized by a finite number of parameters
  i.e. continuous group where 1. If g(ai) ?
  g(bi) g(ci) then - ci is an analytical fn. of
  ai and bi . 2. The group manifold is
  differentiable. ( 1 and 2 are actually
  equivalent)
• Group Generators Because of above conditions,
  any element can be generated by a Taylor
  expansion and expressed as
• (where we have generalized for N parameters).
• Convention Call A1, A2 ,etc. As the generators
  (local behavior determined by these).

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Lie Algebras

• Commutation is def as A,B AB - BA
• If generators (A i) are closed under commutation,
  i.e. then they form a Lie Algebra.

Generators and physical reality

• Hermitian conjugate A? take transpose of
  matrix and complex conjugate of elements
• U ei?A ------ if U is unitary , A must be
  hermitian

U? U 1
A? A
Hermitian operators observables with real
eigenvalues in QM
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Symmetry restated in terms of Group Theory
State of a system ?? Dirac
notation Transformation U?? ?? Action on
state Linear Transformation U ( ?? ?? )
U?? U?? distributive Composition U1U2(
?? ) U1(U2 ?? ) U1 ?? Transformation
group If U1 , U2 , ... , Un obey the group
rules, they form a group (under
composition) Action on operator U? U -1
(symmetry transformation) Again, What is
Symmetry? Symmetry is the invariance of a system
under the action of a group U? U -1 ?
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Why use Symmetry in physics?

• 1. Conservation Laws (Noethers Theorem)
• 2. Dynamics of system
• Hamiltonian total energy operator
• Many-body problems know Hamiltonian, but full
  system too complex to solve
• Low energy modes All microscopic interactions
  not significant Collective modes more
  important
• Need effective Hamiltonian

For every continuous symmetry of the laws of
physics, there must exist a conservation law.
Use symmetry principles to constrain general form
of effective Hamiltonian strength parameters
usually fitted from experiment
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High TC Superconductivity

• The Cuprates (ex. Lanthanum Strontium doping)

CuO4 lattice

• BCS or New mechanism? - d-wave pairing with
  long-range order.

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The procedure - 1
1. Find relevant degrees of freedom for
system 2. Associate second-quantized operators
with them (i.e. Combinations of creation and
annihilation operators) 3. If these are closed
under commutation, they form a Lie Algebra which
is associated with a group symmetry group of
system. ? Subgroup A subset of the group that
satisfies the group requirements among
themselves G ? A . ? Direct product
subgroup chain G A1? A2 ? A3 ... if
(1) elements of different subgroups commute
and (2) g a1 a2 a3 ... (uniquely )
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The Procedure - 2
4. Identify the subgroups and subgroup chains
these define the dynamical symmetries of the
system. (next slide.) 5. Within each subgroup,
find products of generators that commute with
all generators these are Casimir operators -
Ci. Ci ,A 0 ? CiA ACi ? ACiA-1 Ci
6. Since we know that effective Hamiltonian
must (to some degree of approximation) also be
invariant use casimirs to construct
Hamiltonian 7. The most general Hamiltonian is a
linear combination of the Casimir invariants of
the subgroup chains - ? ? aiCi
where the coefficients are strength parameters
(experimental fit)
? Cis are invariant under the action of the
group !!
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Dynamical symmetries and Subgroup Chains
Hamiltonian Physical implications
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• Good experimental agreement with phase diagram.

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Extra Slides
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Casimirs and the SU(4) Hamiltonian
Casimir operators
Model Hamiltonian Effect of parameter (p)
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High TC Superconductivity - SU(4) lie algebra

• Physical intuition and experimental
  clues Mechanism D-wave pairing Ground
  statesAntiferromagnetic insulators
• So, relevant operators must create singlet and
  triplet d-wave pairs
• So, we form a (truncated) space collective
  subspace whose basis states are various
  combinations of such pairs -

• We then identify 16 operators that are physically
  relevant
• 16 operators U(4) group generators of SU(N)
  N2

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Noethers Theorem

• If ? is the Hamiltonian for a system and is
  invariant under the action of a group ? U? U -1
  ?
• Operating on the right with U, U? U -1 U ? U
• i.e. Commutator is zero ? U? - ? U 0 U , ?
  
• Quantum Mechanical equation of motion
• So, if , then U is a constant of the motion
• Continuous compact groups can be represented by
  Unitary matrices.
• U can be expressed as (i.e. a Taylor
  expansion)
• Since U is unitary, we can prove that A is
  Hermitian
• So, A corresponds to an observable and U
  constant ? A constant
• So, eigenvalues of A are constant Quantum
  numbers ? conserved

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Nature of U and A

• For any finite or (compact) infinite group, we
  can find Unitary matrices that represent the
  group elements
• U ei?A exp(i?A) (A - generator, ? -
  parameter)
• U unitary ? U? U 1 (U? - Hermitian
  conjugate)
• exp(-i?A?) exp(i?A) 1
• exp ( i?(A - A?) ) 1
• (A - A?) 0 ?
  A A?
• So, A is Hermitian and it therefore
  corresponds to an observable
• ex. A can be Px - the generator of 1D
  translations
• ex. A can be Lz - the generator of rotations
  around one axis

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Angular momentum theory
1. System is in state with angular momentum ??
state is invariant under 3D rotations of the
system. 2. So, system obeys lie algebra defined
by generators of rotation group su(2) algebra
SU(2) group simpler to use 3. Commutation
rule Lx,Ly i ? Lz , etc. 4. Maximally
commuting subset of generators only one
generator 5. Cartan subalgebra Lz Stepping
operators L Lx i Ly L- Lx - i Ly
Casimir operator C L2 Lx2 Ly2 Lz2
6. C commutes with all group elements CU UC
UCU-1 C C is invariant under the action of
the group