Download Lecture I: Collective Excitations: From Particles to Fields Free Scalar

Lecture I
1
Lecture I: Collective Excitations: From Particles to Fields
Free Scalar Field Theory: Phonons
The aim of this course is to develop the machinery to explore the properties of quantum
systems with very large or infinite numbers of degrees of freedom. To represent such systems
it is convenient to abandon the language of individual elementary particles and speak about
quantum fields. In this lecture, we will consider the simplest physical example of a free or
non-interacting many-particle theory which will exemplify the language of classical and quantum
fields. Our starting point is a toy model of a mechanical system describing a classical chain of
atoms coupled by springs.
. Discrete elastic chain
RI-1
φI
(I-1)a
Ia
M
ks
(I+1)a
Equilibrium position x̄n ≡ na; natural length a; spring constant ks
Goal: to construct and quantise a classical field theory
for the collective (longitundinal) vibrational modes of the chain
. Discrete Classical Lagrangian:
L=T −V =
N X
n=1
p.e. in spring
k.e.
}|
{
z }| { z
m 2
ks
ẋn − (xn+1 − xn − a)2
2
2
assume periodic boundary conditions (p.b.c.) xN +1 = N a + x1 (and set ẋn ≡ ∂t xn )
Using displacement from equilibrium φn = xn − x̄n
L=
N X
m
n=1
2
φ̇2n
ks
2
−
(φn+1 − φn ) ,
2
p.b.c : φN +1 ≡ φ1
In principle, one can obtain exact solution of discrete equation of motion — see PS I
However, typically, one is not concerned with behaviour on ‘atomic’ scales:
1. for such purposes, modelling is too primitive! viz. anharmonic contributions
2. such properties are in any case ‘non-universal’
Aim here is to describe low-energy collective behaviour — generic, i.e. universal
In this case, it is often permissible to neglect the discreteness of the microscopic entities
of the system and to describe it in terms of effective continuum degrees of freedom.
Lecture Notes
October 2006
Lecture I
2
. Continuum Lagrangian
Describe φn as a smooth function φ(x) of a continuous variable x;
makes sense if φn+1 − φn a (i.e. gradients small)
Z
X
1 L=N a
1/2
3/2
φn → a φ(x)
dx
,
φn+1 − φn → a ∂x φ(x)
−→
,
a 0
x=na
x=na
n
N.B. [φ(x)] = L1/2
φ(x)
φI
RI
Lagrangian functional
{
Z L }|
L[φ] =
dx L(φ, ∂x φ, φ̇),
Lagrangian density
}|
{
m 2 ks a2
L(φ, ∂x φ, φ̇) = φ̇ −
(∂x φ)2
2
2
z
z
0
. Classical action
Z
S[φ] =
Z
dt L[φ] =
Z
dt
0
L
dx L(φ, ∂x φ, φ̇)
• N -point particle degrees of freedom 7→ continuous classical field φ(x)
• Dynamics of φ(x) specified by functionals L[φ] and S[φ]
What are the corresponding equations of motion...?
——————————————–
. Hamilton’s Extremal Principle: (Revision)
Z
Suppose classical point particle x(t) described by action S[x] =
dt L(x, ẋ)
Configurations x(t) that are realised are those that extremise the action
i.e. for any smooth function η(t), the “variation”,
δS[x] ≡ lim→0 1 (S[x + η] − S[x]) = 0 is stationary
; Euler-Lagrange equations of motion
Z t
Z t
S[x + η] =
dt L(x + η, ẋ + η̇) =
dt (L(x, ẋ) + η∂x L + η̇∂ẋ L) + O(2 )
0
Z
δS[x] =
dt (η∂x L + η̇∂ẋ L)
0
by parts
=
Z
=0
z
}|
{
d
dt ∂x L − (∂ẋ L) η = 0
dt
Note: boundary term, η∂ẋ L|t0 vanishes by construction
Lecture Notes
October 2006
Lecture I
3
"#(x,t)
!
!(x,t)
x
t
. Generalisation to continuum field x 7→ φ(x)?
Apply same extremal principle: φ(x, t) 7→ φ(x, t) + η(x, t)
with both φ and η periodic in x, i.e. φ(x + L) = φ(x)
Z t Z L S[φ + η] = S[φ] + dt
dx mφ̇η̇ − ks a2 ∂x φ∂x η + O(2 ).
0
Integrating by parts
0
boundary terms vanish by construction: η φ̇|t0 = 0 = η∂x φ|L
0
Z t Z L dt
dx mφ̈ − ks a2 ∂x2 φ η = 0
δS = −
0
0
Since η(x, t) is an arbitrary smooth function, m∂t2 − ks a2 ∂x2 φ = 0,
i.e. φ(x, t) obeys classical wave equation
General solutions of the
p form: φ+ (x + vt) + φ− (x − vt)
where v = a ks /m is sound wave velocity and φ± are arbitrary smooth functions
x=-νt
φ–
φ+
x=νt
. Comments
• Low-energy collective excitations – phonons – are lattice vibrations
propagating as sound waves at constant velocity v
• Trivial behaviour of model is consequence of simplistic definition:
Lagrangian is quadratic in fields 7→ linear equation of motion
Higher order gradients in expansion (i.e. (∂ 2 φ)2 ) 7→ dispersion
Higher order terms in potential (i.e. interactions) 7→ dissipation
• L is said to be a ‘free (i.e. non-interacting) scalar (i.e. one-component) field theory’
• In higher dimensions, field has vector components 7→ transverse and longintudinal modes
Variational principle is example of functional analysis
– useful (but not essential method for this course) – see lecture notes
Lecture Notes
October 2006
Lecture II
4
Lecture II: Collective Excitations: From Particles to Fields
Quantising the Classical Field
Having established that the low energy properties of the atomic chain are represented by a
free scalar classical field theory, we now turn to the formulation of the quantum system.
. Canonical Quantisation procedure
Recall point particle mechanics:
1. Define canonical momentum, p = ∂ẋ L
2. Construct Hamiltonian, H = pẋ − L(p, x)
3. Promote position and momentum to operators with canonical commutation relations
x 7→ x̂,
p 7→ p̂,
[p̂, x̂] = −i~,
H 7→ Ĥ
Natural generalisation to continuous field:
1. Canonical momentum, π(x) ≡
∂L
, i.e. applied to chain, π = ∂φ̇ (mφ̇2 /2) = mφ̇
∂ φ̇(x)
2. Classical Hamiltonian
Hamiltonian density H(φ, π)
zh
}|
i{
H[φ, π] ≡ dx π φ̇ − L(∂x φ, φ̇)
,
Z
1 2 ks a2
π +
(∂x φ)2
2m
2
H(φ, π) =
i.e.
3. Canonical Quantisation
(a) promote φ(x) and π(x) to operators: φ 7→ φ̂, π 7→ π̂
(b) generalise commutation relations, [π̂(x), φ̂(x0 )] = −i~δ(x − x0 )
N.B. [δ(x − x0 )] = [Length]−1 (Ex.)
Operator-valued functions φ̂ and π̂ referred to as quantum fields
Ĥ represents a quantum field theoretical formulation of elastic chain, but not yet a solution.
As with any function, φ̂(x) and π̂(x) can be expressed as Fourier expansion:
P
k
1 X ±ikx φ̂k
φ̂(x)
= 1/2
e
,
π̂(x)
L
π̂k
k
φ̂k ≡ 1
L1/2
π̂k
Z
L=N a
∓ikx
dx e
0
φ̂(x)
π̂(x)
runs over all discrete wavevectors k = 2πm/L, m ∈ Z, Ex: confirm [π̂k , φ̂k0 ] = −i~δkk0
Advice: Maintain strict conventions(!) — we will pass freely between real and Fourier space.
Lecture Notes
October 2006
Lecture II
5
Hermiticity: φ̂† (x) = φ̂(x), implies φ̂†k = φ̂−k (similarly π̂). Using
δk+k0 ,0
{
z
Z L }|
Z L
X
X
1
0
dx ei(k+k )x =
k 2 φ̂k φ̂−k
dx (∂ φ̂)2 =
(ik φ̂k )(ik 0 φ̂k0 )
L
0
0
0
k
k,k
mωk2 /2
z }| {
i
Xh 1
ks a2 2
π̂k π̂−k +
k φ̂k φ̂−k ,
Ĥ =
2m
2
ωk = v|k|,
v = a(ks /m)1/2
k
i.e. ‘modes k’ decoupled
Comments:
• Ĥ describes low-energy excitations of system (waves)
in terms of microscopic constituents (atoms)
• However, it would be more desirable to develop picture where
relevant excitations appear as fundamental units:
. Quantum Harmonic Oscillator (Revisited)
Ĥ =
Defining ladder operators
r
mω
i
â ≡
x̂ +
p̂ ,
2~
mω
†
â ≡
1
p̂2
+ mω 2 q̂ 2
2m 2
r
mω
2~
If we find state |0i s.t. â|0i = 0 ; Ĥ|0i =
i
x̂ −
p̂
mω
~ω
2 |0i,
;
1
Ĥ = ~ω ↠â +
2
i.e. |0i is g.s.
Using commutation relations [â, ↠] = 1, one may then show |ni ≡ â†n |0i
is eigenstate with eigenvalue ~ω(n + 21 )
!
Comments: Although single-particle, a-representation suggests many-particle interpretation
• |0i represents ‘vacuum’, i.e. state with no particles
• ↠|0i represents state with single particle of energy ~ω
• â†n |0i is n-body state, i.e. operator ↠creates particles
Lecture Notes
October 2006
Lecture II
6
• In ‘diagonal’ form Ĥ = ~ω(↠â + 21 ) simply counts particles (viz. ↠â|ni = n|ni)
and assigns an energy ~ω to each
. Returning to harmonic chain, consider
r
mωk
i
ak ≡
φ̂k +
π̂−k ,
2~
mωk
a†k
r
≡
mωk
2~
φ̂−k −
i
π̂k
mωk
N.B. By convention, drop hat from operators a
−i~δkk0
z }| {
i [π̂−k , φ̂−k0 ] −[φ̂k , π̂k0 ] = δkk0
with [ak , a†k0 ] =
2~
. And obtain (Ex. - PS I)
Ĥ =
X
k
1
†
~ωk ak ak +
2
Elementary collective excitations of quantum chain (phonons)
created/annihilated by operators a†k and ak
Spectrum of excitations is linear ωk = v|k| (cf. relativistic)
Comments:
• Low-energy excitations of discrete model involve slowly varying collective modes;
i.e. each mode involves many atoms;
• Low-energy (k → 0) 7→ long-wavelength excitations,
i.e. universal, insensitive to microscopic detail;
• Allows many different systems to be mapped onto a few classical field theories;
• Canonical quantisation procedure for point mechanics generalises to quantum field theory;
• Simplest model actions (such as the one considered here) are quadratic in fields
– known as free field theory;
• More generally, interactions ; non-linear equations of motion viz. interacting QFTs.
Lecture Notes
October 2006
Lecture II
7
. Other examples? † Quantum Electrodynamics
EM field — specified by 4-vector potential A(x) = (φ(x), A(x))
Z
Classical action :
S[A] =
d4 x L(A),
(c = 1)
1
L = − Fµν F µν
4
Fµν = ∂µ Aν − ∂ν Aµ — EM field tensor
Classical equation of motion:
Euler − Lagrange eqns.
}|
{
z
∂L
β
=0
∂Aα L − ∂
∂(∂ β Aα )
7→
Maxwell0 s eqns.
z }| {
∂α F αβ = 0
Quantisation of classical field theory identifies elementary excitations: photons
for more details, see handout, or go to QFT!
Lecture Notes
October 2006
Lecture III
8
Lecture III: Second Quantisation
We have seen how the elementary excitations of the quantum chain can be presented
in terms of new elementary quasi-particles by the ladder operator formalism. Can this
approach be generalised to accommodate other many-body systems? The answer is provided
by the method of second quantisation — an essential tool for the development of interacting
many-body field theories. The first part of this section is devoted largely to formalism – the
second part to applications aimed at developing fluency. Reference: see, e.g., Feynman’s
book on “Statistical Mechanics”
. Notations and Definitions
Starting with single-particle Schrodinger equation,
Ĥ|ψλ i = λ |ψλ i
how can one construct many-body wavefunction?
!λ
nλ
nλ
!4
!3
0
1
0
1
!2
!1
1
3
1
1
!0
5
1
bosons
fermions
Particle indistinguishability demands symmetrisation:
e.g. two-particle wavefunction for fermions i.e. particle 1 in state 1, particle 2...
state 1, particle 1
1 z }| {
ψF (x1 , x2 ) ≡ √ ( ψ1 (x1 ) ψ2 (x2 ) − ψ2 (x1 )ψ1 (x2 ))
2
1
In Dirac notation: |1, 2iF ≡ √ (|ψ1 i ⊗ |ψ2 i − |ψ2 i ⊗ |ψ1 i)
2
N.B. ⊗ denotes outer product of state vectors
. General normalised, symmetrised, N -particle wavefunction
of bosons (ζ = +1) or fermions (ζ = −1)
X
1
|λ1 , λ2 , . . . λN i ≡ p Q∞
ζ P |ψλP1 i ⊗ |ψλP2 i . . . ⊗ |ψλPN i
N ! λ=0 nλ ! P
• nλ — no. of particles in state λ; (for fermions, Pauli exclusion: nλ = 0, 1)
Lecture Notes
October 2006
Lecture III
•
P
P:
9
Summation over N ! permutations of {λ1 , . . . λN }
required by particle indistinguishability
• Parity P — no. of transpositions of two elements which brings permutation
(P1 , P2 , · · · PN ) back to ordered sequence (1, 2, · · · N )
In particular, for fermions, hx1 , . . . xN |λ1 , . . . λN i is Slater determinant, det ψi (xj )
Evidently, “first quantised” representation looks clumsy!
motivates alternative representation...
. Second quantisation
Define vacuum state: |Ωi, and set of field operators aλ and adjoints a†λ — no hats!
N
Y
1
aλ |Ωi = 0,
pQ∞
λ=0 nλ !
i=1
a†λi |Ωi = |λ1 , λ2 , . . . λN i
cf. ladder operators for phonons N.B. ambiguity of ordering?
Field operators fulfil commutation relations for bosons (fermions)
[aλ , a†µ ]−ζ = δλµ ,
[aλ , aµ ]−ζ = [a†λ , a†µ ]−ζ = 0
where [Â, B̂]−ζ ≡ ÂB̂ − ζ B̂ Â is the commutator (anti-commutator)
• Operator a†λ creates particle in state λ, and aλ annihilates it
• Commutation relations imply Pauli exclusion for fermions: a†λ a†λ = 0
• Any N -particle wavefunction can be generated by application of set of
N operators to a unique vacuum state
e.g.
|1, 2i = a†2 a†1 |Ωi
• Symmetry of wavefunction under particle interchange maintained by
commutation relations of field operators
e.g.
|1, 2i = a†2 a†1 |Ωi = ζa†1 a†2 |Ωi = ζ|2, 1i
(Providing one maintains a consistent ordering convention,
the nature of that convention doesn’t matter)
...
a
F
2
a+
a
F
1
a+
a
F
0
0
. Fock space: Defining FN to be linear span of all N -particle states |λ1 , . . . λN i,
Fock space F is defined as ‘direct sum’ ⊕∞
N =0 FN
operators a and a† connect different subspaces FN
Lecture Notes
October 2006
Lecture III
10
• General state |φi of the Fock space is linear combination of states
with any no. of particles
• Note that vacuum |Ωi (sometimes written as |0i) is distinct from zero!
. Change of basis:
a†λ̃ |Ωi
a†λ |Ωi
z}|{ X z}|{
P
|λi hλ|λ̃i
Using resolution of identity 1 ≡ λ |λihλ|, we have |λ̃i =
λ
i.e. a†λ̃ =
X
λ
hλ|λ̃ia†λ ,
and aλ̃ =
X
λ
hλ̃|λiaλ
e.g. Fourier representation: aλ ≡ ak , aλ̃ ≡ a(x)
√
eikx / L
Z L
X z }| {
1
a(x) =
hx|ki ak ,
ak = √
dx e−ikx a(x)
L 0
k
. Occupation number operator: n̂λ = a†λ aλ measures no. of particles in state λ
e.g. (bosons)
1 + a†λ aλ
z}|{
†
† n
†
aλ aλ (aλ ) |Ωi = aλ aλ a†λ (a†λ )n−1 |Ωi = (a†λ )n |Ωi + (a†λ )2 aλ (a†λ )n−1 |Ωi = · · · = n(a†λ )n |Ωi
Ex: check for fermions
So far we have developed an operator-based formulation of many-body states. However,
for this representation to be useful, we have to understand how the action of first quantised
operators on many-particle states can be formulated within the framework of the second
quantisation. To do so, it is natural to look for a formulation in the diagonal basis and
recall the action of the particle number operator. To begin, let us consider...
Second Quantised Representation of Operators
. One-body operators: i.e. operators which address only one particle at a time
N
X
p̂2n
2m
n=1
n=1
P
Suppose ô diagonal in orthonormal basis |λi, i.e. ô = ∞
oλ = hλ|ô|λi
λ=0 |λioλ hλ|,
e.g. k.e., |λi ≡ |pi and op = p2 /2m
!
N
X
0
0
hλ1 , · · · λN |Ô1 |λ1 , · · · λN i =
oλi hλ01 , · · · λ0N |λ1 , · · · λN i
Ô1 =
N
X
ôn ,
e.g. k.e. T̂ =
i=1
= hλ01 , · · · λ0N |
Lecture Notes
∞
X
λ=0
oλ n̂λ |λ1 , · · · λN i,
October 2006
Lecture III
11
Since this holds for any basis state, Ô1 =
∞
X
oλ n̂λ =
λ=0
∞
X
oλ a†λ aλ
λ=0
i.e. in diagonal representation, simply count number of particles in state λ
and multipy by corresponding eigenvalue of one-body operator
Transforming to general basis (recall aλ =
Ô1 =
P
ν hλ|νiaν )
X
X
hµ|λioλ hλ|νia†µ aν =
hµ|ô|νia†µ aν
µν
λµν
i.e. Ô1 scatters particle from state ν to µ with probability amplitude hµ|ô|νi
. Examples of one-body operators:
R
P
1. Total number operator: N̂ = dx a† (x)a(x) = k a†k ak
P
Sαβ = hα|Ŝ|βi = 21 σαβ
2. Electron spin operator: Ŝ = αβ Sαβ a†α aβ ,
where α =↑, ↓, and σ are Pauli spin matrices
1
1 0
0 1
z
σz =
7→ Ŝ = (n̂↑ − n̂↓ ),
σ+ = σx + iσy =
7→ Ŝ + = a†↑ a↓
0 −1
0
0
2
3. Free particle Hamiltonian
Z L
X p2
(−~2 ∂x2 )
Ex.
†
ap ap =
a(x)
dx a† (x)
2m
2m
0
p
Z
i.e.
L
Ĥ = T̂ + V̂ =
dx a† (x)
0
p̂2
+ V (x) a(x)
2m
where p̂ = −i~∂x
. Two-body operators: i.e. operators which address two-particles
E.g. symmetric pairwise interaction: V (x, x0 ) ≡ V (x0 , x) (such as Coulomb)
acting between two-particle states N.B. 1/2 for double counting
Z
Z
1
V̂ =
dx dx0 |x, x0 iV (x, x0 )hx, x0 |
2
When acting on N -body states,
V̂ |x1 , x2 , · · · xN i =
N
1 X
V (xn , xm )|x1 , x2 , · · · xN i
2
n6=m
In second quantised form, it is straightforward to show that (Ex.)
Z
Z
1
V̂ =
dx dx0 a† (x)a† (x0 )V (x, x0 )a(x0 )a(x)
2
Lecture Notes
October 2006
Lecture III
12
i.e. annihilation operators check for presence of particles at x and x’ – if they exist, asign the
potential energy and then recreate particles in correct order (viz. statistics)
Z
Z
1
dx dx0 V (x, x0 )n̂(x)n̂(x0 ) does not reproduce the two-body operator
N.B.
2
. In non-diagonal basis
Ô2 =
X
λλ0 µµ0
Oµ,µ0 ,λ,λ0 a†µ0 a†µ aλ aλ0 ,
e.g. in Fourier basis: a† (x) =
1
2
Z
1 X
L1/2
Oµ,µ0 ,λ,λ0 ≡ hµ, µ0 |Ô2 |λ, λ0 i
eikx a†k can show that (Ex.)
k
dxdx0 a† (x)a† (x0 )V (x − x0 )a(x0 )a(x) =
k’,σ’
k’+q,σ’
X
V (q)a†k1 a†k2 ak2 +q ak1 −q
k1 ,k2 ,q
V(q)
k,σ
k–q,σ
Feynman diagram:
Lecture Notes
October 2006
Lecture IV
13
Lecture IV: Applications of Second Quantisation
1. Phonons: oscillator states |ki form a Fock space:
for each mode k, arbitrary state of excitation can be created from vacuum
|ki = a†k |Ωi,
ak |Ωi = 0,
[ak , a†k0 ] = δkk0
P
Hamiltonian, Ĥ = k ~ωk (a†k ak + 1/2) is diagonal:
|k1 , k2 , ...i = a†k1 a†k2 · · · |Ωi is eigenstate of Ĥ with energy ~ωk1 + ~ωk2 + · · ·
2. Interacting Electron Gas
(i) Free-electron Hamiltonian
2
XZ
p̂
(0)
†
Ĥ =
+ V (x) cσ (x),
dx cσ (x)
2m
σ=↑,↓
[cσ (x), c†σ0 (x0 )] = δ(x − x0 )δσ,σ0
(ii) Interacting electron gas:
Z
Z
X
1
e2
(0)
Ĥ = Ĥ +
dx dx0
c†σ (x)c†σ0 (x0 )
c (x0 )cσ (x)
0 | σ0
2
|x
−
x
σσ 0
. Comments:
. Phonon Hamiltonian is example of ‘free field theory’:
involves field operators at only quadratic order...
. (whereas) electron Hamiltonian is typical of an interacting field theory
and is infinitely harder to analyze...
To familiarise ourselves with second quantisation, in the remainder of this and the next
lecture, we will explore several case studies: ‘Atomic limit’ of strongly interacting electron gas: electron crystallisation and Mott transition; Quantum magnetism; and weakly
interacting Bose gas
Tight-binding and the Mott transition
According to band picture of non-interacting electrons, a 1/2-filled band of states is metallic.
But strong Coulomb interaction of electrons can effect a transition to a crystalline phase in
which electrons condense into an insulating magnetic state – Mott transition. We will employ
the second quantisation to explore the basis of this phenomenon.
. ‘Atomic Limit’ of crystal
How do atomic orbitals broaden into band states? Show transparencies
Lecture Notes
October 2006
Lecture IV
14
E
s=1
!
s=0
!
"1
"1B
"1A
"0
"0A
"0B
E
1
0
x
V(x)
!/ a
E E
x
0
k
a
(n!1)a
na
(n+1)a
Weak overlap of tightly bound orbital states 7→ narrow band of Bloch states |ψks i,
specified by band index s, k ∈ [−π/a, π/a] in first Brillouin zone.
Bloch states can be used to define ‘Wannier basis’, cf. discrete Fourier decomposition
1
|ψns i ≡ √
N
B.Z.
X
−ikna
e
k∈[−π/a,π/a]
N
1 X ikna
|ψks i ≡ √
e
|ψns i,
N n=1
|ψks i,
k=
2π
m
Na
!n0 (x)
x
(n!1)a
na
(n+1)a
In ‘atomic limit’, Wannier states |ψns i mirror atomic orbital |si on site n
c†ns |Ωi Z
c† (x)|Ωi ψns (x)
z }| {
z}|{ z }| {
Field operators associated with Wannier basis: |ψns i = dx
|xi hx|ψns i
c†ns
and using completeness
≡
Z
dx ψns (x)c† (x)
∗
0
ns ψns (x )ψns (x)
P
c† (x) =
X
∗
ψns
(x)c†ns ,
= δ(x − x0 )
[cns , c†n0 s0 ]+ = δnn0 δss0
ns
Lecture Notes
October 2006
Lecture IV
15
i.e. (if we include spin index σ) operators c†nsσ /cnsσ create/annihilate electrons at site n in
band s with spin σ
. In atomic limit, bands are well-separated in energy.
If electron densities are low, we may focus on lowest band s = 0.
Transforming to Wannier basis,
2
X Z
p̂
Ĥ =
dx c†σ (x)
+ V (x) cσ (x)
2m
σ=↑,↓
Z
Z
X
1
+
dx dx0
c†σ (x)c†σ0 (x0 )V (x − x0 )cσ0 (x0 )cσ (x)
2
σσ 0
X
X
†
=
tmn cmσ cnσ +
Umnrs c†mσ c†nσ0 crσ0 csσ0
mnrs,σσ 0
mn,σ
2
p̂
+ V (x)]|ψn i = t∗nm and “interaction pawhere “hopping” matrix elements tmn = hψm |[ 2m
rameters”
Z
Z
1
e2
∗
Umnrs =
dx dx0 ψm
(x)ψn∗ (x0 )
ψr (x0 )ψs (x)
2
|x − x0 |
(For lowest band) representation is exact:
but, in atomic limit, matrix elements decay exponentially with lattice separation
(i) “Tight-binding” approximation:
(
m=n
−t
mn
neighbours ,
tmn =
0
otherwise
Ĥ (0) '
1
In discrete Fourier basis: c†nσ = √
N
−t
N X
nσ
X
nσ
B.Z.
X
c†nσ cnσ − t
X
c†n+1σ cnσ + h.c.
nσ
e−ikna c†kσ
k∈[−π/a,π/a]
δkk0
z
}|
{
X
X 1 X
0
†
e−i(k−k )na e−ika c†kσ ck0 σ + h.c. = −2t
cos(ka) c†kσ ckσ
cn+1σ cnσ + h.c. = −t
N n
0
kσ
kk σ
Ĥ (0) =
X
( − 2t cos ka)c†kσ ckσ
kσ
(k)
- /a
/a
k
B.Z.
As expected, as k → 0, spectrum becomes free electron-like:
k → − 2t + t(ka)2 + · · · (with m∗ = ~2 /2a2 t)
(ii) Interaction
Lecture Notes
October 2006
Lecture IV
16
• Focusing on lattice sites m 6= n:
P
1. Direct terms Umnnm ≡ Vmn — couple to density fluctuations:
m6=n Vmn n̂m n̂n
; potential for charge density wave instabilities
F ≡U
2. Exchange coupling Jmn
mnmn (Ex. – see handout)
X
X
1
F
Umnmn c†mσ c†nσ0 cmσ0 cnσ = −2
Jmn
Ŝm · Ŝn + n̂m n̂n ,
4
0
1
Ŝm = c†mα σαβ cmβ
2
m6=n
m6=n,σσ
i.e. weak ferromagnetic coupling (JF > 0) cf. Hund’s rule in atoms
spin alignment 7→ symmetric spin state and asymmetric spatial state lowers p.e.
F exponentially small in separation |m − n|a
But, in atomic limit, both Vmn and Jmn
• ‘On-site’ Coulomb or ‘Hubbard’ interaction
X
X
n̂n↑ n̂n↓ ,
Unnnn c†nσ c†nσ0 cnσ0 cnσ = U
nσσ 0
n
U ≡ 2Unnnn
. Minimal model for strong interaction: Mott-Hubbard Hamiltonian
Ĥ = −t
X
nσ
(c†n+1σ cnσ + h.c.) + U
X
n̂n↑ n̂n↓
n
...could have been guessed on phenomenological grounds
Transparencies on Mott-Insulators and the Magnetic State
Lecture Notes
October 2006
Lecture V
17
Lecture V: Quantum Magnetism and the Ferromagnetic Chain
N
1
. Spin S Quantum Heisenberg Magnet
Ĥ = −J
N
X
m=1
2
spin analogue of discrete harmonic chain
Ŝm · Ŝm+1
p.b.c. Ŝn+N = Ŝn
Sign of exchange constant J depends on material parameters c.f. previous lecture.
Our aim is to uncover ground states and nature of low-energy (collective) excitations.
. Classical ground states
• Ferromagnet: all spins aligned along a given (arbitrary) direction
⇒ manifold of continuous degeneracy (cf. crystal)
• Antiferromagnet: Néel state – (where possible) all neighbouring spins antiparallel
. Quantum ground states:
Ĥ = −J
Xh
z z
Ŝm
Ŝm+1 +
1 + −
(Ŝ Ŝ
+ Ŝ − Ŝ + )
2 z m m+1}| m m+1
{
x x
y y
Ŝm
Ŝm+1 + Ŝm
Ŝm+1
i
m
where Ŝ ± = Ŝ x ± iŜ y denotes spin raising/lowering operator
z
• Ferromagnet: as classical, e.g. |g.s.i = ⊗N
m=1 |Sm = Si
No spin dynamics in |g.s.i, i.e. no zero-point energy! (cf. phonons)
Manifold of degeneracy explored by action of total spin lowering operator
P
m
−
Ŝm
+ −
• Antiferromagnet: spin exchange interaction (viz. Ŝm
Ŝm+1 ) ; zero point fluctuations
which, depending on dimensionality, may or may not destroy ordered ground state
.Elementary excitations
Development of ordered state breaks continuous spin rotation symmetry ; low-energy
collective excitations (spin waves or magnons) – cf. phonons in a crystal
Lecture Notes
October 2006
Lecture V
18
Example of general principle known as Goldstone’s theorem: Breaking of a continuous
symmetry accompanied by appearance of gapless excitations
However, as with lattice vibrations, ‘general theory’ is nonlinear.
Fortunately, low-energy excitations described by free theory
To see this, for large spin S, it is helpful to switch to representation in which spin
deviations are parameterised as bosons:
|S z = Si
|S z = S − 1i
..
.
|n = 0i
|n = 1i
..
.
|S z = −Si
|n = 2Si
i.e. a maximum of n bosons per lattice site (“softcore” constraint)
For ferromagnet with spins oriented along z-axis,
the g.s. coincides with vacuum |g.s.i ≡ |Ωi, i.e. am |Ωi = 0
Mapping useful when spin wave excitation involves n 2S
. Mapping of operators:
(Setting ~ = 1) operators obey quantum spin algebra
;
[Ŝ α , Ŝ β ] = iαβγ Ŝ γ
[Ŝ + , Ŝ − ] = 2Ŝ z ,
[Ŝ z , Ŝ ± ] = ±Ŝ ±
cf. bosons: [a, a† ] = 1
†
z
According to mapping, Ŝ = S − a a;
therefore, to leading order in S 1 (spin-wave approximation),
Ŝ − ' (2S)1/2 a† ,
Ŝ + ' (2S)1/2 a
In fact, exact mapping provided by Holstein-Primakoff transformation (Ex.)
Ŝ − = a† 2S − a† a
1/2
,
Ŝ + = (Ŝ − )† ,
Ŝ z = S − a† a
. Applied to ferromagnetic Heisenberg spin S chain, ‘spin-wave’ approximation:
Ĥ = −J
N X
z z
Ŝm
Ŝm+1
m=1
1 + −
− +
+ (Ŝm Ŝm+1 + Ŝm Ŝm+1 )
2
= −J
o
Xn
S 2 − S(a†m am − a†m+1 am+1 ) + S(am a†m+1 + a†m am+1 ) + O(S 0 )
= −J
X
m
m
S 2 − 2Sa†m am + S a†m am+1 + h.c. + O(S 0 )
with p.b.c. Ŝm+N = Ŝm and am+N = am
Lecture Notes
October 2006
Lecture V
19
To leading order in S, Hamiltionian is bilinear in Bose operators;
diagonalised by discrete Fourier transform (Ex.)
√
eikn / N
B.Z.
X z }| {
1 X −ikn †
†
†
†
hn|ki an ,
an = √
ak =
e
ak ,
[ak , a†k0 ] = δkk0
N
n
k
Noting
δkk0
z
}|
{
X 1 X
X
X
−i(k−k0 )m −ika †
†
e
e
ak ak0 + h.c. =
(am am+1 + h.c.) =
cos k a†k ak
N
m
m
k
kk0
2
Ĥ = −JN S +
B.Z.
X
ωk a†k ak + O(S 0 ),
k
where ωk = 2JS(1 − cos k) = 4JS sin2 (k/2)
At low energy (k → 0), spin waves have free particle-like spectrum
Terms of higher order in S ; spin-wave interactions
N
1
2
. Spin S Quantum Heisenberg Antiferromagnet
Ĥ = J
N
X
m=1
Lecture Notes
Ŝm · Ŝm+1 ,
J > 0,
p.b.c. Ŝm+N = Ŝm
October 2006
Lecture V
20
Classical (Néel) ground state no longer an eigenstate;
nevertheless, it serves as useful reference for spin-wave expansion
In this case, useful to rotate spins on one sublattice, say B, through 180o about x,
i.e.
ŜBx 7−→ ŜBx ,
ŜBy 7−→ −ŜBy ,
ŜBz 7−→ −ŜBz
Transformation is said to be canonical in that it respects spin commutation relations
Under mapping ŜB± 7−→ ŜB∓
X
1 + +
− −
z z
Ĥ = −J
Ŝm Ŝm+1 − (Ŝm Ŝm+1 + Ŝm Ŝm+1 )
2
m
In rotated frame, classical ground state is ferromagnetic
− −
but Ŝm
Ŝm+1 ; zero-point fluctuations (ZPF)
z
−
= S − a†m am , Ŝm
' (2S)1/2 a†m , etc.
Applying spin wave approximation: Ŝm
i
Xh
a†m am + a†m+1 am+1 + am am+1 + a†m a†m+1 + O(S 0 )
Ĥ = −N JS 2 + JS
m
; processes that do not conserve particle number! (ZPF)
P
Turning to Fourier representation: am = N 11/2 k eikm ak , etc., and using
N
X
am am+1
m=1
δk+k0 ,0
z
}|
{
N
X 1 X
X
X
0
=
ei(k+k )m eik ak0 ak =
a−k ak eik ≡
a−k ak
N m=1
k
k
kk0
γk = cos k
z
}|
{
1 ik
(e + e−ik )
2
ak a†k − 1
z}|{
X †
[ak ak + a†k ak +γk (a−k ak + a†k a†−k )]
Ĥ = −N JS 2 + JS
k
= −N JS(S + 1) + JS
X
k
( a†k
a−k )
1
γk
γk
1
ak
a†−k
+ O(S 0 )
To diagonalise Ĥ, we must implement only operator transformations that preserve
canonical commutation relations:
ak
i.e. setting A =
(k index suppressed), we must implement transformations
a†−k
1
0
†
†
e = LA such that [A
ei , A
e ] = [Ai , A ] = gij , with g =
A 7→ A
j
j
0 −1
e = LA; we require
Consider operator transformation A 7→ A
ei , A
e† ] =! gij = Lim L∗nj [Am , A†n ] = (L g L† )ij
[A
j
i.e. L belongs to the group of Lorentz transformations. For real elements,
cosh θk sinh θk
L=
Bogoliubov transformations
sinh θk cosh θk
Lecture Notes
October 2006
Lecture VI
21
Lecture VI: Bogoliubov Theory
Inverse transformation
−1 e
A = L A,
ak
a†−k
=
cosh θk
− sinh θk
− sinh θk
cosh θk
αk
†
α−k
Applied to Hamiltonian,
1 γk
1
γ
k
†
†
−1
e L
e
A
A=A
L−1 A
γk 1
γk 1
cosh(2θ
)
−
γ
sinh(2θ
)
γ
cosh(2θ
)
−
sinh(2θ
)
k
k
k
k
k
k
†
e
e
A
=A
as “12”
as “11”
if tanh(2θk ) = γk , off-diagonal elements vanish.
1
1
=
2
(1 − γk2 )1/2
(1 − tanh (2θk ))1/2
diagonal elements given by (1 − γk2 )1/2 = | sin k|, i.e.
X
†
| sin k| αk† αk + α−k α−k
Ĥ = −N JS(S + 1) + JS
+ O(S 0 )
With cosh(2θk ) =
k
= −N JS(S + 1) + 2JS
X
k
| sin k|
αk† αk
1
+
+ O(S 0 )
2
Ground state defined by αk |g.si
and spectrum of excitations are linear (i.e. relativistic), (cf. phonons, photons, etc.)
Experiment?
. Do ZPF destroy long-range order?
Referring to sublattice magnetisation
1 X †
1 X
(−1)n Ŝnz |g.s.i = S − hg.s.|
a ak |g.s.i
N n
N k k
1 X
†
=S−
hg.s.|(− sinh θk α−k + cosh θk αk† )(cosh θk αk − sinh θk α−k
)|g.s.i
N k
Z
Z 1/a
1
1 X
dd k 1 2
2 −1/2
=S−
sinh θk = S −
(1 − γk )
−1 ∼
k d−1 dk
d
N k
(2π) 2
k
0
hg.s.|
Lecture Notes
October 2006
Lecture VI
22
i.e. quantum fluctuations destroy long range AFM order in 1d – spin liquid
. Frustration
On “bipartite” lattice, AF LRO survives ZPF in d > 1
For non-bipartite lattice (e.g. triangular), system is said to be frustrated
; spin liquid phase in higher dimension
Bogoliubov Theory of weakly interacting Bose gas
Although strong interactions can lead to the formation of unusual ground states of electron
system, the properties of the weakly interacting system mirror closely the trivial behaviour of the
non-interacting Fermi gas. By contrast, even in the weakly interacting system, the Bose gas has
the capacity to form a correlated phase known as a Bose-Einstein condensate. The aim of this
lecture is to explore the nature of the ground state and the character of the elementary excitation
spectrum in the condensed phase.
Consider N bosons confined to volume Ld . If non-interacting, at T = 0 all bosons
1
condensed in lowest energy state of single-particle system, viz. |g.s.i0 = √ (a†0 )N |Ωi
N!
How is g.s. and excitation spectrum influenced by weak (repulsive) interaction?
(0)
k
ĤI
z }| {
z Z
}|
{
X ~2 k2 †
1
d
d 0 †
† 0
0
0
Ĥ =
a ak +
d x d x a (x)a (x )V (x − x )a(x )a(x)
2m k
2
k
1 X
ĤI =
V (q) a†k0 a†k ak−q ak0 +q
d
2L
0
k,k ,q
If interaction is sufficiently weak, g.s. still condensed
with lowest single-particle state macroscopically occupied, i.e.
Nk=0
= O(1)
N
Therefore, since N̂0 = a†k=0 ak=0 = O(N ) 1 and a0 a†0 − a†0 a0 = 1,
√
a0 and a†0 can be approximated by C-number N0
Taking (for simplicity) V (q) = V const.,
i.e. a contact interaction V (x − x0 ) = V δ d (x − x0 ), expansion in N0 obtains
X †
V
V
1
1/2
† †
2
ĤI =
N +
N0
2ak ak +
a−k ak + ak a−k + O(N0 )
2
2Ld 0 Ld
k6=0
cf. quantum AF in spin-wave approximation
3/2
N.B. Momentum conservation eliminates terms at O(N0 )
. Physical interpretation of components:
Lecture Notes
October 2006
Lecture VI
23
• V a†k ak represents the ‘Hartree-Fock energy’ of excited particles interacting with condensate
N.B. Contact interaction disguises presence of direct and exchange contributions
• V (a−k ak + a†k a†−k ) represents creation or annihilation of particle pairs from condensate
Note that, in this approximation, total no. of particles is not conserved
X †
N
Finally, using N = N0 +
ak ak to trade N0 for N , and defining density, n = d
L
k6=0
Ĥ =
V nN X (0)
Vn
+
k + V n a†k ak +
a−k ak + a†k a†−k
2
2
k6=0
As with quantum AF, Ĥ diagonalised by Bogoluibov transformation:
Vn
ak
αk
cosh θk − sinh θk
=
, with tanh(2θk ) = (0)
†
†
−
sinh
θ
cosh
θ
a−k
α−k
k
k
k + V n
1X
V nN
−
Ĥ =
2
2
(0)
(k
k6=0
+ nV ) +
z
X (0)
k
+Vn
k6=0
k
}|
2
2
{
1/2 − (V n)
αk† αk
1
+
2
In particular, for |k| → 0, low-energy excitations have linear (relativistic)
dispersion, k =
(0)
[k (2V
n+
(0)
k )]1/2
' ~c|k| with ‘sound’ speed c =
Vn
m
1/2
.
At high energies (|k| > k0 = mc/~), spectrum becomes free particle-like.
. † Ground state wavefunction: defined by condition αk |g.s.i = 0
Since Bogoluibov transformation can be written as αk = Û ak Û −1 where (exercise)


X θk † †
Û = exp 
(a a − ak a−k )
2 k −k
k6=0
may infer true g.s. from non-interacting g.s. as |g.s.i = Û |g.s.i0
. Experiment? transparencies
When cooled to T ∼ 2K, liquid 4 He undergoes transition to Bose-Einstein condensed state
Neutron scattering can be used to infer spectrum of collective excitations
In Helium, steric interactions are strong and at higher energy scales
an important second branch of excitations known as rotons appear
A second example of BEC is presented by ultracold atomic gases:
By confining atoms to a magnetic trap, time of flight measurements
can be used to monitor momentum distribution of condensate
Moreover, the perturbation imposed by a laser due to the optical
dipole interaction provides a means to measure the sound wave velocity
Lecture Notes
October 2006
Lecture VI
Lecture Notes
24
October 2006
Lecture VII
25
Lecture VII: Feynman Path Integral
. Motivation:
• Alternative formulation of QM (cf. canonical quantisation)
• Close to classical construction — i.e. semi-classics easily accessed
• Effective formulation of non-perturbative approaches
• Prototype of higher-dimensional field theories
. Time-dependent Schrödinger equation
i~∂t |Ψi = Ĥ|Ψi
Formal solution: |ψ(t)i = e−iĤt/~ |ψ(0)i =
. Time-evolution operator
i
|Ψ(t0 )i = Û (t0 , t)|Ψ(t)i,
0
Û (t0 , t) = e− ~ Ĥ(t −t) θ(t0 − t)
X
n
e−iEn t/~ |nihn|ψ(0)i
N.B. Causal
• Real-space representation:
Z
dq|qihq|
0
0
0
0
0
0
Ψ(q , t ) ≡ hq |Ψ(t )i = hq |Û (t , t)
i
∧
|Ψ(t)i =
Z
dq U (q 0 , t0 ; q, t)Ψ(q, t),
0
where U (q 0 , t0 ; q, t) = hq 0 |e− ~ Ĥ(t −t) |qiθ(t0 − t) — propagator or Green function:
i~∂t0 − Ĥ Û (t0 − t) = i~δ(t0 − t)
N.B. ∂t0 θ(t0 − t) = δ(t0 − t)
Physically: U (q 0 , t0 ; q, t) describes probability amplitude for particle to propagate
from q at time t to q 0 at time t0
. Construction of Path Integral
Feynman’s idea: divide time evolution into N → ∞ discrete time steps ∆t = t/N
e−iĤt/~ = [e−iĤ∆t/~ ]N
Then separate the operator content so that momentum operators stand to the left
and position operators to the right:
e−iĤ∆t/~ = e−iT̂ ∆t/~ e−iV̂ ∆t/~ + O(∆t2 )
hqF |[e−iĤ∆t/~ ]N |qI i ' hqF |∧e−iT̂ ∆t/~ e−iV̂ ∆t/~ ∧ . . . ∧e−iT̂ ∆t/~ e−iV̂ ∆t/~ |qI i
Lecture Notes
October 2006
Lecture VII
26
Inserting at ∧ resol. of id. =
Z
Z
dqn
dpn |qn ihqn |pn ihpn |, and using hq|pi = √
1
eiqp/~ ,
2π~
e−iV̂ ∆t/~ |qn ihqn |pn ihpn |e−iT̂ ∆t/~ = |qn ie−iV (qn )∆t/~ hqn |pn ie−iT (pn )∆t/~ hpn |,
1 iqn (pn −pn+1 )
e
and hpn+1 |qn ihqn |pn i =
2π~
Z NY
−1
hqF |e−iĤt/~ |qI i =
"
#
N
N
−1 Y
X
dpn
i
qn+1 − qn
dqn
exp − ∆t
V (qn ) + T (pn+1 ) − pn+1
2π~
~
∆t
n=1
n=0
n=1
qN =qF ,q0 =qI
phase space
p
tn
qF
N N–1
t
0
qI
i.e. at each time step, integration over the classical phase space coords. xn ≡ (qn , pn )
Contributions from trajectories where (qn+1 − qn )pn+1 > ~ are negligible
— motivates continuum limit
Z
Z t
D(q, p)
dt0
q(t)=qF ,q(0)=qI
pq̇|t0 =tn
z
}|
{
z 0}| {
H(q, p|t0 =tn )
}|
{
z
Z NY
N
−1
−1
N
h
}|
{
X z
Y dpn
qn+1 − qn i
i
−iĤt/~
hqF |e
|qI i =
dqn
exp − ∆t
(V (qn ) + T (pn+1 ) − pn+1
)
2π~
~
∆t
n=1
n=0
n=1
qN =qF ,q0 =qI
Propagator expressed as functional integral:
Hamiltonian formulation of Feynman Path Integral
−iĤt/~
hqF |e
|qI i =
Z
D(q, p) exp
q(t)=qF ,q(0)=qI
hi
~
z
Z
0
Action
}|
{
Lagrangian
t
z
}|
{i
dt0 (pq̇ − H(p, q))
Quantum transition amplitude expressed as sum over all possible phase space
trajectories (subject to appropriate b.c.) and weighted by classical action
Lecture Notes
October 2006
Lecture VII
27
. Lagrangian formulation: for “free-particle” Hamiltonian H(p, q) =
hqF |e−iĤt/~ |qI i =
Z
Dq e−(i/~)
Rt
0
dt0 V (q)
Z
q(t)=qF ,q(0)=qI
p2
2m
+ V (q)
Gaussian integral on p
{
Z }|
i t 0 p2
dt
− pq̇
Dp exp −
~ 0
2m
z
2
p0
p2
1 z }| {2 1 2
− pq̇ 7→
(p − mq̇) − mq̇
2m
2m
2
Functional integral justified by discretisation
−iĤt/~
hqF |e
Z t 2
i
mq̇
0
|qI i =
Dq exp
dt
− V (q)
~ 0
2
q(t)=qF ,q(0)=qI
Z
e = lim
Dq → Dq
N →∞
Nm
it2π~
N/2 NY
−1
dqn
n=1
. Connection of path integral to classical statistical mechanics
Consider flexible string held under constant tension, T ,
and confined to ‘gutter-like’ potential, V (u)
x
V(u)
u
i.e. u(x) is displacement from potential minimum
Potential energy stored in spring due to line tension:
extension
z
}|
{ T
2
2 1/2
from x to x + dx, dVT = T [(dx + du ) − dx]' dx (∂x u)2
2
Z
Z L
1
VT [∂x u] ≡ dVT =
dx T (∂x u(x))2
2 0
RL
and from external (gutter) potential: Vext [u] ≡ 0 dx V [u(x)]
According to Boltzmann principle,
equilibrium partition function of periodic system (β = 1/kB T )
Z
Z L T
−βF
2
Z = tr e
=
Du(x) exp −β
dx
(∂x u) + V (u)
2
u(L)=u(0)
0
Lecture Notes
October 2006
Lecture VII
28
“tr” denotes sum over configurations, cf. quantum transmission amplitude
. Mapping:
0
−iĤt/~
hq |e
|qi =
Z t 2
i
mq̇
0
Dq(t) exp
dt
− V (q)
~ 0
2
Z
Wick rotation t → −iτ 7→ imaginary (Euclidean) time path integral
Z τ
Z t
Z τ
Z t
0
2
0
2
0
idt (∂t0 q) −→ −
dτ (∂τ 0 q) ,
−
idt V (q) −→ −
dτ 0 V (q)
0
0
hq |e
−iĤt/~
0
|qi =
Z
1
Dq exp −
~
0
Z
τ
dτ
0
m
0
2
0
2
(∂τ 0 q) + V (q)
N.B. change of relative sign!
(a) Classical partition function of 1d system coincides with QM amplitude
Z
Z = dq hq|e−iĤt/~ |qi
t=−iτ
where time is imaginary, and ~ play role of temperature, 1/β
Generally, path integral for quantum field φ(q, t) in d space dimensions
corresponds to classical statistical mechanics of d + 1-dim. system
(b) Quantum partition function
Z = tr(e
−β Ĥ
Z
)=
dq hq|e−β Ĥ |qi
i.e. Z is transition amplitude hq|e−iĤt/~ |qi evaluated at imaginary time t = −i~β.
(c) Semi-classics
R
As ~ → 0, PI dominated by stationary config. of action S[p, q] = dt(pq̇ − H(p, q))
Z
δS = S[p + δp, q + δq] − S[p, q] = dt [δp q̇ + p δ q̇ − δp ∂p H − δq ∂q H] + O(δp2 , δq 2 , δpδq)
Z
= dt [δp (q̇ − ∂p H) + δq (−ṗ − ∂q H)] + O(δp2 , δq 2 , δpδq)
i.e. Hamilton’s classical e.o.m.: q̇ = ∂p H, ṗ = −∂q H with b.c. q(0) = qI , q(t) = qF
Similarly, with Lagrangian formulation: δS = 0 ⇒ ∂t (∂q̇ L) − ∂q L = 0
What about contributions from fluctuations around classical paths?
Usually, exact evaluation of PI impossible — must resort to approximation schemes...
. saddle-point and stationary phase analysis
Lecture Notes
October 2006
Lecture VII
29
Principle: consider integral over single variable,
Z ∞
dz e−f (z)
I=
−∞
Expect integral to be dominated by minima of f (z); suppose unique i.e. f 0 (z0 ) = 0
7→ 0
z }| { 1
f (z) = f (z0 ) + (z − z0 ) f 0 (z0 ) + (z − z0 )2 f 00 (z0 ) + · · ·
2
s
Z ∞
2π −f (z0 )
2 00
e
I ' e−f (z0 )
dz e−(z−z0 ) f (z0 )/2 =
00
f (z0 )
−∞
Example :
= s! if s ∈ Z Z
z }| {
Γ(s + 1) =
0
f 0 (z) = 1 −
∞
s −z
dz z e
Z
=
0
∞
dz e−f (z) ,
f (z) = z − s ln z
s
1
s
i.e. z0 = s, f 00 (z0 ) = 2 =
z
z0
s
√
i.e. Γ(s + 1) ' 2πse−(s−s ln s) — Stirling’s formula
If minima not on integration contour – deform contour through saddle-point
e.g. Γ(s + 1), s complex
What if exponent pure imaginary? Fast phase fluctuations ; cancellation
i.e. expand around region of slowest (i.e. stationary) phase and use identity
r
Z ∞
2π iπ/4
2
e
dz eiaz /2 =
a
−∞
. Can we apply same approach to analyse PI? Yes
but we must develop basic tool of QFT – Gaussian functional integral!
Lecture Notes
October 2006
Lecture VIII
30
Lecture VIII: Quantum Harmonic Oscillator
. Free particle propagator: Difficult to obtain from PI, but useful for normalization,
and easily obtained from equation for Green function, (i~∂t − Ĥ)Ĝfree (t) = i~δ(t),
which in Euclidean time t = −iτ becomes a diffusion equation,
~2 ∇2
~∂τ −
Gfree (qF , qI , t) = ~δ(qF − qI )δ(τ )
2m
Solution: (PS3)
Gfree (qF , qI ; t) ≡ hqF |e
−ip̂2 t/2m~
m 1/2
i m(qF − qI )2
θ(t)
|qI iθ(t) =
exp
2πi~t
~
2t
. Quantum particle in single (symmetric) well: V (q) = V (−q)
V
ω
q
e.g. QM amplitude
−iĤt/~
G(0, 0; t) ≡ h0|e
Z t 2
i
mq̇
0
|0i θ(t) =
Dq exp
dt
− V (q)
~ 0
2
q(t)=q(0)=0
Z
. Evaluate PI by stationary phase approx: general recipe
(i) Parameterise path as q(t) = qcl (t) + r(t) and expand action in r(t)
r2 00
0
V
(q
)
+
rV
(q
)
+
V (qcl ) + · · ·
cl
cl
q˙cl ṙ + ṙ
2
Z t h q˙cl z+ 2}|
{
i
z }| {
2
0 m
S[q̄ + r] =
dt
(q̇cl + ṙ)
−
V (qcl + r)
2
0
2
Z
= S[qcl ] +
0
t
2
δ2 S
δq(t0 )δq(t00 )
δS
=0
δq(t0 )
z
}|
{ 1
dt0 r(t0 ) [−mq̈cl − V 0 (qcl )] +
2
Z
0
t
z
}|
{
dt0 r(t0 ) −m∂t20 − V 00 (qcl ) r(t0 ) + · · ·
(ii) Classical trajectory: mq̈cl = −V 0 (qcl )
Many solutions – choose non-singular qcl = 0, i.e. S[qcl ] = 0 and V 00 (qcl ) = mω 2 const.
Z t
Z
0
i
0
0 m
2
2
G(0, 0; t) '
Dr exp
dt r(t )
−∂t0 − ω r(t )
~ 0
2
r(0)=r(t)=0
N.B. if V was quadratic, expression trivially exact
Lecture Notes
October 2006
Lecture VIII
31
More generally, qcl (t) non-trivial 7→ non-vanishing S[qcl ] — see PS3
Fluctuations? — example of a...
. Gaussian functional integration: mathematical interlude
R∞
R∞
2
2
• One variable Gaussian integral:
( −∞ dv e−av /2 )2 = 2π 0 r dr e−ar /2 =
r
Z ∞
2π
− a2 v 2
dv e
=
,
Re a > 0
a
−∞
2π
a
• Many variables:
Z
1
dv e− 2 v
T Av
= (2π)N/2 det A−1/2
A is +ve definite real symmetric N × N matrix
Proof: A diagonalised by orthogonal trans: D = OAOT
Change of variables: v = OT w (Jacobian det(O) = 1) ; N decoupled Gaussian
QN
P
2
integrations: vT Av = wT Dw = N
i=1 di = det D = det A
i di wi and
• Infinite number of variables; interpret {vi } 7→ v(t) as continuous field and
Aij 7→ A(t, t0 ) = ht|Â|t0 i as operator kernel
Z
Z
Z
1
0
0
0
Dv(t) exp −
dt dt v(t)A(t, t )v(t ) ∝ (det Â)−1/2
2
(iii) Applied to QW, A(t, t0 ) = − ~i mδ(t − t0 )(−∂t20 − ω 2 ) and
−1/2
G(0, 0; t) ' J det −∂t20 − ω 2
where J absorbs constant prefactors (im, ~, etc.)
What does ‘det’ mean? Effectively, we can expand trajectories r(t0 )
in eigenbasis of  subject to b.c. r(t) = r(0) = 0
cf. PIB
−∂t2 − ω 2 rn (t) = n rn (t),
0
i.e. Fourier series expansion: rn (t0 ) = sin( nπt
),
t
det
−∂t2
−ω
2 −1/2
=
∞
Y
n=1
−1/2
n
=
∞ Y
nπ 2
n=1
n = ( nπ
)2 − ω 2
t
n = 1, 2, . . . ,
t
−ω
2
−1/2
. For V = 0, G = Gfree known — use to eliminate constant prefactor J
"
2 #−1/2 ∞
Y
G(0, 0; t)
ωt
m 1/2
G(0, 0; t) =
Gfree (0, 0; t) =
1−
Θ(t)
Gfree (0, 0; t)
nπ
2πi~t
n=1
Lecture Notes
October 2006
Lecture VIII
32
Applying identity
Q∞
n=1 [1
x 2 −1
− ( nπ
)] =
G(0, 0; t) '
r
x
sin x
mω
Θ(t)
2πi~ sin(ωt)
(exact for harmonic oscillator)
Singular behaviour is a feature of ladder-like states of harmonic oscillator leading to
periodic coherent superposition and dynamical echo (see PS3).
Double Well: Tunneling and Instantons
How can QM tunneling be described by path integral? No semi-classical expansion!
. E.g transition amplitude in double well: G(a, −a; t) ≡ ha|e−iĤt/~ | − ai
V
q
. Feynman PI:
Z t
i
0 m 2
dt
q̇ − V (q)
G(a, −a; t) =
Dq exp
~ 0
2
q(0)=−a
Z
q(t)=a
Stationary phase analysis: classical e.o.m. mq̈ = −∂q V
7→ only singular (high energy) solutions Switch to alternative formulation...
. Imaginary (Euclidean) time PI: Wick rotation t = −iτ
N.B. (relative) sign change! “V → −V ”
Z
q(τ )=a
1
G(a, −a; τ ) =
Dq exp −
~
q(0)=−a
Z
τ
dτ
0
0
m
2
2
q̇ + V (q)
Saddle-point analysis: classical e.o.m. mq̈ = +V 0 (q) in inverted potential!
solutions depend on b.c.
(1) G(a, a; τ ) ; qcl (τ ) = a
(2) G(−a, −a; τ ) ; qcl (τ ) = −a
(3) G(a, −a; τ ) ; qcl : rolls from −a to a
Combined with small fluctuations, (1) and (2) recover propagator for single well
(3) accounts for tunneling – known as “instanton” (or “kink”)
Lecture Notes
October 2006
Lecture VIII
33
q
V
a
a
-a
q
t
-a
ω –1
. Instanton: classically forbidden trajectory connecting two degenerate minima
— i.e. topological, and therefore particle-like
τ →∞
For τ large, q˙cl ' 0 (evident), i.e. “first integral” mq˙cl 2 /2 − V (qcl ) = → 0
precise value of fixed by b.c. (i.e. τ )
R
Saddle-point action (cf. WKB dqp(q))
Z τ
Z a
Z a
Z τ
0 m 2
0
2
Sinst. =
dτ
q̇cl + V (qcl ) '
dτ mq̇cl =
dqcl mq̇cl =
dqcl (2mV (qcl ))1/2
2
0
0
−a
−a
τ →∞
Structure of instanton: For q ' a, V (q) = 21 mω 2 (q − a)2 + · · ·, i.e. q̇cl ' ω(qcl − a)
τ →∞
qcl (τ ) = a − e−τ ω , i.e. temporal extension set by ω −1 τ
Imples existence of approximate saddle-point solutions
involving many instantons (and anti-instantons): instanton gas
q
a
τ5
τ4
τ3
τ2
τ1
τ
–a
. Accounting for fluctuations around n-instanton configuration
G(a, ±a; τ ) '
X
Kn
n even / odd
Z
τ
Z
τ1
dτ1
0
0
dτ2 · · ·
Z
τn−1
An,cl. An,qu.
z
}|
{
dτn An (τ1 , . . . , τn ),
0
constant K set by normalisation
An,cl. = e−nSinst. /~ — ‘classical’ contribution
An,qu. — quantum fluctuations (cf. single well): Gs.w. (0, 0; t) ∼
An,qu. ∼
n
Y
i
√ 1
sin ωt
n
Y
1
p
e−ω(τi+1 −τi )/2 ∼ e−ωτ /2
∼
sin(−iω(τi+1 − τi ))
i
τ n /n!
G(a, ±a; τ ) '
Lecture Notes
X
n even / odd
K n e−nSinst. /~ e−ωτ /2
zZ
τ
Z
dτ1
0
0
τ1
}|
dτ2 · · ·
Z
τn−1
{
dτn
0
October 2006
Lecture VIII
34
X
=
e−ωτ /2
n even / odd
Using ex =
P∞
n=0 x
n
1 τ Ke−Sinst. /~
n!
n /n!,
N.B. non-perturbative in ~!
G(a, −a; τ ) ' Ce−ωτ /2 sinh τ Ke−Sinst. /~
G(a, a; τ ) ' Ce−ωτ /2 cosh τ Ke−Sinst. /~ ,
Consistency check: main contribution from
P
nX n /n!
n̄ = hni ≡ Pn n
= X = τ Ke−Sinst. /~
X
/n!
n
no. per unit time, n̄/τ exponentially small, and indep. of τ , i.e. dilute gas
V
A
S
q
ψ
q
. Physical interpretation: For infinite barrier, oscillators independent,
coupling splits degeneracy – symmetric/antisymmetric
G(a, ±a; τ ) ' ha|Sie−S τ /~ hS| ± ai + ha|Aie−A τ /~ hA| ± ai
2
Setting A/S = ~ω/2 ± ∆
2 , and noting |ha|Si| = ha|SihS| − ai =
C
2
= |ha|Ai|2 = −ha|AihA| − ai
C −(~ω−∆)τ /2~
cosh(∆τ /~)
G(a, ±a; τ ) '
.
e
± e−(~ω+∆)τ /2~ = Ce−ωτ /2
sinh(∆τ /~)
2
. Remarks:
(i) Legitimacy? How do (neglected) terms O(~2 ) compare to ∆?
In fact, such corrections are bigger but act equally on |Si and |Ai
i.e. ∆ = ~Ke−Sinst. /~ is dominant contribution to splitting
V
–V
q
qm
q
q
τ
qm
(ii) Unstable States and Bounces: survival probability: G(0, 0; t)? No even/odd effect:
h
i
Γ
−ωτ /2
−Sinst /~ τ =it
−iωt/2
G(0, 0; τ ) = Ce
exp τ Ke
= Ce
exp − t
2
True decay rate has additional factor of 2: Γ ∼ |K|e−Sinst /~ (i.e. K imaginary)
see Coleman for details
Lecture Notes
October 2006
Lecture IX
35
Lecture IX: Coherent States
Generalisation of PI to many-body systems problematic due to particle indistinguishability. Can second quantisation help? automatically respects particle statistics
Require complete basis on Fock space to construct PI
R
i.e. analogue of dq dp |qihq|pihp| = id.
Such eigenstates exist and are known as...
reference: Negele and Orland
. Coherent States (Bosons)
What are eigenstates of Fock space operators: ai and a†i with [ai , a†j ] = δij ?
As a state of the Fock space, an eigenstate |φi can be expanded as
|φi =
(a† )n1 (a†2 )n2
Cn1 ,n2 ,··· √1
· · · |0i
√
n
n
1
2
,···
X
n1 ,n2
N.B. notation |0i for vacuum state!
(i) a†i |φi = φi |φi? — clearly, eigenstate of a†i can not exist:
if minimum occupation of |φi is n0 , minimum of a†i |φi is n0 + 1
P
(ii) ai |φi = φi |φi? — can exist and given by: |φi ≡ exp[ i φi a†i ]|0i i.e. φ ≡ {φi }
Proof: since ai commutes with all a†j for j 6= i — focus on one element i
φa†
ae
φa†
|0i = [a, e
]|0i =
∞
X
φn
n=0
n!
† n
[a, (a ) ]|0i =
∞
X
nφn
n=1
n!
(a† )n−1 |0i = φ exp(φa† )|0i
a(a† )n = aa† (a† )n−1 = (1 + a† a)(a† )n−1 = (a† )n−1 + a† a(a† )n−1 = n(a† )n−1 + (a† )n a
i.e. |φi is eigenstate of all ai with eigenvalue φi
. Properties of coherent state |φi
• Hermitian conjugation:
∀i :
hφ|a†i = hφ|φ̄i
φ̄i is complex conjugate of φi
• By direct application of ∂φi (and operator commutativity):
∀i :
Lecture Notes
a†i |φi = ∂φi |φi
October 2006
Lecture IX
36
P
• Overlap: with hθ| = |θi† = h0|e
i θ̄i ai
"
hθ|φi = h0|e
P
i θ̄i ai
i θ̄i φi
h0|φi = exp
#
X
θ̄i φi
i
i.e. states are not orthogonal! operators not Hermitian
#
"
• Norm: hφ|φi = exp
|φi = e
P
X
φ̄i φi
i
• Completeness — resolution of id. (for proof see notes)
Z Y
dφ̄i dφi
π
i
e−
P
i
φ̄i φi
|φihφ| = 1F
where dφ̄i dφi = dRe φi dIm φi
. Coherent States (Fermions)
Following bosonic case, seek state |ηi s.t.
ai |ηi = ηi |ηi,
η = {ηi }
But anticommutativity [ai , aj ]+ = 0 (i 6= j) demands that ai aj |ηi = −aj ai |ηi
i.e. eigenvalues ηi must anticommute!!
ηi ηj = −ηj ηi
ηi can not be ordinary numbers — in fact, they obey...
. Grassmann Algebra
In addition to anticommutativity, defining properties:
(i) ηi2 = 0 (cf. fermions) but note: these are not operators, i.e. [ηi , η̄i ]+ 6= 1
(ii) Elements ηi can be added to, and multiplied, by ordinary complex numbers
c + ci η i + cj η j ,
ci , cj ∈ C
(iii) Grassmann numbers anticommute with fermionic creation/annihilation operators
[ηi , aj ]+ = 0
. Calculus of Grassmann variables:
(iv) Differentiation: ∂ηi ηj = δij
N.B. ordering matters ∂ηi ηj ηi = −ηj ∂ηi ηi = −ηj for i 6= j
(v) Integration:
Lecture Notes
R
dηi = 0,
R
dηi ηi = 1
i.e. differentiation and integration have the same effect!!
October 2006
Lecture IX
37
. Gaussian integration:
Z
Z
Z
Z
−η̄aη
dη̄dη e
= dη̄dη (1 − η̄aη) = a dη̄ η̄ dηη = a
Z Y
T
dη̄i dηi e−η̄ Aη = detA
(exercise)
i
cf. ordinary complex variables
. Functions of Grassmann variables:
Taylor expansion terminates at low order since η 2 = 0, e.g.
F (η) = F (0) + ηF 0 (0)
Using rules
Z
Z
dηF (η) =
dη [F (0) + ηF 0 (0)] = F 0 (0) ≡ ∂η F [η]
i.e. differentiation and integration have same effect on F [η]!
Usually, one has a function of many variables F [η], say η = {η1 , · · · ηN }
∞
X
1 ∂ n F (0)
ηj · · · ηi
F (η) =
n! ∂ηi · · · ∂ηj
n=0
but series must terminate at n = N
with these preliminaries we are in a position to introduce the
. Fermionic coherent state: |ηi = exp[−
P
i
ηi a†i ]|0i i.e. η = {ηi }
Proof (cf. bosonic case)
a exp(−ηa† )|0i = a(1 − ηa† )|0i = ηaa† |0i = η|0i = η exp(−ηa† )|0i
Other defining properties mirror bosonic CS — problem set
. Differences:
(i) Adjoint: hη| = h0|e−
P
i
ai η̄i
Z
(ii) Gaussian integration:
≡ h0|e
P
i
η̄i ai
but N.B. η̄i not related to ηi !
dη̄dη e−η̄η = 1
N.B. no π’s
Completeness relation
Z Y
i
Lecture Notes
dη̄i dηi e−
P
i
η̄i ηi
|ηihη| = 1F
October 2006
Lecture X
38
Lecture X: Many-body (Coherent State) Path Integral
Having obtained a complete coherent state basis for the creation and annihilation operators, we could proceed by constructing path integral for the quantum time evolution
operator. However, since we will be interested in application involving a phase transition,
it is more convenient to begin with the quantum partition function.
. Quantum partition function
X
Z=
hn|e−β(Ĥ−µN̂ ) |ni,
F = −kB T ln Z
{n}∈Fock Space
β=
1
,
kB T
µ — chemical potential
In coherent state basis
Z
X
P
Z = d[ψ̄, ψ]e− i ψ̄i ψi
hn|ψihψ|e−β(Ĥ−µN̂ ) |ni
n
Elimination of |ni requires identity: hn|ψihψ|ni = hζψ|nihn|ψi
Proof: for, e.g., |ni = a†1 a†2 · · · a†n |0i
hn|ψi = h0|an · · · a2 a1 |ψi = ψn · · · ψ2 ψ1 h0|ψi = ψn · · · ψ2 ψ1
hψ|ni = ψ̄1 ψ̄2 · · · ψ̄n
hn|ψihψ|ni = ψn · · · ψ2 ψ1 ψ̄1 ψ̄2 · · · ψ̄n = ψ1 ψ̄1 ψ2 ψ̄2 · · · ψn ψ̄n
= (ζ ψ̄1 ψ1 )(ζ ψ̄2 ψ2 ) · · · (ζ ψ̄n ψn ) = hζψ|nihn|ψi
Note that Ĥ and N̂ even in operators allowing matrix element to be commuted through
Z
P
Z = d[ψ̄, ψ]e− i ψ̄i ψi hζψ|e−β(Ĥ−µN̂ ) |ψi
. Coherent State Path Integral
Applied to many-body Hamiltonian of fermions or bosons
X
X
Ĥ − µN̂ =
(hij − µδij )a†i aj +
Vij a†i a†j aj ai
ij
ij
N.B. operators are normal ordered
Follow general strategy of Feynman:
(i) Divide ‘time’ interval, β, into N segments of length ∆β = β/N
hζψ|e−β(Ĥ−µN̂ ) |ψi = hζψ|e−∆β(Ĥ−µN̂ ) ∧e−∆β(Ĥ−µN̂ ) ∧ · · · e−∆β(Ĥ−µN̂ ) |ψi
Lecture Notes
October 2006
Lecture X
39
(ii) At each position ‘∧’ insert resolution of id.
Z
1F = d[ψ̄n , ψn ]e−ψ̄n ·ψn |ψn ihψn |
i.e. N -independent sets N.B. each ψn is a vector with elements {ψi }n
(iii) Expand exponent in ∆β
h
i
hψ 0 |e−∆β(Ĥ−µN̂ ) |ψi = hψ 0 | 1 − ∆β(Ĥ − µN̂ ) |ψi + O(∆β)2
= hψ 0 |ψi − ∆βhψ 0 |(Ĥ − µN̂ )|ψi + O(∆β)2
= hψ 0 |ψi [1 − ∆β (H(ψ 0 , ψ) − µN (ψ 0 , ψ))] + O(∆β)2
0
0
0
' eψ ·ψ e−∆β(H(ψ ,ψ)−µN (ψ ,ψ))
with
H(ψ 0 , ψ) =
X
hψ 0 |Ĥ|ψi X
0
ψ̄
ψ
+
Vij ψ̄i0 ψ̄j0 ψj ψi
=
h
j
ij
i
0
hψ |ψi
ij
ij
similarly N (ψ 0 , ψ) N.B. hψ 0 |ψi bilinear in ψ, i.e. commutes with everything
Z=
N
Y
Z
d[ψ̄n , ψn ]e−
PN
n=1
[ψ̄n ·(ψn −ψn−1 )+∆β(H(ψ̄n ,ψn−1 )−µN (ψ̄n ,ψn−1 ))]
n=0
ψ̄N =ζ ψ̄0 ,ψN =ζψ0
Continuum limit N → ∞
Z β
N
X
ψn − ψn−1
→
dτ,
∆β
→ ∂τ ψ ,
∆β
τ =n∆β
0
n=0
N
Y
n=0
d[ψ̄n , ψn ] → D(ψ̄, ψ)
comment on “small” Grassmann nos.
Z=
Z
−S[ψ̄,ψ]
ψ̄(β)=ζ ψ̄(0)
ψ(β)=ζψ(0)
D(ψ̄, ψ)e
Z
,
S[ψ̄, ψ] =
0
β
dτ ψ̄ · ∂τ ψ + H(ψ̄, ψ) − µN (ψ̄, ψ)
With particular example:
#
Z β "X
X
S[ψ̄, ψ] =
dτ
ψ̄i (τ ) [(∂τ − µ)δij + hij ] ψj (τ ) +
Vij ψ̄i (τ )ψ̄j (τ )ψj (τ )ψi (τ )
0
ij
ij
quantum partition function expressed as path integral over fields ψi (τ )
. Matsubara frequency representation
Often convenient to express path integral in frequency domain
Z β
1 X
1
−iωn τ
ψ(τ ) = √
ψn e
,
ψωn = √
dτ ψ(τ )eiωn τ
β ω
β 0
n
Lecture Notes
October 2006
Lecture X
40
where, since ψ(τ ) = ζψ(τ + β)
2nπ/β,
ωn =
(2n + 1)π/β,
bosons,
,
fermions
n∈Z
ωn are known as Matsubara frequencies
1
Using
β
β
Z
dτ ei(ωn −ωm )τ = δωn ωm
0
X
S[ψ̄, ψ] =
ijωn
+
ψ̄iωn [(−iωn − µ) δij + hij ] ψjωn +
1X
β ij ω
X
Vij ψ̄iωn1 ψ̄jωn2 ψjωn3 ψiωn4 δωn1 +ωn2 ,ωn3 +ωn4
n1 ωn2 ωn3 ωn4
P
e.g. Harmonic chain: Ĥ = k ~ωk (a†k ak + 1/2)
Z β X
ψ̄k (∂τ + ~ωk − µ)ψk
S=
dτ
0
P R
e.g. Electron gas: Ĥ =
β
Z
S=
dτ
0
XZ
2
p̂
cσ (r)−
dr c†σ (r) 2m
drψ̄σ (r, τ )(∂τ +
σ
Z
−
σ
k
β
dτ
0
XZ
P
σσ 0
R
2
e
0
dr dr0 c†σ (r)c†σ0 (r0 ) |r−r
0 | cσ 0 (r )cσ (r)
p̂2
− µ)ψσ (r, τ )
2m
dr dr0 ψ̄σ (r, τ )ψ̄σ0 (r0 , τ )
σ,σ 0
e2
ψσ0 (r0 , τ )ψσ (r, τ )
|r − r0 |
. Connection between coherent state and Feynman Path integral
e.g. QHO: Ĥ = ~ω(a† a + 1/2),
Z
−β Ĥ
Z = tr e
=
ψ(β)=ψ(0)
Setting ψ(τ ) =
Z=
mω 1/2
2~
Z
p.b.c
[a, a† ] = 1, i.e. bosons!
e−β~ω/2 in D(ψ̄, ψ)
Z β
D(ψ̄, ψ) exp −
dτ ψ̄(∂τ + ~ω)ψ
0
i
[q(τ )+ mω
p(τ )],
Z
D(p, q) exp −
0
with p, q real, and noting
0
β
dτ
p2
1
ipq̇
+ mω 2 q 2 −
2m 2
~
i
t,
~
Z t
Z
i
0
Z = D(p, q) exp
dt (pq̇ − H(p, q))
~ 0
cf. (Euclidean time) FPI; β =
Lecture Notes
β
Z
dτ q ṗ = −
Z
β
dτ pq̇
0
τ=
i 0
t,
~
i ∂q
∂q
= 0
~ ∂τ
∂t
October 2006
Lecture X
41
. Evaluation of Z from field integral
(i) ‘Bosonic’ oscillator: Ĥ = ~ω(a† a + 1/2)
J det(∂τ + ~ω)−1
zZ
{ Z Y
Z}|β
P
dτ ψ̄ (∂τ + ~ω) ψ = ( dψ̄ωn dψωn ) e− n ψ̄ωn (−iωn +~ω)ψωn
ZB = D(ψ̄, ψ) exp −
0
n
"
2 #−1
∞
∞
0 Y
Y
J
J
~ωβ
−1
=J
[β(−iωn + ~ω)]−1 =
(~ωβ)2 + (2nπ)2
=
1+
~ωβ
~ωβ
2πn
ωn
n=1
n=1
∞ x 2 −1
Y
J0
x
=
where
1+
=
2 sinh(~ωβ/2)
πn
sinh x
n=1
Y
Normalisation: as T → 0, ZB dominated by g.s., i.e. limβ→∞ ZB = e−β~ω/2
i.e. J 0 = 1,
ZB =
1
2 sinh(~βω/2)
(ii) ‘Fermionic’ oscillator: Ĥ = ~ω(a† a + 1/2), [a, a† ]+ = 1
Gaussian Grassmann integration
ZF = J det(∂τ + ~ω) = J
= J0
∞
Y
"
n=1
1+
~ωβ
(2n + 1)π
Y
[β(−iωn + ~ω)] = J
(~ωβ)2 + ((2n + 1)π)2
n=0
ωn
2 #
∞
Y
= J 0 cosh(~ωβ/2),
∞
Y
"
1+
n=1
x
π(2n + 1)
2 #
= cosh(x/2)
Using normalisation: limβ→∞ ZF = e−β~ω/2
J 0 = 2e−β~ω
ZF = 2e−β~ω cosh(~βω/2).
cf. direct computation: ZB = e−β~ω/2
P∞
n=0
e−nβ~ω , ZF = e−β~ω/2
P1
n=0
e−nβ~ω .
Note that normalising prefactor J 0 involves only a constant offset of free energy,
F = − β1 ln Z statistical correlations encoded in content of functional integral
Lecture Notes
October 2006
Lecture XI
42
Lecture XI: Matsubara frequency summations
. Quantum partition function of ideal (i.e. non-interacting) gas (from coherent states)
Useful for “normalisation” of interacting theories
X
e.g. (1) Fermions: Ĥ =
α a†α aα
α
As a warm-up, in coherent state representation:
Z
X
P
−β(Ĥ−µN̂ )
−β(Ĥ−µN̂ )
Z0 = tr e
=
hn|e
|ni = d(ψ̄, ψ)e− α ψ̄α ψα h−ψ|e−β(Ĥ−µN̂ ) |ψi
n
Using identity
e−β(Ĥ−µN̂ ) = e−β
P
†
α (α −µ)aα aα
=
Y
e−β(α −µ)n̂α =
α
Z0 =
Z
=
d(ψ̄, ψ)e−
YZ
YZ
α
=
YZ
α
=
α
h−ψ|ψi
Y n z }| { o
−ψ̄α ψα
−β(α −µ)
α ψ̄α ψα
e
1+ e
− 1 (−ψ̄α ψα )
α
α
=
P
Y
1 + e−β(α −µ) − 1 n̂α
Y
1 − 2ψ̄α ψα
z }| { dψ̄α dψα e−2ψ̄α ψα 1 + e−β(α −µ) − 1 (−ψ̄α ψα )
dψ̄α dψα 1 − 2ψ̄α ψα − e−β(α −µ) − 1 ψ̄α ψα
dψ̄α dψα −ψ̄α ψα (1 + e−β(α −µ) )
1 + e−β(α −µ)
i.e. Fermi − Dirac distribution
α
Exercise: show (using CS) that in Bosonic case
Z0 =
∞
YX
e−nβ(α −µ) =
α n=0
Y
α
1 − e−β(α −µ)
−1
i.e. Bose − Einstein distribution
What about field integral...?
. Quantum partition function of ideal gas:
#
" Z
Z
β
X
Z0 =
D(ψ̄, ψ) exp −
dτ
ψ̄α (∂τ + α − µ)ψα
b.c.
Z
=
0
"
D(ψ̄, ψ) exp −
Lecture Notes
α
#
X
α,ωn
ψ̄α,ωn (−iωn + α − µ)ψα,ωn = J
Y
α,ωn
[β(−iωn + α − µ)]−ζ
October 2006
Lecture XI
43
where J absorbs constant prefactors
From Z0 = tr e−β(Ĥ−µN̂ ) we can obtain thermal occupation number:
n(T ) ≡
1
1
ζ X
1
1
tr[N̂ e−β(Ĥ−µN̂ ) ] =
∂µ Z0 = ∂µ ln Z0 ≡ −∂µ F = −
Z0
βZ0
β
β α,ω iωn − α + µ
n
P
. To perform summations of the form, I = ωn h(ωn ), helpful to
introduce complex auxiliary function g(z) with simple poles at z = iωn

β

, bosons

exp(βz) − 1
e.g. g(z) =
β


, fermions
exp(βz) + 1
In bosonic case: poles when βz = 2πin, i.e. z = iωn ; close to pole,
β
eβ(iωn +δz)
−1
=
β
1
'
−1
δz
eβ δz
noting that g(z) has simple poles with residue ζ,
I
X
ζ
Res [g(z)h(−iz)]|z=iωn
I=
dz g(z)h(−iz) = ζ
2πi γ1
ω
n
where contour encircles poles
γ1
ω
γ2
ω
As long as we don’t to cross singularities of g(z)h(−iz), we are free to distort contour
If g(z)h(−iz) decays sufficiently fast at |z| → ∞ (i.e. faster than z −1 ), useful to
‘inflate’ contour to infinite circle when integral along outer perimeter vanishes and
ζ
I=
2πi
I
γ2
N.B. X
z}|{
h(−iz)g(z) = − ζ
Res [h(−iz)g(z)]|z=zk
k
For problem at hand,
h(ωn ) = −
Lecture Notes
ζX
1
,
β α iωn − α + µ
h(−iz) = −
ζX
1
β α z − α + µ
October 2006
Lecture XI
44
Although h(−iz) seems to scale as 1/z at infinity,
this reflects failure of continuum limit of the action: ψ̄m
(ψm+1 − ψm )
7→ ψ̄∂τ ψ
∆β
Integral made convergent by including infinitesimal
+
(iωn − α + µ) 7→ (iωn e−iωn 0 − α + µ)
Since h(−iz) involves simple poles at z = α − µ,
n(T ) = −ζ
X
Res [g(z)h(−iz)]|z=α −µ =
α
X
α
X nB (α ), bosons,
=
nF (α ), fermions
eβ(α −µ) − ζ
α
1
where nF/B are Fermi/Bose distribution functions
. Applications of Field Integral:
In remaining lectures we will address two case studies which
exhibit phase transition to non-trivial ground state at low temperatures
• Bose-Einstein condensation and superfluidity
• Superconductivity
Lecture Notes
October 2006
Lecture XI
45
Bose-Einstein condensation from field integral
Although we could start our analysis of application of the field integral with the weakly
interacting electron gas, we would find that correlation effects could be considered perturbatively. Our analysis of the field integral would not engage any non-trivial field configurations of the action: the platform of the non-interacting electron system remains
adiabatically connected to that of the weakly interacting system. In the following we will
explore a problem in which the development of a non-trivial ground state — the BoseEinstein condensate — is accompanied by the appearance of collective modes absent in the
non-interacting system.
. Consider Bose gas subject to weak short-ranged repulsive contact interaction:
Z
Z
g
d
†
dd r a† (r)a† (r)a(r)a(r)
Ĥ = d r a (r)Ĥ0 a(r) +
2
R
. Expressed as field integral: Z = tr e−β(Ĥ−µN̂ ) = ψ(β)=ψ(0) D(ψ̄, ψ) e−S[ψ̄,ψ] , where
Z
S=
β
Z
dτ
0
i
h
g
dd r ψ̄(∂τ + Ĥ0 − µ)ψ + (ψ̄ψ)2
2
As a warm-up exercise, consider first the...
. Non-interacting Bose gas (g = 0)
Z
Y
P
=
D(ψ̄, ψ) e− a,ωn ψ̄a,ωn (−iωn +a −µ)ψa,ωn = J
Z0 ≡ Z g=0
ψ(β)=ψ(0)
a,ωn
1
β(−iωn + a − µ)
where eigenvalues of Ĥ0 , a ≥ 0 and 0 = 0
P
While stability requires µ ≤ 0, precise value fixed by condition N = a nB (a )
. Bose-Einstein condensation (BEC)
µ
0
Tc
T
• As T reduced, µ increases until, at T = Tc , µ = 0
• For T < Tc , µ remains zero and a macroscopic number of particles, N0 = N − N1 ,
condense into ground state: BEC
X
i.e. for T < Tc ,
nB (a )
≡ N1 < N
a
Lecture Notes
µ=0
October 2006
Lecture XI
46
. How can this phenomenon be incorporated into path integral?
Although condensate characterised by g.s. component ψ0 ≡ ψa=0,ωn =0 , for T < Tc ,
fluctuations seemingly unbound (i.e. µ = 0 = 0 and action for ψ0 vanishes!)
In this case, we must treat ψ0 as a
Lagrange multiplier which fixes particle number below Tc :
X0
S0 |µ=0− = −β ψ̄0 µψ0 +
ψ̄aωn (−iωn + a − µ) ψaωn
a,ωn
1
β(−iωn + a − µ)
a,ωn
1
1 X0
1
i.e. N = ∂µ ln Z0 |µ=0− = ψ̄0 ψ0 −
= ψ̄0 ψ0 + N1
β
β a,ω iωn − a
Z0 = eβ ψ̄0 µψ0 × J
Y0
n
i.e. ψ̄0 ψ0 = N0 translates to no. of particles in condensate
. Weakly Interacting Bose Gas
Bosons confined to box of size L with p.b.c. and Ĥ0 = p̂2 /2m described by action
Z β Z
p̂2
g
d
2
dτ d r ψ̄(∂τ +
S=
− µ)ψ + (ψ̄ψ)
2m
2
0
Since field integral intractable, turn to mean-field theory
(a.k.a. “saddle-point” approximation – Landau theory) valid for T Tc
Variation of action w.r.t. ψ̄ obtains the saddle-point equation:
p̂2
∂τ +
− µ + g ψ̄ψ ψ = 0
2m
P ik·r
1
0
solved by constant ψ(r, τ ) ≡ Ld/2
ψk (τ ) = Lψd/2
ke
where ψ0 minimises saddle-point action
1
g
g
2
S[ψ̄0 , ψ0 ] = −µψ̄0 ψ0 + d (ψ̄0 ψ0 ) ,
i.e.
−µ + d ψ̄0 ψ0 ψ0 = 0
β
2L
L
Lecture Notes
October 2006
Lecture XI
47
• For µ < 0, only trivial solution ψ0 = 0 – no condensate
• For µ ≥ 0, s.p.e. solved by any configuration with |ψ0 | = γ ≡
p
µLd /g
N.B. interaction allows µ > 0; ψ̄0 ψ0 ∝ Ld reflects macroscopic population of g.s.
• Condensation of Bose gas is example of a continuous phase transition, i.e. “order parameter”
ψ0 grows continuously from zero
• saddle-point solution is “continuously degenerate”, ψ0 = γ exp(iφ), φ ∈ [0, 2π]
• One ground state chosen ; spontaneous symmetry breaking – Goldstone’s theorem:
expect branch of gapless excitations
Taking into account fluctuations, we may address the phenomenon of superfluidity...
Lecture Notes
October 2006
Lecture XII
48
Lecture XII: Superfluidity
Previously, we have seen that, when treated in a mean-field or saddle-point approximation,
the field theory of the weakly interacting Bose gas shows a transition to a Bose-Einstein
condensed phase when µ = 0 where the order parameter, the p
complex condensate wavefunction ψ0 acquires a non-zero expectation value, |ψ0 | = γ ≡ µLd /g. The spontaneous
breaking of the continuous symmetry associated with the phase of the order parameter is
accompanied by the appearance of massless collective phase fluctuations. In the following,
we will explore the properties of these fluctuations and their role in the phenomenon of
superfluidity.
. Starting with the model action for a Bose system, (~ = 1)
Z β Z
∂2
g
d
2
S[ψ̄, ψ] =
dτ d r ψ̄ ∂τ −
− µ ψ + (ψ̄ψ)
2m
2
0
saddle-point analysis revealed that, for µ > 0, ψ
acquires a non-zero expectation value: |ψ0 | = (µLd /g)1/2
Phase transition accompanied by spontaneous symmetry breaking
(of U (1) field associated with phase of global ψ)
To investigate consequence of transition, must explore role of fluctuations
To do so, it is convenient to parameterise ψ(r, τ ) = [ρ(r, τ )]1/2 eiφ(r,τ )
Z
1 β
ρ β
dτ ∂τ (ρ1/2 ρ1/2 ) = − = 0
2 0
2 0
z
}|
{
Z
Z
Z
β
Using
1.
β
1/2
dτ ψ̄∂τ ψ =
dτ ρ
1/2
∂τ ρ
1
1/2 iφ
iφ
1/2
2. ∂(ρ e ) = e
∂ρ + iρ ∂φ
2ρ1/2
Z β
Z β
Z
2
3.
dτ ψ̄∂ ψ = −
dτ ∂ ψ̄ · ∂ψ = −
0
+
0
β
dτ iρ∂τ φ
0
0
Z
S[ρ, φ] =
Z
dτ
0
Lecture Notes
β
0
0
β
dτ
1
(∂ρ)2 + ρ(∂φ)2
4ρ
1
1
gρ2
2
2
d r iρ∂τ φ +
(∂ρ) + ρ(∂φ) − µρ +
2m 4ρ
2
d
October 2006
Lecture XII
49
Expansion of action around saddle-point: ρ(r, τ ) = (ρ0 + δρ(r, τ )) eiφ(r,τ ) ,
Z β Z
g(ρ0 + δρ)2
d
dτ d r −µ(ρ0 + δρ) +
S[δρ, φ] =
2
0
1
1
2
2
(∂(δρ)) + (ρ0 + δρ)(∂φ)
+i(ρ0 + δρ)∂τ φ +
2m 4(ρ0 + δρ)
Z β Z
n z =}|0 {
gδρ2
d
= S0 [ρ0 ] +
dτ d r (−µ + gρ0 ) δρ +
2
0
7→ 0
o
z }| {
1
1
2
2
+ O(δρ3 , δρ, ∂φ)
+ iρ0 ∂τ φ +iδρ∂τ φ +
(∂(δρ)) + ρ0 (∂φ)
2m 4ρ0
Finally, discarding gradient terms involving massive fluctuations δρ,
Z β Z
h
i
g
ρ0
dτ dd r iδρ ∂τ φ + δρ2 +
S[δρ, φ] ' S0 [ρ0 ] +
(∂φ)2
2
2m
0
• First term has canonical structure ‘momentum × ∂τ (coordinate)’, cf. “pq̇”
• Second term describes “massive” fluctuations in “Mexican hat” potential
• Third term measures energy cost of spatially varying massless phase flucutations:
i.e. φ is a Goldstone mode
Gaussian integration over δρ:
2
g
(∂τ φ)2
i
δρ + ∂τ φ +
2
g
2g
z
}|
{
Z β Z
Z
Z
Z
h
β
2
gδρ2 i
d
d (∂τ φ)
dτ d r iδρ ∂τ φ +
= const. × exp −
dτ d r
D(δρ) exp −
2
2g
0
0
; effective action for low-energy degrees of freedom, φ,
Z
Z
1 β
1
ρ0
d
2
2
S[φ] ' S0 +
dτ d r (∂τ φ) + (∂φ) .
2 0
g
m
cf. Lagrangian formulation of harmonic chain (or massless Klein-Gordon field)
Z
Z
Z
m 2 1 2
d
2
S = dt d r
φ̇ − ks a (∂φ) = dx ∂ µ φ∂µ φ
2
2
i.e. low-energy excitations involve collective phase fluctuations with a spectrum ωk =
gρ0
|k|
m
However, action differs from harmonic chain in that phase field φ
is periodic on 2π – i.e. the space is not simply connected
This means that it can support topologically non-trivial
field configurations involving windings – i.e. vortices
Lecture Notes
October 2006
Lecture XII
50
. Physical ramifications: current density
1 †
p̂
p̂ †
ĵ(r, τ ) =
a (r, τ ) a(r, τ ) −
a (r, τ ) a(r, τ )
2
m
m
ρ0
fun. int i −→
(∂ ψ̄(r, τ ))ψ(r, τ ) − ψ̄(r, τ )∂ψ(r, τ ) ' ∂φ(r, τ )
2m
m
Variation of action S[δρ, φ] ;
i∂τ φ = −gδρ,
i.e. ∂φ is measure of (super)current flow
i∂τ δρ =
ρ0 2
∂ φ=∂·j
m
• First equation: system adjusts to fluctuations of density
by dynamical phase fluctuation
• Second equation ; continuity equation (conservation of mass)
Crucially, s.p.e. possess steady state solutions with non-vanishing
ρ0
current flow: if φ independent of τ , δρ = 0 and ∂ 2 φ = ∂ · j = 0
m
For T < Tc , a configuration with a uniform
density profile can support a steady state divergenceless (super)flow
Superflow imposed by boundary conditions, cf. Coulomb: ∂ 2 φ = −
ρ(r)
e.g. φ(r) ' −φ0 ln |x2 + y 2 | translates to a line vortex
Notice that a ‘mass term’ in the phase action (viz. mφ φ2 ) would spoil this property,
i.e. the phenomenon of superflow is intimately linked to the Goldstone mode
. Steady state current flow in normal environments is prevented by the mechanism of
energy dissipation, i.e. particles scatter off imperfections inside the system and thereby
convert part of their energy into the creation of elementary excitations
How can dissipative loss of energy be avoided?
Trivially, no energy can be exchanged if there are no elementary excitations to create
In reality, this means that the excitations of the system should be
energetically inaccessible (k.e. of carriers too small to create excitations)
But this is not the case here! there is no energy gap (ωk = vs |k|)
However, there is an ingenuous argument due to Landau (see notes) showing
that a linear excitation spectrum can stabilize dissipationless transport for v < vs
Lecture Notes
October 2006
Lecture XII
51
Cooper instability of electron gas
In the final section of the course, we will explore a pairing instability of the electron
gas which leads to condensate formation and the phenomenon of superconductivity.
. History:
• 1911 discovery of superconductivity (Onnes)
• 1950 Development of (correct) phenomenology (Ginzburg-Landau)
• 1951 “isotope effect” — clue to (conventional) mechanism
• 1957 BCS theory of conventional superconductivity (Bardeen-Cooper-Schrieffer)
• 1976 Discovery of “unconventional” superconductivity (Steglich)
• 1986 Discovery of high temperature superconductivity in cuprates (Bednorz-Müller)
• ???? awaiting theory?
. (Conventional) mechanism: exchange of phonons induces non-local electron interaction
Ĥ 0 = Ĥ0 +
|Mq |2 ~ωq
c†
c†
c c
2 − (~ω )2 k−qσ k0 +qσ 0 k0 σ 0 kσ
(
−
)
k
k−q
q
0
kk q
X
Electrons can lower their energy by sharing lattice polarisation
As a result electrons can condense as pairs into state with energy gap to excitations
. Cooper instability
Consider two electrons above filled Fermi sea:
Is weak pair interaction V (r1 − r2 ) sufficient to create bound state?
Consider variational state
spin singlet
}|
symm. gk = g−k
{spatial
zX }|
{
1
ik·(r1 −r2 )
ψ(r1 , r2 ) = √ (| ↑1 i ⊗ | ↓2 i − | ↑2 i ⊗ | ↓1 i)
gk e
2
|k|≥k
z
F
Applied to (spin-independent) Schrödinger equation: Ĥψ = Eψ
X
X
gk [2 k + V (r1 − r2 )] eik·(r1 −r2 ) = E
gk eik·(r1 −r2 )
k
Fourier transforming equation: × L1d
X
k0
Lecture Notes
Vk−k0 gk0 = (E − 2k )gk ,
k
RL
0
0
dd (r1 − r2 )e−ik ·(r1 −r2 )
Z
1
0
Vk−k0 = d dd r V (r)ei(k−k )·r
L
October 2006
Lecture XII
52
− LVd
0
{|k − F |, |k0 − F |} < ωD
otherwise
X
X
V X
V X
1
1
V X
− d
gk0 = (E − 2k )gk 7→ − d
gk0 =
=1
gk 7→ − d
L k0
L k E − 2k k0
L
E
−
2
k
k
k
If we assume Vk−k0 =
Z
Z
Z
1 X
dd k
1
Using d
is DoS
=
= ν() d ∼ ν(F ) d, where ν() =
d
L k
(2π)
|∂k k |
Z F +ωD
V X
1
d
ν(F )V
E − 2F − 2ωD
− d
' −ν(F )V
=
ln
=1
L k E − 2k
E − 2
2
E − 2F
F
In limit of “weak coupling”, i.e. ν(F )V 1
− ν( 2 )V
E ' 2F − 2ωD e
F
• i.e. pair forms a bound state (no matter how small interaction!)
• energy of bound state is non-perturbative in ν(F )V
. Radius of pair wavefunction: g(r) =
Using gk =
1
2k −E
× const., ∂k =
P
k
∂k ∂
∂k ∂k
gk eik·r ,
= v ∂∂k , and
δ(r − r0 )
}|
{
z
Z
Z
X
1 X
2
d
d 0
0 1
ik·(r−r0 )
∗ 0
|∂k gk | = d r d r r · r d
e
g(r)g (r ) = dd r r2 |g(r)|2 ,
Ld k
L k
2
R F +ωD
R d 2
P
1
2 ∂
2
2
d
ν()
v
d
r
r
|g(r)|
|∂
g
|
k
k
∂
2−E
hr2 i = R d
= Pk
= F R F +ωD
2
1
|g
|
d r|g(r)|2
d ν() (2−E)
k
k
2
F
R
+ω
F
D
4d
vF2 4
vF2
(2−E)4
' R F F+ωD d
=
3 (2F − E)2
(2−E)2
F
if binding energy 2F − E ∼ kB Tc , Tc ∼ 10K, vF ∼ 108 cm/s, ξ0 = hr2 i1/2 ∼ 104 Å,
i.e. other electrons must be important
. BCS wavefunction
Two electrons in a paired state has wavefunction
1
φ(r1 − r2 ) = √ (| ↑1 i ⊗ | ↓2 i − | ↓1 i ⊗ | ↑2 i)g(r1 − r2 )
2
Drawing analogy with Bose condensate, consider variational state
N/2
ψ(r1 · · · r2N ) = N
Lecture Notes
Y
n=1
φ(r2n−1 − r2n )
October 2006
Lecture XII
53
Is ψ compatible with Pauli principle? For a single pair,
Z L
Z L
1
d
dd r2 g(r1 − r2 )c†↑ (r1 )c†↓ (r2 )|Ωi
d r1
|φi = d
L 0
0
δk+k0 ,0 gk
z Z
{
Z L }|
L
X
X 1
0
d
ik·r1 ik ·r2 † †
d
c
|Ωi
=
=
c
gk c†k↑ c†−k↓ |Ωi
e
d
r
g(r
−
r
)e
d
r
2
1
2
1
k↑ k0 ↓
2d
L
0
0
k
k,k0
where gk =
Then, of the terms in the expansion of
#
"
N
Y
X
|ψi =
gkn c†kn ↑ c†−kn ↓ |Ωi
n=1
1
Ld
R
dd r g(r)eik·r
kn
those with all kn s different survive
Generally, more convenient to work in grand canonical ensemble
where one allows for (small) fluctuations in the total particle number, viz.
cf. coherent state of pairs
z "
}|
# {
Y
X
(uk + vk c†k↑ c†−k↓ )|Ωi ∼ exp
|ψi =
gk c†k↑ c†−k↓ |Ωi
k
k
where normalisation demands u2k + vk2 = 1 (exercise)
1 |k| < kF
In non-interacting electron gas vk =
0 |k| > kF
In interacting system, to determine the variational parameters, (uk , vk ),
one can use a variational principle, i.e. to minimise hψ|Ĥ − F N̂ |ψi
. BCS Hamiltonian
However, since we are interested in both the g.s. energy and spectrum of excitations,
we will follow a different route and explore the model Hamiltonian
Ĥ =
X
k
Lecture Notes
k c†kσ ckσ −
V X † †
c 0 c 0 c−k↓ ck↑
Ld kk0 k ↑ −k ↓
October 2006
Lecture XIII
54
Lecture XIII: BCS theory of Superconductivity
. From Cooper argument, two electrons above Fermi sea can form a bound state
Z L
Z L
1
d
|φi = d
d r1
dd r2 g(r1 − r2 )c†↑ (r1 )c†↓ (r2 )|Ωi
L 0
0
δk+k0 ,0 gk
z Z
{
Z L }|
L
X
X 1
d
d
ik·r1 ik0 ·r2 † †
c
c
|Ωi
=
e
gk c†k↑ c†−k↓ |Ωi
d
r
d
r
g(r
−
r
)e
=
0
1
2
1
2
k↑ k ↓
2d
L
0
0
k
k,k0
where gk =
1
Ld
R
dd r g(r)eik·r denotes pair wavefunction
To develop insight into the many-body system, consider effective theory
involving only interaction between pairs: BCS Hamiltonian
Ĥ =
X
k
k c†kσ ckσ −
V X † †
c 0 c 0 c−k↓ ck↑
Ld kk0 k ↑ −k ↓
Transition to condensate signalled by development of “anomalous average”
b̄k = hg.s.|c†k↑ c†−k↓ |g.s.i, i.e. |g.s.i is not an eigenstate of particle number!
Since we expect quantum fluctuations of b̄k to be small, we may set
c†k↑ c†−k↓
small
}|
{
z
† †
= b̄k + ck↑ c−k↓ − b̄k
(cf. approach to BEC where a†0 replaced by a C-number) so that
ξk
X z }| { †
V X † †
(k − µ) ckσ ckσ − d
Ĥ − µN̂ =
c 0 c 0 c−k↓ ck↑ ,
L kk0 k ↑ −k ↓
kσ
X †
V X
† †
0
0
b̄k c−k0 ↓ ck0 ↑ + bk ck↑ c−k↓ − b̄k bk + O(small)2
'
ξk ckσ ckσ − d
L kk0
kσ
Setting
V X
bk ≡ ∆, obtain the “Bogoliubov-de Gennes” or “Gor’kov” Hamiltonian
Ld k
Ĥ − µN̂ =
=
X
kσ
X
k
Ld |∆|2
† †
¯
−
∆c−k0 ↓ ck0 ↑ + ∆ck↑ c−k↓ +
V
k
X
ck↑
Ld |∆|2
ξk −∆
+
ξ
+
c−k↓
†
k
¯ −ξk
−∆
c−k↓
V
k
ξk c†kσ ckσ
c†k↑
X
For simplicity, let us for now assume that ∆ is real
(soon we will see that global phase is arbitrary...)
Lecture Notes
October 2006
Lecture XIII
55
Bilinear in fermion operators, Ĥ − µN̂ diagonalised by transformation
ck↑
c†−k↓
OT
z
}|
{ γk↑
uk vk
=
†
−vk uk
γ−k↓
where anticommutation relations require OT O = 1,
i.e. u2k + vk2 = 1 (orthogonal transformations)
Substituting, transformed Hamiltonian OHOT diagonalised if (Ex.)
2ξk uk vk + ∆(vk2 − u2k ) = 0
i.e. setting uk = sin θk and vk = cos θk ,
tan 2θk = −
∆
,
ξk
∆
sin 2θk = p 2
,
ξk + ∆2
ξk
cos 2θk = − p 2
ξk + ∆2
(N.B. for complex ∆ = |∆|eiφ , vk = eiφ cos θk )
As a result,
2
γk↑
Ld ∆2 X †
(ξk + ∆2 )1/2
ξk +
+
Ĥ − µN̂ =
γk↑ γ−k↓
†
γ−k↓
−(ξk2 + ∆2 )1/2
V
k
k
X
Ld ∆2 X 2
†
=
(ξk − (ξk2 + ∆2 )1/2 ) +
+
(ξk + ∆2 )1/2 γkσ
γkσ
V
k
kσ
X
†
Quasi-particle excitations, created by γkσ
, have minimum energy ∆
g.s. identified as state annihilated by all the quasi-particle operators γkσ , i.e.
Y
Y
γ−k↓ γk↑ |Ωi =
(uk c−k↓ + vk c†k↑ )(uk ck↑ − vk c†−k↓ )|Ωi
|g.s.i ≡
k
=
Y
k
vk (uk c−k↓ c†−k↓
k
+ vk c†k↑ c†−k↓ )|Ωi = const. ×
Y
(uk + vk c†k↑ c†−k↓ )|Ωi
k
in fact, const. = 1
Note that global phase of ∆ is arbitrary, i.e. |g.s.i continuously degenerate (cf. BEC)
. Self-consistency condition: BCS gap equation
∆≡
Lecture Notes
V X
V X
V X
† †
b̄
=
hg.s.|c
c
|g.s.i
=
uk vk
k
k↑ −k↓
Ld k
Ld k
Ld k
V X
V X
∆
p
=
sin
2θ
=
k
d
d
2
2L k
2L k
ξk + ∆2
October 2006
Lecture XIII
56
Z
V ν(µ) ωD
1
V X
1
p
=
i.e. 1 =
dξ p
= V ν(µ) sinh−1 (ωD /∆)
d
2
2
2
2
2L k
2
ξk + ∆
ξ
+
∆
−ωD
1
if ωD ∆, ∆ ' 2ωD e− ν(µ)V
!
1
ξ
1
k
1− p 2
7→ θ(µ − k ),
. In limit ∆ → 0, vk2 = cos2 θk = (cos 2θ + 1) =
2
2
ξk + ∆2
and |g.s.i collapses to filled Fermi sea with chemical potential µ
For ∆ 6= 0, states in vicinity of µ rearrange into condensate of Cooper pairs
p
. Spectrum of quasi-particle excitations ξk2 + ∆2 shows rigid energy gap ∆
. Density of quasi-particle states:
Z
q
p
1 X
2
2
ρ() = d
δ( − ξk + ∆ ) = dξν(ξ)δ( − ξ 2 + ∆2 )
L kσ
X Z ∞ δ ξ − s(2 − ∆2 )1/2
= 2ν(µ)Θ( − ∆)
dξ ≈ ν(µ)
2
∂
( − ∆2 )1/2
(ξ 2 + ∆2 )1/2 ∂ξ
s=±1 −∞
i.e. spectral weight transferred from Fermi surface to interval [∆, ∞]
. Field Theory of Superconductivity
Starting point is Hamiltonian for local (contact) pairing interaction:
"
#
Z
2
X
p̂
Ĥ = dd r
c†σ (r)
cσ (r) − V c†↑ (r)c†↓ (r)c↓ (r)c↑ (r)
2m
σ
. Quantum partition function: Z = tr e−β(Ĥ−µN̂ )
R
x≡(τ,r)
Z=
Z
ψ(β)=−ψ(0)
n
D(ψ̄, ψ) exp −
z
Z
β
}|Z
dτ
0
0
dx
{
L
d
d r
hX
σ
ψ̄σ
io
p̂2
− µ ψσ − V ψ̄↑ ψ̄↓ ψ↓ ψ↑
∂τ +
2m
where ψσ (r, τ ) denote Grassmann (anticommuting) fields
Options for analysis:
• perturbative expansion in V ? No — transition to condensate non-perturbative in V
• Mean-field (saddle-point) analysis
To prepare for s.p. analysis, it is useful to trade Grassmann fields for
“slow fields” that parameterise the low-energy fluctuations of condensed phase
Lecture Notes
October 2006
Lecture XIII
57
This is achieved by a general technique known as...
. Hubbard-Stratonovich decoupling:
Introduce complex commuting field ∆(r, τ ) whose expectation value
translates to that of “anomalous average” hc†↑ c†↓ i
1
V
e
R
Z
dx ψ̄↑ ψ̄↓ ψ↓ ψ↑
=
Using identity
Rβ
0
¯ ∆) exp
D(∆,
n
dτ ψ̄↓ ∂τ ψ↓ = −
−
Rβ
0
Z
dx
¯
(∆+V
ψ̄↑ ψ̄↓ )(∆+V ψ↓ ψ↑ )−V ψ̄↑ ψ̄↓ ψ↓ ψ↑
zV
h |∆(r, τ )|2
(∂τ ψ̄↓ )ψ↓ =
V
Rβ
0
}|
{
io
¯ ↓ ψ↑ + ∆ψ̄↑ ψ̄↓ )
+ (∆ψ
ψ↓ ∂τ ψ̄↓
(p)
(h)
[Ĝ0 ]−1
[Ĝ0 ]−1
z
z
}|
}|
{
{
Z β Z L
Z β Z L
2 2
2 2
~
∂
~
∂
dd r ψ̄↓ ∂τ −
dτ
dd r ψ↓ ∂τ +
dτ
− µ ψ↓ =
+ µ ψ̄↓
2m
2m
0
0
0
0
(p/h)
where Ĝ0
denotes GF or propagator of free particle/hole Hamiltonian,
Z
Z
R
|∆|2
¯ ∆)e− dx g
Z = D(ψ̄, ψ) D(∆,
Gorkov Hamiltonian Ĝ −1
Nambu spinor Ψ̄
z
}|
{ i
h
z }| { [Ĝ(p) ]−1
ψ↑
∆
0
× exp − dx ( ψ̄↑ ψ↓ )
(h) −1
¯
ψ̄↓
∆
[Ĝ0 ]
Z
Z
Z
R
|∆|2
¯ ∆)e− dx g exp − dx Ψ̄Ĝ −1 Ψ
= D(ψ̄, ψ) D(∆,
Z
Using Gaussian Grassmann field
" integral:
#
Z
X
D(Ψ̄, Ψ) exp −
Ψ̄i Aij Ψj = det A = exp[ln det A] = exp[tr ln A]
ij
Z=
Z
h
¯
D(∆, ∆) exp −
zZ
Effective action S[∆]
}|
{
i
|∆|2
−1
dx
+ tr ln Ĝ [∆]
V
meaning of trace
i.e. Z expressed as functional field integral over complex scalar field ∆(x)
Formal expression is exact; but to proceed, we must invoke some approximation:
Lecture Notes
October 2006
Lecture XIII
58
. Examples of Hubbard-Stratonovich decoupling
−β(Ĥ−µN̂ )
e.g. (1) weakly interacting electron gas: Z ≡ tr e
"Z
β
Z
S=
dτ
d
d r
0
X
σ
1
+
2
Z
dd rdd r0
"Z
β
X
σ,σ 0
dτ
Seff =
dd r
0
X
σ
=
ψ̄(0)=−ψ̄(β)
ψ(0)=−ψ(β)
D(ψ̄, ψ)e−S[ψ̄,ψ]
p̂2
ψ̄σ (r, τ ) ∂τ +
− µ ψσ (r, τ )
2m
#
2
e
ψ̄σ (r, τ )ψ̄σ0 (r0 , τ )
ψσ0 (r0 , τ )ψσ (r, τ )
|r − r0 |
Coulomb interaction decoupled by scalar field, Z =
Z
Z
Z
Z
D(ψ̄, ψ)
Dφ e−Seff
#
p̂2
1
ψ̄σ (r, τ ) ∂τ +
− µ + ieφ ψσ (r, τ ) +
(∂φ)2
2m
8π
Physically: φ represents bosonic photon field that mediates Coulomb interaction
e.g. (2) itinerant ferromagnetism in Hubbard model
Z
S=
dτ
X
kσ
ψ̄kσ (∂τ + k − µ)ψkσ + 3U
where Sm =
1
2
P
αβ
Z
−2S2m
z
}|
{
X
dτ
ψ̄m↑ ψ̄m↓ ψm↓ ψm↑
ψ̄mα σαβ ψmβ (cf. electron spin operator)
Hubbard interaction decoupled by vector field, Z =
Z
Seff =
dτ
X
kσ
ψ̄kσ (∂τ + k − µ)ψkσ +
Z
m
Z
Z
D(ψ̄, ψ)
DM e−Seff
"
#
X M2
X
m
dτ
−
ψ̄mα Mm · σαβ ψmβ
2U
m
αβ
~ represents bosonic magnetisation field
Physically: M
Lecture Notes
October 2006
Lecture XIV
59
Lecture XIV: Field Theory of Superconductivity
Recap: Cast as field integral
Z=
Z
z
n Z
D(ψ̄, ψ) exp −
ψ(β)=−ψ(0)
0
β
(p)
[Ĝ0 ]−1
}|Z
{
z
}|
{
hX
io
L
2
p̂
dd r
ψ̄σ ∂τ +
− µ ψσ − V ψ̄↑ ψ̄↓ ψ↓ ψ↑
dτ
2m
0
σ
x≡(τ,r)
R
dx
local pair interaction may be decoupled by Hubbard-Stratonovich field, ∆(x)
Z
¯ ∆)e−S[∆] with
Integrating over the Grassmann fields, ψ̄σ , and ψσ , Z = D(∆,
Z
p/h
[Ĝ0 ]−1
−1
dx hx|tr2 ln Ĝ [∆]|xi
Z
z
}|
{
|∆|2
−1
S[∆] = dx
− tr ln Ĝ [∆] ,
Ĝ −1 =
V
= ∂τ
(p)
[Ĝ0 ]−1
¯
∆
+
p̂2
/−
−µ
2m ∆
(h) −1
[Ĝ0 ]
To proceed further, it was necessary to invoke some approximation
. I. Mean-field theory: far from critical temperature, Tc , we expect
field integral to be dominated by saddle-point:
Z
1 ¯
¯ ∆ + |δ∆|2 )
δ∆ + δ ∆
δS ≡ S[∆ + δ∆] − S[∆] = dx (∆
V
h
i
0 δ∆
−1
−1
+ tr ln Ĝ
−tr ln Ĝ +
¯
δ∆
0
0 δ∆
= (· · ·) − tr ln 1 + Ĝ
¯
δ∆
0
Z
1 ¯
0 δ∆
¯
= dx (∆ δ∆ + δ ∆ ∆) − tr Ĝ
+ O(|δ∆|2 )
¯
δ∆
0
V
Z
¯
= (· · ·) − dx G21 (x, x)δ∆(x) + G12 (x, x)δ ∆(x)
+ O(|δ∆|2 )
where tr[Ĝ21 δ∆] =
i.e. ∆(x) obeys the saddle-point condition:
Z
dx hx|Ĝ21 δ∆|xi =
Z
dx G21 (x, x)δ∆(x)
δS
∆(x)
=
− G12 (x, x) = 0
¯
V
δ∆
With the Ansatz ∆(x) = ∆ const., Ĝ|ki = G(k)|ki, with |ki ≡ |ωn , ki and
~2 k 2
−iωn + ξk
∆
−1
G (k) =
,
ξ
=
−µ
k
¯
∆
−iωn − ξk
2m
1
−iωn − ξk
−∆
G(k) =
¯
−∆
−iωn + ξk
−ωn2 − ξk2 − |∆|2
Lecture Notes
October 2006
Lecture XIV
60
i.e. ∆ obeys the gap equation:
p
e−ik·x / βLd/2
z }| {
X
∆
1 X
∆
1 X
= hx|Ĝ12 |xi =
hx|ki
G12 (k)hk|xi =
G
(k)
=
12
V
βLd k
βLd ω ,k ωn2 + Ek2
k
n
with Ek =
p
ξk2 + |∆|2 and k · x = ωn τ − k · r
Using (fermionic) Matsubara frequency summation
X
X
β
h(ωn ) =
Res h(−iz) βz
e + 1 z=zp
ω
p
n
1
1
and
, zp = ±Ek with residue h(zp ) = ±
(z − Ek )(−z − Ek )
2Ek
∆
1 X
1
1
∆
1 X
∆
−
= d
=
tanh(βEk /2)
−βE
βE
d
k + 1
V
L k
e
e k + 1 2Ek
L k
2Ek
with h(−iz) =
For T = 0, β → ∞,
Z
Z
1 X 1
dξν(ξ)
ν(0) ~ωD
dξ
1
~ωD
−1
p
p
= d
=
'
= ν(0) sinh
,
V
L k 2Ek
2 −~ωD ξ 2 + |∆|2
|∆|
2 ξ 2 + |∆|2
1
i.e. |∆| ' 2~ωD exp −
ν(0)V
For T = Tc , ∆ = 0,
Z ~ωD
tanh(βc ξ/2)
1
1
' ν(0)
dξ
' ν(0) ln(1.14βc ~ωD ),
kB Tc ' 1.14~ωD exp −
V
2ξ
ν(0)V
−~ωD
. II. Ginzburg-Landau theory: since ∆ develops continuously from zero,
close to Tc , we may develop perturbative expansion in (small) ∆(x)
0 ∆
−1
−1
Noting :
Ĝ [∆] = Ĝ0 1 + Ĝ0 ¯
,
Ĝ0 ≡ Ĝ(∆ = 0)
∆ 0
tr ln Ĝ
−1
[∆] =
tr ln Ĝ0−1
2
1
0 ∆
− tr Ĝ0 ¯
+ ···,
∆ 0
2
ln(1 + z) = −
∞
X
(−z)n
n=1
n
−1
• Zeroth order term in ∆ ; ‘free particle’ contribution, viz. Z0 = etr ln Ĝ0 = det Ĝ0−1
• First (and all odd) order term(s) absent
Lecture Notes
October 2006
Lecture XIV
61
• Second order term:
(p/h)
(p/h)
Noting Ĝ0
G0 (k)
|ωn , ki
}|
{
z}|{ z
|ki =(−iωn + (~2 k2 /2m − µ))−1 |ki
Z
X
1
and using id. =
|kihk|,
∆k = p
dx e−ik·x ∆(x)
d
βL
k
√
βLd
z }| {
X (p)
(h)
¯
G0 (k) hk|∆|k 0 i G0 (k 0 )hk 0 |∆|ki
∆k0 −k /
(p)
(h) ¯
tr[Ĝ0 ∆Ĝ0 ∆]
=
kk0
pairing susceptibility Π(q)
z
}|
{
X
X (p)
1
(h)
¯q
∆q ∆
G (k)G0 (k + q)
βLd k 0
q
q=k0 −k
=
i.e. Π(ωm , q) =
1
1
1 X
,
d
βL ω ,k −iωn + ξk −i(ωn + ωm ) − ξk+q
ξk =
n
~2 k2
−µ
2m
Combined with bare term,
S[∆] =
X 1
V
q
¯ q ∆q + O(|∆|4 )
+ Π(q) ∆
In principle, one can evaluate Π(q) explicitly;
however we can proceed more simply by considering a...
. ‘Gradient expansion’:
τ
}|
=0
z }| {
∂
∂
1
Π(q, ωm ) = Π(0) + iωm
Π(0) +qα
Π(0) + qα qβ
∂(iωm )
∂qα
2
K
2
= Π(0) + iωm τ + q2 + O(ωm
, q4 )
2
z
{
1
K = ∂q2 Π(0)
}| d {
∂2
2
Π(0)
+O(ωm
, q4 )
∂qα ∂qβ
Kδαβ ,
z
At large enough temperatures, kB Tc 1/τ , dynamics may be neglected
altogether (viz. ∆(x) ≡ ∆(r)) and one obtains
. Ginzburg-Landau action
Z
S[∆] =
=β
Lecture Notes
β
dτ
Xt
0
q
Z
2
+ Kq
2
¯ q ∆q + O(|∆|4 )
∆
t
K
d r |∆|2 + |∂∆|2 + u|∆|4 + · · ·
2
2
d
October 2006
Lecture XIV
where
62
t
1
= + Π(0), and K, u > 0
2
V
(cf. weakly interacting Bose gas)
. Landau Theory: If we assume that dominant contribution to Z = e−βF arises from
minumum action, i.e. spatially homogeneous ∆ that minimises
S[∆]
t
= |∆|2 + u|∆|4
d
βL
2
2
one obtains |∆| t + 4u|∆|
|∆| =
= 0,
0p
t>0
−t/4u t < 0
i.e. for t < 0, spontaneous breaking of continuous U(1) symmetry associated
with phase ; gapless fluctuations — Goldstone modes
S [!]
|!|
Im [!]
Re [!]
Tc
t
1
With Π(0) ' −ν(0) ln(1.14β~ωD ) (as before), Tc fixed by condition ≡ + Π(0)|T =Tc = 0,
2
V
1
i.e.
= ν(0) ln(1.14βc ~ωD )
V
Therefore
t
1
= + Π(0, T ) = ν(0) ln
2
V
βc
β
= ν(0) ln
T
Tc
T − Tc
= ν(0) ln 1 +
Tc
' ν(0)
T − Tc
Tc
i.e. physically t is a ‘reduced temperature’
Lecture Notes
October 2006
Lecture XV
63
Lecture XV: Superconductivity and Gauge Invariance
. Recall: Starting with Hamiltonian for electrons with local (contact) pairing interaction:
#
"
Z
2
X
p̂
†
†
cσ (r) − V c↑ (r)c↓ (r)c↓ (r)c↑ (r)
Ĥ = dd r
c†σ (r)
2m
σ
quantum partition function can be expressed as field integral involving complex field
Z
X 1
¯
−S[
∆,∆]
¯ ∆]e
Z = D[∆,
,
S=
+ Π(q) |∆q |2 + O(∆4 )
V
q
where pair susceptibility
Π(q) =
1 X (p)
(h)
G (k)G0 (k + q),
βLd k 0
(p/h)
G0
(k) =
1
−iωn + /− (~2 k2 /2m
− µ)
Gradient expansion of action ; Ginzburg-Landau theory
Z
K
t
2
2
4
d
S[∆] = β d r |∆| + |∂∆| + u|∆| + · · ·
2
2
where
t
1
T − Tc
= + Π(0) ' ν(0)
, and constants K, u > 0
2
V
Tc
. What about the physical properties of the condensed phase?
To establish origin of perfect diamagnetism (and zero resistance),
one must accommodate electromagnetic field in Ginzburg-Landau action
. Inclusion of EM field into action requires minimal substitution: p̂ → p̂ − eA
and addition of action for photon field (~ = 1, c = 1, 4π0 = 1, µ0 = 1/0 c2 = 4π. )
Z
1
SEM = − dx
Fµν F µν ,
Fµν = ∂µ Aν − ∂ν Aµ
4µ0
Repitition of field theory in presence of vector field obtains
generalised Ginzburg-Landau theory: Z =
Z
Z
DA
¯ −S
D[∆, ∆]e
L
Z
S=β
EM
z
}|
{i
K
1
d
2
2
4
d r |∆| + |(∂ − i2eA)∆| + u|∆| +
(∂ × A)2
2
2
8π
ht
Factor of 2 due to pairing (focusing only on spatial fluctuations of A)
. Gauge Invariance: Action invariant under local gauge transformation
A 7→ A0 = A − ∂φ(r),
Lecture Notes
∆ 7→ ∆0 = e−2ieφ(r) ∆
October 2006
Lecture XV
64
(∂ − i2eA)∆ 7→ (∂ − i2e(A − ∂φ))e−2ieφ(r) ∆ = e−2ieφ(r) (∂ − i2eA)∆
i.e. |(∂ − i2eA)∆|2 (as well as ∂ × A) invariant
. “Anderson-Higgs mechanism”: phase of complex order parameter ∆ = |∆|e−2ieφ(r)
can be absorbed into A 7→ A0 = A − ∂φ(r)
Z
t
K
m2ν 2
1
d
2
2
4
2
S = β d r |∆| + (∂|∆|) +
A + u|∆| +
(∂ × A)
2
2
2
8π
where m2ν = 4e2 K|∆|2
i.e. massless phase degree of freedom φ(r) has disappeared
and photon field A has acquired a ‘mass’ !
Example of a general principle:
“Below Tc , Goldstone bosons (φ) and gauge field A conspire to create massive
excitations, and massless excitations are unobservable”, cf. electroweak theory
p
Coherence (healing) length ξ = K/t describes scale over which fluctuations
are correlated – diverges on approaching transition
. Meissner effect: minimisation of action w.r.t. A
B
z }| {
1
∂× (∂ × A) −m2ν A = 0
4π
7→
(∂ 2 − 4πm2ν )B = 0
B = 0 is the only constant uniform solution ; perfect diamagnetism
1/mν provides the length scale (London penetration depth),
over which a magnetic field can penetrates the superconductor at the boundary
Free energy of superconductor first proposed on phenomenological grounds — how?
...& why is crude gradient expansion so successful?
. Statistical Field Theory
Superconducting transition is an example of a “critical phenomena”
Close to critical point Tc , the thermodynamic properties of a system
are dictated by “universal” characteristics
To understand why, consider a simpler prototype:
the classical Ising (i.e. one-component) ferromagnet:
X
X
H = −J
Siz Sjz + B
Siz ,
Siz = ±1
hiji
i
Equilibrium Phase diagram?
Lecture Notes
October 2006
Lecture XV
65
S
1
2
H
Tc
T
What happens in the vicinity of critical point?
1. 1st order transition — order parameter (magnetisation) changes discontinuously;
correlation length (scale over which fluctuations correlated) remains finite
2. 2nd order transition — order parameter changes continuously;
correlation length diverges (ξ ∼ 1/t1/2 )
...motivates consideration of “hydrodynamic” theory which surrenders information
about microscopic length scales and involves a coarse-grained order parameter field
Z
−βF
Z=e
= DS(r) e−βHeff [S(r)]
with βHeff [S] constrained (only) by fundamental symmetry (translation, rotation, etc.)
Z
t 2 K
2
4
d
S + (∂S) + uS + · · · + BS
βHeff [S(r)] = d r
2
2
cf. Ginzburg-Landau Theory
. Landau theory: S(r) = S const.
βF min t 2
4
= S
S + uS ,
Ld
2
etc.
. Continuous phase transitions separate into Universality classes
with the same characteristic critical behaviour
E.g. (1) Ising model – liquid/gas: S → density ρ, B → pressure P
E.g. (2) Superconductivity – classical XY ferromagnet: ∆0 + i∆00 → (Sx , Sy )
Lecture Notes
October 2006