Download Luttinger Liquids

Luttinger Liquids
Thors Hans Hansson
Stockholm University
Quantum Field Theory for Condensed Matter
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Outline
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Why is 1d special?
A particle moving in 1d will by necessity collide with its neighbors.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Why is 1d special?
A particle moving in 1d will by necessity collide with its neighbors.
Thus we expect collective excitations
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Why is 1d special?
A particle moving in 1d will by necessity collide with its neighbors.
Thus we expect collective excitations
We expect bosons with repulsive interactions to behave similar to
fermions.
To see this even for free particles - Evaluate the partition function!!
Thus, statistics cannot be distinguished from interactions.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
How to realize 1d fermions
Quantum wires.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
How to realize 1d fermions
Quantum wires.
Carbon nano-tubes
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
How to realize 1d fermions
Quantum wires.
Carbon nano-tubes
Constrictions in 2d electron gases in semiconductor interfaces
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
How to realize 1d fermions
Quantum wires.
Carbon nano-tubes
Constrictions in 2d electron gases in semiconductor interfaces
Trapped cold fermionic atoms
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
How to realize 1d fermions
Quantum wires.
Carbon nano-tubes
Constrictions in 2d electron gases in semiconductor interfaces
Trapped cold fermionic atoms
Spin chains (by continuum limit and mapping)
...
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Theoretical tools
Perturbation theory
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Theoretical tools
Perturbation theory
Renormalization Group
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Theoretical tools
Perturbation theory
Renormalization Group
Bosonization
In this lecture I will introduce a phenomenological approach, du to
Haldane, to 1d harmonic liquids, that applies to both bosons and fermions,
and which stresses their similarity.
In the next lecture I will use this to explain bosonization of 1d Fermi
systems and the concept of Luttinger Liquid, which is the generic state of
a 1d system, in a similar way as Fermi Liquids are the generic state of
Fermi systems in higher dimensions.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The Luttinger Hamiltonian
The model introduced by Luttinger, is given by the Hamiltonian
ˆ
ˆ
X
1
†
H=
vF dx ψα (iα∂x − kF )ψα −
dxdx 0 ρ(x)V (x − x 0 )ρ(x 0 )
2
α=±1
where α lables the two Fermi points, vF is the Fermi velocity, and
†
†
ρ = ψ+
ψ+ + ψ−
ψ− is the total density.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Next subject
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Bosons in 1d
Bosonic field theory in 1+1 dimensions,
ˆ
ˆ
S = dtdx φ? i∂t φ − dt H
which satisfies the periodic boundary condition φ(x + L) = φ(x). Varying
φ? gives the equation of motion
δH
i∂t φ(x, t) = ?
.
δφ (x, t)
and taking
ˆ
ˆ
1
1
?
2
dx φ (−∂x ) φ +
dxdx 0 φ? (x)φ? (x 0 )V (x − x 0 )φ(x)φ(x 0 ) ,
H=
2m
2
we get the non-linear Schrödinger equation
ˆ
1 2
i∂t φ(x, t) = −
∂ φ + dx 0 φ? (x 0 )V (x − x 0 )φ(x)φ(x 0 )
2m x
appropriate for spin-less bosons with mass m interacting via a
density-density interaction.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Quantization
From the action we get the canonical commutation relations,
[φ(x, t), φ† (x 0 , t)] = δ(x − x 0 ) ,
where the delta funtion is understood to be periodic with period L. In the
following, we shall use a phase-density representation,
√
φ† = ρ e iϕ ,
and a direct substitution gives,
0
[ρ(x), e iϕ(x ) ] = e iϕ(x) δ(x − x 0 )
0
assuming that [e iϕ(x) , e iϕ(x ) ] = 0 for x 6= x 0 . This relation is however a bit
tricky to use since the density operator for non-relativistic point particles is
a sum of delta functions at the particle positions xn ,
ρ(x) =
N
X
δ(x − xn )
n=1
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Quantization
It is instructive to start with a lattice version of
[φi (t), φ†j (t)] = δij .
Assuming that variables on different sites commute, we immediately gets
the equal time commutation relations
[ρi , e iϕj ] = e iϕj δij
or
[ϕi , ρj ] = iδij
Since we are going to study the low momentum, or long wave length
theory, it is tempting to take a naive continuum limit to get
[ϕ(x), ρs (x 0 )] = iδ(x − x 0 )
???
But this is not compatible with ρ(x) being a sum of delta functions!!
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The low momentum field ρs
We shall regard ρs as a smoothened field which only contains the long
wave length components of ρ. The short distance cutoff is set by the
average distance between the particles, a = L/N0 , or, equivalently, the
momentum cutoff is the inverse mean density ρ0 = N0 /L.
Since ϕ is an angle we have
ϕ(x + L) = ϕ(x) + Jπ
with J an even integer
Since
[N̂, e iϕ ] = e iϕ
The ”soliton” operator e iϕ(x) creates a unit charge at x, just as on the
lattice.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The low momentum field θ
Introduce a new low momentum field θ by,
ρs (x) = ρ0 + Π(x) =
1
∂x θ(x)
π
Π is the deviations of ρs from the constant density ρ0 .
Clearly, ϕ and Π are conjugate variables, but we also get
[ϕ(x), θ(x 0 )] =
iπ
(x − x 0 )
2
where (x) is the sign function.
What is θ ??
ˆ
N̂ =
L
dx
0
1
1
∂x θ(x) = [θ(L) − θ(0)]
π
π
which is half the winding number of the map from the circle in real space,
parametrized by the polar angle α = 2πx/L, to the circle in the space of
field configurations, parametrized by the angular field variable θ.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Field expansions and zero modes
Thus expand,
θ(x) = θ̂ +
πx
N̂ + θ̃(x)
L
where θ̃(x + L) = θ̃(x) and the operator θ̂0 is independent of x.
In analogy with the above, we write
ϕ(x) = ϕ̂ +
πx ˆ
J + ϕ̃(x)
L
so J is the zero mode of the total current.
The ”zero mode” operators θ̂ and ϕ̂ are defined by the relations
ˆ =i,
[ϕ̂, N̂] = [θ̂, J]
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Putting the particles back!
Haldane’s ansatz for the granular density operator ρg
ρg (x) = ∂x θ(x)
N
X
δ(θ(x) − nπ) =
n=1
N
X
δ(x − xn )
n=1
´
where θ(xn ) = n π. The integrated charge dx ρ is not changing
continuously, but jumps one unit at each of the N points xn .
To understand the prefactor ∂x θ, first consider the case of a constant
density so that θ(x) = πρ0 x = πNx/L, and
ρ0,g =
N
N
X
πN X πN
n
δ(
x − nπ) =
δ(x − L)
L
L
N
n=1
n=1
i.e. a sum over evenly spaced delta functions which we can interpret as the
density profile of a perfect crystal of point particles.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Putting the particles back
Check of consistency,
1
ρ̄(x) =
≈
ˆ
x+ 2
x− 2
N
1X
dx ρg (x) =
n=1
ˆ
θ(x+ 2 )
dθ δ(θ − nπ)
(1)
θ(x− 2 )
1
1 θ(x + 2 ) − θ(x − 2 )
≈ ∂x θ(x) = ρs (x)
π
π
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Another representation for ρg
The periodic delta function in the expression for ρg can be rewritten as,
ρg (x) =
∞
X
1
∂x θ(x)
e i2mθ(x)
π
m=−∞
This expression is useful if a couple of low order terms give a good
approximation to the full density operator ρ.
If there are many particles is the system, we can replace the operator N̂
with the average value N0 , and approximate
e i2mθ(x) ≈ e i2mkF x e i2mθ̃(x) .
when averaging, the m 6= 0 terms are oscillating rapidly, and can be
neglected when θ̃ only has momentum components much smaller that kF .
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Summary
For the granular density ρg to give a good description of the microscopic
density:
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Summary
For the granular density ρg to give a good description of the microscopic
density:
θ̃ must be small
and for the the sum to be dominated by the leading term(s)
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Summary
For the granular density ρg to give a good description of the microscopic
density:
θ̃ must be small
and for the the sum to be dominated by the leading term(s)
θ̃ must be a low momentum field
These restrictions are important. One can of course find a field θ so that
the expression for ρg is true for arbitrary particle positions, but this field
would not satisfy the commutation relations and cannot be used to
construct an effective low energy theory.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Next subject
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The Boson field operator
Recall
φ† =
√
ρ e iϕ ,
But what is the square root of a delta function??
Use the representation
r
1 − x2
e δ (x) = lim
→0
π
and thus
1
p
√
1
1 4 − x2
e 2 = (π) 4 δ (x/ 2)
δ (x) = lim
→0 π
so up to a normalization the square root is again a delta function. Thus,
Boson field operator
X
p
√
Ψ†B = A ρ0 + Π(x)
e i2mθ(x) e iϕ(x) ∼ ρ0 e iϕ
m
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Commutation relations
Following Haldane’s notation, I use Ψ†B rather than φ† to emphasize that
it depends on the low momentum fields θ and ϕ.
Using the notation φm = e i2mθ(x) e iϕ(x) we have
φm (x)φm (x 0 ) = e imπ(x−x
0 )−imπ(x 0 −x)
φm (x 0 )φm (x) = φm (x 0 )φm (x)
and the other combinations of φ†m (x) and φm (x 0 ) can be checked in the
same manner.
This shows that the commutator [ΨB (x), ΨB (x 0 )] = 0 for x 6= x 0 etc.
The δ-function in the commutator [ΨB (x), ΨB (x 0 )] also follows formally,
but care must be taken in the interpretation, since we are dealing with low
momentum fields.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The Fermion field operator
Defining
Fermion field operator
X
p
√ Ψ†F = A ρ0 + Π(x)
e i(2m+1)θ(x) e iϕ(x) ∼ ρ0 e iθ + e −iθ e iϕ
m
the formula from the previous slide is modified to
1
φm (x)φm (x 0 ) = e i(m+ 2 )π(x−x
0 )−i(m+ 1 )π(x 0 −x)
2
φm (x 0 )φm (x) = φm (x 0 )φm (x)
and we get the desired anti-commutation relation
{ΨF (x), ΨF (x 0 )} = 0
for
x 6= x 0
Thus, we have expressed fermion operators in terms of bosonic fields!!
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Next subject
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The quadratic Hamiltonian
Expand the interaction energy to second order in the deviations from
constant density and zero current to get,
ˆ
1
H=
dx vJ (∂x ϕ)2 + vN (∂x θ − πρ0 )2
2π
where vJ = πρ0 /m and vN are velocity parameters that depends on the
interaction.
Using the notation
ρ0 = N0 /L
vs =
√
r
vN vJ
g=
vJ
vN
we get
The Harmonic Liquid Hamiltonian
H=
vs
2π
ˆ
1
π
dx g (∂x ϕ̃)2 + (∂x θ̃)2 + [vJ Jˆ2 + vN (N̂ − N0 )2 ]
g
2L
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The meaning of g
H=
vs
2π
ˆ
1
π
dx g (∂x ϕ̃)2 + (∂x θ̃)2 + [vJ Jˆ2 + vN (N̂ − N0 )2 ]
g
2L
r
√
vJ
ρ0 = N0 /L vs = vN vJ
g=
vN
Recall N = N+ + N− and J = N+ − N− , so only for g = 1 do the right
movers decouple from the left movers. At this point vN = vJ as must be
the case for free particles.
g =1 corresponds to free fermions!!
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Lagrangian and equations of motion
vs
1
1
2
2
g (∂x ϕ̃) + (∂x θ̃)
L = ∂x θ̃∂t ϕ̃ −
π
2π
g
To decouple the two fields θ̃ and ϕ̃, we define,
χ± =
√
1
g ϕ̃ ∓ √ θ̃
g
to get
L=±
1 X
χ± ∂x (∂t ± vs ∂x )χα
4π ±
The dispersion relation is
(∂t ± vs ∂x )χ± = 0
which shows that χ+ describes a right-moving, and χ− a left moving
wave. It is now also clear that for given values of N and J, the spectrum is
that of a free boson, with the two chiralities described by χ± .
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Next subject
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Bosonization technology I
The first step in the calculation of the correlations functions is to express
Ψ†F ± in terms of the independent fields χ± ,
Ψ†F ± ∼
√
i
ρ0 e ∓iπρ0 x e 2
q
1 √
± g
g
q
1 √
χ+ + 2i
∓ g χ−
g
Note that for the free theory corresponding to g = 1, Ψ†F ± ∼ e iχ± .
introduce the imaginary time variable τ = it and use the formula
he iaχ± (x,τ ) e ibχ± (x
0 ,τ 0 )
i = e ab G± (x,t;x
0 ,t 0 )
where we defined the Euclidean Greens function
∓2π
.
G± =
∂x (−i∂τ ∓ vs ∂x )
or more explicitly,
ˆ
ˆ
dk
dω e iωτ +ikx
G± (x, τ ) = ∓2π
.
2π
2π ik(ω ∓ ivs k)
A direct evaluation of this integral gives
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Bosonization technology II
G± (x, τ ) = ln(x ± ivs τ ) .
and the corresponding Eucledian correlation functions
he iaχ± (x,τ ) e ibχ± (x
0 ,τ 0 )
i = (x ± ivs τ )ab
We can now calculate correlation functions, and the simplest example is,
hΨF ± (x, τ )Ψ†F ± (0, 0)i ∼
e ±iπρ0 x
,
(x + ivs τ )γ± (x − ivs τ )γ∓
where γ± = (1/g + g ± 2)/4. For τ = 0 we get the simple result,
hΨF ± (x, 0)Ψ†F ± (0, 0)ig =1 ∼
e ±iπρ0 x
1
+g
x 2g 2
We see that for g = 1 we have vN = vJ = vs = vF where vF is the Fermi
velocity, and, recalling that kF = πρ0 , we get
hΨF ± (x, 0)Ψ†F ± (0, 0)ig =1 ∼
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
e ±ikF x
(x ± ivs τ )
Quantum Field Theory for Condensed Matter
/ 34
Next subject
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Absence of quasiparticle poles
Note that in the expression,
hΨF ± (x, τ )Ψ†F ± (0, 0)i ∼
e ±iπρ0 x
,
(x + ivs τ )γ+ (x − ivs τ )γ−
there is no pole except for g = 1 !!
To actually calculate the Fourier transform of this expression to extract the
momentum distribution N(p) discussed in the context of Fermi liquids, is a
bit complicated. The result is however, as expected, that the discontinuity
totally disappears, which shows that there are no stable gapless charge
excitations in the system.
Although there are no gapless fermions in the spectrum, the bosonic
excitations are gapless. In this respect the Luttinger model differs
qualitatively from a superconductor. In an RG treatment one can in fact
see that there is a very delicate balance between a tendency towards
superconducting and charge density wave ordering.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Next subject
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The concept of Luttinger Liquid
In Haldane’s paper from 1981 two facts are stressed:
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The concept of Luttinger Liquid
In Haldane’s paper from 1981 two facts are stressed:
the above results would be essentially unchanged in a more realistic
system where the dispersion is no longer linear, corresponding to
non-harmonic terms in the energy
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The concept of Luttinger Liquid
In Haldane’s paper from 1981 two facts are stressed:
the above results would be essentially unchanged in a more realistic
system where the dispersion is no longer linear, corresponding to
non-harmonic terms in the energy
many other systems, e.g. certain spin chains, show the same
qualitative behavior in the low energy region.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
The concept of Luttinger Liquid
In Haldane’s paper from 1981 two facts are stressed:
the above results would be essentially unchanged in a more realistic
system where the dispersion is no longer linear, corresponding to
non-harmonic terms in the energy
many other systems, e.g. certain spin chains, show the same
qualitative behavior in the low energy region.
He coined the term ”Luttinger liquid” to describe all such system which
has low energy characteristics in common with the simple Luttinger model.
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Next subject
1
Fermons in 1d – Why, What, and How?
2
The Luttinger Hamiltonian
3
Haldane’s theory of Harmonic Liquids
The mean field approximation
Field Operators and Commutators
Harmonic Liquids
4
Bosonization and the concept of Luttinger Liquid
Fermionic correlation functions
Absence of quasiparticle poles
The concept of Luttinger Liquid
A non-trivial example
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34
Effect of a periodic substrate
So far we only used the simplest approximation to the operators ρg , ΨF
etc. An important example where terms with m 6= 0 must be kept is when
a periodic potential with wave vector q = 2nkF is added to the
Hamiltonian,
ˆ
Hper = V
dx cos(2nkF x)ρ(x)
corresponding to a commensurate filling of one particle in every n minima
of the potential. Substituting ρ with ρg , we get
ˆ
√
Hper = V ρ0 dx cos(2nθ̃(x))
where we picked up the n = ±m terms in the expression for ρg .
The result is a Sine-Gordon model, which describes a gapped phase. The
physics of the Sine-Gordon type systems is rich and interesting!
Thors Hans Hansson (Stockholm University)
Luttinger Liquids
Quantum Field Theory for Condensed Matter
/ 34