Download Effective Field Theories for Topological states of Matter

Effective Field Theories for
Topological states of Matter
Lectures given at the IIP-UFRN summer shool at
Natal/RN, Brazil, August 2015
T. H. Hansson
Fysikum
Stockholm University
Contents
1 Introduction
3
2 Non-interacting systems
2.1 The Integer Quantum Hall Effect . . . . . . . . . . . . . . . .
2.1.1 Quantization of the Hall conductance I . . . . . . . .
2.1.2 Quantization of the Hall conductance II . . . . . . . .
2.2 Topological effective actions . . . . . . . . . . . . . . . . . . .
2.2.1 The effective response action . . . . . . . . . . . . . . .
2.2.2 The topological field theory . . . . . . . . . . . . . . .
2.2.3 Topological action from functional bosonization . . . .
2.3 Topological band insulators . . . . . . . . . . . . . . . . . . .
2.3.1 Chern insulators . . . . . . . . . . . . . . . . . . . . .
2.3.2 TR invariant topological insulators in d=2 . . . . . . .
2.3.3 Topological classification from edge modes . . . . . . .
2.3.4 The topological field theory for the Quantum Spin Hall
effect . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Topological index for the TRI topological insulators . .
2.3.6 Topologicl insulators in d 6= 2 . . . . . . . . . . . . . .
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3 Weakly interacting systems
3.1 p-wave superconductors . . . . . . . . . . . . . . .
3.1.1 The Kitaev chain . . . . . . . . . . . . . . .
3.1.2 The two dimensional p-wave superconductor
3.2 Fluctuating s-wave superconductors . . . . . . . . .
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3.2.1
3.2.2
3.2.3
BF theory of s-wave superconductors - heuristic approach 39
The 3+1 dimensional BF theory . . . . . . . . . . . . .
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Microscopic derivation of the BF theory . . . . . . . .
41
4 Fractional Quantum Hall Liquids
4.1 The Laughlin states . . . . . . . . . . . . . . . . . . . . .
4.1.1 The Chern-Simons-Ginzburg-Landau theory . . .
4.1.2 From the CSGL theory to the effective topological
4.2 The Abelian hierarchy . . . . . . . . . . . . . . . . . . .
4.3 Non-abelian Quantum Hall states and CFT . . . . . . .
. . .
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theory
. . .
. . .
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49
A How to normalize the current
53
B An elementary derivation of eq. (50)
54
C The parity anomaly in 2+1 dimensions
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2
1
Introduction
I assume that you have learned how states of matter are classified according to
their symmetries. The basic tool in this classification is the order parameter
that was originally introduced by Landau. The order parameter ψa transforms
according to some representation of a symmetry group, and the ground states
are characterized by the expectation value hψa i. Typical examples are a ferromagnet, where G is the rotation group and ψa is the magnetization m,
~ and
superfluids, where ψa is the complex ”condensate wave function”, χ, which
is an eigenfunction corresponding to a macroscopic eigenvalue of the density
matrix.
There are, however, states with the same symmetry properties, which nevertheless are distinct in a way that I will now describe. Two states are defined
to be in the same universality class if one can be obtained from the other by
quasi-adiabatically changing the parameters of the Hamiltonian without the
(bulk) gap to the excited states ever closing. This condition has been put in
a more formal mathematical form by Hastings and Wen[1], but in these lectures we will not be concerned with questions of rigor. We shall only deal with
gapped states, where the notion of quasi-adiabatic continuation is most clearly
formulated and understood.1 We shall also restrict ourselves to fermionic systems, although there are many interesting possibilities for topological phases
of bosons both in condensed matter spin systems – ”spin liquids” – and in
various configurations of cold atoms.
We define trivial states as those that can be adiabatically connected to the
vacuum (which from the point of view of condensed matter physics, is a perfect
insulator). The non-trivial states are called topological. The topological states
can further be divided into subgroups depending on their properties. One
division, based on the properties of the excited states, is between integer and
fractional states. The latter are characterized by excitations with fractional
quantum numbers; examples are fractional quantum Hall liquids with charge
e/q, q odd integer, or superconductors with spin zero fermionic excitations.
Another division, due to Chen, Gu and Wen[2], is based on entanglement
properties of the ground state. They define a state to be short-range entangled
if and only if it can be transformed into a direct-product state through a
local unitary evolution,2 and examples are usual symmetry breaking states,
such as ferromagnets, as well as topological insulators. States which are not
short-range entangled are, by definition, long-range entangled, or topologically
1
Hastings and Wen also discusses systems without a bulk gap, but the arguments are
more complicated.
2
For a mathematical definition of ”local unitary evolution” see. Ref. [2].
3
ordered (TO) states.3
The integer topological phases can be understood in terms of non-interacting
fermions, which are classified according their dimension and symmetry properties[3,
4]. The classes can be trivial or non-trivial. The latter are characterized by
a non-trivial value of a topological index that can either take integer values, a
Z index, or the values ±1 which is a Z2 index. If we change the Hamiltonian,
without changing its symmetry, the index can change only at points where the
gap to excited states vanishes. A symmetry-protected topological phase (SPT),
is a nontrivial phase which however can be connected to the trivial phase if we
allow adiabatic changes (or local unitary evolutions) that do not respect the
symmetry in question. The topological index is related to physical characteristics of the system, which thus are robust against (not too drastic) symmetry
preserving changes in the Hamiltonian Note that this classification holds also
in the presence of disorder, again as long as the relevant symmetries are not
broken. It is also generally assumed that in many cases the presence of weak
interactions will not change the classification, but this is in general a difficult
issue to resolve.
A couple of comments are in order: Some of the ”symmetries” referred
to in this classification are not real symmetries in the sense of unitary operators under which the Hamiltonian is invariant. Instead, they are anti -unitary
transformations, and in this case the invariance of the Hamiltonian implies
conditions on the spectrum. There are also topologically non trivial integer
states integer that are not invariant under any anti-unitary symmetry - the
integer Hall liquid and the Chern insulators, which we discuss below, are prime
example. Note that since the topological properties of these states do not rely
on any symmetry, they are more robust than those of the SPT-states.
In these lectures, I will not dwell upon the general mathematical definitions
of TO or SPT states, but rather give several examples of topological states
of matter, and develop the mathematical formalism as it is needed. Since the
archetypical examples of topologically ordered state are the fractional quantum
Hall liquids, one easily gets the impression that only strongly correlated systems are topologically ordered. That this is not true has been known for a long
time, since weakly coupled superconductors in the presence of a fluctuating
electromagnetic field were known to be topologically ordered. Symmetry protected phases have also been known, at least as a theoretical possibility, since
the work of Haldane on topological effects in 1d spin chains. It is, however,
only with the fairly recent discovery of the time reversal invariant topologi3
This terminology is not universally adhered to in the literature. Often ”fractional” and
”topologically ordered” are used synonymously.
4
cal insulators in both two and three dimensions that the importance of SPT
states has been widely appreciated. From this new perspective it is natural
to first study the topological properties of the non-interacting fermi systems,
then move to weakly interacting systems, and finally to strongly coupled ones.
In the course of three lectures, I can pick just a few simple examples from each
group.
Topological states of matter have been studied by a great variety of methods. Important lessons can be learned from exactly soluble models. These are
mostly in 1d, but there are also important examples of topologically ordered
lattice models in higher dimensions, such as Kitaev’s honeycomb model and
its generalizations by Levin and Wen[5] and Walker and Wang[6].
Integer states of free fermions can be studied by directly diagonalizing the
Hamiltonian in question to get the full energy spectrum; for any but the simplest cases, this has to done numerically. Needless to say, a direct numerical
approach to interacting systems is much harder, and one is limited to a small
number of particles. Much is however happening in this area because of the
development of computer codes based on the density matrix renormalization
group and tensor networks. For a readable review with references to the original works, see Ref. [7]. In the study of the quantum Hall effect, an important
road to progress has been to construct explicit, ”representative”, many-body
wave functions. These have been obtained by a variety of methods, but only
in some simple cases they are known to be ground states of a model Hamiltonian. In the last section, I will briefly describe one important approach based
on conformal field theory.
The methods just mentioned are all describing the microscopic physics,
but there are complementary approaches based on effective low-energy quantum field theories.4 These theories are of different types, depending on how
much information about the system they encode. The first type closely resembles the Ginzburg-Landau theory used to describe the usual symmetry
breaking non-topological phases, but differ from these in that they have dynamical gauge fields. Examples are the Chern-Simons-Ginzburg-Landau theories for quantum Hall liquids[8], and the Ginzburg-Landau-Maxwell theories
for superconductors.5 These theories have information both about topological
4
In a sense of course all theories used in condensed matter physics are effective since
even the most ”microscopic” approaches only model a small part of the full Hilbert space
of nuclei and electrons.
5
Most textbooks in condensed matter theory will cover the Ginzburg-Landau-Maxwell
theory. For a modern text see [9]. Ref. [10], by S. Weinberg, one of the founders of effective
field theory, gives a good presentation from the field theoretic point of view. There are also
several excellent recent textbooks[11, 12] on the general subject of these lectures
5
quantities, such as charges and statistics of quasiparticles, and of collective
bosonic excitations such as plasmons or magnetorotons. The second type, are
the topological field theories, which do not describe any dynamics in the bulk,
but do carry information about topology, and also about excitations at the
boundaries of the system. Typical examples are the Wen-Zee Chern-Simons
theories for quantum Hall liquids[13, 14] and the BF-theories for superconductors and topological insulators[15, 16]. The third type, the effective actions for
external fields, or effective response actions, are in a strict sense not effective
theories, since they does not have any dynamical content, but encodes the
response to the system to external
R µνσ perturbations. A simple example is the
Chern-Simons term (σH /2) Aµ ∂ν Aσ , which encodes the Hall response of
a 2d system. Another example is the effective action for gravitational response
which encodes response to changes in the energy and momentum currents and
thus is related to finite temperature response. In these lectures the common
theme will be topological field theories. This is a vast subject with a large core
of results in mathematical physics, but with important applications both in
string theory and condensed matter physics. Here you will meet only the simplest of these theories, namely those containing the abelian Chern-Simons and
BF terms, which are closely connected, in their actions. On the other hand
you will see these topological terms appear in several connected, but distinct,
context and by so, hopefully, get a deeper understanding of the explanatory
power of topological field theory in condensed matter physics.
I have tried to make the notes reasonably self-contained, but of course I
will often refer to original articles or reviews for proof of various statements
and derivations of many of the equations. The list of references is by no mean
exhaustive; when there are good reviews I often cite these rather than the
original papers.
2
Non-interacting systems
We now consider two seemingly very simple systems. The first is a two dimensional electron gas in a strong magnetic field, and at properly tuned particle
density. The second is a band insulators with strong spin-orbit interaction.
These systems are of course not free in the sense that the electron-electron
interactions should have somehow mysteriously disappeared, but since there
is a gap to any excited state in the bulk for all values of the couplings,6 we
can imagining starting from the non-interacting system, and then adiabati6
Mathematically we can consider the state on a compact manifold, where we can ignore
the problems related to gapless edge states.
6
cally turning on the interaction without inducing a level-crossing (i.e. the gap
to the ground state does not close). This implies that the system remains
in the same universality class as the non-interacting system. In both these
examples, the topological properties are related to topological properties of
the ground-state wave functions, which in the simplest case are just a single
Slater determinant corresponding to a number of completely filled bands or
Landau levels. It is the object of topological band theory to characterize and
classify different states of matter based on various topological invariants that
characterize the mappings from the Brillouin zone to various spaces of ground
state wave functions, or equivalently various families of Hamiltonians. As already mentioned, the topological properties of these states, do not rely on any
special microscopic Hamiltonian. Instead they are captured in effective topological field theories, and the response to external fields is encoded in effective
response action.
2.1
The Integer Quantum Hall Effect
The integer quantum Hall effect (IQHE) that was discovered in 1980[17], is observed when a very clean (mobility about 105 cm2 /V s) two dimensional electron
gas is cooled below two K and subjected to a perpendicular magnetic field of
the order of 20 T. The basic observation is that the conductivity is quantized in
integer units of the fundamental quantum of conductance, σ0 = e2 /h = e2 /2π
(I will always put c = 1, and often ~ = 1).
If we neglect electron-electron interactions, this is the famous Landau problem and we know that the energy is quantized as En = n~ωc with the cyclotron
frequency ωc = eB/m. Each of these Landau levels has a macroscopic degeneracy such that there is one quantum state in the area 2π`2B = 2π/eB which
corresponds to a unit flux, φ0 = h/e = 2π/e, and where we introduced the
magnetic length `B . To get wave functions, we must pick a gauge, and the
choice should be dictated by the symmetries of the problem. A simple choice
is to take a cylindrical geometry, and pick the gauge Ax = −By, where x is
the coordinate around the circumference of the cylinder that we take to be
L = 2πR.
Let us first assume that the density of electrons is adjusted so that it
precisely fills the lowest Landau Level (LLL). (Experimentally this is usually
done by changing the magnetic field, rather than the density.) The many body
wave function is now a Slater determinant formed from the functions,
x
2 /2`2
B
ψm (x, y) = N eim R e−(y−ym )
= Ñ eimz e−y
2 /2`2
B
,
(1)
where z = (x + iy)/R, and ym = `2B /R. Thus, the single-particle functions are
7
plane waves around the cylinder and Gaussian wave packets centered around
the positions ym , where m is the angular momentum around the cylinder. Experimentally one cannot realize a cylindrical geometry, but rather the closely
related Corbino geometry obtained by flattening the cylinder. It is easy to
translate results between these two geometries. Note that the area of the
cylinder between two adjacent Gaussians is ∆A = ∆ym L = 2π`2B , corresponding to a unit flux, ∆Φ = B∆A = 2π/e = φ0 , i.e. there is one state per unit
flux.
2.1.1
Quantization of the Hall conductance I
To calculate the conductance, we imagine slowly turning on a magnetic flux
Φ(t) through the cylinder. By Farady’s law, this will induce an electric field
in the x direction, given by
Z
L
dxEx = LEx =
0
dΦ
.
dt
(2)
The Hall conductance is defined by
Iy = σH ∆x V = σH LEx = σH
dΦ
.
dt
(3)
where ∆x V is the voltage drop over the distance L in the x direction. Notice
that from this relation we ask get for the current density, Jx = Ix /L = σH Ex ,
so the conductivity, which is a local property, equals the conductance, which
is the quantity measured in experiments[18].
If we consider a long and very thin cylinder, it is easy to visualize what
is happening when the flux is turned on. The Gaussians, describing the individual electrons, will simply move along the cylinder, and inserting an integer
unit of flux, which corresponds to a proper gauge transformation, will simply shift all the Gaussians back to an identical pattern. The minimal shift
of this kind is to move the charges one step along the cylinder. By integrating (3) to ∆Q = σH ∆Φ, we can then extract the Hall conductivity as
σH = ∆Q/∆Φ = e/(2π/e) = e2 /2π. This argument, although deceivingly simple, is in fact very powerful. By a clever thought experiment, where a region on
the cylinder contains impurities while the regions around it are clean, one can
give a strong argument for why the IQHE is insensitive to impurities[19, 20].
Although basically correct, the above derivation is somewhat hand-waving
in that we considered an infinite system. If we instead take a finite length
cylinder we can make the argument precise, but at the cost of dealing with
8
the complications of boundary conditions. The simplest is to use periodic
boundary conditions
Tx (Lx )ψ(z) = eiφx ψ(z)
Ty (Ly )ψ(z) = eiφy ψ(z) ,
;
(4)
where Ti are magnetic translation operators. The angles φi can be interpreted
as fluxes going through the holes in the torus since an electron that is transported around the cycle x will pick up an Aharonov-Bohom phase eΦy = φx
etc. From the above, it should be clear that we can calculate the conductivity
from the adiabatic response to a change in the angles φi . It is beyond the scope
of these lectures to develop the theory of adiabatic response (for a discussion of
this, see [21]), but the same result can be obtained by using the Kubo formula
for linear response. This will be done below in a slightly different setting, so
I will for now just cite the result,
σH =
ie2 ij ∂φi Ψ|∂φj Ψ .
~
(5)
Naively, this does not look right. It seems to imply that the conductivity, which
is an intensive quantity, depends on the boundary condition via the phases φi .
This is not possible on physical grounds, so can equally well integrate of the
fluxes to get the more appealing formula,
Z 2π
Z 2π
ie2 1
ij
σH =
dφ
dφ
∂
Ψ|∂
Ψ
.
(6)
x
y
φ
φ
i
j
~ (2π)2 0
0
It now turns out, that viewed from the correct angle, this expression is very
simple. To see this we introduce the following notation,
Ai (φi ) = i hΨ|∂φi Ψi ,
(7)
which you recognize as the Berry potential corresponding to a change of the
external parameters φi . Thus we get,
Z
e2 1
σH =
B,
(8)
~ (2π)2 T
where B = ij ∂φi Aj (φj ) is the Berry field strength, and where the surface integral is over the torus, T .7 For a detailed discussion of (8), and the calculation
of the integral, I refer to the original paper [22] where it is also argued in more
detail why it is proper to average over the boundary conditions.
7
Be careful not to mix up the Berry potential (7), and the corresponding Berry field
strength B, with the usual electromagnetic quantities Aµ and B.
9
The crucial point is, however, that σH is a topological invariant, i.e. it
is insensitive to (not too large) changes in the Hamiltonian, typically adding
impurities or interactions. I will not here give the proof that σH as given
by (8) is quantized, but notice that it amounts to calculating a magnetic flux
through a closed surface, so in analogy with the case of a magnetic monopole in
ordinary electromagnetism it is expected to be quantized. A detailed analysis
shows that
Z
1
B = C1 ,
(9)
2π T
where the first Chern number, C1 , which is known from the mathematics of
fibre bundles, is an integer. We can thus calculate the actual value of σH for
the simplest case - a filled Landau level of non-interacting electrons without
any impurities, and still get the correct result for a realistic system.
2.1.2
Quantization of the Hall conductance II
We now turn to a more direct method that only applies to free electrons, but
which gives a relevant background for our further discussion of topological
insulators. First recall that from linear response theory, we have for a perturbation by a weak, spatially homogeneous, harmonic, external electromagnetic
vector potential δAy (t) = A(ω)e−i(ω+i) ( is a infinitesimal constant used to
define the adiabatic switching of the perturbation),
i
δJx (t) =
~
t
Z
dt0 h0|[Jx (t), Jy (t0 )]|0iδAy (t0 ) ,
(10)
−∞
where Ji is the electric current density. The dc Hall conductivity is defined
by
Jx = σH Ey ,
(11)
and recalling that Ey = iωAy , a little bit of algebra (do it!) allows us to
rewrite (10) as
σH
1
= lim
ω→0 ~ω
=
ie2
~
∞
X
Eα <EF <Eβ
X
Eα <EF <Eβ
Z
t
dt e−i(ω+i) [hα|Jx (t)|βihβ|Jy (0)|αi − hα|Jy (0)|βihβ|Jx (t)|αi]
−∞
1
[(Jx )αβ (Jy )βα − (Jy )αβ (Jx )βα ]
(Eα − Eβ )2
10
(12)
where Eα are the energy eigenvalues corresponding to the eigenstates |αi of
the Hamiltonian, and we used the notation (Jx )αβ = hα|Jx (t)|βi etc.. The
α-states are in the filled Landau levels, and the β states in the unfilled ones.
Using the ”Kondo formula” (12), it is not too hard to derive the result (6)
by relating the current to the change in bounday conditions[22], but here we
will take another route and consider the case of free electrons in the absence
of disorder. For simplicity, let us assume that the lowest Landau level is
completely filled, and all the others are empty. Any single-particle excitation
(these are the only ones of relevance in the linear response of a free system)
amounts to lifting an electron from a filled level to one of the empty ones, i.e.
it is a particle-hole excitation, and the energy denominator in (12) is ∼ ωc .
This is very important, and for a partially filled Landau level, which is the
the case for the fractional quantum Hall effect (FQHE), the non-interacting
ground state is degenerate, and the above derivation fail. In the last section we
shall see how electron-electron interactions will save the situation and explain
the FQHE.
We shall now treat the simplest case, where not only disorder is neglected,
but also the lattice potential. The first thing to do is to find a way to label
the massively degenerate states in the Landau levels. To do this we first recall
that in a constant magnetic field the relevant (magnetic) translation operators,
which commute with the Hamiltonian, do not commute among themselves.
The reason is that going between two points by different paths are not inverse
operations, since the combined operation of first moving somewhere, and then
moving back to the same point by another path, involves enclosing a flux
which by the Aharonov-Bohm argument gives a phase to the wave function.
But from this we also learn that if we pick a lattice which is such that the flux
through a unit cell is an integer number of flux quanta, the lattice translation
operators will all commute and can be simultaneously diagonalized. Thus we
can evoke Bloch’s theorem and express the wave functions as,
~
ψ~k,n = eik·~r u~k,n (~r)
(13)
where n is a band index (here the Landau level index) and ~k the crystal, or
quasi,
√ momentum that lives in a ”magnetic Brillouin zone”, |ki | ≤ π/`B =
π eB (the shape of this zone is gauge dependent, but the area is fixed to
support n units of magnetic flux). Thus, in this case the index α in (12) is
short for (n, ~k). The Bloch functions u~k,n (~r) are eigenfunctions of the Bloch
Hamiltonian,
HBl =
~2
~ + eA
~ + ~k)2
(−i∇
2m
11
(14)
that depends parametrically on the crystal momentum. This means that we
have a map from the ~k space, which is topologically a torus, into the space of
Hamiltonians, and thus ground state wave functions.
In the absence of impurities and electron-electron interactions, the ground
state is just a Slater determinant of the single particle wave functions (13) and
by substituting
J i (~r) = −e
∂HBl
∂ki
(15)
in (12), and some algebra, we get the following formula for the conductivity[23],
Z
ie2 X
d2 k
σH =
B(~k)
~ n BZ (2π)2
(16)
where B = ij ∂ki Aj (~k) and
Aj (~k) = i u|∂kj u .
(17)
Note that these expressions look very similar to (7) and (5), but the derivatives are with respect to the crystal momentum, not the phases encoding the
boundary conditions, and, very importantly, the bras and kets in (17) are
single particle states, while in (7) it was the full multi-electron state.
To actually calculate the integral in (16), one rewrites it using Stokes theoH
~ where the integral is around the magnetic Brillouin zone. If
rem as ∼ d~k A,
the phase of the wave function was well defined for all ~k in the zone, this integral would be zero, but this is not the case. For the detailed argument needed
to establish this, I refer to [23], where the value of the integral is related to
the presence of zeros in the single particle states at u~k (~r) at fixed ~r.
We already noted that the previous result (5) is much more general, since
it does not rely on a single-particle picture, and thus is valid for a general
interacting system with impurities, as long as the bands are filled and do not
cross. The derivation given here is interesting, since it opens the possibility
of having a topological phase in a crystalline system even in the absence of a
magnetic field and demonstrates that topological band theory can be used to
determine the actual values of the topological invariants.
The above discussion is a simplified version of the arguments originally
given by Thouless, Kohmoto, Nightingale and den Nijs[24], who considered
the general case of lattice potential, Vlat (~r) with a periodicity commensurate
12
with the periodicity of the magnetic lattice.8 The connection to fiber bundles
you find in Refs. [25] and [23] where the latter gives a detailed derivation
starting from linear response theory. The more general formulation in terms
of fluxes presented in Sect. 2.1.1 was given later by Niu, Thouless and Wu in
Ref. [22].
A final comment. Later we shall discuss the fractional quantum Hall effect
where σH are rational fractions of the quantum of conductivity, e2 /2π, and
you should worry about the above result that seems to indicate that fractions
are not allowed. The resolution to this quandary is that in deriving the result
one must assume that the wave function is single valued on a torus. This is
not true for a fractional state. For example, a Laughlin state at filling fraction
1/q has a q-fold degenerate ground state on the torus.
2.2
2.2.1
Topological effective actions
The effective response action
We now show how to encode the quantum Hall response, derived above, in
an effective response action. This concept is very similar to quantities well
known to you, namely the free energy F (T ) and the grand canonical potential
Ω(T, µ). From these thermodynamical potentials, we can calculate all relevant
thermodynamical quantities by taking appropriate derivatives. The effective
action Γ is a generalization of Ω which is not only a function of a chemical
potential, but also a functional of various space-time dependent external fields.
In these lectures we shall not consider finite temperature, and also specialize
to the (most important) example of an electromagnetic field Aµ . Thus the
quantity of interest to us is Γ[Aµ ] which encodes the response of the ground
state to changes in the external electromagnetic field.
For a system with a Hall conductance σH we have
σH
ΓH [A] =
2
Z
d3 x µνσ Aµ (x)∂ν Aσ (x).
(18)
Although we shall only consider non-relativistic systems, it will be convenient
to use a relativistic notation so that in 2+1 dimensions, relevant to the quantum Hall effect, we write x = (~r, t) and d3 x = dtdxdy. Recalling that the
current is just the derivative of the action with respect to the electromagnetic
8
This has the effect of breaking the degeneracy of the Landau levels and expand them
into bands of finite widths.
13
field, we easily verify that (18) implies
Ji =
δΓH
= σH ij E j
δAi (x)
(19)
Notice that this effective action is quite different from the one you know from
the usual theory of dielectric and diamagnetic media, where we have
Z
χ
χm ~ 2 e ~2
Γmed. = d3 x
E −
B .
(20)
2
2
R 2
~ −B
~ 2 ) we get, ∇ · D
~ =ρ
Adding this to the usual Maxwell action SM = 21 (E
~ = E
~ = (1 + χe )E,
~ and similarly for the macroscopic field H.
~
with D
While Γmed breaks Lorentz invariance (since the presence of a medium
defines a frame) it does preserve all other symmetries of the Maxwell theory
including the U (1) gauge symmetry. ΓH on the other hand violates time
reversal symmetry, T , and parity P . This is however quite natural since the
quantum Hall system assumes the presence of a background magnetic field
which breaks these symmetries explicitly. It is more troublesome that for a
system with boundaries, ΓH is not gauge invariant, since under the gauge
transformation
Aµ → Aµ + ∂µ λ
we get the variation
Z
Z
3
µνσ
δ d x Aµ (x)∂ν Aσ (x) =
A
(21)
dxi Ei (x)λ(x)
(22)
∂A
where ∂A is the boundary of the area A. This is puzzling, since it implies that
the current is not conserved at the boundary, which is clearly not allowed.
2.2.2
The topological field theory
The resolution to the above quandry is that there is an extra piece in the
effective action, that only resides on the boundary, and describes an edge
current in the Quantum Hall sample. Such edge currents are known to be
present and it is important to find a formulation of the effective low energy
theory that incorporates them in a natural way. The basic tool will be that
of topological field theory. We shall return to this concept several times later,
but for now we shall just look at the simplest example and see that it has the
desired properties. We take the Lagrangian
L(b; A, j) = −
e µνσ
1 µνσ
bµ ∂ν bσ −
bµ ∂ν Aσ − jqµ bµ
4π
2π
14
(23)
where b is an auxiliary gauge field, and jq a quasiparticle current. The first
term in (23) is called the Chern-Simons (CS) term, and this particular topological field theory is thus usually called (abelian) CS theory. To understand
the meaning of the topological vector field, b, we calculate the electric current
J,
Jµ =
e
δL
= − µνσ ∂ν bσ
δAµ
2π
(24)
so b is just a way to parametrize J. Note that J, which by definition is
conserved, is invariant under the gauge transformation
bµ → bµ + ∂µ χ ,
(25)
where χ is a scalar, since it is the field strength corresponding to the vector
potential b. Since b is related to the conserved current, we shall refer to it as
”hydrodynamical”.9
There are (at least) two reasons for why this theory is referred to as topological. First you notice that it does not depend on the metric tensor. A
normal kinetic term has the general covariant form ∼ g µν Dµ φDν φ, and thus
depends on the geometry of the space on which it is defined. If the action
does not depend on the metric, correlation functions of operators that also do
not depend on the metric, will be topological objects. It would take us too
far to develop this point, which was originally stressed by Witten in a very
influential (but also quite difficult) paper [26]. The other reason is much easier
to understand. The equation of motion for the b field is,
µνσ ∂ν bσ = −eµνσ ∂ν Aσ − 2πjqµ
(26)
that is the field strength is completely determined by the external sources. This
means that, as opposed to usual Maxwell electrodynamics, there are no propagating photons - the equations of motion are just constraints. For instance, the
zeroth component of (26) is 2πρ = ij ∂i bj ≡ B (b) , which relates the topological
magnetic field, B (b) , to the charge density ρ = j 0 , of the external current.
The analysis just given is, however, true only for a system on an infinite
plane, the case of boundaries, hosting gapless degrees of freedom will be discussed below. Also, if the system lives on a closed surface with holes (higher
genus) there will be a finite number of dynamical degrees of freedom.
9
Note that the conventions for this field differ. I use the notation from [8], while in the
work by Wen[14] the hydrodynamical field is denoted by a.
15
Since the Lagrangian (23) is quadratic in b we can integrate it out to get
an effective action for A only. In path integral language we write,
Z
R 3
iΓ[A,j]
e
= D[~b]ei d r L(b;A,j)
(27)
performing the integral we get,
Z
hσ
i
H µνσ
3
µ
Γ[A, j] =
dx
Aµ (x)∂ν Aσ (x) + ej (x)Aµ (x)
2
Z
π +
d3 xd3 y j µ (x)
(x − y)j ν (y)
d µν
(28)
where we recall that σH = e2 /2π, and where (1/d)µν (x − y) is the inverse of
the Chern-Simons operator kernel µνσ ∂σ . The first term in this expression is
just the term (18) derived earlier, while the last term is a statistical interaction
between the particles described by the source j. We can think of the sources
as holes in the filled Landau level. These holes have the same properties as
positive electrons, and in particular they are fermions. The last term provides
the minus sign that the wave function aquires when two identical fermions are
exchanged. A simple way to understand this phase is to first recall that the
equation of motion (26) associates charge with flux and then realize that and
the resulting charge - flux composites will pick up an Aharanov-Bohm like
phase when encircling each other.10
We now show how the theory (23) in a natural way incorporates the presence of edge excitations. For a more thorough discussion you should consult
the paper by Stone[27] and the review by Wen[28]. We specify the action by
integrating the Lagrangian (23) over a bounded and simply connected region
D,
Z
S[b; A] =
d3 x L(b, A) .
(29)
D
Since the coupling to the external field A is only via the field strengt Fµν ,
it is invariant under the usual gauge transformation (21), but, by the same
argument used to derive (22) we see that under the transformation (25) we
get a non zero variation at the boundary ∂D. What this means is that the
pure gauge mode, ∂µ χ which in the bulk has no physical meaning, and has to
be removed by some suitable gauge fixing, will at the edge manifest itself as a
10
The story is a little more subtle and I will come back to in the last section on the
fractional quantum Hall effect.
16
physical degree of freedom. To see this explicitly we substitute bµ = ∂µ χ into
the Lagrangian (29) to get,
Z
Z
1
3
µνσ
S[b, χ; A] = −
d x bµ ∂ ν bσ +
dtdx χ ∂x (∂t − v∂x )χ(x, t) (30)
4π D
∂D
where for simplicity we neglected the external field A (which is easily reintroduced), and where the field χ(x, t) has support only on the boundary ∂D
parametrized by the coordinate x. We also added an extra term ∼ χ ∂x2 χ that
does not follow from (29), but which will be present if there is an electrostatic
confining potential[28], which is needed to define the quantum Hall droplet.
The meaning of this term is clear from the equation of motion for the χ-field,
(∂t − v∂x )χ(x, t) = 0 ,
(31)
which shows that v is the velocity of a gapless edge mode propagating in one
~ ×B
~
direction. The physical origin of this velocity is obvious - it is the E
drift velocity of the electrons in the external magnetic field and the confining
electric field at the boundary. If we reintroduce the electromagnetic field and
study the current conservation at the boundary, we will see that the nonconservation of the bulk current, which follows from (22) is compensated by a
corresponding non-conservation of the boundary current, so the total charge is
conserved[27]. The mathematics related to this result is quite interesting. The
boundary theory should after all just be a model for electrons moving in one
direction along a line, and as such we would expect the theory just to be that
of a Fermi liquid, or if interactions are present, a Luttinger liquid. In both
cases we would expect the boundary charge to be conserved. What is special
here is that the mode is chiral, i.e. it only propagates in one direction. From
the theory of Luttinger liquids, we learn that in the presence of an electric
field, the right and left moving currents are not separately conserved, but
only their sum, which is the electromagnetic current. The difference, which
defines the axial current, which in Dirac notation reads jµA = ψ̄γ3 γµ ψ, is not
conserved because of the axial anomaly. The subject of anomalies in quantum
field theory is fascinating, but will not be discussed in these lectures.
2.2.3
Topological action from functional bosonization
The way we obtained the topological action (23) was a bit indirect. We first
derived the effective response action Γ[Aµ ], and then showed that it could be
derived from (23). Surely it would be more satisfactory to directly derive (23)
from the microscopic theory. This is indeed possible, and we shall do it in two
17
quite different ways. In this section you will be introduced to the method of
functional bosonisation, and later, in the context of the FQHE, the method of
flux attatchment.
We start by a general exposition of the method, and will then specialize to
various insulating topological phases. For this section, which closely follows
Ref. [29], we start with the usual expression for the partition function,
Z
Z[A] = D[ψ̄, ψ]eiS[ψ̄,ψ,A]
(32)
where the action S describes the fermionic system in d spatial dimensions, and
Aµ is an external U (1) field; usually describing electromagnetism. Knowing Z
we can calculate the current response by taking derivatives with respect to A.
We shall furthermore assume S to be gauge invariant which means that
Z[A + a] = Z[A]
(33)
for any a being a pure gauge, i.e. satisfying,
fµν = ∂µ aν − ∂µ aν = 0 .
Thus we can express Z as
Z
Y
µνλ...αβ δ[fαβ (a(x))] .
Z[A] = D[a] Z[A + a]
(34)
(35)
µν···
where the delta functionals under the product sign enforce constraint (34).11
In (35), x is a point in D = d + 1 dimensional space-time, and µνλ...αβ is
the D-dimensional totally anti-symmetric Levi-Civita symbol. Introducing
an auxiliary D − 2 dimensional tensor field bµ1 µ1 ...µD−2 to express the delta
functional as a functional Fourier integral, we get,
Z
R D µνλ...αβ
i
bµνλ... fαβ (a)
,
(36)
Z[A] = D[a]D[b] Z[A + a]e− 2 d x and by the shift a → a − A, finally,
Z
R D µνλ...αβ
i
bµνλ... [fαβ (a)−Fαβ (A)]
Z[A] = D[a]D[b] Z[a]e− 2 d x .
11
(37)
In the case of manifolds which are topologically non-trivial, such as a torus, the condition
(34) does not fully specify the gauge and the integral (35) sums over all values of the nontrivial Wilson loops which amounts to summing over all ”twisted” boundary conditions for
the fermions.
18
Given this we can calculate the expectation value of the U (1) current as
hj µ (x)i = i
δ ln Z[A]
= hµνλρ... ∂ν bλρ... (x)i
δAµ (x)
(38)
and similarly for higher order correlation functions. Note that by construction,
the current is conserved. To appreciate the meaning of the field bµνλ... , let us
look at the simplest special cases. For D = 2, b is a scalar, and the relation
(38) reads,
hj µ (x)i = hµν ∂ν b(x)i ,
(39)
which you might recognize if you are familiar with the method of bozonization
in 1+1 dimension. This case is special, in the sense that (39) holds even if the
average is removed, that is it holds as an operator identity. A concise account
of the fascinating physics and mathematics of 1+1 D systems can be found in
Ref. [11].
For D = 3, b is vector field and
hj µ (x)i = hµνσ ∂ν bσ (x)i .
(40)
Up to a normalization, which we will discuss below, you recognize this as the
previously derived relation (24) for the electric current.
Clearly the formula (37) is useful only if we can, at least approximately,
evaluate the fermionic functional integral to get Z[a]. In 1+1 dimensions this
can sometimes be done exactly. In higher dimensions this is not possible, but
we can find an approximation by assuming there is a gap and making a derivative expansion. Because of gauge invariance, the result can only depend on
the field strength, fµν . (For a non-abelian gauge field, which carries charge, Z
can also depend on the covariant derivatives of f .) There are general methods
for carrying out the derivative expansions, that we will not discuss here.
In the case of the IQHE, we already know one piece in Z[a] that will for
sure be present namely the one related to the Hall conductivity. Thus,
Z
σH
2
d3 x µνσ aµ ∂ν aσ + O(fµν
).
(41)
Γ[a] =
2
As we already discussed, this expression is gauge invariant up to boundary
terms, and we also notice that it is the only possible such term with only one
derivative. It is instructive to derive this relation by a direct calculation of
the relevant bubble diagram. This calculation, gives the right coefficient, but
does not explain why it is a topological invariant. You are encouraged to do
this exercise!
19
Thus to understand the low-energy and momentum behavior of the IQHE,
we can use the effective Lagrangian,
LIQHE = −
1 µνσ
C1 µνσ
bµ ∂ν (aσ − Aσ ) +
aµ ∂ ν aσ ,
2π
4π
(42)
where we expressed the hall conductivity in the Chern number of the filled
Landau levels, and renormalized the field b so that it, up to the factor (−e), is
identical to the previous expression (24) for the electromagnetic current. As
you will see in the next section, essentially the same formula will be applicable
also to a 2+1 dimensional topological insulator.
If we put C1 = 1, corresponding to a single filled Landau level, and integrate
over the a field, we finally get the result,
LIQHE = −
1 µνσ
1 µνσ
bµ ∂ν bσ +
A µ ∂ ν bσ .
4π
2π
(43)
which, again up to the factor (−e), is identical to the previously derived (23).
Above we used a seemingly arbitrary argument to fix the the normalization
of the field b, and you should worry about this since a different convention
would give a different value for σH when b is integrated over to obtain the
effective response action (28).
To understand this point, we must look closer at the first term in (23)
which is the 2d incarnation of the topological BF theory which is defined in
any dimension by,
1
LBF = − µνλ...αβ bµνλ... fαβ (a) .
2
(44)
In the next section we shall briefly discuss the 3d case in the connection with
fluctuating superconductors. There is a rich mathematical literature on BF
theory[30, 31], but here we shall only cover material of direct relevance for
physics. One such point is the question of normalization brought up above.
With the chosen normalization the ground state is unique, as is required for a
number of filled Landau levels. A derivation of this result is given in Appendix
A.
2.3
Topological band insulators
One of the early successes of quantum mechanics was the division of crystalline materials into conductors, semiconductors and insulators depending on
whether or not the fermi level is inside a band gap (there is no sharp distinction between insulators and semi-conductors; only a conventional classification
20
depending on the size of the gap). The insulators seem to be the most boring,
and it was only rather recently that it was recognized that not all band insulators are the same. We have already learned that an integer quantum Hall
state is an unusual isolator in that its usual, Ohmic, conductivity is zero, while
it has a quantized Hall conductance. Although the arguments in the previous
section hint at the possibility of having a non-trivial topology related to mappings from the Brillouin zone to the space of single particle wave functions,
this seems to rely heavily on the presence of a magnetic field and the related
Landau level structure.
2.3.1
Chern insulators
In an important paper from 1988, Haldane showed that one can have a quantum Hall effect without any net magnetic field[32]. He constructed an effective
model of electrons hopping on a hexagonal lattice, penetrated by a staggered
magnetic field that is on average zero. The model did, however, break time
reversal invariance (T RI) and using current terminology, it would be called a
Chern insulator that exhibits a quantized anomalous Hall effect. A model that
is slightly simpler than the one used by Haldane is free electrons hopping on
a square lattice, with a π-flux on each elementary plaquette. Since the main
theme of these lectures are continuum field theory descriptions, I will not give
the position space lattice Hamiltonian, which you can find in the original work
Ref. [33]. For the present purpose, it suffices to say that the Chern insulator
can be modelled by the following two band momentum space Hamiltonian,
X †
c~k,α hαβ (~k)c~k,β ,
(45)
HC =
~k
with
a
hαβ (~k) = da (~k)σαβ
,
(46)
where d~ = (sin kx , sin ky , M + cos kx + cos ky ), and we measure both energy and
M in units of some hopping strength, t.
Following the same arguments as for the IQHE, the Hall conductivity is
given by (8), i.e. by the Berry flux through the Brillouin zone,
Z
d2 k
ie2 X
B(~k) ,
(47)
σH =
~ n BZ (2π)2
where now the Berry potential is calculated from the occupied eigenstates
|Ek i of hαβ (~k). To calculate the flux, we note that HC is nothing but the
21
~ The spectrum
Hamiltonian for a spin-half particle moving in a magnetic field d.
is given by the Zeeman energy is thus ±|d|, and the eigenfunctions satisfy,
dˆ · ~σ |Ek ; ±i = ±|Ek ; ±i .
(48)
~ > 0, and taking the Fermi
Assuming that there is no gap closing, i.e. |d|
energy at 0, the Berry potential for the two bands is
A±
j = hEk ; ±|i∂j |Ek ; ±i
(49)
and a calculation gives,
1 ˆ
ˆ
ˆ
(50)
B ± (~k) = ij ∂i A±
j = ∓ ij d · ∂i d × ∂j d ,
2
where the lower sign corresponds to the filled band. This result can be obtained
in various ways without using the general methods employed in Berry’s original
paper[34]. The most direct, although a bit tedious, is to first find |Ek ; −i and
then just calculate. An alternative derivation that does not require the explicit
wave functions is given in Appendix B.
The expression in the left hand side of (50) is the Jacobian of the function
ˆ
~
d(k), and we define the integer valued Pontryagrin index by
Z
1
Q=
d2 k ij dˆ · ∂i dˆ × ∂j dˆ,
(51)
8π
which measures how many times the surface of the unit sphere on which dˆ is
defined, is covered by the map from the compact manifold where ~k is defined.
Thus it follows from (47) and (50) that σH = ne2 /2π just as for integer Hall
effect.
It remains to determine the value of n = Q. For large |M |, where the
hopping can be neglected, the eigenfunctions |Ek i become ~k-independent and
the Pontryagin density is identically zero. (M 1 is the atomic limit where,
hopping can be neglected, and the wave functions are sharply localized at the
lattice sites.) Since Q is a topological quantity, it can only change when the
gap in the Fermi spectrum closes and the derivation of (47) breaks down. Since
the energies of the filled states are Ek = −|d(~k)|, we see that for M = −2 the
gap closes at ~k = 0, for M = 2 at ~k = (π, π) and for M = 0 at ~k = (0, π) and
~k = (π, 0)
Let us now analyze what happens when M increases from a large negative
value towards 2. Putting M = −2 + m, and linearizing the Hamiltonian we
get,
Hlin = kx σ1 + ky σ2 + mσ3 ,
22
(52)
Figure 1: Fig. 1 Diagram giving rise tot the CS action as the lowest term in a
derivative expansion.
which we recognize as the Hamiltonian for a D = 2+1 Dirac particle.
Since the topological nature of a phase is a low-energy property, we should
be able to capture the change in phase by analyzing the continuum Dirac
theory in the vicinity of m = 0. This motivates the following,
Digression: The continuum D=2+1 Dirac theory
Consider the Lagrangian,
LD = ψ̄ (γ µ (i∂µ − eAµ ) − m) ψ
(53)
with γ µ = (σ3 , iσ2 , −iσ1 ). We calculate the electromagnetic response by integrating out the fermions, which, to lowest order i Aµ , amounts to calculating
the loop diagram in Fig. 1. This was originally done by Redlich[35] with the
result,
Z
m e2
d2 r µνσ Aµ ∂ν Aσ .
(54)
ΓD [A] =
|m| 8π
In Appendix C we give an alternative derivation of this result by calculating
the response to a constant magnetic field. Just as in the Schrodinger
√ case, the
eigenvalues of the Dirac equation fall into Landau levels, En = ± neB + m2 ,
for n > 0, and the contribution from these states to σH cancel. Only the lowest
Landau level, with n = 0 and energy E0 = m, contributes.
How does this analysis relate to the previous expression (47) for σH in terms
of the integral over the Berry flux? By calculating the Dirac wave functions
(putting Aµ to zero), we can calculate A ,and thus B, from (49), and the result
for the response is precisely the one obtained above with the direct field theory
approach (see e.g. Sect. 8.2 in Ref. [12]).
Note that the coefficient being half of that in (47) translates into a Hall
conductivity σH = ±e2 /4π which is half of the one calculated above. This
23
is surprising since we have argued that the Hall conductivity, for topological reasons, must be an integer times e2 /2π. The solution to this apparent
contradiction, is that it is not possible to consistently formulate the Dirac
equation on a lattice without ”doubling” the number of low-energy fermions.
This result, first obtained in the context of high energy physics by Nielsen and
Ninomiya[36], basically says that the low-energy physics of fermions in a band
of finite width, cannot be faithfully represented by a single Dirac field.
End digression
We now return to the model (45). Recall that we put M = −2+m, and we want
to know what happens as m is tuned from a small negative value to a small
positive value. Doing this changes the spectrum only in the close vicinity of
~k = 0, so the change in σH should be faithfully represented by the linearized
Dirac theory. From (54) we get the the change ∆σH = (1 − (−1))e2 /4π =
e2 /2π. A similar analysis can be made at the other gap closing points. The
result is that σH , in units of e2 /2π changes as 0 → 1 → −1 → 0 as M is tuned
from −∞ to ∞. It should now also be clear, that the effective topological
theory for the Chern insulator is identical to that for the IQHE given by (42).
Note that at M = 0 the gap closes at two points, so to model this transition
we need two Dirac fields, and thus the change of two units in σH .
2.3.2
TR invariant topological insulators in d=2
A breakthrough was made in two papers by Kane and Mele[37, 38], who considered an effective theory of graphene. As you might know, the electronic
structure of graphene is characterized by two Dirac cones for each spin. This
so-called valley degeneracy comes about because of the symmetries of the band
structure. In undoped graphene the fermi level is at the tip of the Dirac cones.
This means that there is no gap, but since the density of states vanishes, the
material is a semi-metal rather than a metal. The simplified hamiltionian used
by Kane and Mele is the following 4 × 4 matrix,
H0 = −i~vF ψ † (σx τz ∂x + σy ∂y )ψ ,
(55)
where the Pauli matrices ~σ act in the pseudo-spin space, related to the sublattices, and ~τ in the valley space. It is, in fact, two copies of the model used by
Haldane, (but without any staggered magnetic field) who considered spin-less
electrons. Kane and Mele’s important observation was that in the presence of
spin, one can add extra interactions that open up gaps without breaking the
T invariance. These are the spin orbit coupling,
HSO = ∆so ψ † σz τz sz ψ ,
24
(56)
where ~σ is yet another set of Pauli matrices that describe the spin. This term
preserves the symmetry under rotations around the z-axis perpendicular to
the graphene plane. The other term is the Rashba coupling
HR = λR ψ † (σx τz sy − σy sx )ψ ,
(57)
which breaks this symmetry, so Sz is no longer conserved. By directly diagonalizing the Hamiltonian, one finds that adding the term (56) opens a mass gap,
2∆so , which will remain even in the presence of (57) as long as 0 ≤ λR ≤ ∆so .
2.3.3
Topological classification from edge modes
One way to find possible topological phases of this model is to study the mappings from the Brillouin zone into the space of wave functions, or equivalently
Hamiltonians, along the same lines as described above for the quantum Hall
states. This is possible, and has been done in various ways by different authors, but is somewhat complicated and we shall not pursue it here. Instead
we shall follow Kane and Mele’s original paper and explain the topology in a
very simple way by analyzing the edge states.
We first recall some properties of the time reversal operator T . In spite of its
name, one should not think of the T transformation as changing the direction
of time, but rather as changing the velocities, and hence the momenta, of all
particles so that they will trace their trajectories in the opposite direction.
Think of it as showing a movie backwards. A Bloch Hamiltonian, HB , which
is the relevant dynamical operator for a periodic system, depends explicitly on
the (crystal) momentum ~k, so under time reversal it transforms as
T HB (~k)T −1 = HB (−~k)
(58)
and is thus in general T invariant only at special points in the Brillouin zone,
namely those where ~k and −~k differ by a reciprocal lattice vector. This typically happens at ~k = 0 and at the zone boundaries (Γ-point and K-points
in the lingo of band theory). Now also recall that Kramers theorem states
that the eigenvalues of a T invariant fermionic Hamiltonian are always doubly
degenerate. In view of this, let us look at the spectrum of the model Hamiltonian for grapheme. In Fig. 2 you see the spectrum, obtained by Kane and
Mele, for a lattice version of H0 + HSO , solved in a strip geometry. In addition
to the bulk bands, you also see two isolated states that transverse the gap.
In the absence of the Rashba coupling, these are helical edge states, meaning
that they are spin separated; at each edge the spin up and spin down currents
propagate in the opposite direction. This is the quantized spin Hall effect,
25
because of the presence of a non-zero edge spin current in the ground state,
just as a quantum Hall system has a non-vanishing charge current.
The quantum spin Hall effect is however not stable, since any interaction
that breaks the spin rotation symmetry, such as HR would destroy it. The
great insight of Kane and Mele was that the topology of the edge states differ
between a trivial insulator and a topological insulator. This point is explained
in Fig. 3, that is taken from the review article by Hasan and Kane[39]. Γa = 0
and Γb = π/a lable TRI points in the Brillouin zone (only half is shown,
−π/a < k < 0 is just a mirror imagage), where there are degenerate Kramers
pairs. The left picture shows an ordinary insulator, i.e. a state that is adiabatically connected to the vacuum, while the right is a topological insulator.
The distinguishing feature for the two phases is whether an odd or even number of Kramers pairs cross the band gap. Clearly, if there is an odd number,
no continuous deformation of the spectrum can remove the edge mode - it is
topologically protected. The presence of such a protected edge mode is intimately connected to the edge being the border line between two topologically
distinct gapped bulk states that can not be adiabatically connected without
the gap collapsing somewhere. This topologically non-trivial state commonly
referred to as a 2d topological insulator, but the term quantum spin Hall effect
is sometimes also used even in cases where the spin current is not conserved.
2.3.4
The topological field theory for the Quantum Spin Hall effect
Just as in case of the integer QH effect, we can construct a topological field
theory to describe the spin Hall effect (remembering that it is not a very stable
phase). Since there are now two conserved currents, corresponding to spin up
and spin down, we expect a topological action with two gauge fields b↑ and
b↓ .
LQSH = −
1 µνσ ↓ ↓
1 µνσ ↑ ↑
bµ ∂ ν bσ +
bµ ∂ν bσ .
4π
4π
(59)
This is an example of a doubled Chern-Simons theory. There is no QH effect,
since the contributions to σH obtained by coupling to an external electromagnetic field come with different signs and cancel each other. Similarly, there is
no chiral electric edge current, but instead a chiral spin current. But again, this
is only relevant in the cases where one component of the spin is conserved.12
12
Using the technique of functional bosonization, one can derive an alternative hydrodynamic theory with only a single conserved U (1) current[29].
26
Figure 2: Edge states in graphene. One dimensional energy bands for a strip of
graphene (shown in inset). The bands crossing the gap are spin filtered edge states.
Figure from Ref. [37]
Figure 3: Edge states in 2d insulators Edge states in trivial (left panel) and
topological (right panel) insulators. Γa and Γb lables TIR points where the spectrum
is degenerate. Figure from Ref. [39]
27
2.3.5
Topological index for the TRI topological insulators
The topological classification of the T invariant insulators differs from that of
the integer quantum Hall states. The former are characterized by an integer
n ∈ Z while the oddness or evenness of the edge modes of the topological
insulator corresponds to just a sign, ζ ∈ Z2 . The mathematical description of
the Z2 topological invariants in the bulk, you will find in the reviews, [39], and
[40] and references therein.
2.3.6
Topologicl insulators in d 6= 2
Although we mainly deal with 2+1 dimensions, it is instructive to look at the
case of 1+1 dimension, since it is more closely related to the interesting case
of 3+1 dimension. From the above it is clear that we again need to compute
Z[a], and we will do this for the continuum Dirac equation, using the path
integral formulation,
Z
R 2
µ
Z[a] = D[ψ̄, ψ]ei d x ψ̄(γ (i∂µ −aµ )−m)ψ .
(60)
We parametrize the gauge field as aµ = µν ∂ν ξ + ∂µ λ, so that F = µν ∂µ aν =
−∂ 2 ξ; the term containing λ is just a gauge transformation which does not
contribute to the action (provided we are on a simply connected manifold).
One can now verify (do it!) that the chiral transformation
ψ → e−iγ3 ξ ψ ,
(61)
where γ3 = iγ0 γ1 , eliminates the transverse gauge field µν ∂ν ξ from the action,
while the mass term changes,
ψ̄(γ µ (i∂µ − aµ ) − m)ψ → ψ̄(γ µ i∂µ − me−2iγ3 ξ )ψ .
(62)
This looks very strange, since for the mass less case it looks like we transformed
away a nontrivial external field! The resolution of the apparent contradiction
is that the path integral measure is not invariant under the transformation.
Using techniques pioneered by Fujikawa[41], one can show that under the
transformation (61),
i
D[ψ̄, ψ] → D[ψ̄, ψ]e− 2π
R
d2 x ξ∂ 2 ξ
.
(63)
which is the path integral incarnation of the axial anomaly referred to at
the end of Section. 2.2.2. In particular we shall be interested in a (spacetime) constant chiral transformation ξ(x) = −θ/2, which does not change the
28
coupling to aµ but only effects the mass term, and introduces a ”θ-term” in
the action,
Z
iθ
Sθ =
d2 x µν ∂µ aν .
(64)
2π
This term, which is the integral of the electric field strength, is topological
since it does not depend on the metric. Also note that, as opposed to the
Chern Simons term, it is fully gauge invariant, and it does not contribute
to the equations of motion. Since in 1+1 dimensions, the field strength is
simply the electric field, Ex , the parameter θ, has a natural interpretation as
a polarization, and the fact that on lattice a polarization is only defined up to
a lattice translation, implies that θ is only defined modulo 2π.13
For the case of the continuum Dirac equation we shall follow the same logic
as in the 2+1 dimensional case, and only calculate how the value of θ differs
between different phases. From (62) we see that taking 2ξ = θ = π amounts
to changing the sign of the fermion mass. Taking the gamma matrices,
γ 0 = σ1
,
γ 1 = iσ 3
γ 3 = iσ 2
,
(65)
we have
H=
0
m − ik
m + ik
0
=
0 Q†
Q 0
.
(66)
It is straightforward to obtain the wave functions, and calculate the Berry
potential,
A=
1 m
.
2 k 2 + m2
(67)
We can now form the Chern-Simons invariant by integrating over the filled
states labeled by k,
Z ∞
1
CS1 [A] =
dk A
(68)
2π −∞
Plugging (67) into (68) gives for the filled Dirac sea,
Z ∞
1
1 m
1 m
CS1 =
dk
=
.
2
2
2π −∞
2k +m
4 |m|
13
For a precise discussion of this point, see Ref. [33].
29
(69)
An alternative way to characterize the topology is by the winding number
defined by,
Z
Z
i
i
−im
1 m
−1
.
(70)
w=
dk Q ∂k Q =
dk 2
=
2π
2π
k + m2
2 |m|
which is twice the invariant CS1 . In fact, on can show that the normalization
is such that the Wilson loop, e2πCS1 , it is invariant under regular gauge transformations where the winding number is integer[42]. Note that the winding
number changes by one unit when the sign of the mass changes, so we conclude
that ∆θ = π.
Just as in the discussion of the Chern number for the Dirac sea, you might
wonder how something that is called a winding number can be non-integer.
The resolution is again related to the regularization of the continuum Dirac
theory. If we instead consider the lattice version,
Hlat = sin(k)σ 2 + (m − 1 + cos(k))σ 1
so that Q = −i sin(k) − (m − 1 + cos(k)) we get
Z π
i
dk ∂k ln(m − 1 + e−ik ) .
w=
2π π
(71)
(72)
For m < 0 the curve m − 1 + e−ik does not wind around the origin, so the
logarithm can be picked to be single valued and thus w = 0. For 0 < m < 2
it winds one turn in the negative direction and w = 1. The step at m = 0 is
the same as in the continuum model.
All of the above generalizes, mutatis mutandis to any space of odd dimension. You can find the general formulae in Ref. [42].
It is only in 1+1 dimensions that Z[a] can be calculated exactly, but using
perturbation theory, the lowest derivative term in odd space-time dimensions
is a generalization of the Chern-Simons action. Defining S = −i ln Z we get
in D=4+1,
Z
C2
dx5 µνσλρ aµ ∂ν aσ ∂λ aρ
(73)
SCS =
2
24π
where the second Chen number, C2 can be calculated by evaluating a fermion
bubble with three external photon lines. Note that this response function is
non-linear !
For even D, there is a generalization of the θ-term, which for the interesting
case of D = 3 + 1, becomes
Z
θ
Sθ = 2 d4 x µνσλ ∂µ aν ∂σ aλ .
(74)
8π
30
This term is very interesting, since substituting it in (37) and carrying out the
integrals over the fields a and b results in the following term in the effective
electromagnetic response function,
Z
θ
~ ·E
~.
Γ[A] = 2 d4 x B
(75)
8π
For a TRI insulator in 3+1 dimensions the theta parameter can take the
values 0 or π mod 2π, corresponding to a trivial or topological insulator, and
Γ[A] is very interesting since it gives a magnetic response to an electric field,
and vice versa[33]. One can also construct effective topological field theories
which again are of the BF type[16, 29] but that is beyond the scope of this
presentation.
There is a lot of interesting mathematics related to the response functions
in different dimensions. Fore example, starting from the D=2+1 CS theory
on thin cylinder one can obtain the topological term in D=1+1, and similarly
one can start from the effective field theory in D = 5 + 1, and by dimensional
reductions generate effective field theories in lower dimensions [33, 29].
3
Weakly interacting systems
We now move to interacting systems, but of a kind that is well understood,
namely superconductors. It is known that even a weak attractive interaction
will turn a fermi liquid into a superconductor at sufficiently low temperature.
The mechanism is the formation of Cooper pairs composed of two electrons
with equal but opposite momenta. Such pairs form, not because of the strength
of the interaction, but because of the large available phase space, given by the
fermi surface. This means that even though the theory is weakly coupled (in
conventional superconductors by an electron-phonon interaction) the ground
state is non perturbative.
Most common superconductors have Cooper pairs where the spins form
a singlet, which forces the orbital wave function to be symmetric. The simplest possibility is an s-wave, and this is in fact the symmetry of the order
parameter in most conventional, or low Tc , superconductors. The high Tc
superconductors discovered in the late 1980s are of the d-wave type. From
quantum field theory point of view, the BCS approach to superconductivity,
amounts to a self-consistent mean field approximation, based on the pairing
field ∆(x) = ψ(x)ψ(x). In the superconducting phase, ∆ aquires a ground
state expectation value, that formally breaks the electromagnetic U (1) gauge
invariance to a Z2 invariance related to the sign of ψ. For many purposes
31
one can neglect the self consistency requirement and just replace the attractive electron-electron interaction with the a coupling term ∼ (∆ψ † ψ † + h.c.),
where ∆ is a fixed background field. Diagonalizing the resulting quadratic
Hamiltonian shows that ∆ induces a gap that totally removes the fermi surface. The excitations are quasiparticles that close to the fermi surface are
equal weight linear combinations of particles and holes, and as such on the
average neutral.
This is all conveniently described by introducing the Nambu spinor Ψ†k =
(a†k , a−k ), in terms of which,
1X †
1X †
ak
H ∆
Ψk H(k)Ψk =
(ak , a−k )
(76)
H=
?
∆ −H
a†k
2
2
k
k
where a†k is the electron creation operator, and where we suppressed all relevant
spin or band indices. We can rewrite the BdG Hamiltonian H(k) as
H(k) = H(k)τz − Im(∆(k)τy + Re(∆(k)τx
(77)
where the Pauli-matrices τi act in the Nambu space, and H and ∆ are 2 × 2
matrices acting in spin space. For a translationally invariant
p system of free
particles where H = (k)1, the spectrum becomes Ek = ± (k)2 + |∆(k)|2 so
for strictly positive ∆ the spectrum is gapped. In the s-wave case ∆ is simply
a constant, while for pairing in higher partial waves there is a ~k dependence.
Note that the Nambu spinor structure really amounts to an artificial doubling of the spectrum, so the associated particle-hole symmetry is just an expression of using a redundant description; the doubling of the spectrum does
not mean that there are two physical states for each degenerate eigenvalue, but
the quasiparticles are, as already mentioned, linear combinations of particles
and holes. It follows that this particle-hole symmetry cannot be broken.
Since in the BdG approximation, superconductors are described as systems
of free electrons, albeit with ”anomalous” terms, they can be treated with the
methods used for the non-interacting systems analyzed in the previous section.
(In fact, the BdG superconductors is part of the general topological classification of phases of free fermions[43, 3, 4]). We are thus again led to study the
topological properties of maps from the momentum space to the space of single
particle wave functions. In the case of p-wave pairing, these maps can be nontrivial as will be discussed in the next section. For s-wave pairing, the maps
are always trivial and thus an s-wave superconductor in the BdG description
is trivial. This is however not the case for a real superconductor, which is
coupled to a fluctuating electro-magnetic field. Such a state is topologically
ordered as will be discuss in Sect. 3.2 below.
32
We saw in the previous section that the insulating topological states were
characterized by protected edge modes, and their long distance topological
properties could be coded in a topological field theory. Here we will se that
superconductors share several of these characteristics but also that new properties are added to the list. In particular, there are two distinct types of gapped
local excitations in a superconductor - quasiparticles and vortices - and these
have very different properties depending on type of pairing, and on whether
or not the coupling to electromagnetism is included.
3.1
p-wave superconductors
We shall now use the BdG approximation to study the simplest cases of spinless
one- and two-dimensional superconductors with a p-wave pairing terms ∼ kx
and ∼ ∆(px + ipy ) respectively.14 Note that ∆? ∆ = |∆|2 |~p|2 , so as opposed to
the d-wave, there are no nodal lines, and thus no gapless nodal excitations.
In the case of two dimensions, an interesting phenomenon occur when one
considers vortices or edges. By a direct (but somewhat involved) analysis of
the BdG equations one finds that for each vortex there is a single zero energy
fermion solution that is self-conjugate, i.e. it is its own antiparticle. This
is to be contrasted with the s-wave case, where all the solutions to the BdG
equation have finite energy.
Self-conjugate fermion operators, or Majorana fermions, are constructed
from from a Dirac fermion ψ and its conjugate ψ † , by forming the combinations
c1 = ψ + ψ †
1
c2 = (ψ − ψ † ) .
i
and
(78)
Clearly these Majorana fermions satisfy the reality condition c†i = ci and it is
also easy to derive the commutation relation
{ci , cj } = 2δij ,
(79)
and show that the number operator can be expressed as n̂ = ψ † ψ = (1 +
ic1 c2 )/2. It is important that it takes a pair of Majorana operators to constitute a degree of freedom; it is not possible to construct a number operator
from only one Majorana fermion. The extension to many operators is straightforward, and it should be clear that any physical system must contain an even
number of Majorana modes. For the p-wave superconductor, this means that
14
There is a close connection to the physically more relevant case of triplet paired spinfull
electrons or 3 He atoms.
33
if there is an odd number of vortices in the bulk, there must be an extra Majorana mode bound to the edge. Notice that the Majorana operators do not
transform properly under the general U (1) gauge transformation ψ → eiα ψ,
but it does under the Z2 subgroup, ψ → ±ψ. This is closely related to that
the Majorana fermions appear together with superconductivity which, as we
discussed above, breaks U (1) to Z2 .
3.1.1
The Kitaev chain
Before turning to the p-wave superconductor in two dimensions, we briefly
describe a related 1d model, due to Kitaev[44], which in a very simple way
demonstrates how fermion number can fractionalize in the presence of superconducting order. This model, which resembles its two-dimensional counterpart in having Majorana edge modes, is of large current interest, since it can be
experimentally realized in quantum wires with strong spin-orbit coupling[45],
or in chains of magnetic atoms on top of a superconductor[46]. Of course,
there is no genuinely one-dimensional superconductor, so in an experiment
the pairing is always induced by proximity to a 3d superconductor.
The following presentation is a shortened and somewhat simplified adoption of Ref. [44]. The Kitaev chain is a model for spinless fermions hopping
on a 1d lattice consisting of L 1 sites, and is given by the Hamiltonian
i
Xh
HK =
−t(a†j aj+1 + a†j+1 aj ) − µ(a†j aj − 21 ) + ∆? aj aj+1 + ∆a†j+1 a†j . (80)
j
Here t is a hopping amplitude, µ a chemical potential, and ∆ an induced
superconducting gap that we by convention chose as real and positive. In
terms of the Majorana fields,
c2j−1 = aj + a†j
1
c2j = (aj − a†j ) ,
i
and
the Hamiltonian becomes
iX
HK =
[−µc2j−1 c2j + (t + ∆)c2j c2j+1 + (−t + ∆)c2j−1 c2j+2 ] .
2 j
(81)
(82)
Let us now consider two two special cases.
1. The trivial case: ∆ = t = 0, µ < 0.
H1 = −µ
X
j
X
1
i
(a†j aj − ) = (−µ)
c2j−1 c2j
2
2
j
34
(83)
The Majorana operators c2j−1 , c2j related to the fermion ψj on the site j are
paired together to form a ground state with the occupation number 0.
2. ∆ = t > 0, µ = 0. In this case
X
H2 = it
c2j c2j+1 .
(84)
j
and here the Majorana operators c2j , c2j+1 from different sites are paired. We
can define new annihilation and creation operators
1
ãj = (c2j + i c2j+1 )
2
and
1
ã†j = (c2j − i c2j+1 ) ,
2
(85)
which are shared between sites j and j + 1. The Hamiltonian becomes
H2 = 2t
L−1
X
1
(ã†j ãj − ) .
2
j=1
(86)
Note that neither c1 nor c2L is part of the Hamiltonian, and as a consequence
the ground state is degenerate, since these two Majorana operators can be
combined to a fermion,
i
Ψ = (c1 + i c2L ) ,
2
(87)
which can be occupied or unoccupied, corresponding to a two-fold degeneracy.
The fermion number corresponding to this field, is, however, delocalized at the
two ends of the chain. Thus, no local perturbation can change the occupation
of this state. It is this property that has made fractionalized fermions an
interesting object for quantum information technology. If a qubit could be
stored in a pair of separated Majorana fermions, it would be very robust against
noise[47].
Although the above analysis was only for two very special points in the
parameter space, Kitaev established that the whole parameter region ∆ 6= 0,
and |µ| < 2|t| is topological. It is illustrative to see how a topological index
appears in this simple case. To this end, we introduce the Nambu spinor
Ψ†k = (a†k , a−k ) in terms of which the Hamiltonian (80), assuming a constant
real ∆,15 in momentum space becomes,
X †
HK =
Ψk HK (k)Ψk ,
(88)
k
15
A real delta means that in the classification of Ref. [3], the Hamiltonian belongs to the
symmetry class BDI, while for a complex ∆ the symmetry class is D.
35
⌧y
a
2⇡
0
⌧z
b
⌧y
2⇡
0
⌧y
c
⌧z
0
2⇡
⌧z
~
Figure 4: Winding numbers ν of d(k)
for the full Kitaev chain, in (a) trivial
phase with w = 0, for 0 < t < µ/2, ∆ > 0, (b) topological phase with w = 1
for µ = 0, 0 < t = ∆ and (c) topological phase with w = −1 for µ = 0,
0 < t = −∆. The arrows denote the direction in which k increases.
with HK (k) given by
~ · ~τ
HK (k) = −µ/2 − t cos(k) τz − ∆ sin(k)τy = −d(k)
(89)
where the Pauli-matrices τi act in the particle-hole spinor space. Just as in
the case of the 1d insulator discussed in Sect. 2.3.6, the topological invariant
takes the form of a winding number. To show this, consider the curve traced
~
out by the vector d(k)
= (0, ∆ sin(k), µ/2 + t cos(k)) in the (τy , τz )-plane (i.e.,
the space of Hamiltonians), as k sweeps through the full Brillouin zone. This
~
is illustrated in Fig. 4, where we (schematically) show the curve d(k)
in the
trivial phase, with winding w = 0, and the two different topological phases,
with winding w = ±1. It is a good exercise to show that the same result is
obtained by using the general formula (70).
To get this result, it was important that we could take ∆ to be real. In a
more general situation this is not always the case. For example, if a current
flows through the s-wave superconductor that induces ∆ by proximity, the
phase of ∆ will vary, and the winding number will no longer be the relevant
topological characteristics, but it will be replaced by a Z2 index[44]. This has
a direct consequence for the spectrum of Josephson junctions. In the first
(real) case, the junction between two topological states with winding number
±1, will host two Majorana zero modes, which amounts to a single Dirac zero
mode, while in the second (complex) case such a junction will have no zero
mode. In both cases there will be a single Majorana zero mode at a junction
between the topological and the trivial phase[48].
36
3.1.2
The two dimensional p-wave superconductor
As explained in an important paper by Read and Green[49], a two-dimensional
spinless p-wave paired BCS superconductor can be in two distinct topological
phases, depending on the strength of the coupling. Just as in the case of the
non-interacting systems we studied in Sect. 2, these phases can be distinguished
by studying maps from the momentum space into the space of ground state
wave functions, which in this case is of the BCS form,
|Ωi =
Y
(u~k + v~k c~†k )|0i ,
(90)
~k
where the two spinor, z = (u~?k , v~k? )T , is a normalized solution to the BdG
equations. We can think of u~k and v~k , as coordinates on a unit Bloch sphere
by introducing a unit vector by n̂ = z †~σ z, where ~σ are the Pauli matrices.
If we furthermore compactify the momentum space by adding the point at
infinity, it also has the topology of a two-sphere. Thus, the functions u~k and
v~k define a map from S 2 to S 2 , and the usual homotopy arguments tell us that
this is characterized by an integer winding number, n.
The detailed analysis in Ref. [49] of the solutions to the BdG equations
shows that for an s-wave paired state, n = 0 corresponding to a topologically
trivial state. For p-wave pairing, there are two possibilities depending on
whether the chemical potential is positive or negative. The latter case, µ < 0
is a trivial state corresponding to strong coupling, in the sense that for small ~k
the pairing function g~k = v~k /u~k ∼ kx − iky , corresponding to an exponentially
decaying pair wave function in real space. For the topologically non-trivial
weak pairing case µ > 0, we instead have g~k ∼ 1/(kx + iky ), corresponding to
a weakly bound Cooper pair with g(~r) ∼ 1/(x + iy).
In between these phases at µ = 0, there is a phase transition, and one
can show that, just as in the quantum Hall system discussed earlier, there is a
gapless boundary state. Taking the simple geometry of an infinite half-plane it
is not too hard to find the exact edge solution[49]. Just as in the quantum Hall
case this solution is chiral, (this is possible since the p-wave pairing function
is chiral). The way this comes about is that the relevant edge solutions to the
BdG equations satisfy the condition, v~k = u~?k , which translates into the the
condition γ~k† = γ~k for the quasiparticle field. Or in other words, the edge mode
is described by a Majorana field. At this point boundary conditions become
interesting. Quantizing on a finite length edge, say a circle with circumference,
L, the spectrum is ∼ π2n/L or ∼ π(2n + 1)/L depending on the boundary
conditions being anti-periodic or periodic. In the periodic case, there is a zero37
mode that describes a single non-paired Majorana mode, analogous to the edge
Majorana modes in the Kitaev chain.
Naively on would assume that the boundary condition that gives an unpaired Majorana mode at the edge would be unphysical, but surprisingly
enough, this is not the case. Just as in the s-wave case, there are vortex solutions, and by explicitly solving the BdG equations in a vortex background, one
can establish that there is also a single, zero-energy mode localized at the vortex. A detailed analysis[49, 50] shows that for an odd number of vortices in the
bulk, one must use periodic boundary conditions on the edge, while for an even
number of vortices, one must take take anti-periodic boundary conditions. In
both cases, there is an even number of Majorana zero energy modes that can
be used to construct a finite dimensional Hilbert space for a fixed boundary and
fixed vortex positions. And now comes the most amazing feature of this system: Quasi-adiabatic braidings of the vortices correspond to non-commuting
unitary operators acting on this finite-dimensional Hilbert space. Since the
vortices should be considered as identical particles, this amounts to having
non-abelian fractional statistics.
The 2d p-wave state we just discussed is fractional in the sense that Majorana modes can be viewed as a half fermion. Also, the ground state on the
torus is degenerate. There is a very close resemblance between this state and
the strongly correlated Moore-Read pfaffian quantum Hall state[51], that is
likely to be the one observed at filling fraction ν = 5/2, and which will be
discussed briefly in the last section. The great interest in these systems shown
by the quantum computing community is again because of the prospects for
providing a topologically robust coding of quantum information[52].
3.2
Fluctuating s-wave superconductors
The Ginzburg-Landau theory for type II superconductors support vortex solutions. The elementary vortex has a core that captures half a unit of magnetic
flux, since the Cooper pair has charge 2e. This means that a quasiparticle
will pick up a phase eiπ = −1 when encircling a vortex at a distance large
enough for it not to penetrate the vortex core. This is an example of a topological braiding phase, which can, as we shall show, readily be captured by
a field theory. Before proceeding we should clarify the distinction between a
real, ”fluctuating”, superconductor coupled to electromagnetism, and a model
superconductor described by a BdG theory without a dynamical electromagnetic field. The difference, which was first clearly pointed out by Kivelson and
Rokshar[53] is that the former totally screens the electromagnetic current while
the second does not. Thus, in the fluctuating superconductor, the only low
38
energy degrees of freedom are the vortices, and electrically neutral fermionic
quasi particles. The topological field theory that describes the low-energy
properties of the s-wave superconductor is the very same BF theory that we
already discussed in connection with the IQHE and the Chern insulator. For
simplicity we will focus on the 2d case, but with a short comment on the 3d
case in Sect. 3.2.2. For pedagogical reasons I shall first give a heuristic derivation of the BF theory, using an analogy with the Chern-Simons theory for the
IQHE discussed in Sect. 2.2.2. Later, in Sect. 3.2.3, we outline a derivation of
the BF theory from a microscopic model.
3.2.1
BF theory of s-wave superconductors - heuristic approach
We first consider the 2+1 dimensional case where both quasiparticles and
vortices are particles so we can proceed in close analogy to the bosonic ChernSimons theory for the quantum Hall effect. Recall that in that case the equations of motion related charge and flux, and the statistics of the quasiparticles
(which in this case is simply holes in the filled Landau level) followed from the
coupling to the gauge field. The present case differ from the above in that we
have two distinct excitations, quasiparticles and vortices, and we will describe
them with two conserved currents, jqµ and jvµ , which we couple to two different
gauge fields, a and b, by the Lagrangian,
Lcurr = −aµ jqµ − bµ jvµ .
(91)
A simple calculation shows that in order to get a phase π when moving a jq
quantum around a jv quantum we should take,
LBF =
1 µνσ
(a)
bµ fνσ
,
2π
(92)
(a)
where fµν = ∂µ aν − ∂ν aµ . This we recognize as the BF action, but with
a coefficient that differ from (42) derived in IQHE case. Putting the parts
together we have the topological action,16
Ltop =
1 µνσ
bµ ∂ν aσ − aµ jqµ − bµ jvµ .
π
16
(93)
The symmetry properties of the Lagrangian (93) are worth a comment. Under the
parity transformation (x, y) → (−x, y) the two potentials transform as (a0 , ax , ay ) →
(a0 , −ax , ay ) and (b0 , bx , by ) → (−b0 , bx , −by ), while under time reversal the transformations
are, (a0 , ax , ay ) → (a0 , −ax , −ay ) and (b0 , bx , by ) → (−b0 , bx , by ), respectively. The unusual
transformation properties of the potential bµ follows from that of the vortex current. It is
easy to check that the BF action is invariant under both P T and CP T .
39
The topological nature of Ltop is clear from the equations of motion,
1 µνσ (a)
1 µνσ
∂ν aσ =
fνσ
π
2π
1 µνσ
1 µνσ (b)
=
∂ν bσ =
fνσ ,
π
2π
jvµ =
(94)
jqµ
(95)
which show that the gauge invariant field strengths are fully determined by
the currents, just as in the Chern-Simons theory. These equations both
have a very direct physical interpretation. For instance, if we write (95) as
jqµ + (1/π)µνσ ∂ν bσ = 0 this expresses that the quasiparticle current is totally
screened by the superconducting condensate if we interpret (−1/π)µνσ ∂ν bσ as
the screening current. This observation can indeed be used to give an alternative derivation, or rather motivation, for our topological field theory; the
potential aµ is nothing but a Lagrange multiplier that enforces the constraint
of total screening of the current jq . For a more detailed discussion, see [15].
It is interesting to consider the quantization and conservation of charge
in our topological field theory. There are two known mechanism for charge
quantization. The first is the presence of monopoles, where the condition of
invisibility of the Dirac string forces quantization of the charge. The second
route, which we shall take here, is to postulate that the gauge fields are compact, or expressed differently, they are angular variables. If we want both
currents to be integer valued, this leads us to require that the gauge fields aµ
and bµ are compact. In the continuum, this means that they transform as
ai → ai + ∂i Λa
bi → bi + ∂i Λb ,
(96)
with gauge functions Λa/b ≡ Λa/b + 2π. The question of current conservation
is more subtle. Since the world line of a point-like vortex can be thought of
as a vortex line in space-time, non-conservation of the vortex charge would
amount to having such world lines ending at a point. This could happen
if there were unit charge magnetic monopoles in space-time, on which two
such world lines could terminate. Such monopoles in space-time are called
instantons and are known to exist in many field theories. Since we do not
have any magnetic monopoles, this is however not a realistic option, and the
vortex charge is conserved. The situation is quite different when it comes to
the electric charge. Here the Cooper-pair condensate acts as a source of pairs
of electrons, and in our topological theory such processes, corresponding to
formation or breaking of pairs, could be incorporated by having instantons in
the b field.
40
3.2.2
The 3+1 dimensional BF theory
Turning to the case of 3+1 dimensions, we have essentially the same construction, but with the difference that the vortices are now strings, and the vector
potential bµ is an antisymmetric tensor, bµν . The action still again of the BF
type and reads,
LBF =
1 µνσλ
bµν ∂σ aλ .
π
(97)
The gauge transformations of the b field are given by
bµν → bµν + ∂µ ξν − ∂ν ξµ
(98)
where ξµ is a vector-valued gauge parameter. The minimal coupling of the b
potential to the world sheet, Σ, of the string is given by the action,
Z
Z
µ
ν d(x
,
x
)
bµν ,
Svort = − dτ dσ µν bµν = − dτ dσ (99)
d(τ, σ) Σ
Σ
where (τ, σ) are time and space like coordinates on the worldsheet. This is a
direct generalization of the coupling of a to the world line, Γ, of a spinon,
Z
Z
dxµ
µ
Ssp = − dx aµ = − dτ
aµ .
(100)
dτ
Γ
Γ
Combining these elements we get the topological action for the 3+1 dimensional superconductor,
Z
Stop = d4 x LBF + Ssp + Svort .
(101)
The proof that this action indeed gives the correct braiding phases can be
found e.g. in Ref. [31], and a discussion of this action in the context of superconductivity has appeared before in Ref. [54].
3.2.3
Microscopic derivation of the BF theory
So far, we did not derive the BF theory, but rather constructed it, or guessed
it, from the the braiding properties of quasiparticles and vortices. It would
obviously be more reassuring if the theory could be derived from a microscopic model. Here we outline such a derivation starting not from the original
fermionic theory, but from an effective Ginzburg-Landau model coupled to a
quasi particle source. (The derivation of this theory from is a standard exercise
41
that can be found e.g. in Ref. [9].) For simplicity, we shall follow ref. [15]
and consider a toy version of the Ginzburg-Landau theory, namely the 2+1 D
relativistic Abelian Higgs models defined by the Lagrangian,17
Lah =
λ
m2 †
1 2
1
|iDµ φ|2 − (φ† φ)2 −
φ φ − Fµν
− eAµ jqµ ,
2M
4
2
4
(102)
where we used a standard particle physics notation where φ is the charge
−2e scalar field representing the Cooper pair condensate, iDµ = i∂µ + 2eAµ
is the covariant derivative, Fµν is the electromagnetic field strength and the
conserved current jqµ , with charge e, is introduced to describe the gapped quasiparticles discussed above.18 Just as in the usual Ginzburg-Landau theory for
type II superconductors, the Abelian Higgs model supports vortex solutions,
which are characterized by a singularity in the phase of the Cooper-pair field.
√
Defining φ = ρ eiϕ , and writing ϕ = ϕ̃ + η, where η is regular, and can be removed by a proper gauge transformation, the vorticity is encoded in the gauge
field aµ = 12 ∂µ ϕ, which depends on the vortex positions. The corresponding
conserved vortex current is parametrized as jv = (1/π)µνσ ∂µ aσ . It is now a
matter of algebraic manipulations, involving integrating out gapped degrees
of freedom, to derive the topological theory (93)[15]. For this to work, it is
crucial that we include a dynamical electromagnetic field, it is only then that
the external currents are completely screened and all bulk modes are gapped
and can be integrated over. Making a derivative expansion, and keeping terms
to second order results in the effective Lagrangian,
2
1 (a) 2 1
e
1 µνσ
(b) 2
(fµν
) − aµ jqµ − bµ jvµ . (103)
Lef f = bµ ∂ν aσ − 2 (fµν ) −
π
4e
4 ms π
Note that the topological theory emerges as the leading term in this expansion!
The higher derivative terms, which are of Maxwell form, are not topological,
and have the effect of introducing bulk degrees of freedom. These are however
gapped, and can physically be identified with the plasmon mode. At low
energies the plasmons can be neglected and we can retain only the topological
part. Another physical effect captured by the Maxwell terms is the London
penetration lenght λL which is the size of magnetic flux tube associated to a
vortex. In the purely topologcal theory, the vortices are strictly point like.
17
This is a toy model not only because we use a relativistic form for the kinetic energy,
but also because we use 2+1 D Maxwell theory, which amounts to a logarithmic Coulomb
interaction. The generalization to the more realistic case is straightforward, and the result
is qualitatively the same. The derivation, however, becomes less transparent.
18
Note that in spite of the relativistic form we normalize the kinetic term such that |φ|2
has the dimension of density.
42
In this section we strictly dealt with s-wave superconductors. The extension to the d-wave case is relatively straightforward, but since there are gapless
quasiparticles associated to the nodal lines, the effective theory must include
these, and becomes quite a bit more complicated[55]. The p-wave case is considerably more difficult because of the Majorana modes associated to vortices.
The effective theory has to encode these in a way that properly describes their
topological properties[56, 57].
4
Fractional Quantum Hall Liquids
We now finally come to the strongly interacting systems, and we shall here
only consider the archetypical examples which are the quantum Hall liquids.
We start with the most celebrated one.
4.1
The Laughlin states
In 1982, not long after the discovery of the IQHE, Tsui, Stormer and Gossard
observed similar plateaux in the conductance at filling fractions, ν = 1/3[58].
Later, many more states were discovered at rational filling fractions, ν = p/q,
the vast majority with with an odd denominator q. This is the celebrated
fractional quantum Hall effect (FQHE). As we briefly discussed in section
2.2.2, the FQHE poses a much more difficult theoretical challenge than the
integer one. The basic difficulty is the massive degeneracy of the free electron
states in an unfilled Landau level. Neglecting the lattice potential, the only
energy scale is that of the Coulomb interaction EC ∼ e2 ρ1/2 , where ρ−1/2 is
the mean distance between the particles. (We assume that the cyclotron gap,
EB ∼ eB/m is large, so that for all practical purposes EC /EB = 0.) As a
consequence there is no small parameter, and thus no hope to understand the
FQHE by using perturbation theory.
The first, and in a sense most successful, approach to the FQH problem
was due to Laughlin[59], who, by an ingenious line of arguments, managed to
guess a many-electron wave function, that has proven to give an essentially
correct description of the states at filling fraction ν = 1/m,
N
Y
P
2
2
Ψm (zi ) = N
(zi − zj )m e− i |zi | /4`B ,
(104)
i<j
where the complex coordinate zj = xj +iyj denotes the position ~rj = (xj , yj ) of
the electron j, N the number of electrons, and N a normalization constant. It
43
is immediately clear that this wave function is fully antisymmetric and has no
components in higher Landau levels. Since it vanishes very quickly when two
electrons approach each other, it does a god job in minimizing the repulsive
Coulomb energy, and since there is no preferred points or directions it describes
a homogeneous, isotropic quantum liquid. It is easy to generalize to the case
of a state containing M quasi holes at the positions ηa ,
N
M Y
N
Y
Y
P
2
2
m
ΨL (ηa ; zi ) = N (η1 . . . ηM ) (zi − zj )
(zi − ηa )e− i |zi | /4`B . (105)
a=1 i=1
i<j
We see that wave function vanishes at the points ηa , which thus correspond
to depletions of the electron liquid, or in other word, holes. It is also not too
difficult to prove that these holes have a sharp19 fractional charge q = e/m
and also obey fractional statistics with an angle θ = π/3. There are many
excellent texts explaining in details how to derive these results, for instance
[61]. We shall also return to the question of statistics in the coming sections.
4.1.1
The Chern-Simons-Ginzburg-Landau theory
Another approach, which is closer to the methods we discussed earlier in the
context of superconductors, is to try to find an effective low energy theory,
and I will outline how this can be done. The starting point is the microscopic Hamiltonian for N electrons in a constant transverse magnetic field B,
interacting via a two-body potential V (r), which should be thought of as a
(suitably screened) Coulomb potential,
N
H=
N
X
1 X
(~pi − eA(~ri ))2 +
V (|~ri − ~rj |) ,
2m i=1
i<j
(106)
~ r) = B (−y, x). Our aim is to find an equivalent bosonic formulation
where A(~
2
of this theory, which should be amenable to a mean-field description. A direct application of the method of functional bosonization described in Section
2.2.3, will not work since we would not be able to compute the partition function Z[a], for a partially filled Landau level. Another approach that might
come to mind is to evoke pairing and introduce a Cooper pair field. This will
19
The qualification ”sharp” is important. This means not only that the (properly defined)
expectation of the total charge, hQi, is (exponentially) localized to a region of the size of the
relevant correlation length, but also that the dispersion hQ2 i − hQi2 can be made arbitrarily
small[60]. Fractional charges which are not sharp is an ubiquitous polarization phenomenon.
44
however also not work since it would describe a superconductor, not an insulating QH state. Instead we proceed by first performing a statistics changing
transformation on the electrons.
The idea is to relate the fermionic wave functions to their bosonic counterparts by the unitary transformation
ΨF (~r1 , . . . ~rN ) = Φk (~r1 . . . ~rN )ΨB (~r1 , . . . ~rN )
(107)
where the phase factor Φk is given by20
k
Φk (~r1 . . . ~rN ) =
Y zb − za 2
a<b
z̄b − z̄a
= ei
P
a<b
αab
,
(108)
with k an odd integer, and αab the polar angle between the vectors ~ra and ~rb ,
which is given by 2αab = ln(za − zb ) − ln(z̄a − z̄b ). The corresponding bosonic
Hamiltonian is identical to the fermionic one, except that it includes a coupling
to a statistical, or Chern-Simons, gauge potential,
X
~
αab .
(109)
~a(~ra ) = k ∇
b6=a
Thus,
HB =
N
i2 X
1 Xh
~ ra ) + ~a(~ra ) −
V (|~ra − ~rb ) .
p~a − eA(~
2m a=1
a<b
(110)
To proceed we first notice that although ~a looks like a pure gauge, it is singular
at the positions of the particles, so that the statistical magnetic field is given
by,
X
ij ∂i aj ≡ b(a) = 2πk
δ 2 (~ra − ~rb ) ,
(111)
b6=a
which amounts to attaching a singular flux tube of strength k to each particle.
Viewed from this angle, the statistical exchange phase, θ = kπ can be seen as
an Aharanov-Bohm effect.21
20
Here I use a and b to label the particles to avoid confusion with the coordinate indices
i, j.
21
The naive picture of the ”composite bosons” as flux-charge composites, is however
slightly misleading since that would imply that you get a phase 2 × k2π when taking one
particle a full turn around another; there are equal contributions from the charge circling
the flux and the flux circling the charge. This is not what happens, the correct phase is k2π
corresponding to the exchange phase kπ[62].
45
We are now ready to construct a quantum field theory describing our
bosonized electron in a path integral formulation. The variables will be a nonrelativistic boson field, φ, describing the electrons, and the statistical gauge
field ~a. The relation (111) implies that ~a is such that there is a flux tube tied
to each particle, i.e.
2πkρ = 2θφ? φ = ij ∂i aj .
(112)
This local constraint is implemented by a lagrange multiplier field a0 , and
the result is the non-relativistic Chern-Simons-Ginzburg-Landau (CSGL) field
theory,
LB = Lφ +
1
a0 ij ∂i aj
2πk
(113)
1
~ − ~a)φ|2 − V (ρ) ,
|(~p + eA
2m
(114)
where
Lφ = φ? (i∂0 − a0 + eA0 )φ −
and where Aµ is an external electromagnetic field, that includes the constant
background magnetic field B, and ρ = φ? φ is the density. The term ∼ a0 ij ∂i aj
~ · ~a = 0, version of the full CS action
is nothing but the Coulomb gauge, ∇
µνσ
∼ aµ ∂ν aσ , so we can finally write the partition function as
Z
R 3
Z[Aµ ] = D[φ? ]D[φ]D[aµ ] ei d x LCSGL (φ,a;A)
(115)
with
LCSGL = Lφ +
1 µνσ
aµ ∂ν aσ .
4θ
(116)
To proceed, we first find a mean-field solution for the ground state using the
simple choice
µ
λ
V (ρ) = − |φ|2 + |φ|4
2
4
r
φ = φ0 =
µ
λ
;
~.
~a = eA
(117)
Combining this with constraint (112), we immediately get for the average
density,
ρ̄ =
1
ρ0
1 (a)
b =
eB =
,
2θ
2πk
k
46
(118)
where ρ0 is the density of a filled Landau level. Or in other terms, the filling
fraction is ν = 1/k.
Just as in the usual Ginzburg-Landau theory, the GLCS theory supports
classical vortex solutions, which for a unit strength vortex is,
√ iϕ
φ ∼
ρ̄ e
(119)
r→∞
aϕ
∼
r→∞
1
r
;
ar = 0 .
Since the constraint (112) relates flux to charge, we can calculate the excess
charge related to the vortex by integrating (118),
Z
Z
ν
ν
2
Qv = d r ρ(~r) =
eΦ =
e d~r · ~a = νe
(120)
2π
2π
so that the vortex describes a quasiparticle with fractional charge νe. In Refs.
[63] and [8] you find a more detailed analysis of the CSGL theory including
the demonstration that the quasiparticles are Abelian anyons with a statistics
angle νπ. Here we shall show how it can be used to extract an effective
topological theory for the Laughlin states.
4.1.2
From the CSGL theory to the effective topological theory
In the presence of a collection of vortices, we parametrize the field φ as
p
(121)
φ = ρ(~r)eiθ(~r) ξv (~r) ,
where θ is a smooth phase, and ξv a singular phase factor defined as to give a
singular ~a as in (119). More precisely, the vortex current is given by,
jvµ =
1 µνσ
∂ν (ξv? ∂σ ξv ) ,
2πi
(122)
which, in spite of the appearance, is non-zero due to the singular nature of ξv .
Substituting (121) in (116) and expanding around the mean-field solution,
yields,
Z
R 3
Z[A] = D[δρ]D[θ]D[ji ]D[aµ ] ei d x δLCSLG (δρ,θ,jµ ,a;A)
(123)
with
δLCSLG = iρ(i∂t θ + ξv? ∂t ξv ) − a0 δρ + j i (∂i θ − ξv? ∂i ξv − ai + eAi ) (124)
Z
m i
1
1 µνσ
+
j ji −
d2 rδρ(~r)V (~r − ~r 0 )δρ(~r 0 ) +
aµ ∂ ν aσ .
2ρ̄
2
4kπ
47
and where we introduced the Hubbard-Stratonovich field ji to linearize the
term quadratic in ∂i θ. Varying θ we get ∂t ρ − ∂ i ji = 0, showing that the
current jµ = (ρ, ji ) is conserved, and thus can be parametrized as
1 µνσ
∂ν bσ .
(125)
2π
This means that we can trade the integration over δρ and ji for integration
over the vector field bµ , and by finally performing the gaussian integration over
aµ (do it!) one obtains,
Z
R 3
Z[A] = D[bµ ] ei d x Ltop (b;A) ,
(126)
jµ =
with
e µνσ
1 µνσ
bµ ∂ν bσ −
Aµ ∂ν bσ + bµ jvµ ,
(127)
4πν
2π
where we used (122) and (125) to rewrite the term ∼ j µ ξv? ∂µ ξ in terms of jv
and b. We also neglected the density fluctuations which give terms with higher
order in derivatives since δρ ∼ ij ∂i aj .
Note that for a filled Landau level, i.e. ν = 1, we exactly reproduced the
previous results (23) and (43), and just as before, the gauge field b parametrizes
the current. For the Laughlin states ν = 1/3, 1/5, . . . the theory superficially
looks very similar, but the fractional value of ν has far reaching consequences,
such as fractional charge and statistics for the quasiparticles, characteristic
ground state degeneracy on higher genus surfaces, and chiral bosonic edge
modes with ν-dependent correlation functions[28].
Ltop = −
4.2
The Abelian hierarchy
In experiments on very clean samples, one sees a large number of FQH states[64].
Most of them fit beautifully into a hierarchical scheme[65], where ”daughter”
states are formed by the condensation of anyonic quasiparticles in a ”parent” FQH state, just as the Laughlin states can be thought of as a condensation of the original electrons. These hierarchy states are theoretically
fairly well-understood using a number of different techniques such as composite fermions[66] and model wave functions constructed using conformal field
theory[67].
All topological information about an abelian QH state can be coded in the
following effective action based on Chern-Simons gauge fields[13, 14],
L=−
e
1
Kαβ µνσ bαµ ∂ν bβσ −
tα Aµ µνσ ∂ν bασ + lα bαµ j µ ,
4π
2π
48
(128)
where the ”K-matrix” K, the ”charge vector” t, and the vector l which describes the quasi particles, are all integer valued, and where Aµ is an external
electromagnetic field. For a hierarchy state at level n, Kαβ is a symmetric,
rank n matrix with odd integers along the diagonal and with all other entries
being even integers.
The CS gauge fields bαµ are chosen in a basis such that they couple minimally to the quasiparticle currents lα jµ , and the charge vector, t, will in the
following be taken as tα = 1. It is explained in detail in [14] how to extract
the topological information from this Lagrangian, and here we just quote the
following important results for the filling fraction ν and the charge Qα and
statistics angle θα for the αth quasiparticle:
ν = tT K−1 t
Qα = −etT K−1 l
θα = πlT K−1 l .
(129)
This version of the theory is correct only on manifolds without curvature, such
as the plane or a torus. On the sphere there are additional terms that codes
the response to curvature, and carry information about the orbital spin of
the electrons and the quasi-particles[14]. Using the connection to conformal
field theory, one can construct a model wave function given the topological
information given in the triplet (K, t, l)[67]. Similar results have been obtained
on the sphere[68], and on the torus[69].
4.3
Non-abelian Quantum Hall states and CFT
As advertised at the end of Sect. 3.1.2, there is a close connection between the
2d p-wave superconductor, and the most famous of the proposed non-abelian
quantum Hall states, namely the Moore-Read (MR), or Pfaffian state[51],
which is a strong candidate for the experimentally observed state at filling
fraction ν = 5/2. In fact the notion of non-abelian statistics of quasiparticles
in a paired state originate in the analysis of the MR state, and at it is a major
experimental challange to test the theoretical predictions.
To explain how you get non-abelian QH states, you should first understand
something about the connection between QH states and conformal field theory
(CFT) in two Eucledian dimensions. You can think of a CFT as a field theory
without any dimensional parameters, and the simplest one is just a massless
scalar field theory. In two dimensions the Lagrangian is
Z
Z
1
2
d2 x ∂µ ϕ∂ µ ϕ ,
(130)
S[ϕ] = d x L =
8π
49
and we shall furthermore assume that the field variable ϕ is periodic with
a period 2πR, where, quite naturally, R is referred to as the boson radius.
(Such a field is called a compactified boson, and
√ R is often also called the
compactification radius.) Let us now pick R = m, and define the following
vertex operators,
Vm (z) = : ei
Hm (η) = : e
√
mϕ(z)
√i ϕ(η)
m
:
(131)
:,
(132)
which are invariant under the shift ϕ → ϕ + 2πR, and where : : denotes a
normal ordering. The two-point function following from (130) is hϕ(z)ϕi(w) =
− ln(z − w), and, using Wick’s theorem, it is now not to hard to show,
M
N
M Y
N
Y
Y
Y
P
1
2
2
m
Ψm (ηa ; zi ) ∼
(ηa − ηb ) k
(zi − zj )
(zi − ηa )e− i |zi | /4`
a=1 i=1
i<j
a<b
N
M
Y
Y
= h V (zi )
H(ηa )Obg i .
i=1
(133)
a=1
where Obg is a neutralizing background charge (which has to be properly chosen
in order to reproduce the Gaussian factor in the wave function[51, 67]). Compared with (105) this expression has an explicit dependence on the quasihole
coordinates ηa which is such that when two of them are interchanged, the wave
iπ
function acquires a phase e k , which precisely mean that the quasiholes are
anyons with statistical parameter π/k. In fact, the original demonstration that
the quasiholes were anyons amounted to showing that the full ηa -dependence
of the normalization constant in (105) is precisely that given by (133). According to a famous conjecture put forward by Moore and Read, this is not a
coincidence, but to be expected in all cases where a FQH wave function can
be written as a correlator of vertex operators in a CFT[51]. This conjecture is
not based just on observing similarities between different formulas, but has a
deep theoretical underpinning that we cannot dwell further upon.
It is natural to ask whether other FQH states than the Laughlin ones, also
can be expressed in terms of correlators in CFTs. With some modifications,
this turns out to be true for the hierarchy states described in the previous
section[67]. Most interestingly, however, it opens for the construction of a
completely new set of FQH states - the non-abelian ones. The idea is to use
correlators in other CFT’s and interpret them as FQH wave functions, and
we now briefly describe how this is done in the important example of the MR
state that we referred to in the discussion of the 2d p-wave superconductor.
50
The MR state and its quasihole excitations are written in terms of correlators of the operators used in the CFT description of the Ising model.22
This is a theory which comprises two ”primary” field, a Majorana (i.e. real)
fermion, ψ, characterized by the two-point functions hψ(z)ψ(w)i = 1/(z − w),
and a spin field σ to which we return below. Just as the electron operator for
a Laughlin states is defined by (131), we can write an electron operator for
the MR state as,
VM R (z) = ψ(z)ei
√
2ϕ(z)
,
(134)
The ground state trial wave function is obtained by computing the correlator
of an even number of electron operators:
ΨM R
N
N
√
Y
Y
i 2ϕ(zi )
=h ψ(zi )ih e
Obg i = Pf
i=1
i=1
1
zi − zj
Y
1 PN
2
(zi − zj )2 e− 4`2 i=1 |zi | ,
i<j
(135)
where
the Pfaffian of an antisymmetric matrix, A, is defined by Pf(A) =
p
Det(A). Note that the second factor in (135) is a Laughlin state for bosons,
at filling fraction ν = 21 , and the Pfaffian factor makes the wave function fully
anti-symmetric without changing the filling fraction. Experimentally there is
no FQH state at this filling fraction, but there is a candidate at ν = 5/2
that amounts to first having filled the two lowest Landau levels completely.
Numerical calculations indicate that the effective electron repulsion between
the electrons in the next level is such that the wave function (135) is favoured.
Before moving on the quasiholes, you should notice that the Pfaffian factor
is precisely of the form of a BCS wave function for a fixed number of particles,
where 1/(zi −zj ) plays the role of the pairing function g(~ri −~rj ). But, referring
back to Sect. 3.1.2, this is precisely what was found for the weakly paired phase
of the 2d p-wave superconductor[49].
To understand the quasihole excitations, we must first take a step back and
understand the relation between holes and electrons in the Laughlin states.
For the ν = 1/3 state, a collection of three holes can, seen from afar, not be
distinguished from the absence of an electron,
lim
ηa ,ηb ,ηc →η
22
N
N
N
N
Y
Y
Y
Y
(zi − ηa ) (zi − ηb ) (zi − ηc ) =
(zi − η)3
i=1
i=1
i=1
For a standard text on CFT, see [70].
51
i=1
(136)
where η is the position of the missing electron. In the CFT, this is expressed
by the operator product expansion (OPE),
lim
ηa ,ηb ,ηc →η
H3 (ηa )H3 (ηa )H3 (ηa ) ∼ V3 (η) .
(137)
This and all other OPEs relevant for the Laughlin states, can be obtained by
repeted use of the basic OPE for vertex operators,
: eiαϕ(z) :: eiαϕ(w) : ∼ (z − w)αβ : ei(α+β)ϕ(z) :
(138)
where ∼ denotes the leading term in a derivative expansion. Another way of
expressing this is by the fusion rule Vα × Vβ = Vα+β . Although in this simple
case, the fusion rule and the OPE follow form the two-point function implied
by the Lagrangian, for a general CFT the fusion rules are more fundamental,
and a Lagrangian might not even exist. In the MR state, the hole excitations
are described by,
HM R (η) = σ(η)e
√i ϕ(η)
8
,
(139)
where σ is a spin operators. The fusion rules for this CFT are,
ψ×ψ =1 ;
1×ψ =ψ
;
σ×σ =1+ψ
(140)
which shows that two holes can fuse to either the trivial state, 1 (which has
the quantum numbers of the vacuum) or a fermion. The presence of different
fusion channels implies that to specify a multi-hole state, it is not sufficient
to know the positions of the holes, one must also have information about
the fusion channels. For 2M holes, each of the first M − 1 pairs (order is
unimportant) can fuse to either 1 or ψ. From the two first relations in (140)
it follows that the first M − 1 pairs will fuse to either 1 or ψ. The last pair
must then be in the channel that makes the full correlator 1, since otherwise
it can be shown to vanish. From this we conclude that the Hilbert space for
2M holes at fixed positions has the the dimension 2M −1 .
To explain in detail how the non-abelian statistics comes about would takes
us to far, so I will jus tell you the result. In the Laughlin case, braiding the
world lines of the holes will result in a phase factor einπ/k , where n is the total
number of turns the particles make around each other, counting clockwise and
anti-clockwise with different signs. In the present case, a braiding operation
will amount to acting on the (multi-component) wave function with a unitary
transformation U , and in general two U s corresponding to different braidings
do not commute - thus the name non-abelian fractional statistics. The rules
for calculating the U for a particular braid can be found in e.g. Ref.[71].
52
Acknowledgement: I thank Maria Hermanns for making useful comments
on a preliminary version of these notes. The derivation of the parity anomaly
given in Appendix C was shown to me by the late professor Ken Johnson of
MIT.
A
How to normalize the current
Here we shall study the first term ∼ µνσ bµ ∂ν aσ in (42) a bit more carefully.
Just as the Chern-Simons term this is a topological action, and the two fields a
and b have no bulk dynamics. This is however not the full story. If the system
is defined on a finite area, Lx × Ly , with periodic boundary conditions, the
zero modes of the fields do acquire dynamics. To understand this, we write
the action on the Hamiltonian form,
Z
k
d3 x µνσ bµ ∂ν aσ
(141)
SBF =
2π
Z
k
d3 x [ij ȧi bj + a0 (ij ∂i bj ) + b0 (ij ∂i aj )] .
=
2π
Note that although the Hamiltonian formally vanishes in the a0 = b0 = 0
gauge, these fields are Lagrange multiplier fields that impose the ”Gauss law”
constraints ij ∂i aj = ij ∂i bj = 0, which can be solved by,
2π
āi (t)
Li
2π
b̄i (t)
= ∂i λb (~r, t) +
Li
ai = ∂i λa (~r, t) +
bi
(142)
where āi and b̄i are spatially constant, and λa/b are periodic functions on the
torus. Inserting this into (141) gives the Lagrangian
Z
SBF = k 2π dt ij ā˙ i b̄j .
(143)
From this we can read the canonical commutation relations23 .
[āi , b̄j ] =
i ij
.
2πk
23
(144)
In case you do not know how to handle actions that are first order in time derivatives,
you can learn in e.g. [72].
53
The Wilson lines along the cycles of the torus given by,
Ai = ei
H
dxi ai
= e2πiā
Bi = ei
H
dxi bi
(145)
are invariant under the large gauge transformations āi → ai + nai , and b̄i →
bi + nbi , corresponding to threading unit fluxes though the holes in the torus.
To assume that the charges that couple to the gauge fields a and b are conserved is equivalent to taking the these fields to be compact, which means that
field configurations differing by the large gauge transformations are identified.
Put differently, the dynamical variables are not the constant U (1) fields āi and
b̄i , but the Wilson loop operators Ai and Bi , which satisfy the commutation
relations,
Al Bm − e
2πi
k
Bm Al = 0
l 6= m .
(146)
For k = 1 all operators commute, and there is a unique ground state, while
for k = 2 we have the algebra,
Ax By + By Ax = 0 and
Ay Bx + Bx Ay = 0 .
(147)
Each of these algebras have a two dimensional representation (think of the
Pauli matrices!) and thus the ground state is 2 × 2 = 4 fold degenerate.
B
An elementary derivation of eq. (50)
Using the notation, dˆ · ~σ ξ± = ±ξ± and ξ ≡ ξ+ we have
(∂i ξ † )∂j ξ = ∂i (ξ † dˆ · ~σ )∂j ξ = (∂i ξ † )dˆ · ~σ ∂j ξ + ξ † (∂i dˆ · ~σ )∂j ξ
= (∂i ξ † )dˆ · ~σ ∂j ξ + ξ † (∂i dˆ · ~σ )∂j (dˆ · ~σ ξ)
(148)
= (∂i ξ )dˆ · ~σ ∂j ξ − ξ † (∂i dˆ · ~σ )∂j ξ + ξ † (∂i dˆ · ~σ )(∂j dˆ · ~σ )ξ
†
ˆ i dˆ = 0. Making
where we in the last equality used σ a σ b = −σ b σ a +2δ ab and d·∂
similar manipulation we get,
1
(∂i ξ † )∂j ξ = (∂i ξ † )dˆ · ~σ ∂j ξ + ξ † (∂i dˆ · ~σ )(∂j dˆ · ~σ )ξ
2
(149)
†
ˆ σ = ξξ † −ξ− ξ−
Next we need the spectral decomposition of the Hamiltonian, d·~
†
and the resolution of unity 1 = ξξ † + ξ− ξ− to get the two identities,
†
†
(∂i ξ † )dˆ · ~σ ∂j ξ = (∂i ξ)(ξξ † − ξ− ξ−
)ξ = Ai Aj + (ξ † ∂i ξ− )(ξ−
∂j ξ) (150)
†
†
)ξ = Ai Aj − (ξ † ∂i ξ− )(ξ−
∂j ξ)
(∂i ξ † )∂j ξ = (∂i ξ)(ξξ † + ξ− ξ−
54
where we used ξ † ∂i ξ = −(∂i ξ † )ξ etc.. Combining (149) and (150) gives,
1
(∂i ξ † )∂j ξ = Ai Aj + ξ † (∂i dˆ · ~σ )(∂j dˆ · ~σ )dˆ · ~σ ξ
4
(151)
Again using the σ matrix algebra, we finally get,
1
ˆ · dˆ
B(~k) = iij ∂i ξ † ∂j ξ = − (∂i dˆ × ∂j d)
2
C
(152)
The parity anomaly in 2+1 dimensions
The most direct way to extract the effective action (54) is to simply calculate
the Feynman diagram in Fig.1 with a suitable regularization[35]. An alternative, and quite instructive, way is to pick a specific background field where
the Dirac equation can be solved exactly and then evoke gauge and Lorentz
invariance to find the general result.
We define the Dirac α-matrices by,
(β, α1 , α2 ) = (σ 3 , −σ 2 , σ 1 )
(153)
and will use complex coordinates
r
z=
eB
(x + iy)
2
~ = B (−y, x),
and the notation ∂ = ∂z and ∂¯ = ∂z̄ . In the symmetric gauge, A
2
where B is a constant magnetic field. The Hamiltonian for the relativistic
massless Landau problem becomes,
~
H = α
~ · (~p − eA)
!
√
√
0
− √12 (∂ − z̄)
0 a†
=
eB
= eB
¯ + z)
√1 (∂
0
a 0
2
(154)
~ and [a, a† ] = 1. Introducing the corresponding number
where p~ = −i∇,
operator state a† a|ni = n|ni, we easily find the following solutions to the
Schrodinger equation,
|0i
|Ψ0 i =
; E0 = 0
(155)
0
√
1
|ni
|Ψn± i = √
; E± = ± neB
2 ±|n − 1i
55
Note that after adding a mass term, Hm = βm, |ψ0 i is still a solution but with
the eigenevalue E0 = m.
The energy levels are however massively degenerate, and in the radial gauge
the relevant extra quantum number, is the angular momentum. Defining,
√
b† = a + 2z
(156)
we have [b, b† ] = [a, a† ] = 1 with all other commutators vanishing, so the full
Schrodinger spectrum is obtained from (155), by the replacement
(b† )k (a† )n
|ni → √ √ |0, 0i .
k! n!
(157)
Next we expand the Dirac field operator in the eigenfunctions ψn,k± = hx|Ψk,n± i
ψ̂(x) =
i X
Xh
ψn,k+ e−iEn t cn,k + ψn,k− eiEn t d†n,k +
ψ0,k e−imt e
n,k
(158)
k
where we regularized the zero mode by adding a small mass m2 eB, and
introduced the fermionic operators cn,k and dn,k satisfying {c†ñ,k̃ } = δn,ñ δk,k̃
etc., and the Majorana operator e which satisfies e2 = 0 and anti-commutes
with all other fermi operators.
For simplicity we considered the m = 0 case, but it is not too hard to find
the full solution even for m 6= 0. The zero modes we have already found, and
the only thing we shall need for the following is that the rest of the spectrum
is gapped and symmetric around E = 0√which follows from charge conjugation
symmetry; fact the energies are En = neB + m2 .
The next step is to calculate the vacuum expectation value of the current
operator
e
j µ = [ψ̄, γ µ ψ]
2
(159)
and for simplicity we shall just consider the time component, i.e. the charge.
Realling that γ 0 = β, we get
e
j 0 = ρ = (ψ † ψ − ψψ † )
2
(160)
and because of the charge conjugation symmetry, only the zero modes contribute to the expectaton value
eX
hρi = ±
|ψ0,k |2 .
(161)
2 k
56
where the positive sign corresponds to a negative m meaning that the (almost)
zero modes are all filled so the contribution comes from the first term in (160);
the negative sign amounts to these modes all being empty. A direct calculation
of the zero-mode wave functions gives,
r r
X
X
eB 2k k −|z|2 2
2
|ψ0k | =
|
z e
|
(162)
2π k!
k
k
X eB
eB
2
=
(2|z|2 )n e−|z| =
(163)
2π
2π
k
which just shows that the density of state in the lowest Landau level is the
same as in the non-relativistic case. Inserting this in (160) gives
hρi =
m e2 B
|m| 4π
(164)
and by Lorentz and gauge invariance.
hj µ i =
m e2 µνσ
δΓD [A]
,
∂ν Aσ = −
|m| 4π
δAµ
(165)
Integrating this expression to get the effective response action, we get the
result (54) quoted in the main text. Note that although the Chern-Simons
term is not gauge invariant, its variation, given by (165), is.
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