Gauge Dynamics and Topological Insulators

# Gauge Dynamics and Topological Insulators

Benjamin Béri, David Tong and Kenny Wong
TCM Group, Cavendish Laboratory, University of Cambridge, UK
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK
bfb26@cam.ac.uk, d.tong@damtp.cam.ac.uk, k.wong@damtp.cam.ac.uk
###### Abstract:

A non-abelian magnetic field in Yang-Mills theory induces the formation of a “W-boson” vortex lattice. We study the propagation of fundamental fermions in the presence of this lattice in dimensions. We show that the spectrum for massless fermions contains four topologically-protected Dirac points with non-zero Bloch momentum. For massive fermions, we compute topological invariants of the band structure and show that it is possible to realise a topological insulator within Yang-Mills theory.

## 1 Introduction

It was discovered long ago that the presence of a background magnetic field in a non-abelian gauge theory results in a tachyonic mode for the gauge bosons [2]. Upon condensation, they form a vortex lattice. Many properties of this instability and the resulting ground state have been studied, most notably in a series of papers by Ambjørn and Olesen [3]-[8].

We know from condensed matter physics that the presence of a lattice gives rise to a range of beautiful phenomena, from elementary band structure to the emergence of novel topologically protected quantities. The purpose of the present paper is simply to understand some of these properties in Yang-Mills theory by studying the propagation of fundamental fermions in the background of the magnetically-induced gauge boson lattice. We will see how a number of aspects of topological insulators naturally arise in non-Abelian gauge theories.

In pursuing this problem, there is one immediate obstacle: the full solution for the background lattice is known only numerically. Fortunately, the topological nature of our quest comes to the rescue. We will show that a number of the key features of the fermion dynamics can be computed exactly, even though one doesn’t have full control over the background in which they move. These properties, which are protected by a combination of topology and symmetry, will be the focus of this work.

While much of the work on topological insulators focusses on theories with a gap, there are also interesting things to say when the gap closes. In our case, we will show that, in the absence of anomalies, Dirac points in the W-boson lattice necessarily come in multiples of four. Perhaps somewhat surprisingly, these Dirac points do not lie at zero Bloch momenta . Instead, they are shifted to , where is the magnitude of the magnetic field. We will show that the position of these Dirac points is one of properties that can be computed exactly: assuming certain symmetries, their locations are immune to both deformations of the lattice and quantum corrections.

For massive fermions propagating in the background of the lattice, the band structure has a gap. We are now firmly in the realm of topological insulators, where a number of topological invariants are associated to the band structure of gapped systems. (See, for example, [9, 10] for reviews). The presence of a background lattice emerging naturally in Yang-Mills theory provides an opportunity to explore these ideas in a context familiar to high-energy theorists. To this end, we provide explicit calculations of the first Chern numbers, known as TKNN invariants [11], as well as the more recently discovered invariant [12] of the lowest bands. We will see that the resulting phenomenology of this phase is essentially that of quantum spin-Hall physics. Moreover, we will show that Yang-Mills coupled to fundamental fermions contains within it the simplest topological insulator111It has recently been argued that QCD in dimensions also lies in a topological phase [13]..

We start in Section 2 by reviewing the gauge boson instability. We present a linearised approximation to the full lattice solution. As we explain, this linearised approximation is not a particularly good approximation. Nonetheless, it will prove sufficient for our needs, acting as a springboard for the exact results that follow.

Section 3 studies the dynamics of massless fermions in this lattice. We diagonalise the fermions in the background of the lattice and compute the location of the Dirac points, before showing that the existence and location of these points is exact. We also find something rather cute in the linearised regime: the off-diagonal gauge boson field — which we refer to as the “W-boson” — plays a dual role. It not only describes the spatial lattice, but also, with a suitable rescaling, determines the energy over the Brillouin zone, .

Section 4 deals with massive fermions. We explain in detail how to quantize the fermions, pay particular attention to the discrete symmetries of the theory and provide detailed calculations of the topological invariants. We also pause in number of places to offer a translation between high-energy and condensed matter language. In particular, we will see how one can vary the chemical potential of the system, to move between trivial and non-trivial topological insulators. Finally, a number of more involved calculations are relegated to a series of appendices.

## 2 Non-Abelian Magnetic Fields

Throughout this paper we discuss dimensional Yang-Mills coupled to a background Higgs field and Dirac fermions222Our conventions: We use signature . The non-Abelian field strength is where the generators are . When we introduce fermions in Section 3, we use the representation of gamma matrices .. The fermions are our primary interest and will be discussed in Sections 3 and 4. The purpose of this section is to introduce the bosonic background in which the fermions propagate. This is a solution to the action

 Sbosonic=−∫d3x  12e2TrFμνFμν+ TrDμϕDμϕ+g4(Trϕ2−v22)2 (2.1)

where is a real, adjoint-valued scalar field.

We will restrict ourselves to the regime where the theory is weakly coupled, and we assume that where fluctuations of the Higgs field are suppressed. The gauge group is broken to and the low-energy excitations consist of a photon and charged W-bosons of mass . We decompose the gauge field as

 Aμ=12(aμWμW⋆μ−aμ)

Throughout this paper, we are interested in what becomes of excitations in the presence of a background magnetic field,

 F12=B2σ3 (2.2)

Before we get into details, it’s useful to first build a simple, intuitive picture for what’s going on. In a magnetic field all charged states form Landau levels. Consider a scalar field of charge and mass in a magnetic background . The familiar Landau level quantization yields the energy levels

 E2scalar=qB(2n+1)+m2     n=0,1,2,…

Now consider a fermion of charge and mass . The Landau levels take the same form, but with an extra contribution analogous to Zeeman splitting of different spin states. With gyromagnetic factor , the energy levels are

 E2fermion=q[B(2n+1)+2Bs]+m2

where the spin takes values . For massless fermions, with , the Zeeman splitting for spin down fermions exactly cancels the zero point energy and the resulting Landau level famously has vanishing energy. We will discuss the fate of such modes in the next Section.

Of more immediate concern is the effect of the magnetic field on the W-bosons. These have spin 1 and charge . One can show that the gyromagnetic factor is again and the Zeeman splitting now overcompensates for the Landau level zero point energy. The states have energy

 E2W−boson=B(2n+1)+2Bs+m2W

where the spin takes values . When the magnetic field exceeds the critical value , the lowest Landau level modes with are tachyonic: the constant background magnetic field (2.2) is unstable to condensation of W-bosons.

The instability of the non-Abelian magnetic fields was first noticed in [2]. Upon condensation, the W-bosons form a lattice structure, breaking continuous translational symmetry on the plane. Various properties of this lattice were explored in a number of papers. See, for example, [3]-[8] and [16].

More recently, the gauge boson lattice has appeared in several situations. The same mechanism of instability is at play in the proposed condensation of rho-mesons in a strong magnetic field [17, 18, 19] and has been suggested as a phenomenological model of the QCD vacuum [20]. The gauge-boson lattice has been employed in the holographic context where it now captures the instability of a strongly correlated system in the presence of a magnetic field [21, 22]. Moreover, it is closely related to the monopole wall configuration discussed [23] and, in the holographic context, in [24, 25].

### 2.1 The W-Boson Lattice

We will now provide a more pedestrian derivation of the tachyonic mode that we described above. To this end, we focus on the equations of motion for the gauge fields. The Yang-Mills action can be decomposed as

 SYM=−14e2∫d3x  |DμWν−DνWμ|2+2m2W|Wμ|2+|fμν−i2(WμW⋆ν−WνW⋆μ)|2

where is the U(1) field strength and , reflecting the fact that the W-bosons have charge under this unbroken .

The equations of motion arising from this action admit the solution and . However, as described above, this solution is unstable. To see this it is sufficient to examine the linearised equations of motion. Working in gauge, these read

 ∂2tWx = −Dy(DyWx−DxWy)+iBWy+m2WWx ∂2tWy = −Dx(DxWy−DyWx)−iBWx+m2WWy (2.3)

supplemented by the Gauss’ law constraint

 ∂t(DxWx+DyWy)=0

Taking suitable linear combinations , it is simple to see that the solutions to (2.3) include modes with . When , these modes become tachyonic.

To describe these modes more explicitly, we must pick a gauge. We choose Landau gauge,

 ax=0   and   ay=Bx (2.4)

The tachyonic, lowest Landau level modes are then given by

 Wx=−iWy=exp(−iky−B2(x+kB)2) (2.5)

These modes are labelled by the continuous parameter .

When these modes condense, they break both rotational and translational invariance in the plane. The result is a lattice structure, similar to the familiar Abrikosov vortex lattice that arises in superconductors. The lattice spacing is set by the magnetic field itself. The W-bosons form currents, screening the magnetic field near the vortex cores while enhancing the magnetic field in the space between between the cores [8]. (This is the opposite to a conventional superconductor).

At the onset of the instability, we can trust the linearised approximation (2.3) and construct the W-boson lattices from the lowest Landau level modes above. A one-parameter family of rhombic lattices is given by the ansatz , where is the linear superposition

 W(x,y)=¯W∞∑n=−∞exp⎛⎝−iπn22−2πin√Byλ−B2(x+2πnλ√B)2⎞⎠ (2.6)

Here, is a constant which is undetermined at the linearised level. The condensation of W-bosons acts as a Higgs mechanism for the (until now) unbroken and the associated photon gets a mass at the scale .

The parameter determines the geometry of the lattice. For example, a square lattice arises from the choice ; a triangular lattice from . This can be seen by noting that the W-boson lattice (2.6) is invariant under the translations by the lattice vectors and . As usual in a magnetic field, translational invariance is only assured up to phase,

 W(→x+→d1)=W(→x)    and    W(→x+→d2)=iexp(2πi√Byλ)W(→x)

For all values of , the unit cell has area in accord with flux quantization.

The parameter is arbitrary at the linearised level. It will be determined by non-linear terms which are expected to energetically prefer the triangular lattice. (The square lattice is, apparently a local, but not global, minimum). The lowest Landau level ansatz (2.6) for the triangular lattice is shown in Figure 1. The phase of winds once around each of the vortex cores, located at with . Naturally, at the core itself.

The linearised approximation (2.6) is only justified near the onset of instability: it is no longer trustworthy when . Nonetheless, in what follows we will work exclusively with the linearised form of the lattice (2.6). The reason for this is that we are ultimately interested in computing topological properties of our system. In the absence of a phase transition, these cannot change as the higher order corrections to the W-boson lattice are taken into account. In Sections 3 and 4, we will identify a number of such properties which, although computed using the linearised lattice, remain true in the full non-linear solution.

Finally, we note that although the W-boson lattice is much studied, there remain open issues about the question of stability. Indeed, it seems unlikely that the lattice is globally stable. We discuss some of these issues in Appendix A.

## 3 Massless Fermions

We now turn to the fermionic excitations. We add to the action (2.1) two Dirac fermions,

 SDirac=−∫d3x  i¯ψ1/Dψ1+i¯ψ2/Dψ2 (3.7)

Both Dirac fermions and transform in the fundamental representation of the gauge symmetry. The parity anomaly prohibits an odd number of fundamental fermions [14, 15], which means that the pair of fermions above is the minimum number possible333The action (3.7) can be obtained by dimensionally reducing dimensional Yang-Mills theory coupled to a single fundamental Dirac fermion , Upon dimensional reduction, the Dirac spinor becomes and above. The presence of a Dirac, as opposed to a Weyl, spinor ensures that the theory is free from the Witten anomaly.. The theory admits a flavour symmetry, with transforming as a doublet.

As we explained above, in a constant magnetic field the Landau level of the fermions has zero energy. We start by describing some properties of this Landau level before moving on to study the effect of the W-boson lattice on these states444For earlier studies of fermions in fixed, background gauge fields, see [26, 27, 28]..

The Landau level has one special property: it contains only half the states of higher Landau levels. Our theory has 8 fermionic excitations. This counting arises because the Dirac spinor has two components while our fermions are doublets of both the gauge symmetry and the flavour symmetry. While higher Landau levels are populated by all 8 species, the Landau level contains only 4 species of fermions. This can be seen by explicitly solving the zero modes of the Dirac equation. We write the gauge doublet as

 ψi=(ψ−iψ+i)

where is the species index. Each component is a Dirac spinor. In Landau gauge (2.4), an orthonormal basis for the the zero modes is provided by

 ψ−i = 4√B2π∫dk2π exp(−iky−B4(x+2kB)2)(ξ−i(k)0) (3.8) ψ+i = 4√B2π∫dk2π exp(+iky−B4(x+2kB)2)(0ξ+i(k)) (3.9)

Here are four Grassmann valued collective coordinates for the Landau level. The argument reflects the degeneracy of this Landau level. Because the spinors transform in the fundamental representation of the gauge symmetry, the spinors have charge . This means that the degeneracy of each species is .

### 3.1 Fermions on the Lattice

The expressions above describe the zero energy states in a constant background magnetic field. However, as we have seen, the magnetic field also causes the W-bosons to condense into a lattice structure. We would like to understand the effect of this W-boson lattice on the fermions. Because both have the same origin, the magnetic and lattice lengths are commensurate. In particular, as fermions have charge under , the area of the fermion magnetic unit cell is twice that of the W-boson lattice unit cell.

Substituting our expressions for the lattice (2.6) and fermion zero modes (3.8) and (3.9) into the action (3.7) gives us an expression for the shift in energy of the fermions,

 (3.10)

Here we have neglected terms coupling the Landau level to higher Landau levels. The effect of such terms on the energy spectrum is suppressed by . The function is a convolution of the W-boson lattice and fermionic zero modes

 ϕ(k,k′) = √B2π∫dxdy exp(i(k′+k)y−B4(x+2kB)2−B4(x+2k′B)2)W(x,y)

The lattice perturbation has the effect of coupling the different modes in the Landau level. Our task is to diagonalise this Hamiltonian while simultaneously keeping the kinetic terms diagonal. This is achieved by the linear combination which is compatible with the magnetic translational symmetry,

 ζ+i(p1,p2) = 14√π2λ2B∞∑n=−∞exp(4πinp2λ√B)ξ+i(+p1+2πn√Bλ) (3.11) ζ−i(p1,p2) = 14√π2λ2B∞∑n=−∞exp(4πinp2λ√B)ξ−i(−p1+2πn√Bλ) (3.12)

We have replaced the index labelling the Landau level modes with a pair of indices . From the definitions, it is simple to check that these variables are periodic, taking values in the range

 pi∈[−Δpi/2,+Δpi/2) with Δp1=2π√Bλ   ,   Δp2=λ√B2 (3.13)

In Appendix B, we demonstrate that are (up to a trivial relabelling) the magnetic Bloch momenta and the torus that they parameterize is the magnetic Brillouin zone. For now, let us simply note that the area of this torus is , which follows from the fact that the area of the magnetic unit cell in real space is and the area of the corresponding unit cell in reciprocal space should be .

In terms of the new variables , the Hamiltonian (3.10) becomes

 H=1√22∑i=1∫T2d2p [iW(2p1B,2p2B)ζ−i(→p)⋆ζ+i(→p)+h.c.] (3.14)

A proof of this result is provided in Appendix C. Something rather nice has happened here. The function , which describes the profile of the W-boson lattice in real space, has a dual role as the energy function over the magnetic Brillouin zone.

The Hamiltonian (3.14) is diagonal in momentum modes, but still requires a trivial diagonalisation in the index. The result is that each species gives rise to a pair of states with energy . This spectrum is drawn in Figure 2. In the ground state, the lower band with negative energies is filled. The excitation spectrum then consists of particles and holes, both with energy . We describe the quantisation of the fermions in more detail in the next section. For now note that, for each species, there exist two gapless points within the magnetic Brillouin zone where . Using the form of the linearised W-boson lattice (2.6), these Dirac points are seen to lie at555A repeated word of caution: a simple relabelling is needed before interpreting these as Bloch momenta. The true Bloch momenta are . This is explained in Appendix B.666This result generalises to . Applying a background non-Abelian magnetic field in the direction, the W-bosons now condense to form a lattice of vortex flux tubes which are translationally invariant in . Fermion propagation in the - plane is governed by (3.14), while propagation in the -direction is trivial. The result is that there are again four Dirac points with momenta

 →p⋆=±(Δp14,−Δp24) (3.15)

### 3.2 Robust Dirac Points

The structure of fermions that we have uncovered in this section is that of a semi-metal. There are four points – two for each flavour – at which the spectrum is gapless, with the fermions described by a Dirac cone. The positions of these Dirac points lie at momenta (3.15). As with all our analysis, this result was derived using the linearised W-boson lattice. The purpose of this section is to show that these results are robust. They continue to hold in the full non-linear lattice. Moreover, they are unaffected by quantum corrections (at least in the absence of a phase transition).

The robust nature of the Dirac points follows the arguments given in [29, 30, 31] (many of which of which can be traced back to [32, 33]). It hinges on the winding of , which, in real space, is shown in Figure 1. Around each of the Dirac points, the phase of winds once. These Dirac points are vortices in momentum space and, due to their topological nature, are protected against perturbations.

To see this winding in more detail, consider the transformation of under reciprocal lattice translations. It is simple to check that is invariant under while, under a translation along , it picks up a phase

 W(2(p1+Δp1)B,2p2B)=e4πip2/Δp2W(2p1B,2p2B) (3.16)

Computing the winding around the magnetic Brillouin zone we have

 ΔargW=−i∮W−1∇W=4π

This result, which follows solely from the translational properties of , tells us that has to have zeroes: these are the two Dirac points.

One may wonder how can have a non-zero winding over , a compact manifold without boundary. The answer lies in the fact that it is the Hamiltonian (3.14) which must be single valued over the torus. The winding of is cancelled by an opposite winding of the operators. Indeed, the vortices in momentum space are effectively inherited from the well-known winding of magnetic Bloch states. The latter can be seen in our operators which, under translation, transform as

 ζ+i(p1+Δp1,p2) = e−2πip2/Δp2ζ+i(p1,p2) ζ−i(p1+Δp1,p2) = e+2πip2/Δp2ζ−i(p1,p2) (3.17)

In Section 4.2, we will see that this winding underlies the nontrivial TKNN invariants (or first Chern numbers) of the fermion bands.

Let us now discuss how these windings persist as the W-boson lattice relaxes to its fully non-linear configuration. It can be shown that the invariance of the system under time reversal, charge conjugation, magnetic lattice translations and global transformations ensures that the quadratic part of the Hamiltonian takes the form

 H=1√22∑i=1∫T2d2p [iWNL(2→pB)ζ−i,NL(→p)⋆ζ+i,NL(→p)+h.c.] (3.18)

where the fermions are adiabatically connected to functions defined in (3.11) and (3.12) and the function is smoothly connected to the function defined in (2.6). Furthermore, this form must also hold in the presence of loop corrections.

In the absence of a phase transition, the winding of cannot change as the W-boson lattice relaxes to its non-linear configuration. The pair of Dirac points for each species of fermion persist.

#### Location of the Dirac Points

We will now show that, given the existence of two Dirac points for each fermion, the symmetries are also sufficient to dictate where they sit. We will need to make a couple of technical assumptions which boil down to the requirement that inherits certain discrete symmetries from from . The first is translational symmetry. The linearised lattice obeys

 W(2p1B+Δp1B,2p2B+Δp2B)=ie2πip2/Δp2W(2p1B,2p2B) (3.19)

The existence of this symmetry can be traced to the fact that the lattice unit cell (in real space) has only half the area of the magnetic unit cell. We assume that the function continues to transform covariantly under translations. The second is an inversion symmetry

 W(2→pB)=W(−2→pB) (3.20)

We assume that also enjoys this property.

The translational symmetry (3.19) and inversion symmetry (3.20) ensure that the Dirac points, defined by , sit at one of two positions: either

 →p⋆=±(Δp14,−Δp24)

or

 →p⋆=±(Δp14,Δp24)

Comparing to (3.15), we see that our lattice realises the first of these possibilities. This choice cannot change as the lattice is smoothly deformed777While this statement is true, in Appendix B we will see that there is an ambiguity in the definition of the choice of Bloch momenta (essentially a choice of origin) which means that the physics is left unchanged for lattices which realise the second choice above..

To conclude this section, let us finish by describing a property which is not robust to corrections: the speed of propagation of the Dirac modes. For the linearised lattice (2.6), it is clear that the dispersion relation near the Dirac point takes the form where the speed is given by . As the ratio increases, the speed of propagation is expected to increase.

## 4 Massive Fermions

In this section, we would like to explore in more detail the structure of the fermionic modes. To do so, we first open up a gap in the spectrum. This is achieved by simply returning to our original action (3.7) and adding a mass for the fermions. We choose to give the different species and equal and opposite masses,

 Smass=∫d3x im(¯ψ1ψ1−¯ψ2ψ2) (4.21)

The significance of this choice lies in the observation that it preserves time reversal, a fact that will prove to be important later. To see this, note that in dimensions the fermionic bilinear is odd under time reversal. However, we can define a new version of this symmetry in which we simultaneously exchange and . The operator (4.21) is invariant under this new symmetry and it also preserves charge conjugation. Full details of the action of time reversal invariance as well as charge conjugation are provided in Appendix D.

The addition of the mass term breaks the flavour symmetry down to a subgroup. Under the pair of Abelian symmetries , has charge while has charge .

### 4.1 Quantisation

We now look a little closer at the quantum states in the Landau level. The effect of the mass is, unsurprisingly, to open up a gap in the spectrum. In terms of the Bloch momenta, the Hamiltonian becomes

 H=∫T2d2p [ mζ−⋆1ζ−1−mζ+⋆1ζ+1+iW√2ζ−⋆1ζ+1−iW⋆√2ζ+⋆1ζ−1 −mζ−⋆2ζ−2+mζ+⋆2ζ+2+iW√2ζ−⋆2ζ+2−iW⋆√2ζ+⋆2ζ−2]

Quantising the fermions gives rise to a pair of particles and a pair of anti-particles, each with energy

 E(→p)2=m2+12|W(2→p/B)|2

The resulting band structure is shown in the figure. To look at these states in more detail, we diagonalise the components. This is simply achieved by introducing annihilation and creation operators and with . Assuming that , we write the creation operators as

 a⋆1 = √E+m2Eζ−⋆1−iW⋆√4E(E+m)ζ+⋆1 b⋆1 = iW⋆√4E(E+m)ζ−1+√E+m2Eζ+1 a⋆2 = iW√4E(E+m)ζ−⋆2+√E+m2Eζ+⋆2 b⋆2 = √E+m2Eζ−2−iW√4E(E+m)ζ+2 (4.22)

where . (Strictly speaking, the collective coordinates should be evaluated at in the above expressions). This choice of basis has the advantage that it remains non-degenerate at the points . It is simple to check that the canonical equal time anti-commutation relations for or, equivalently, , translate into the canonical anti-commutation relations for and ,

 {ai(→p),a⋆j(→p′)}=δijδ(→p−→p′)    and     {bi(→p),b⋆j(→p′)}=δijδ(→p−→p′)

In terms of these new operators, the lowest Landau level Hamiltonian is, after normal ordering, simply

 H=2∑i=1∫BZd2p  E(a⋆iai+b⋆ibi) (4.23)

The ground state of the system obeys and is neutral under the global symmetry.

The excited states fall into one of four bands: a pair of particles and a pair of anti-particles. Each is labelled by the Bloch momenta and carries flavour charge,

+1 +1
+1

However, there are further quantum numbers that can be associated to each band that follow from topological considerations. In the next two sections we turn to these, first dealing with the TKNN invariants and then moving on to the more subtle invariant.

### 4.2 TKNN Invariants

In the presence of a gap, bands can be classified by a number of topological invariants. These arise from the winding of the states as we move around the Brillouin zone . The simplest of these invariants is the first Chern number or TKNN invariant [11].

We start by focussing on the band of states . As we move around the magnetic Brillouin zone, parameterised by , the phase of these states changes. This is captured by the Berry connection, an Abelian 1-form over defined by

 γ(→p)=i⟨Ω|a1∂∂→pa⋆1|Ω⟩⋅d→p

The TKNN invariant is then defined to be the first Chern number,

 C1=12π∫T2dγ

Analogous definitions also hold for the other three bands of states, , and .

To evaluate the first Chern number, we need to know how the states transform under translations through reciprocal lattice vectors and defined in (3.13). We assume that the ground state is invariant. One can check that the creation operators defined in (4.22) are invariant under . However, under a translation along , they pick up a phase.

 a⋆1(p1+Δp1,p2) = e−2πip2/Δp2a⋆1(p1,p2) a⋆2(p1+Δp1,p2) = e+2πip2/Δp2a⋆2(p1,p2) b⋆1(p1+Δp1,p2) = e−2πip2/Δp2b⋆1(p1,p2) b⋆2(p1+Δp1,p2) = e+2πip2/Δp2b⋆2(p1,p2) (4.24)

To proceed, we cover the magnetic Brillouin zone with four charts, , . These charts are defined in Figure 4 where we have also given names through to a number of overlap regions.

By Stokes’ theorem, we can express the Chern number as a sum of integrals of the connection around the boundaries of the four charts.

 C1=12π(∮∂U1γ+∮∂U2γ+∮∂U3γ+∮∂U4γ)

Naively one might think that this vanishes vanishes because we’re integrating along each edge twice, once in each direction. However, the fact that the creation operators transform by a phase under translations by reciprocal lattice vectors means that the Berry connections evaluated on the edges of different charts differ by a gauge transformation. This is responsible for the non-vanishing Chern number.

Let’s focus once again on the states. In overlap region , we have

 γp2∣∣U2=γp2∣∣U1+2πΔp2

Similarly, in overlap region , we have

 γp2∣∣U4=γp2∣∣U3+2πΔp2

On all other overlap regions, the transformation properties are trivial. The resulting TKNN invariant for the band is therefore

 C1=12π∫B(γp2∣∣U2−γp2∣∣U1)dp2+12π∫F(γp2∣∣U4−γp2∣∣U3)dp2=+1

Very similar calculations hold for the other bands. We can now extend the table listing the and flavour charges of the different bands to also include the Chern number,

It is worth seeing how these Chern numbers fit in with the discrete symmetries defined in Appendix D. Under charge conjugation, the states map into . Correspondingly, both have the same Chern number. Meanwhile, under time reversal invariance the states map into . This is reflected in the fact that they carry opposite Chern number.

These TKNN invariants can be related to the Dirac points that arise when the gap closes. We saw in Section 3.2 that the winding of the phase acquired by the fermions as they traverse the Brillouin zone (3.17) is directly related the phase acquired by the W-bosons (3.16). Moreover, the phase acquired by the W-bosons is responsible for the pair of Dirac points that lie in the Brillouin zone. The upshot is that the Chern numbers in the gapped phase are directly related to the number of Dirac points in the gapless phase. Specifically, for each species of fermion,

 \hashDirac points=|C1(a⋆i|Ω⟩)+C1(b⋆i|Ω⟩)|=2

### 4.3 Z2 Invariants

In a theory with a time reversal symmetry with the property that , there is a more subtle topological invariant that one can compute: this is the invariant [12, 34, 35, 36].

We describe the time reversal invariance of our system in Appendix D. There are, in fact, two different time reversal symmetries that one can define. We call these with , defined in in (D.36), and with , defined in (D.37). It is the existence of the latter that allows us to define the invariant in our system.

The index characterises pairs of bands related by time reversal. For our system, we can associate a index to the pair and another index to the pair. While is usually difficult to compute, it can be easily obtained when a symmetry is present. In this case, it is simply the parity of the Chern number of either of the two bands within the pair [37, 38, 39]

 ν=C1    mod 2 (4.25)

For our system, this gives

 ν(ai|Ω⟩)=ν(b⋆i|Ω⟩)=1

The same result can also be obtained using the inversion symmetry of the theory. This is discussed in Appendix E.

The above invariants characterise pairs of bands. Usually, however, one is interested in the invariant for the whole system. The relationship is straightforward: one simply sums over pairs of filled bands

 νtot=∑j∈ filled pairsνj       mod 2 (4.26)

In order to make sense of which bands are filled and which are unfilled, we need to return to the 1920’s and a Dirac sea picture of the fermions, in which the anti-particles are holes in filled negative energy states. To this end, we introduce the “zero-energy” state,

 |0⟩=∏→pb⋆1(→p)b⋆2(→p)|Ω⟩

Then the table of the different bands of excitations can be rewritten as

In the vacuum, the time-reversed pair of bands is filled. Using the definition of the index in terms of the Chern number (4.25), we have . In the same picture, higher Landau level antiparticle states are also all filled. However, these higher Landau levels have twice the states of the level. This means that they consist of two time-reversed pairs of bands, each with , and do not contribute to the total sum (4.26). We thus have

 νtot=1 (4.27)

strongly suggesting that the Dirac-Yang-Mills theory is a topological insulator. We will return to this conclusion below.

From the discussion above, it appears that there is no more information in the invariant than in the original TKNN invariants. This is not quite true. We will review why the invariant is the more robust than the TKNN invariants in Section 5.

### 4.4 Quantum Spin-Hall Effect

The existence of a non-trivial invariant in a theory with a global symmetry is usually associated with the quantum spin-Hall effect [40, 41]. Here the global symmetry is identified with electromagnetism while the “spin” is the conserved charge associated to the symmetry. We now discuss how the quantum spin-Hall effect arises in the Dirac-Yang-Mills theory.

Both the Hall conductivity and the spin-Hall conductivity are directly related to the TKNN invariants. In our system, the Hall conductivity for both and vanishes, a fact which is guaranteed by time reversal symmetry. However, a mixed Hall conductivity survives. This is the spin-Hall effect: a background electric field for results in a “spin” current for . The TKNN formula [11, 42] for the spin-Hall conductivity is888There is a slight subtlety here associated to the difference between particle physics and condensed matter language. In condensed matter, the sum over filled bands of the Fermi sea is always finite. In the high-energy language, the sum over bands in the Dirac sea is infinite. A slightly better formulation in the high-energy context, in form that is manifestly invariant under charge conjugation, is (4.28) For our system, this agrees with (4.29).

 σxy=−e22π∑filled bandsQAQFC1 (4.29)

where and are the charges under and . In our theory, this is an infinite sum due to the filled higher Landau levels for anti-particle states. However, as in the calculation of the invariant, the contribution from these states vanishes. This is because the higher Landau levels have twice the states of the level and, for each sector, these carry opposite TKNN invariants. Hence, the only non-vanishing contributions come from which give

 σxy=e2π (4.30)

While it is nice to see the derivation of the spin-Hall conductivity through TKNN invariants, we should stress that, despite appearances, the existence of the lattice played little role in the above discussion. To highlight this, we note that the spin-Hall conductivity can be much more easily derived (at least from a particle physicists perspective) without mention of the lattice or background magnetic field. We need simply couple background gauge fields and to the and currents respectively. The mass term for the fermions prevents the appearance of zero energy states as these gauge fields are introduced. Integrating out the massive fermions then results in mixed Chern-Simons coupling (see, for example, [43] for a review),

 LCS=σxyA(A)∧F(F).

This is the low-energy effective action for the spin-Hall effect. A standard calculation reproduces (4.30).

### 4.5 Topological Classification and Boundary Modes

In the previous sections, we have seen that our system carries non-zero invariant and exhibits the quantum spin-Hall effect. We might be tempted to conclude that we have ourselves a topological insulator of the simplest type (see, for example [9, 10]). In fact, we shouldn’t be too hasty. In this section we explain why it’s a little premature to refer to our system as a topological insulator. In the next, we describe the deformation that is necessary to make this claim.

There is a well-known classification of topological insulators based on the discrete symmetries of the problem. This is, at heart, a classification of Hamiltonians of free fermions [44, 45], and forms the basis of the “periodic table of topological insulators” [46, 47, 48]. It is natural to ask: where in this classification table does our Dirac-Yang-Mills theory sit?

In Appendix D we describe the discrete symmetries of our theory that leave the background W-boson lattice invariant999There is a subtle difference in the language used in high-energy physics and condensed matter physics when describing discrete symmetries. We provide a translation service in Appendix D.. The theory enjoys a charge conjugation symmetry that we call . Importantly, the presence of the W-boson condensate means that this must be defined such that . There is also time reversal invariance. The existence of the global symmetry means that there are two possibilities for time reversal: with or with .

The existence of these symmetries place the fermions in Yang-Mills theory in either symmetry class CI or CII. According to [46, 47, 48] neither of these classes can realise topologically nontrivial massive fermionic phases in dimensions. Yet, as we have just seen, our theory has a non-vanishing invariant and exhibits quantum spin-Hall physics. How to reconcile this with its position in the periodic table?

The resolution lies in the observation that the adjective “topologically nontrivial” is meaningful only if there is a “topologically trivial” reference system; the classification table only counts the number of topologically distinct phases without singling one of them out as “trivial”. The table merely tells us that there is only one phase in class CI/CII and dimensions. From our results above, we conclude that any form of a Dirac-Yang-Mills system must have a nontrivial index, , as long as charge conjugation and time reversal symmetries are preserved.

A similar issue arises if we focus on just one of the sectors, say . This sector alone has charge conjugation, but not time reversal symmetry placing it in class . The classification scheme of [48] states that the Chern number should lie in . However, this should not be interpreted to mean that is even; indeed, the explicit calculation above shows that . Rather, it tells us that Chern numbers of topologically distinct phases differ by an even integer.

The upshot of this is that we may have a “non-trivial” topological insulator, but this only really makes sense if there is a “trivial” insulator to compare with! In the next section we will describe the deformation of our theory that is necessary to realise this. First, however, let us explain how one can see these results from a slightly different perspective.

#### The View from the Boundary

Many of the results from the story of topological insulators can be most simply derived by looking at the modes that propagate on the boundary between two phases. This also provides a useful viewpoint on the discussion above.

The boundary modes are intimately related to the different topological characterisations of the bulk, a relationship which sometimes goes by the name “bulk-boundary correspondence”. This correspondence states:

• For a domain wall interpolating between regions with different indices, there are an odd number of counter-propagating pairs of modes which cannot be given a mass if time reversal invariance is preserved.

• For a domain wall interpolating between regions with Chern numbers and , the difference between the number of left- and right-moving modes is 101010Here, we are using the Dirac sea picture of quantum field theory. The Chern numbers that we refer to are the Chern numbers of the filled bands..

For more details see, for example, [9, 10, 29].

Let us consider a system that respects charge conjugation and time reversal symmetries. Suppose that this system has a domain wall separating two distinct regions, and suppose that this domain wall supports massless propagating excitations. The dynamics of the fermions within the domain wall are described by a dimensional low energy action of the form

 S1D=∫dtdx iψ†∂tψ−∫dtdx ψ†(x)(−iV)∂xψ(x), (4.31)

with a Hermitian velocity matrix .

The discrete symmetries impose certain restrictions on the boundary modes. To see how they arise, let us examine the transformation properties of the velocity matrix under discrete symmetries, starting with charge conjugation. Charge conjugation induces the relation

 ^C[(−iV)∂x]^C−1=(iV)∂x

where is anti-unitary111111The statement that is anti-unitary is probably familiar to condensed matter theorists and foreign to particle theorists. We explain the difference in language in Appendix D where we also give the concrete form of for our problem. and . Equivalently,

 ^CV^