From the adiabatic theorem of quantum mechanics to topological states of matter

Jan Carl Budich1,2 and Bjorn Trauzettel21Department of Physics, Stockholm University, Se-106 91 Stockholm, Sweden;

2Institute for Theoretical Physics and Astrophysics,University of Wurzburg, 97074 Wurzburg, Germany

(Dated: February 14, 2013)

Owing to the enormous interest the rapidly growing field of topological states of matter (TSM)has attracted in recent years, the main focus of this review is on the theoretical foundations of TSM.Starting from the adiabatic theorem of quantum mechanics which we present from a geometricalperspective, the concept of TSM is introduced to distinguish gapped many body ground states thathave representatives within the class of non-interacting systems and mean field superconductors,respectively, regarding their global geometrical features. These classifying features are topologicalinvariants defined in terms of the adiabatic curvature of these bulk insulating systems. We review thegeneral classification of TSM in all symmetry classes in the framework of K-Theory. Furthermore,we outline how interactions and disorder can be included into the theoretical framework of TSMby reformulating the relevant topological invariants in terms of the single particle Green’s functionand by introducing twisted boundary conditions, respectively. We finally integrate the field of TSMinto a broader context by distinguishing TSM from the concept of topological order which has beenintroduced to study fractional quantum Hall systems.

I. INTRODUCTION

Stimulated by the theoretical prediction1–3 andexperimental discovery4 of the quantum spin Hall(QSH) state, tremendous interest has been recentlyattracted by the study of topologocical properties ofnon-interacting band structures5–7. A topological stateof matter (TSM) can be understood as a non-interactingband insulator which is topologically distinct from aconventional insulator. This means that a TSM cannotbe adiabatically, i.e., while maintaining a finite bulkgap, deformed into a conventional insulator withoutbreaking its fundamental symmetries. This classifica-tion approach provides a new paradigm in condensedmatter physics which goes beyond the mechanism oflocal order parameters associated with spontaneoussymmetry breaking8: Different TSM differ in the valueof their defining topological invariant but concur in allconventional symmetries. Interestingly, these globaltopological features are not always immediately visiblein the microscopic equations of motion. However, thebulk topology leads to unique finite size effects at theboundary of a finite sample which has been coined bulkboundary correspondence7,9,10. This general mechanismgives rise to peculiar holographic transport propertiesof topologically non-trivial systems. Predicting andprobing the rich phenomenology of these topologicalboundary effects in mesoscopic samples has become oneof the most rapidly growing fields in condensed matterphysics in recent years.

The focus of this review is threefold. First, we view thetopological invariants characterizing TSM as the globalanalogues of geometric phases associated with the adi-abatic evolution of a physical system11–13. Geometricphases are well known to reflect the local inner-geometricproperties of the Hilbert space of a system with a param-

eter dependent Hamiltonian14,15. The topological invari-ants represent global features of a Bloch Hamiltonian,i.e., of a system whose parameter space is its k-space.This approach integrates TSM and physical phenomenarelated to geometric phases into a common theoreticalcontext. Second, we present the topological classifica-tion of TSM in the framework of K-Theory16 in a moreaccessible way than in the pioneering work by Kitaev17.Third, we discuss very recent developments concerningthe generalization of the relevant topological invariantsto interacting and disordered systems several of whichwere published after existing reviews on TSM.

II. THE ADIABATIC THEOREM

The gist of the adiabatic theorem can be understoodat a very intuitive level: Once prepared in an instanta-neous eigenstate with an eigenvalue which is separatedfrom the neighboring states by a finite energy gap ∆, thesystem can only leave this state via a transition whichcosts a finite excitation energy ∆. A simple way to esti-mate whether such a transition is possible is to look atthe Fourier transform H(ω) of the time dependent Hamil-tonian H(t). If the time dependence of H is made suffi-

ciently slow, H(ω) will only have finite matrix elementsfor ω ∆. In this regime the system will stick to thesame instantaneous eigenstate. This behavior is knownas the adiabatic assumption.

A. Proof due to Born and Fock

The latter rather intuitive argument is at the heart ofthe adiabatic theorem of quantum mechanics which hasbeen first proven by Born and Fock in 192818 for non-degenerate systems. Let |n(t)〉n be an orthonormal

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set of instantaneous eigenstates of H(t) with eigenvaluesEn(t)n. The exact solution of the Schrodinger equationcan be generally expressed as

|Ψ(t)〉 =∑n

cn(t)|n(t)〉e−iφnD(t), (1)

where the dynamical phase φnD(t) =∫ tt0En(τ)dτ has

been separated from the coefficients cn(t) for later conve-nience. Plugging Eq. (1) into the Schrodinger equationyields

cn = −cn〈n|d

dt|n〉 −

∑m 6=n

cm〈n|(ddtH

)|m〉

Em − Eneiφ

nmD (t) (2)

with φnmD (t) = φnD(t) − φmD(t). The salient consequenceof the adiabatic theorem is that the last term in Eq.(2) can be neglected in the adiabatic limit since its de-nominator |En − Em| ≥ ∆ is finite whereas the ma-trix elements of d

dtH become arbitrarily small. Moreprecisely, if we represent the physical time as t = Ts,where s is of order 1 for a change in the Hamiltonianof order ∆ and T is the large adiabatic timescale, thenddt = 1

Tdds . Now, d

dsH (t(s)) is by construction of order

∆. The entire last term in Eq. (2) is thus of order 1T .

Under these conditions19, Born and Fock18 showed thatthe contribution of this second term vanishes in the adi-abatic limit T →∞. Note that this is not a trivial resultsince the differential equation (2) is supposed to be in-tegrated from t = 0 to t ∼ T , so that one could naivelyexpect a contribution of order 1 from a coefficient thatscales like 1/T . The coefficient of cn in the first term onthe right hand side of Eq. (2) is purely imaginary since0 = d

dt 〈n|n〉 = ( ddt 〈n|)|n〉 + 〈n| ddt |n〉 and hence does notchange the modulus of cn when the differential equationcn = −cn〈n| ddt |n〉 is solved as

cn(t) = cn(t0) e−

∫ tt0〈n| ddτ |n〉dτ . (3)

Born and Fock18 argue that 〈n| ddt |n〉 = 0 ∀t amounts toa choice of phase for the eigenstates and therefore ne-glect also the first term on the right hand side of Eq. (2).

This review article is mainly concerned with physicalphenomena associated with corrections to this in generalunjustified assumption.

B. Notion of the geometric phase

By neglecting the first term on the right hand sideof Eq. (2), Ref. 18 overlooks the potentially nontrivialadiabatic evolution, known as Berry’s phase12, associ-ated with a cyclic time dependence of H. After a period[0, T ] of such a cyclic evolution, Eq. (3) yields

cn(T ) = cn(0)e−∮ T0〈n| ddτ |n〉dτ . (4)

To understand why the phase factor e−∮ T0〈n| ddτ |n〉dτ can

in general not be gauged away, we remember thatthe Hamiltonian depends on time via the time depen-dence R(t) of some external control parameters. Hence,

〈n| ddt |n〉 = 〈n|∂µ|n〉Rµ, where ∂µ = ∂∂Rµ . To reveal the

mathematical structure of the latter expression, we define

AB(d

dt

)= ABµ Rµ = −i〈n|∂µ|n〉Rµ, (5)

where AB = ABµ dRµ is called Berry’s connection.

AB clearly has the structure of a gauge field: Under thelocal gauge transformation |n〉 → eiξ|n〉 with a smoothfunction R 7→ ξ(R), Berry’s connection transforms like

AB → AB + dξ.

Furthermore, the cyclic evolution defines a loop γ : t 7→R(t), t ∈ [0, T ] , R(0) = R(T ) in the parameter manifoldR. If γ can be expressed as the boundary of some pieceof surface, then, using the theorem of Stokes, we cancalculate

−i∮ T

0

〈n| ddτ|n〉dτ =

∫γ

AB =

∫SdAB =

∫SFB , (6)

where in the last step Berry’s curvature FB = FBµνdRµ∧dRν is defined as

FBµν = −i (〈∂µn|∂νn〉 − 〈∂νn|∂µn〉) = 2Im 〈∂µn|∂νn〉

with the shorthand notation |∂µn〉 = ∂µ|n〉. Note thatFB is a gauge invariant quantity that is analogous to thefield strength tensor in electrodynamics. Defining theBerry phase associated with the loop γ as ϕBγ =

∫γAB =∫

S FB we can rewrite Eq. (4) as

cn(T ) = cn(0)e−iϕBγ . (7)

The manifestly gauge invariant Berry phase ϕBγ can haveobservable consequences due to interference effects be-tween coherent superpositions that undergo different adi-abatic evolutions. The analogue of this phenomenologydue to an ordinary electromagnetic vector potential isknown as the Aharonov-Bohm effect20. The geometri-cal reason why Berry’s connection AB cannot be gaugedaway all the way along a cyclic adiabatic evolution is thesame as why a vector potential cannot be gauged awayalong a closed path that encloses magnetic flux, namelythe notion of holonomy on a curved manifold. We willcome back to the concept of holonomy shortly from amore mathematical point of view. For now we only com-ment that the Berry phase ϕBγ is a purely geometricalquantity which only depends on the inner-geometrical re-lation of the family of states |n (R)〉 along the loop γ andreflects an abstract notion of curvature in Hilbert spacewhich has been defined as Berry’s curvature FB .

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C. Proof due to Kato

For a degenerate eigenvalue, Berry’s phase is promotedto a unitary matrix acting on the corresponding degener-ate eigenspace13. The first proof of the adiabatic theoremof quantum mechanics that overcomes both the limita-tion to non-degenerate Hamiltonians and the assumptionof an explicit phase gauge for the instantaneous eigen-states was reported in the seminal work by Tosio Kato11

in 1950. We will review Kato’s results briefly for thereader’s convenience and use his ideas to illustrate thegeometrical origin of the adiabatic phase. The explicitproofs are presented at a very elementary and self con-tained level in Ref. 11. Our notation follows Ref. 21which is convenient to relate the physical quantities toelementary concepts of differential geometry.

Let us assume without loss of generality that the sys-tem is at time t0 = 0 in its instantaneous ground state|Ψ0(0)〉 or, more generally, since the ground state mightbe degenerate, in a state |Ψ〉 satisfying

P (0)|Ψ〉 = |Ψ〉, (8)

where P (t) is the projector onto the eigenspace associatedwith the instantaneous ground state energy E0(t) whichis defined as

P (t) =1

2πi

∮c

dz

z −H(t),

where the complex contour c encloses E0(t) which is againassumed to be separated from the spectrum of excita-tions by a finite energy gap ∆ > 0. To understand theadiabatic evolution, we are not interested in the dynam-

ical phase φD(t) =∫ t

0E0(τ)dτ . We thus define a new

time evolution operator U(t, 0) = eiφD(t)U(t, 0). Clearly,

U represents the exact time evolution operator of a sys-tem which has the same eigenstates as the original systembut has been subjected to a time dependent energy shiftthat transforms E0(t) → E0(t) = 0 ∀t. Kato proved theadiabatic theorem in a very constructive way by writingdown explicitly the generator A of the adiabatic evolu-tion:

A(d

dt

)= −

[P , P

]. (9)

In the adiabatic limit, U(t, 0)P (0) was shown11 to con-verge against the adiabatic Kato propagator K, i.e.,

U(t, 0)P (0)adiabatic limit−→ K(t, 0) = T e−

∫ t0A( d

dτ )dτ . (10)

The adiabatic assumption is now a direct corollary fromEq. (10) and can be elegantly expressed as21

P (t)K(t, 0) = K(t, 0)P (0), (11)

implying that a system, which is prepared in an instan-taneous ground state at t0 = 0, will be propagated to a

state in the subspace of instantaneous ground states att by virtue of Kato’s propagator K. Note that K is acompletely gauge invariant quantity, i.e., independent ofthe choice of basis in the possibly degenerate subspace ofground states. The Kato propagator K(T, 0) associatedwith a cyclic evolution in parameter space thus yields theBerry phase12 and its non-Abelian generalization13, re-spectively. We will call this general adiabatic phase thegeometric phase (GP) in the following. The GP Kγ rep-resenting the adiabatic evolution along a loop γ in pa-rameter space can be expressed in a manifestly gaugeinvariant way as

Kγ = T e−∫γA. (12)

Kato’s propagator is the solution of an adiabatic ana-logue of the Schrodinger equation, an adiabatic equationof motion that can be written as(

d

dt+A

(d

dt

))|Ψ(t)〉 = 0, (13)

for states satisfying P (t)|Ψ(t)〉 = |Ψ(t)〉, i.e., statesin the subspace of instantaneous groundstates. Beforeclosing the section, we give a general and at least numer-ically always viable recipe to calculate the Kato propaga-tor K(t, 0). We first discretize the time interval [0, t] inton steps by defining ti = i tn . The discrete version of Eq.(13) for the Kato propagator reads (see Eq. (9))

K(ti, 0)−K(ti−1, 0) = (P (ti)− P (ti−1))P (ti−1)−P (ti) (P (ti)− P (ti−1))K(ti−1, 0). (14)

Using P (ti−1)K(ti−1, 0) = K(ti−1, 0) and P 2 = P , Eq.(14) can be simplified to

K(ti, 0) = P (ti)K(ti−1, 0),

which is readily solved by K(ti, 0) =∏ij=0 P (tj). Taking

the continuum limit yields13,14,21

K(t, 0) = limn→∞

n∏i=0

P (ti), (15)

which is a valuable formula for the practical calculationof the Kato propagator.

III. GEOMETRIC INTERPRETATION OFADIABATIC PHASES

We now view the adiabatic time evolution as an ab-stract notion of parallel transport in Hilbert space and re-veal the GP associated with a cyclic evolution as the phe-nomenon of holonomy due to the presence of curvaturein the vector bundle of ground state subspaces over themanifold R of control parameters. Interestingly, Kato’sapproach to the problem provides a gauge invariant, i.e.,a global definition of the geometrical entities connection

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and curvature, whereas standard gauge theories are de-fined in terms of a complete set of local gauge fields alongwith their transition functions defined in the overlap oftheir domains. This difference has an interesting physi-cal ramification: Quantities that are gauge dependent inan ordinary gauge theory like quantum chromodynamics(QCD) are physical observables in the theory of adia-batic time evolution. To name a concrete example, onlygauge invariant quantities like the trace of the holonomy,also known as the Wilson loop, are observable in QCDwhereas the holonomy itself, in other words the GP de-fined in Eq.(12), is a physical observable in Kato’s theory.This subtle difference has been overlooked in standardliterature on this subject15,22 which we interpreted as anincentive to clarify this point below in greater detail.

A. Adiabatic time evolution and parallel transport

To get accustomed to parallel transport, we firstexplain the general concept with the help of a veryelementary example, namely a smooth piece of twodimensional surface embedded in R3. If the surface isflat, there is a trivial notion of parallel transport oftangent vectors, namely shifting the same vector in theembedding space from one point to another. However,on a curved surface, this program is ill-defined, sincea tangent vector at one point might be the normalvector at another point of the surface. Put shortly,a tangent vector can only be transported as parallelas the curvature of the surface admits. On a curvedsurface, parallel transport along a curve is thus definedas a vanishing in-plane component of the directionalderivative, i.e., a vanishing covariant derivative of avector field along a curve. The normal component ofthe directional derivative reflects the rotation of theentire tangent plane in the embedding space and isnot an inner-geometric quantity of the surface as a twodimensional manifold.

The analogue of the curved surface in the context ofadiabatic time evolution is the manifold of control pa-rameters R, parameterizing for example external mag-netic and electric fields. The analogue of the tangentplane at each point of the surface is the subspace of de-generate ground states of the Hamiltonian H(R) at eachpoint R in parameter space. An adiabatic time depen-dence of H amounts to traversing a curve t 7→ R(t) inR at adiabatically slow velocity. A cyclic evolution isuniquely associated with a loop γ in R. We will now ex-plicitly show that the adiabatic equation of motion (13)defines a notion of parallel transport in the fiber bundle ofground state subspaces over R in a completely analogousway as the ordinary covariant derivative ∇ on a smoothsurface defines parallel transport in the tangent bundle ofthe smooth surface. We first note that d

dt = Rµ∂µ is re-

ferring to a particular direction Rµ in parameter space,which depends on the choice of the adiabatic time de-

pendence of H. We can get rid of this dependence byrephrasing Eq. (13) as

D|Ψ〉 = (d+A) |Ψ〉 = 0, (16)

where the adiabatic derivative D = d + A has beendefined, A = − [(dP ), P ] and here as in the followingP |Ψ〉 = |Ψ〉. The R-dependence has been dropped fornotational convenience. The adiabatic derivative D takesa tangent vector, e.g., d

dt , as an argument to boil down to

the directional adiabatic derivative ddt + A

(ddt

)appear-

ing in Eq. (13). For the following analysis the identitiesP 2 = P and P |Ψ〉 = |Ψ〉 are of key importance. It is nowelementary algebra to show

P (dP )P = 0. (17)

Eq. (17) has a simple analogue in elementary geometry:Consider the family of unit vectors n(t)t where t pa-rameterizes a curve on a smooth surface. Then, since1 = 〈n|n〉, we get 0 = d

dt 〈n|n〉 = 2〈n|n〉, i.e., the changeof a unit vector is perpendicular to the unit vector itself.Using Eq. (17), we immediately derive PA|Ψ〉 = 0 andwith that

D|Ψ〉 = 0⇔ Pd|Ψ〉 = 0. (18)

This makes the analogy of our adiabatic derivative D tothe ordinary notion of parallel transport manifest: |Ψ〉 isparallel-transported if the in-plane component of itsderivative vanishes.

Curvature and holonomy

Let us again start with a very simple example of acurved manifold, a two dimensional sphere S2, whichhas constant Gaussian curvature. Parallel-transportinga tangent vector around a geodesic triangle, say theboundary of an octant of the sphere gives a defect anglewhich is proportional to the area of the triangle or, moreprecisely, the integral of the Gaussian curvature over theenclosed area. This defect angle is called the holonomyof the traversed closed path. This elementary examplesuggests that the presence of curvature is in some senseprobed by the concept of holonomy. This intuition isabsolutely right. As a matter of fact, the generalizedcurvature at a given point x of the base manifold ofa fiber bundle is defined as the holonomy associatedwith an infinitesimal loop at x. More concretely, thecurvature Ω is usually defined as Ωµν = [∇µ,∇ν ] whichrepresents an infinitesimal parallel transport around aparallelogram in the µν-plane.

In total analogy, we define

Fµν |Ψ〉 = [Dµ, Dν ] |Ψ〉 = P [Pµ, Pν ]P |Ψ〉, (19)

with the shorthand notation Pµ = ∂µP . Restricting thedomain of F to states which are in the projection P , we

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can rewrite Eq. (19) as the operator identity

F = FµνdRµ ∧ dRν = P [(dP ) ∧ (dP )]P. (20)

In the general case of a non-Abelian adiabatic connec-tion, i.e., if the dimension of P is larger than 1, we cannotsimply use Stokes theorem to reduce the evaluation of Eq.(12) to a surface integral of F over the surface boundedby γ, as has been done in the case of the Abelian Berrycurvature in Eq. (6). However, the global one to one cor-respondence between curvature and holonomy still existsand is the subject of the Ambrose-Singer theorem23.

Relation between Kato’s and Berry’s language

In order to make contact to the more standard lan-guage of gauge theory, we will now express Kato’s mani-festly gauge invariant formulation11 in local coordinatesthereby recovering Berry’s connection AB12,14 and itsnon-Abelian generalization13, respectively. For this pur-pose, let us fix a concrete basis |α(R)〉α, R ∈ O ⊂ R inan open subset O of the parameter manifold. We assumethe loop γ to lie inside of O. Otherwise we would haveto switch the gauge while traversing the loop. We willdrop the R-dependence of |α〉 right away for notationalconvenience. The projector P can then be represented asP =

∑α|α〉〈α|. Let us start the cyclic evolution without

loss of generality with |Ψ(0)〉 = |α(0)〉. From Eq. (11) weknow that the solution |Ψ(t)〉 = K(t, 0)|α(0)〉 of Eq. (13)satisfies P (t)|Ψ(t)〉 = |Ψ(t)〉 at every point in time duringthe cyclic evolution. Hence, we can represent |Ψ(t)〉 inour gauge as

|Ψ〉 =∑β

〈β|Ψ〉|β〉 = UBβα|β〉, (21)

where the t-dependence has been dropped for brevity.From Eq. (18), we know that P d

dt |Ψ〉 = 0 which implies

〈γ| ddt |Ψ〉 = 0. Plugging this into Eq. (21) yields

d

dtUBγα = −

∑β

〈γ| ddt|β〉UBβα. (22)

Redefining AB for the non-Abelian case as a matrix val-ued gauge field through ABαβ = −i〈α|∂µ|β〉dRµ, Eq. (22)is readily solved as

UB(t) = T e−i∫ t0AB( ddτ )dτ .

The representation matrix of the GP associated with theloop γ then reads

UBγ = T e−i∫γAB . (23)

By construction, UBγ is the representation matrix of theGP Kγ , i.e., (

UBγ)α,β

= 〈α(0)|Kγ |β(0)〉,

or, more general, for any point in time along the path(UB(t)

)α,β

= 〈α(t)|K(t, 0)|β(0)〉. (24)

Eq. (24) makes the relation between Kato’s formulationof adiabatic time evolution and the non-Abelian Berryphase manifest. In contrast to the gauge independenceof Kato’s global connection A, AB behaves like a localconnection (see Ref. 24 for rigorous mathematical def-initions) and depends on the gauge, i.e., on our choiceof the family |α(R)〉α of basis states. Under a smoothfamily of basis transformations U(R)R acting on thelocal coordinates AB transforms like23,24

AB → AB = U−1ABU + U−1dU (25)

resulting in the following gauge dependence of Eq. (23),

UBγ → UBγ = U−1UBγ U, (26)

which only depends on the basis choice U = U (R(0)) atthe starting point of the loop γ.

Inserting our representation P =∑α|α〉〈α| into the

gauge independent form of the curvature, Eq.(20), wereadily derive

FB,αβµν = 〈α| [Pµ, Pν ] |β〉 = (dAB)αβµν + (AB ∧AB)αβµν ,

which defines FB as the usual curvature of a non-Abeliangauge field23, i.e.,

FB = dAB +AB ∧ AB , (27)

which transforms under a local gauge transformationU like

FB → U−1FBU.

B. Gauge dependence and physical observability

The gauge dependence of the non-Abelian Berryphase UBγ (see Eq. (26)) has led several authors15,22 tothe conclusion that only gauge independent features likethe trace and the determinant of UBγ can have physicalmeaning. However, working with Kato’s manifestlygauge invariant formulation, it is understood that theentire GP Kγ is experimentally observable. In theremainder of this section we will try to shed some lighton this ostensible controversy.

In gauge theory, it goes without saying that explicitlygauge dependent phenomena are not immediately physi-cally observable and that only the gauge invariant infor-mation resulting from a calculation performed in a spe-cial gauge can be of physical significance. At a formallevel this is a direct consequence of the fact that the La-grangian of a gauge theory is constructed in a manifestly

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gauge invariant way by tracing over the gauge space in-dices. The physical reason for this is quite simple: Aconcrete gauge amounts to a local choice of the coordi-nate system in the gauge space. Under a local changeof basis, a non-abelian gauge field A transforms like (seealso Eq. (25))

A→ A = U−1AU + U−1dU,

where U(x) is a smooth family of basis transformations,with x labeling points in the base space of the theory,e.g., in Minkowski space. Now, since the gauge space isan internal degree of freedom, the basis vectors in thisspace are not associated with physical observables. Thissituation is fundamentally changed in Kato’s adiabaticanalogue of a gauge theory. Here, the non-Abelian struc-ture is associated with a degeneracy of the Hamiltonian,e.g., Kramers degeneracy in the presence of time rever-sal symmetry (TRS). For a system in which spin is agood quantum number, Kramers degeneracy is just spindegeneracy, which makes the spin the analogue of thegauge degree of freedom in an ordinary gauge theory.However, the magnetic moment associated with a spin isa physical observable which can be measured. The basisvectors, e.g., |↑〉, |↓〉 have an objective meaning for the ex-perimentalist (a magnetic moment that points from thelab-floor to the sky which we call z- direction). For con-creteness, let us assume that we have calculated a GPKγ = |↑〉〈↓|+ |↓〉〈↑|. The representation matrix of Kγ inthis basis of Sz eigenstates is clearly the Pauli matrixσx. Choosing a different gauge, i.e., a different basis forthe gauge degree of freedom at the starting point of thecyclic adiabatic evolution, we of course would have ob-tained a different representation matrix UBγ for Kγ , e.g.,σz, had we chosen the basis as eigenstates of Sx (see Eq.(26)). However, the fact that Kγ rotates a spin which isinitially pointing to the lab-ceiling upside down is gaugeindependent physical reality.

IV. FROM GEOMETRY TO TOPOLOGY

In Section III, we worked out the relation betweenthe GP and the notion of curvature as a local geometricquantity. The topological invariants characterizing TSMare in some sense global GPs. They measure globalproperties which cannot be altered by virtue of localcontinuous changes of the physical system. Continuous isat this stage of the analysis synonymous with adiabatic,i.e., happening at energies below the bulk gap. Lateron, we will additionally require local continuous changesto respect the fundamental symmetries of the physicalsystem, e.g., particle hole symmetry (PHS) or timereversal symmetry (TRS).

Let us illustrate the correspondence between local cur-vature and global topology of a manifold with the helpof the simplest possible example. We consider a two di-mensional sphere S2 with radius r. This manifold has a

constant Gaussian curvature of κ = 1r2 . The integral of

κ over the entire sphere obviously gives 4π, independentof r. The Gauss-Bonnet theorem in its classical form(see, e.g., Ref. 25) relates precisely this integral of theGaussian curvature of a closed smooth two dimensionalmanifoldM to its Euler characteristic χ in the followingway:

1

2π

∫Mκ = χ(M) (28)

Note that χ is a purely algebraic quantity which is de-fined as the number of vertices minus the number of edgesplus the number of faces of a triangulation26 of M. χ isby construction of simplicial homology26 a topological in-variant which can only be changed by poking holes intoM and gluing the resulting boundaries together so asto create closed manifolds with different genus. Hence,Eq. (28) nicely demonstrates how the integral of the localinner-geometric quantity κ over the entire manifold yieldsa topological invariant. Concretely, for our example S2, atriangulation is provided by continuously deforming thesphere into a tetrahedron. Simple counting of vertices,edges, and faces yields χ(S2) = 4−6+4 = 2, in agreementwith Eq. (28). More generally speaking, e = κ

2π is our

first encounter with a characteristic class27, the so calledEuler class of M, which upon integration over M yieldsthe topological invariant χ. Similar mathematical struc-tures will be ubiquitous when it comes to the classifica-tion of TSM. The simplest example in this context is thequantum anomalous Hall (QAH) state28–30, a 2D insu-lating state in symmetry class A (see Section V A) whichis characterized by its first Chern number (see SectionVI A for a detailed discussion)

C1 =

∫BZ

iTr [F ]

2π. (29)

The formal analogy between Eq. (28) and Eq. (29) isstriking. The parameter space over which the adiabaticcurvature F is integrated in Eq. (29) is the k-space ofthe physical system, i.e., the Brillouin zone (BZ) for aperiodic system, which has the topology of a torus. InSection II, we showed that the GP can be viewed as theflux through the surface in parameter space which isbounded by the corresponding cyclic adiabatic evolution.Along similar lines, the first Chern number is analogousto a monopole charge enclosed by the entire BZ of theQAH insulator.

The close correspondence between adiabatic evolutionand the first Chern number has also been discussed inRefs.31–33: For a 2D system in cylinder geometry, a cir-cumferential electric field can be modeled by a time de-pendent magnetic flux threading the cylinder in axial di-rection. In this scenario, the first Chern number canbe viewed as the shift of the charge polarization in ax-ial direction associated with the adiabatic threading ofone flux quantum. Along these lines, Laughlin31 had in-terpreted the quantum Hall effect as an adiabatic charge

7

pumping process, even before the formal relation betweenthe Hall conductivity and the first Chern number was es-tablished in Refs. 34–36.

V. BULK CLASSIFICATION OF ALLNON-INTERACTING TSM

Very generally speaking, the understanding of TSMcan be divided into two subproblems. First, findingthe group that represents the topological invariant fora class of systems characterized by their fundamentalsymmetries and spatial dimension. Second, assigningthe value of the topological invariant to a representativeof such a symmetry class, i.e., measuring to whichtopological equivalence class a given system belongs. Weaddress the first problem in this section and the secondproblem in VI.

The general idea that yields the entire table of TSMis quite simple: In addition to requiring a bulk insulat-ing gap, physical systems of a given spatial dimensionare divided into 10 symmetry classes distinguished bytheir fundamental symmetries, i.e., TRS, PHS, and chiralsymmetry (CS)37. The topological properties of the cor-responding Cartan symmetric spaces of quadratic candi-date Hamiltonians determine the group of possible topo-logically inequivalent systems. We outline the mathemat-ical structure behind this general classification scheme insome detail. First, we briefly review the construction ofthe ten universality classes37. Then, we present the asso-ciated topological invariants for non-interacting systemsof arbitrary spatial dimension giving a complete list of allTSM6 that can be distinguished by virtue of this frame-work. Finally, we discuss in some detail the origin ofcharacteristic patterns appearing in this table using theframework of K-Theory along the lines of the pioneeringwork by Kitaev17.

A. Cartan-Altland-Zirnbauer symmetry classes

A physical system can have different types of symme-tries. An ordinary symmetry6 is characterized by a set ofunitary operators representing the symmetry operationsthat commute with the Hamiltonian. The influence ofsuch a symmetry on the topological classification canbe eliminated by transforming the Hamiltonian into ablock-diagonal form with symmetry-less blocks. Thetotal system then consists of several uncoupled copiesof symmetry-irreducible subsystems which can be clas-sified individually. In contrast, the “extremely genericsymmetries”6 follow from the anti-unitary operations ofTRS and PHS. Involving complex conjugation accordingto Wigner, they impose certain reality conditions onthe system Hamiltonian. In total, the behavior of thesystem under these operations, and their combination,the CS operation, defines ten universality classes which

we call the Cartan-Altland-Zirnbauer (CAZ) classes.For disordered systems, these classes correspond toten distinct renormalization group (RG) low energyfixed points in random matrix theory37. The spaces ofcandidate Hamiltonians within these symmetry classescorrespond to the ten symmetric spaces introduced byCartan in 192638 defined in terms of quotients of Liegroups represented in the Hilbert space of the system.For translation-invariant systems, the imposed realityconditions are inherited by the Bloch Hamiltonianh(k) (see Eq. (33) below).

In the following, the anti-unitary TRS operation willbe denoted by T and the anti-unitary PHS will be de-noted by C. The Hamiltonian H of a physical systemsatisfies these symmetries if

T HT −1 = H, (30)

and

CHC−1 = −H, (31)

respectively. According to Wigner’s theorem, these anti-unitary symmetries can be represented as a unitary op-eration times the complex conjugation K. We defineT = TK, C = CK. Using the unitarity of T,C alongwith H = H† we can rephrase Eqs. (30-31) as

THTT † = H,CHTC† = −H. (32)

There are two inequivalent realizations of these anti-unitary operations distinguished by their square whichcan be plus identity or minus identity. For example,T 2 = ±1 for the unfolding of a particle with integer/half-integer spin, respectively. Clearly, T 2 = ±1 ⇔ TT ∗ =±1 and C2 = ±1 ⇔ CC∗ = ±1. In total, there are thusnine possible ways for a system to behave under the twoanti-unitary symmetries: each symmetry can be absent,or present with square plus or minus identity. For eightof these nine combinations, the behavior under the com-bination T C is fixed. The only exception is the so calledunitary class which breaks both PHS and TRS and caneither obey or break their combination, the CS. This classhence splits into two universality classes which add up toa grand total of ten classes shown in Tab. I. For a pe-riodic system, symmetry constraints similar to Eq. (32)hold for the Bloch Hamiltonian h(k), namely

ThT (−k)T † = h(k),

ChT (−k)C† = −h(k), (33)

where T,C now denote the representation of the unitarypart of the anti-unitary operations in band space.

For a continuum model, the real space HamiltonianH(x) is defined through

H =

∫ddx Ψ†(x)H(x)Ψ(x),

8

Class TRS PHS CS

A (Unitary) 0 0 0

AI (Orthogonal) +1 0 0

AII (Symplectic) -1 0 0

AIII (Chiral Unitary) 0 0 1

BDI (Chiral Orthogonal) +1 +1 1

CII (Chiral Symplectic) -1 -1 1

D 0 +1 0

C 0 -1 0

DIII -1 +1 1

CI +1 -1 1

TABLE I. Table of the CAZ universality classes. 0 denotesthe absence of a symmetry. For PHS and TRS,±1 denotes thesquare of a present symmetry, the presence of CS is denotedby 1. The last four classes are Bogoliubov de Gennes classesof mean field superconductors where the superconducting gapplays the role of the insulating gap.

where Ψ is a vector/spinor comprising all internal degreeslike spin, particle species, etc.. The k-space on which theFourier transform H(k) of H(x) is defined does not havethe topology of a torus like the BZ of a periodic system.However, the continuum models one is concerned with incondensed matter physics are effective low energy/large

distance theories. For large k, H(k) will thus genericallyhave a trivial structure so that the k-space can be en-dowed with the topology of the sphere Sd by a one pointcompactification which maps k → ∞ to a single point.The symmetry constraints on H(k) have the same formas those on the Bloch Hamiltonian h(k) shown in Eq.

(33). By abuse of notation, we will denote both H(k) andh(k) by h(k). Nevertheless, we will point out several dif-ferences between periodic systems and continuum modelsalong the way.

B. Definition of the classification problem forcontinuum models and periodic systems

For translation-invariant insulating systems with n oc-cupied and m empty bands and continuum models withn occupied and m empty fermion species, respectively,the projection P (k) =

∑nα=1|uα(k)〉〈uα(k)| onto the oc-

cupied states is the relevant quantity for the topologicalclassification. The spectrum of the system is not of in-terest for adiabatic quantities as long as a bulk gap be-tween the empty and the occupied states is maintained.We thus deform the system adiabatically into a flat bandinsulator, i.e., a system with eigenenergy ε− = −1 for alloccupied states and eigenenergy ε+ = +1 for all emptystates. The eigenstates are not changed during this defor-mation. The Hamiltonian of this flat band system thenreads39,40

Q(k) = (+1) (1− P (k)) + (−1)P (k) = 1− 2P (k)

Obviously, Q2 = 1,Tr [Q] = m − n. Without furthersymmetry constraints, Q is an arbitrary U(n+m) matrixwhich is defined up to a U(n) × U(m) gauge degree offreedom corresponding to basis transformations withinthe subspaces of empty and occupied states, respectively.Thus, Q is in the symmetric space

Gn+m,m(C) = Gn+m,n(C) = U(n+m)/(U(n)× U(m)).

Geometrically, the complex Grassmannian Gk,l(C) is ageneralization of the complex projective plane and is de-fined as the set of l-dimensional planes through the ori-gin of Ck. The set of topologically different translation-invariant insulators is then given by the group g of homo-topically inequivalent maps k 7→ Q(k) from the BZ T d ofa system of spatial dimension d to the space Gn+m,m(C)of possible Bloch Hamiltonians. For continuum modelsT d is replaced by Sd and g is by definition given by

g = πd (Gn+m,m(C)) , (34)

where the n-th homotopy group πn of a space is bydefinition the group of homotopically inequivalentmaps from Sd to this space. For translation-invariantsystems defined on a BZ, the classification can bemore complicated than Eq. (34) if the lower homo-topy groups πs, s = 1, . . . , d − 1 are nontrivial. Forthe quantum anomalous Hall (QAH) insulator28, a2D translation-invariant state which does not obeyany fundamental symmetries, we can infer fromπ2 (Gn+m,m(C)) = Z, π1 (Gn+m,m(C)) = 0 that an inte-ger topological invariant must distinguish possible statesof matter in this symmetry class, i.e., possible mapsT 2 → Gn+m,m(C). The condition π1 (Gn+m,m(C)) = 0 isnecessary because the π2 classifies maps from S2, whichis only equivalent to the classification of physical mapsfrom T 2 if the fundamental group π1 of the target spaceis trivial. The difference between the base space of aperiodic systems which is a torus and of continuummodels which has the topology of a sphere has interestingphysical ramifications: The so called weak topologicalinsulators are only topologically distinct over a torusbut not over a sphere. Physically, this is visible in thelacking robustness of these TSM which break down withthe breaking of translation symmetry.

Requiring further symmetries as appropriate for theother nine CAZ universality classes is tantamount toimposing symmetry constraints on the allowed mapsT d → Gn+m,m(C), k 7→ Q(k) for translation-invariantsystems and Sd → Gn+m,m(C), k 7→ Q(k) for continuummodels, respectively. The set of topologically distinctphysical systems is then still given by the set of homo-topically inequivalent maps within this restricted space,i.e., the space of maps which cannot be continuously de-formed into each other without breaking a symmetry con-straint. For example, for the chiral classes characterized

9

by CS = 1, Q can be brought into the off diagonal form40

Q =

(0 q

q† 0

)

with qq† = 1, which reduces the corresponding targetspace to U(n). For the chiral unitary class AIII with-out further symmetry constraints, the calculation ofg amounts to calculating g = πd (U(n)) = Z for odd d andg = πd (U(n)) = 0 for even d, respectively, providedn ≥ (d+ 1)/2. Additional symmetries will again imposeadditional constraints on the map T d → U(n), k 7→ q(k).

This procedure rigorously defines the group g of topo-logical equivalence classes for non-interacting translation-invariant insulators in arbitrary spatial dimension andCAZ universality class. However, the practical calcula-tion of g can be highly non-trivial and has been achievedfor continuum models via various subtle detours, for ex-ample the investigation of surface nonlinear σ-models, inRefs. 6 and 41. In the following, we outline a mathemat-ical brute force solution to the classification problem interms of K-Theory which has been originally introducedin the seminal work by Kitaev in 200917. This methodwill naturally explain the emergence of weak topologicalinsulators. We will not assume any prior knowledge onK-Theory. In Tab. II, we summarize the resulting groupsof topological sectors g for all possible systems41. We no-tice an interesting diagonal pattern relating subsequentsymmetry classes to neighboring spatial dimensions. Fur-thermore, the pattern of the two unitary classes shows aperiodicity of two in the spatial dimension. If we hadshown the classification for higher spatial dimensions, wewould have observed a periodicity of eight in the spatialdimension for the eight real classes. The following dis-cussion is dedicated to provide a deeper understandingof these fundamental patterns as pioneered in Ref. 17.The mentioned periodicities have first been pointed outin Refs. 39 and 41.

C. Topological classification of unitary vectorbundles

In order to prepare the reader for the application of K-Theory and to motivate its usefulness, we first formulatethe classification problem in the language of fiber bun-dles. The mathematical structure of a non-interactinginsulator of spatial dimension d is that of a vector bundle

Eπ→M (see Ref. 42 for a pedagogical review on vector

bundles in physics). Roughly speaking, a vector bundleconsists of a copy of a vector space V which is called thetypical fiber over each point of its so called base manifoldM. The projection π of the bundle projects the entirefiber Vp ' V over each point p ∈ M to p. Conversely,Vp can be viewed as the inverse image of p under π, i.e. ,Vp = π−1 [p]. The base manifoldM of E is the d dimen-sional k-space of the system, and the fiber over a point

k given by the (projective) space of occupied states P (k).The gauge group of the bundle is U(n), where n is the di-mension of P . Locally, say in a neighborhood U ⊂M ofa given point p ∈ M, any vector bundle looks like thetrivial product U × V . Globally however, this productcan be twisted which gives rise to topologically distinctbundles. If no further symmetry conditions are imposed(see class A in Tab. I), the question of how many topo-logically distinct insulators in a given dimension exist istantamount to asking how many homotopically differentU(n) vector bundles can be constructed over M. Thisquestion can be formally answered for arbitrary smoothmanifolds M as we will outline now. The general idea

is the following. There is a universal bundle ξΠ→ X into

which every bundle E can be embedded through a bundle

map23 f : E → ξ such that

f∗ξ = E, (35)

where f :M→ X is the map between the base manifolds

associated with the bundle map f . That is to say everybundle can be represented as a pullback bundle f∗ξ23 ofthe universal bundle ξ by virtue of a suitable bundle map

f . The key point is now that homotopically different bun-dles E are distinguished by homotopically distinct mapsf . Thus, the set of different TSM is the set π [M,X ] ofhomotopy classes of maps from the k-spaceM to the basemanifold X of the universal bundle. X is also called theclassifying space of U(n) and is given by the GrassmanianGN,n(C) = U(N)/(U(n)×U(N−n)) for sufficiently large

N , i.e., N > dd2 + ne. To be generic in the dimension ofthe system d, we take the inductive limit X = Gn(C∞) =limN→∞GN,n(C) = limN→∞ U(N)/(U(n)× U(N − n)).We thus found for the set Vectn(M,C) of inequivalentU(n) bundles over M the expression

Vectn(M,C) = π [M, Gn(C∞)] ,

which is known for some rather simple base manifolds.In particular for spheres Sd, there is a trick to calcu-late Vectn(Sd,C): Sd can always be decomposed intotwo hemispheres which are individually trivial. The ho-motopy of a bundle over Sd is thus determined by theclutching function fc defined in the overlap Sd−1 of thetwo hemispheres, i.e., along the equator of Sd. Phys-ically, fc translates a local gauge choice on the upperhemisphere into a local gauge choice on the lower hemi-sphere and is thus a function fc : Sd−1 → U(n). Thegroup of homotopy classes of such functions is by defini-tion πd−1(U(n)). Interestingly, for n > d−1

2 , these groupsare given by

Vectn(Sd,C) = πd−1(U(n)) =

Z, d− 1 odd

0 , d− 1 even(36)

This periodicity of two in (d − 1) is known as thecomplex Bott periodicity. The physical meaning of Eq.(36) is the following: In the unitary universality class A,

10

Class constraint d = 1 d = 2 d = 3 d = 4

A none 0 Z 0 ZAIII none on q Z 0 Z 0

AI QT (k) = Q(−k) 0 0 0 ZBDI q∗(k) = q(−k) Z 0 0 0

D τxQT (k)τx = −Q(−k), m = n Z2 Z 0 0

DIII q(k)T = −q(−k),m = n even Z2 Z2 Z 0

AII iσyQT (k)(−iσy) = Q(−k), m, n even 0 Z2 Z2 Z

CII iσyq∗(k)(−iσy) = q(−k), m = n even Z 0 Z2 Z2

C τyQT (k)τy = −Q(−k), m = n 0 Z 0 Z2

CI q(k)T = q(−k),m = n 0 0 Z 0

TABLE II. Table of all groups g of topological equivalence classes. The first column denotes the CAZ symmetry class, dividedinto two unitary classes without anti-unitary symmetry (top) and eight “real” classes with at least one anti-unitary symmetry(bottom). The second column shows the symmetry constraints on the flat band maps, where we have chosen the representationT = K for T 2 = 1, T = iσyK for T 2 = −1, as well as C = τxK for C2 = 1, C = τyK for C2 = −1. Here, σy denotes thePauli matrix in spin space, τx, τy denote Pauli matrices in the particle hole pseudo spin space of Bogoliubov de Gennes Hilbertspaces. In the last four columns, g is listed for d = 1, . . . , 4.

there is an integer topological invariant in even spatialdimension (e.g., QAH in d = 2) and no TSM in oddspatial dimension.

This classification has two shortcomings. First, it can-not be readily generalized to other CAZ classes at thissimple level. Second, only systems with the same num-ber of occupied bands n can be compared. However,adding some topologically trivial bands to the systemshould yield a system in the same equivalence class, ifthose bands can be considered as inert, i.e., if they do notchange the low energy physics close to the Fermi surface.Both shortcomings can be overcome in the framework ofK-Theory16,43.

D. K-Theory approach to a complete classification

K-Theory16,43 is concerned with vector bundles whichhave a “sufficiently large” fiber dimension. This means,that topological defects which can be unwound by justincreasing the fiber dimension are not visible in theresulting classification scheme. This is physically reason-able, as trivial occupied bands from inner localized shellsfor example increase the number of bands as comparedto the effective low energy models under investigation.Models of different number of such trivial bands shouldbe comparable in a robust classification scheme. In Ref.44, K-Theory was used to discuss analogies between theemergence of D-branes in superstring theory and thestability of Fermi-surfaces in non-relativistic systems.The use of K-Theory for the classification of TSM hasbeen pioneered in Refs. 1 and 2 for the QSH state andmore systematically been discussed for general TSM inRef. 17. This analysis in turn can be understood as aspecial case of a twisted equivariant K-Theory as hasbeen reported very recently45.

Crash-course in K-Theory

The direct sum of two vector bundles E⊕F is the directsum of their fibers over each point. This addition has onlya semi-group structure, since E ⊕G = F ⊕G; E ' F .A minimal counterexample is given by E = TS2, F =S2 × R2. F is clearly trivial, whereas TS2, the tangentbundle of S2, i.e., the disjoint union of all tangent planesof S2, is well known to be non-trivial. However, addingNS2, the bundle of normal vectors to S2, to both bundlesE,F we obtain the same trivial bundle S2 × R3. Thismotivates the concept of stable equivalence

Es' F ⇔ E ⊕ Zm ' F ⊕ Zn, (37)

where Zn =M×Kn, K = R,C is the trivial bundle overthe fixed base manifoldM, which plays the role of an ad-ditive zero as far as stable equivalence is concerned. Wedenote the set of K-vector bundles overM by VK(M) inthe following. Note that stably equivalent bundles canhave different fiber dimension, as m 6= n in general inEq. (37). The benefit of this construction is:

E ⊕G = F ⊕G⇒ Es' F (38)

This is because for vector bundles on a smooth manifoldevery bundle can be augmented to a trivial bundle, i.e.,

∀G∃H,l G⊕H = Zl. (39)

Eq. (38) naturally leads to the notion of a subtractionon VK(M) by virtue of the Grothendieck construction:Consider the pairs (E1, E2) ∈ VK(M)×VK(M) and de-fine the equivalence relation

(E1, E2) ∼ (F1, F2)⇔ ∃H F1 ⊕ E2 ⊕H ' E1 ⊕ F2 ⊕H.(40)

11

Looking at Eq. (40), we can intuitively think of theequivalence class (E1, E2)∼ as the formal difference E1−E2. We now define the K-group as the quotient

K(M) = (VK(M)× VK(M)) / ∼, (41)

which identifies all formal differences that are equiv-alent in the sense of Eq. (40). Due to Eq. (39),every group element in K(M) can be represented inthe form (E,Zn). However, (E,Zn) (E,Zm) forn 6= m. We define the virtual dimension of (E,F ) asdv = rk(E) − rk(F ), where rk denotes the rank, i.e.,the fiber dimension of a vector bundle. By restrictingK(M) to elements with dv = 0, we obtain the restricted

K-group K(M) = g ∈ K(M)|dv(g) = 0. K(M) isisomorphic to the set of stable equivalence classesof VK(M). Up to now, the construction has beenindependent of the field over which the vector spacesare defined. In the following, we will distinguish thereal and complex K-groups KR(M),KC(M). Physically,KC will be employed to characterize systems withoutanti-unitary symmetries, whereas KR is relevant forsystems in which at least one anti-unitary symmetryimposes a reality constraint on the k-space.

A crucial notion in K-Theory which is also our mainphysical motivation to study it is that of the stable range.The idea is that at sufficiently large fiber dimension nno “new” bundles can be discovered by looking at evenlarger fiber dimension. Sufficiently large in terms of thedimension d of M means n ≥ nC = d/2 + 1 for thecomplex case and n ≥ nR = d + 1 for the real case,respectively. More formally, every bundle E with n >nK can be expressed as a sum

E ' F ⊕ Zn−nK (42)

of a bundle F with fiber dimension nK and a trivial bun-

dle for K = R,C. Since clearly Es' F (see Eq. (38)),

this means that all stable equivalence classes have rep-resentatives in fiber dimension n ≤ nK . Furthermore, asituation like our counterexample above where we aug-mented two non-isomorphic bundles by the same trivialbundle NS2 to obtain the same trivial bundle cannot oc-cur in the stable range. That is to say F as appearingin Eq. (42) is uniquely defined up to isomorphisms. Thestable range hence justifies the approach of K-theory ofignoring fiber dimension when defining the stable equiva-lence. The key result in the stable range which connectsK-Theory to our goal of classifying all inequivalent vectorbundles with sufficiently large but arbitrary fiber dimen-sion on equal footing reads43

KK(M) = Vectn(M,K) = π [M, Gn(K∞)] ∀n≥nK .(43)

The complex Bott periodicity Eq. (36) with period pC =2 has a real analogue concerning the homotopy groups ofO(n) with period pR = 8. This immediately implies in

the language of K-Theory

KK(Sd+pK ) = KK(Sd), K = R,C. (44)

We define

K−dK (M) = KK(SdM), (45)

where S is the reduced suspension (see Refs. 26 and 43for a detailed discussion) which for a sphere Sk indeedsatisfies SSk = Sk+1. The stronger version of the Bottperiodicity in K-Theory now reads43

K−d−pKK (M) = K−dK (M), K = R,C, (46)

which only for M = Sl trivially follows from Eq. (44).

Using this periodicity, the definition of K−dK in Eq. (45)can be formally extended to d ∈ Z.

Class Classifying Space

A C0 = U(n+m)/ (U(n)× U(m))

AIII C1 = U(n)

AI R0 = O(n+m)/ (O(n)×O(m))

BDI R1 = O(n)

D R2 = U(2n)/U(n)

DIII R3 = U(2n)/Sp(2n)

AII R4 = Sp(n+m)/ (Sp(n)× Sp(m))

CII R5 = Sp(n)

C R6 = Sp(2n)/U(n)

CI R7 = U(n)/O(n)

TABLE III. Table of all classifying spaces Cq, Rq of complexand real K-Theory, respectively. The first column denotesthe CAZ symmetry class. From top to bottom, the next com-plex/real classifying space is the loop space of its predecessor,i.e., Cq+1 = ΩCq(mod 2), Rq+1 = ΩRq(mod 8)

The Bott clock

From the very basic construction of homotopy groupsthe following identities for the homotopy of a topologicalspace X are evident:

πd(X) = π[Sd, X

]= π

[SSd−1, X

]= π

[Sd−1,ΩX

],

where ΩX denotes the loop space43 of X, i.e., the spaceof maps from S1 to X. Iterating this identity givesπd(X) = π0(ΩnX) using the complex Bott periodicity(36), we immediately see that counting the connectedcomponents π0 (U(n)) , π1 (U(n)) = π0 (ΩU(n)) of theunitary group and its first loop space, we can classifyall U(n) vector bundles over Sd in the stable range, i.e.,with n > d

2 . The real analogue of the Bott periodic-ity with period pR = 8 leads to analogous statementsfor O(n) bundles over Sd which depend only on the con-nected components of O(n) and its first seven loop spaces

12

ΩiO(n), i = 1, . . . , 7 (see Tab. III). This defines a Bottclock with two ticks for the complex case and eight ticksfor the real case, respectively. Interestingly, these tenspaces, for the complex and real case together, are pre-cisely the ten Cartan symmetric spaces in which the timeevolution operators associated with Hamiltonians in theten CAZ classes lie. After this observation, only twopoints are missing until a complete classification of allTSM of continuum models can be achieved. The firstpoint is a subtlety related to the interdependence of thetwo wave vectors k and −k as shown in Eq. (33), whichmakes the real Bott clock tick counter clockwise. The sec-ond point is the inclusion of symmetry constraints intothe scheme which leads to the clockwise ticking Cliffordclock (see Eq. (50) below). The combination of both im-plies that the topological invariant of a continuum modelof dimension d in the CAZ class q only depends on thedifference q − d (mod 8) for the eight real classes and onq − d (mod 2) for the two complex classes A and AIII,respectively.

Reality and k-space topology

For systems which obey anti-unitary symmetries thereal structure of the Hamiltonian H(x) is most con-veniently accounted for in its Majorana representation.H(x) can in this representation be expressed in terms ofa real antisymmetric 2n× 2n-matrix B,

Ψ†(x)H(x)Ψ(x) =i

4Bijcx,icx,j , (47)

where cx,i, i = 1, . . . , 2n are the Majorana operators rep-resenting the n fermion species at x. On Fourier trans-form, H =

∫Ψ†HΨ can be written as17

H =i

4

∫ddkAij(k)c−k,ick,j , (48)

where A is skew hermitian and satisfies

A∗(k) = A(−k). (49)

Eq. (49) naturally leads to a real vector bundle structureas defined in Ref. 46 for the bundle of eligible A-matricesover the k-space (Rd, τ), where the involution τ (see Ref.46) is given by k 7→ −k. On one-point compactification,this real k-space becomes a sphere Sd = (Sd, τ) with thesame involution17. Whereas the ordinary sphere Sd canbe viewed as a reduced suspension S of Sd−1 over thereal axis, Sd can be understood as the reduced suspen-sion S of Sd−1 over the imaginary axis. This picture isalgebraically motivated by comparing the involution τ tothe ordinary complex conjugation which, restricted to theimaginary axis of the complex plane is of the same form.Interestingly, in the language of definition (45), S playsthe role of an inverse to S17,46, i.e.,

KR(M) = K−1R (SM) = KR(SSM).

This means that the Bott clock over Sd is reversed ascompared to its analogue over Sd.

Real K-theory and the Clifford clock

The main reason for the real construction of Eq. (48)is that the anti-unitary symmetry constraints yieldingthe eight real CAZ classes (all except A and AIII) canbe distinguished in terms of anti-commutation relationsof the A-matrix with real Clifford generators16,17. Ata purely algebraic level, these constraints can be trans-formed so as to be expressed only in terms of positiveClifford generators17, i.e., generators that square to plusidentity. We call the restricted K-group of a vector bun-dle of A-matrices overM that anti-commute with q pos-itive Clifford generators Kq

R(M). Interestingly16,17,

KqR(M) ' K−qR (M). (50)

Eq. (50) defines a Clifford clock that runs in the oppo-site direction as the Sd Bott clock. This algebraic phe-nomenon explains the full periodic structure of the tableof TSM of continuum models (see Tab. II). The classify-ing spaces of A-matrices for systems that anti-commutewith q Clifford generators are shown in Tab. III.

Periodic systems

The classification of periodic systems is much morecomplicated from a mathematical point of view. Theirbase space is the real Brillouin zone T d = (T d, τ), wherethe involution τ giving rise to the real structure is againgiven by k 7→ −k. For T d the reduced suspension doesnot provide a trivial relation between the K-Theory ofdifferent spatial dimension like SSd = Sd+1 for the basespace of continuum models. The general calculation ofall relevant K-groups over T d has been reported in Ref.17. Interestingly, the resulting groups always contain therespective classification of continuum models in the samesymmetry class as an additive component. Additionally,the topological invariants of weak TSM, i.e., TSM whichare only present in translation-invariant systems, can beinferred. The Clifford clock defined in Eq. (50) is in-dependent of the base space and hence still applicable.Here, we only review the general result calculated in Ref.17

K−qR (T d) ' K−qR (Sd)⊕

(d−1⊕s=0

(d

s

)K−qR (Ss)

). (51)

The second term on the right hand side of Eq. (51) en-tails the notion of so called weak topological insulatorswhich are obviously due to TSM in lower dimensions.To name the most prominent example, the Z2 invariantcharacterizing the QSH insulator in d = 2 in the presenceof TRS, CAZ class AII, yields a 3Z2 topological invari-ant characterizing the weak topological insulators withthe same symmetry in d = 3.

13

Lattice systems with disorder

In a continuum model, disorder that is not too shortranged so as to keep the k-space compactification forlarge k valid, can be included into the model system with-out changing the classification scheme. However, per-turbing a translation-invariant lattice system with disor-der also gives its k-space (now defined in terms of a dis-crete Fourier transform) a discrete lattice structure whichis not directly amenable to investigation in the frameworkof K-Theory which we only defined over smooth base-manifolds. Ref. 17 shows that a Hamiltonian featuringlocalized states in the energy gap can be transformedinto a gapped Hamiltonian upon renormalization of pa-rameters. The physical consequence of this statement isthat only a mobility gap is needed for the classificationof a TSM and no energy gap in the density of states.Furthermore, Ref. 17 argues without explicit proof thatthe classification problem of gapped lattice systems with-out translation invariance is equivalent to the classifi-cation problem of continuum models. This statementagrees with the physical intuition that the breaking oftranslation-invariance must remove the additional struc-ture of weak TSM as described for periodic systems byEq. (51).

VI. CALCULATION OF TOPOLOGICALINVARIANTS OF INDIVIDUAL SYSTEMS

In Section V, we have shown how many different TSMcan be expected in a given spatial dimension and CAZclass. Now, we outline how insulating systems withinthe same CAZ class and dimension can be assigned atopological equivalence class in terms of their adiabaticconnection defined in Eq. (9) and their adiabatic curva-ture defined in Eq. (20), respectively. A complete caseby case study in terms of Dirac Hamiltonian representa-tives of all universality classes of this problem has beenreported in Ref. 6. We outline the general patterns relat-ing the classification of neighboring (see Tab. II) univer-sality classes following the analysis in Refs. 6, 39, and 40.Interestingly, all topological invariants can be calculatedusing only complex invariants, namely Chern numbersand chiral unitary winding numbers. The anti-unitarysymmetries are accounted for by the construction of adimensional hierarchy in Section VI B starting from a socalled parent state in each symmetry class for which thecomplex classification concurs with the real classification.In Section VI C, we show how the topological invariantscan be defined for disordered systems with the help oftwisted boundary conditions. Furthermore, we discuss ageneralization of the non-interacting topological invari-ants to interacting systems in Section VI D.

A. Systems without anti-unitary symmetries

Chern numbers of unitary vector bundles

Eq. (35) shows that every U(n) bundle E → M canbe represented as a pullback from the universal bun-

dle ξ → Gn(C∞) by some bundle map f . Chernclasses are de Rham cohomology classes, i.e., topologi-cal invariants47 that are defined as the pullback of cer-tain cohomology classes of the classifying space Gn(C∞).The cohomology ring H∗ (Gn(C∞)) consists only of evenclasses and is generated by the single generator cj ∈H2j (Gn(C∞)) , j = 1, . . . , n for every even cohomologygroup48. The Chern classes ci of E are defined as thepullback ci = f∗ci from the classifying space by the map

f : M → Gn(C∞) associated with the bundle map f .Due to the Chern-Weyl theorem23, Chern classes can beexpressed in terms of the curvature, i.e., in our case, theadiabatic curvature F defined in Eq. (20) of E. Explic-itly, the total Chern class c can be expressed as23

c = det

(1 +

iF2π

)= 1 + c1(F) + c2(F) . . . . (52)

The determinant is evaluated in gauge space and prod-ucts of F are understood to be wedge products. cj isthe monomial of order j in F . Obviously, cj is a 2j-formand can only be non-vanishing for 2j ≤ d, where d isthe dimension of the base manifold M, i.e., the spatialdimension of the physical system. Another characteris-tic class which generates all Chern classes is the Cherncharacter23

ch = Tr[eiF2π

]= 1 + ch1(F) + ch2(F) + . . . . (53)

Due to their importance for later calculations, we ex-plicitly spell out the first two Chern characters ch1 =Tr[iF2π

], ch2 = − 1

8π2 Tr [F ∧ F ]. Importantly, for evend = 2p, the integral

Cp =

∫M

chp

yields an integer, the so called p-th Chern number24.These Chern numbers characterize systems in the uni-tary symmetry class A which can only be non-trivial ineven spatial dimension (see Tab. II).

Winding numbers of chiral unitary vector bundles

In Section V, we have shown that the classifying spacefor a chiral unitary (AIII) system is given by U(n) andthat the topological sectors are defined by homotopicallydistinct maps k 7→ q(k) ∈ U(n). Now, we discuss howto assign an equivalence class to a given map q by cal-culating its winding number49–51 following Ref. 6. FromTab. II it is clear that only in odd spatial dimension

14

d = 2j − 1 there can be a non-trivial winding number.We define

wq2j−1 =(−(j − 1)!)

(2j − 1)!(2πi)jTr[(q−1dq)2j−1

], (54)

which has been dubbed winding number density6. Inte-grating this density over the odd-dimensional base man-ifold M representing the k-space of the physical system,we get the integral winding number ν2j−1

ν2j−1 =

∫Mwq2j−1, (55)

which is well known to measure the homotopy of the mapk 7→ q(k).

Relation between chiral winding number and Chern Simonsform

So far, the relation between the adiabatic connection ofa chiral system and its topological invariant has not beenmade explicit. Since characteristic classes like Cherncharacters are closed 2j-forms, they can locally be ex-pressed as exterior derivatives of (2j − 1)-forms. Theseodd forms are called the Chern Simons forms associatedwith the even characteristic class23,52. For the j-th Cherncharacter chj , which is a 2j form, the associated ChernSimons form Q2j−1 reads23

Q2j−1(A,Ft) =1

(j − 1)!

(i

2π

)j ∫ 1

0

dt STr[A,F j−1

t

],

(56)

where Ft = tF + (t2 − t)A ∧ A is the curvature of theinterpolation tA between the zero connection and A andSTr denotes the symmetrized trace. Explicitly, we haveQ1 = i

2πTr [A] , Q3 = − 18π2 Tr

[AdA+ 2

3A3].

It is straightforward to show6, that in a suitable gauge,the Berry connection of a chiral bundle yields AB =12qdq

†, where q ∈ U(n) is again the chiral map char-acterizing the system. This is not a pure gauge due tothe factor 1

2 which entails that the associated curvature

FB does not vanish. Plugging AB and FB into Eq. (56)immediately yields6

Q2j−1(AB ,FBt ) =1

2wq2j−1. (57)

Eq. (57) directly relates the winding number density tothe Chern Simons form. We define the Chern Simonsinvariant of an odd dimensional system as

CS2j−1 =

∫MQ2j−1 (mod 1),

where (mod 1) accounts for the fact that∫MQ2j−1 has

an integer gauge dependence due to π2j−1 (U(n)) = Z forn > j. Looking back at Eq. (55), we immediately get

ν2j−1(mod 2) = 2CS2j−1(mod 2).

FIG. 1. Illustration of the WZW dimensional extension. Thecircle at v = 0 represents the physical system. The polesat v = ±π represent the trivial reference system without k-dependence. The two interpolations are conjugated by ananti-unitary symmetry, here exemplary denoted by TRS.

We note that the (mod 2) can be dropped if we fix thegauge as described above to AB = 1

2qdq−1. This estab-

lishes the desired relation between the winding numberof a chiral unitary system and its adiabatic curvature.

B. Dimensional hierarchy and real symmetryclasses

Until now, we have only discussed how to calculatetopological invariants of systems in the complex sym-metry classes A and AIII. Interestingly, for some realuniversality classes, the classification in the presence ofanti-unitary symmetries concurs with the unitary classi-fication (see Tab. IV). The first known example of thistype is in the symplectic class AII in d = 4 which is char-acterized by the second Chern number of the correspond-ing complex bundle53,54. Another example of this kind isthe p+ ip superconductor in d = 2 and symmetry class Dwhich is characterized by its first Chern number, i.e., inthe same way as the QAH effect in class A. In odd dimen-sions similar examples exist for real chiral classes, e.g., forDIII in d = 3, where the winding number is calculatedusing Eq. (54) in the same way as for the chiral unitaryclass AIII in the same dimension. All the topological in-variants just mentioned are integer invariants. In someuniversality classes, these integers can only assume evenvalues (see Tab. IV). For physically relevant dimensions,i.e., d = 1, 2, 3, these exceptions are CII in d = 1, C ind = 2, and CI in d = 3. All other states where the com-plex and the real classification concur, can be viewed asparent states of a dimensional hierarchy within the samesymmetry class from which all Z2 invariants appearingin Tab. IV can be obtained by dimensional reduction.This approach was pioneered in the seminal work by Qi,Hughes, and Zhang39.

The general idea is more intuitive if we considerthe parent state as a Wess-Zumino-Witten (WZW)dimensional extension55,56 of the lower dimensionaldescendants instead of thinking of a dimensional re-

15

Class d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8

A 0 Z 0 Z 0 Z 0 ZAIII Z 0 Z 0 Z 0 Z 0

AI 0 0 0 2Z 0 Z2 Z2 ZBDI Z 0 0 0 2Z 0 Z2 Z2

D Z2 Z 0 0 0 2Z 0 Z2

DIII Z2 Z2 Z 0 0 0 2Z 0

AII 0 Z2 Z2 Z 0 0 0 2ZCII 2Z 0 Z2 Z2 Z 0 0 0

C 0 2Z 0 Z2 Z2 Z 0 0

CI 0 0 2Z 0 Z2 Z2 Z 0

TABLE IV. Table of all groups of topological equivalence classes. The first column denotes the symmetry class, divided intotwo complex classes without any anti-unitary symmetry (top) and eight real classes with at least one anti-unitary symmetry(bottom). Chiral classes are denoted by bold letters. The parent states of dimensional hierarchies are boxed. For all non-chiralboxed states, the classification concurs with that of class A in the same dimension. For all chiral boxed states, the classificationconcurs with that of class AIII in the same dimension. 2Z indicates that the topological integer can only assume even valuesin some cases. Such states are never parent states.

duction from the d-dimensional parent state to itsdescendants. This works as follows: We fix a localized(d − 1)-dimensional insulator without any hoppingthat satisfies the required anti-unitary symmetriesas a trivial reference state. This reference state isdescribed by the k-independent Bloch Hamiltonian h0.The (d − 1)-dimensional physical system of interest ischaracterized by the Bloch Hamiltonian h(k). Then,we interpolate by varying the parameter v betweenthe (d − 1)-dimensional physical system (v = 0) andthe trivial state (v = π) without closing the insulatinggap. However, the intermediate (d − 1)-dimensionalsystem at fixed v 6= 0, π might well break the requiredanti-unitary symmetries. The crucial step is now todo the interpolation for v ∈ [0, π] and v ∈ [−π, 0] in asymmetry conjugated way (see Fig. 1). That is to say,we require our (d − 1)-dimensional system of interestand the resulting d-dimensional extended system to bein the same CAZ class. This d-dimensional system ischaracterized by the Bloch Hamiltonian h(k, v). Thev ∈ [−π, 0] and the v ∈ [0, π] half of the extendedk-space then are not independent of each other but giveequal contributions to the integer topological invariantof the d-dimensional extended system6,39. One mightnow ask to which extend the resulting integer invariantof the extended system depends on our choice of theinterpolation h(k, v) between h(k) = h(k, v = 0) andh0 = h(k, v = ±π). To answer this question, one

considers two interpolations h(k, v), h(k, v). It is thenelementary to show39 that the difference between theinteger invariants of these two d-dimensional systemsis an even integer. This implies that a Z2 information,namely the parity of the integer invariant associatedwith the extended system, is well defined only in termsof the physical system with spatial dimension (d− 1).

A similar procedure can be repeated a second time to

obtain a Z2 invariant for a (d − 2) dimensional seconddescendant39. From the procedure just sketched, it isobvious why the exceptional phases which are character-ized by an even integer are not parent states of such adimensional hierarchy. The generic constructions for allpossible classes can be found in Refs. 6 and 39. Withthat, we are provided with a general and fairly explicitrecipe for the practical calculation of the topological in-variants for all possible CAZ classes in all spatial dimen-sions.

C. Bulk invariants of disordered systems andtwisted boundary conditions

Our practical calculation of topological invariantsso far has been focused on periodic systems with aBZ, i.e., M = T d and continuum models where thek-space can be compactified to a sphere, i.e., M = Sd.As already pointed out in Section V, the situationis more complicated for disordered lattice models.Seminal progress along these lines was reported for thequantum Hall state by Niu et al. in 198557. Theseauthors use twisted boundary conditions (TBC) todefine the quantized Hall conductivity σxy for a 2Dsystem as a topological invariant only requiring abulk mobility gap. We briefly review their analysisand propose the framework of TBC as a general recipeto calculate topological invariants for disordered systems.

The Hall conductivity σxy resulting from a linear re-sponse calculation at zero temperature yields

σxy =−2

AIm∑n 6=0

〈0|Hx|n〉〈n|Hy|0〉(En − E0)2

,

where |0〉 is the many body ground state, A is the areaof the system, and Hi = ∂H

∂ki . In the presence of a mag-

16

netic field, translation-invariance is defined in terms ofthe magnetic translation operator TB

36 which concurs

with the ordinary translation operator T (a) = eiak inthe absence of a magnetic field. TBC now simply meanthat a (magnetic) translation by the system length Lj inj-direction gives an additional phase factor eiφj . φj iscalled the twisting angle in j-direction. Gauging awaythis additional phase to obtain a wave function with pe-riodic boundaries amounts to a gauge transformation ofthe Hamiltonian which shifts the momentum operatorlike

−i∂j → −i∂j + φj (58)

Using Hi = Hφi = [∂φi ,H] the Hall conductivity canbe expressed as the sensitivity of the ground state wavefunction to TBC.

σxy =2

AIm〈∂φxΨ0|∂φyΨ0〉,

where |Ψ0〉 is the many body ground state after the men-tioned gauge transformation which depends on the twist-ing angles. Defining θ = Lxφx, ϕ = Lyφy,

σxy = 2Im〈∂θΨ0|∂ϕΨ0〉 = iFθϕ. (59)

The main merit of Ref. 57 is to show that this expressionactually does not depend on the value of (θ, ϕ) as longas the single particle Green’s function of the system isexponentially decaying in real space. This condition ismet if the Fermi energy lies in a mobility gap. Hence, atrivial integration can be introduced as follows:

σxy =i

4π2

∫ 2π

0

dθ

∫ 2π

0

dϕFθϕ = G0

∫T 2

iF2π

= G0C1,

(60)

where G0 = e2

h = 12π and the integer C1 is by definition

the first Chern number of the ground state line bundleover the torus of twisting angles. This constructionmakes the topological quantization of the Hall conduc-tivity manifest.

Eq. (58) shows the close relation between momentumand twisting angles. One is thus tempted to just replacethe BZ of each periodic system by the torus of twistingangles for the corresponding disordered system which istopologically equivalent to a fictitious periodic systemwith the physical system as single lattice site58,59. Wewill proceed along these lines below but would like tocomment briefly on the special role played by the quan-tum Hall phase first. Eq. (59) represents the physicalobservable σxy in terms of the twisting angles. Niu et al.argued rigorously57 that σxy of a bulk insulating systemcan actually not depend on the value of these twistingangles which allows them to express σxy as a manifestlyquantized topological invariant in Eq. (60). For ageneric TSM, the topological invariant of the clean sys-tem does in general not represent a physical observable.

Furthermore, the integration over the twisting angleswill not be trivial, i.e., the function to be integrated willactually depend on the twisting angles. Employing thepicture of a periodic system with the physical systemas a single site is problematic inasmuch as the bulkboundary correspondence at the “boundary” of a singlesite is hard to define mathematically rigorously. In thequantum Hall regime for example, it is well known thatin disordered systems a complicated landscape of local-ized states and current carrying regions produces theunchanged topologically quantized Hall conductivity9.However, the edge states of the disordered quantum Hallstate are in general not strictly localized at the boundary.

Replacing the BZ of a translation-invariant system bythe torus of twisting angles in the disordered case yieldsa well defined topological invariant which adiabaticallyconnects to the topological invariant of the clean systemwhere the relation

∂kj = ∂φj (61)

follows from Eq. (58). This is because from a purelymathematical perspective it cannot matter which toruswe consider as the base space of our system. In thissense, the framework of TBC is as good as it gets concern-ing the definition of topological invariants for disorderedsystems. When calculating the topological invariant of asymmetry protected TSM through dimensional extension(see Section VI B), a hybrid approach between TBC inthe physical dimensions and periodic boundaries in theextra dimension can be employed to define a topologicalinvariant for the disordered system which can be calcu-lated more efficiently60. The fact that in some symme-try protected topological phases the topological invari-ants are not directly representing physical observablesis a not a problem of the approach of TBC but is a re-markable difference between these TSM and the quantumHall state at a more fundamental level. Recently, an S-matrix approach to calculating topological invariants ofnon-interacting disordered TSM has been reported61

D. Taking into account interactions

Up to now, our discussion has only been concernedwith non-interacting systems. As a matter of fact,the entire classification scheme discussed in Sec. Vmassively relies on the prerequisite that the Hamiltonianis a quadratic form in the field operators. The violationof this classification scheme for systems with two particleinteractions has been explicitly demonstrated in Ref. 62.

As we are not able to give a general classification ofTSM for interacting systems, we search for an adiabaticcontinuation of the non-interacting topological invariantsto interacting systems. This procedure does from itsoutset impose certain adiabaticity constraints on the

17

interactions that can be taken into account. The topo-logical invariants for non-interacting systems are definedin terms of the projection P on the occupied singleparticle states defining the ground state of the system.The main assumption is thus that the gapped groundstate of the non-interacting system is adiabaticallyconnected to the gapped ground state of the interactingsystem. A counter-example of this phenomenology arefractional quantum Hall states63,64, where a gap due tonon-adiabatic interactions emerges in a system which isgapless without interactions. However, it is clear thatthe phase space for low energy interactions will be muchlarger in a gapless than in a gapped non-interactingsystem. We thus generically expect the classificationscheme at hand to be robust against moderate inter-actions. However, beyond mean field interactions, theHamiltonian cannot be expressed as an effective singleparticle operator. Hence, we need to find a formulationof the topological invariants that adiabatically connectsto the non-interacting language and is well definedfor general gapped interacting systems. The key toachieving this goal is to look at the single particleGreen’s function G instead of the Hamiltonian. Thisapproach has been pioneered in the field of TSM by Qi,Hughes, and Zhang39 who formulated a topological fieldtheory for TSM in the CAZ class AII.

Chern numbers and Green’s function winding numbers

The role model for this construction is again the Hallconductivity σxy of a gapped 2D system. In Ref. 49,σxy has been expressed in terms of G by perturbative ex-pansion of the effective action of a gauge field A that iscoupled to the gapped fermionic system in the frameworkof quantum electrodynamics in (2+1)D. The leading con-tribution stemming from a vacuum polarization diagramyields the Chern Simons action

SCS =σxy2

∫d2xdt εµνσAµ∂νAσ =

σxy2

∫A ∧ dA.

The prefactor σxy in units of the quantum of conductanceassumes the form

σxy =1

24π2

∫d2kdωTr

[(GdG−1)3

], (62)

where d now denotes the exterior derivative in combinedfrequency-momentum space and G is the time orderedGreen’s function, or, equivalently as far as the calcula-tion of topological invariants is concerned, the contin-uous imaginary frequency Green’s function as used inzero temperature perturbation theory. Eq. (62) hasfirst been identified as a topological invariant and beenproven in a non-relativistic condensed matter context inRefs. 65–67. An analogous expression has been derivedby Volovik using a semi-classical gradient expansion50.

The similarity between Eq. (54) and the integrand ofEq. (62) is striking. Obviously, Eq. (62) representsσxy as a winding number in 3D frequency-momentumspace. If this construction makes sense, we should byintegration of Eq. (62) over ω recover the representa-tion of σxy as the first Chern number in the 2D BZ forthe special case of the non-interacting Green’s functionG0(ω, k) = (iω− h(k))−1 . This straightforward calcula-tion relies on the residue theorem and has been explicitlypresented in Ref. 39. Its result can be readily general-ized to higher even spatial dimensions and higher Chernnumbers, respectively. In Ref. 51, a perturbative expan-sion similar to Ref. 49 has been presented for fermionscoupled to a gauge field in arbitrary even spatial dimen-sion 2n. The resulting analogue of the Hall conductivity,i.e., the prefactor of the Chern Simons form in (2n+ 1)D(see Eq. (56)) can be expressed as10,39,51

N2r+1 [G] = N (r)

∫BZ×Rω

Tr[(GdG−1

)2r+1], (63)

N (r) =−r!

(2r + 1)!(2πi)r+1.

Performing again the integration over the frequency ana-lytically for the noninteracting Green’s functionG0 yields

N2r+1 [G0] = Cr. (64)

Eq. (64) makes manifest that N2r+1 [G], whichcan be formulated for an interacting system, repro-duces the non-interacting classification for the freeGreen’s function G0 of the non-interacting system.The topological invariance of N2r+1 [G] is clear byanalogy with Eq. (55): Whereas the winding num-ber ν2j−1 measures the homotopy of the chiral mapk 7→ q(k) ∈ U(n) which, properly normalized, yieldsan integer due to π2j−1 (U(n)) = Z, n > j, Eq. (63)measures the homotopy of G ∈ GL(n + m,C) in the(2r + 1)D frequency-momentum space which is alsointeger due to π2r+1 (GL(n+m,C)) = Z, n+m > r.

The dimensional hierarchy for symmetry protected de-scendants of a parent state which is characterized by aChern number (see Section VI B) can be constructed ina completely analogous way for the interacting general-ization N2r+1 of the Chern number Cr56. The resultingtopological invariants for the descendant states have beencoined topological order parameters in Ref. 56. Disor-der can again be accounted for by imposing TBC andreplacing the k-space of the system by the torus of twist-ing angles (see Section VI C). Our discussion is limitedto insulating systems. A detailed complementary anal-ysis of the topological properties of different quantumvacua can be found in Ref. 68.

Interacting chiral systems

The integer invariant of chiral unitary systems (classAIII) in odd spatial dimension 2r − 1 is not a Chern

18

number but a winding number (see Section VI A). For allthese systems and dimensional hierarchies with a chiralparent state, i.e., all chiral TSM (see Tab. IV), a similarinteracting extension of the definition of the invariants interms of G(iω, k) has been reported in Ref. 69:

I2r [G] = n(r)

∫BZ×Rω

Tr[Q (dQ)

2r], (65)

where n(r) is a normalization constant, and Q(iω, k) =G−1(iω)UCHG(iω, k), with the unitary representationmatrix UCH of the chiral symmetry operation. In thenon-interacting limit, I2r reduces to ν2r−1 as defined inEq. (55)69.

Fluctuation driven topological transitions

Thus far, we have shown that for a non-interactingsystem the integration over ω reproduces the band struc-ture classification scheme formulated in terms of the adi-abatic curvature. However, the additional frequency de-pendence of the single particle Green’s function can causephenomena without non-interacting counterpart. To seethis, we represent the single particle Green’s function ofan interacting system as

G(ω, k) = (iω − h(k)− Σ(ω, k))−1,

where Σ is the self-energy of the interacting system. InRef. 69, it has been pointed out, that the value ofN2r+1 [G] cannot only change due to gap closings in theenergy spectrum as in the case of the Chern number Cr.This is due to the possibility of poles in the ω-dependenceof the self-energy which give rise to zeros of the Green’sfunction, whereas gap closings correspond to poles of G.From the analytical form of N1 it is immediately clearthat both poles and zeros ofG can change the value ofN1.More generally, the G↔ G−1 symmetry of N2r+1 makesclear that poles of G can be seen as zeros of G−1 andvice versa on an equal footing. In Ref. 70, it has beendemonstrated that the ω-dependence of Σ can change anon-trivial winding number into a trivial one. The emer-gence of a topologically nontrivial phase due to dynami-cal fluctuations which has no non-interacting counterparthas been presented in Refs. 71 and 72.

Chern numbers of effective single particle Hamiltonians

Due to its additional ω-integration, the practical cal-culation of N2r+1 can be numerically very challengingonce the single particle Green’s function of the interact-ing system has been calculated. A major breakthroughalong these lines has been the observation that an effec-tive single particle Hamiltonian defined in terms of theinverse Green’s function at zero frequency can be definedto effectively reduce the topological classification to the

non-interacting case. This possibility has first been men-tioned in Refs. 73 and 74 and been generally proven inRef. 75. The authors of Ref. 75 show, using the spectralrepresentation of the Green’s function, that one can al-ways get rid of the ω-dependence of G. We only reviewthe physical results of this analysis. The both accessibleand explicit proof can be found in Ref. 75. The physicalconclusion is as elegant as simple: Instead of calculatingN2r+1 we can just calculate the Chern number Cr asso-ciated with the fictitious Hamiltonian

h(k) = −G−1(0, k), (66)

the occupied states of which are just its eigenstates withnegative eigenvalues which have been dubbed R-zeros75

since they are positive energy eigenstates of G−1(0, k).Obviously, −G−1

0 (0, k) = h(k) for the non-interacting

Green’s function. Hence, h(k), which has recently beencoined topological Hamiltonian76, adiabatically connectsto h(k) in the non-interacting limit. Note that the pos-sibility of eliminating the ω-dependence is not in con-tradiction to the relevance of this ω-dependence for thetopology of the interacting system. All it shows is thatthe relevant changes due to a different pole structure ofG as a function of ω can be inferred from its value atω = 0.

Topological Hamiltonian for chiral interacting systems

In principle, the construction of the topological Hamil-tonian h(k) can be readily generalized to chiral interact-ing systems as has been mentioned in Ref. 77. To seethis, we note that the crucial argument for the construc-tion of the topological Hamiltonian brought forward inRef. 75 is the following: The continuous interpolation

G(iω, k, λ) = (1− λ)G(iω, k) + λ[iω +G−1(0, k)

]−1

does not contain any singularities or gap closings. Thus,as long as the calculation of a topological invariantin terms of G(iω, k) is concerned, we can also use

G(iω, k) = G(iω, k, λ = 1) =[iω +G−1(0, k)

]−1. Ob-

viously, G(iω, k) =[iω − h(k)

]−1

is the Green’s func-

tion of a fictitious non-interacting system which is gov-erned by the topological Hamiltonian h(k). The mereexistence of the topological invariant for chiral systemsin terms of G(iω, k) as presented in Ref. 69 (see also Eq.(65)) hence suffices to argue that one can equally well

investigate the topology of h(k) and its symmetry pro-tected descendants (see Section VI B) instead of directlyevaluating Eq. (65). Since the single particle Green’sfunction inherits the fundamental symmetries from theHamiltonian69, h(k) will also obey these symmetries. Inparticular, for an interacting system with chiral symme-try, the topological Hamiltonian can be brought into the

19

flat band off-diagonal form

h(k) '

(0 q(k)

q†(k) 0

), (67)

where q(k) ∈ U(n) for the topologically equivalentflat-band system. This construction generically definesa topological invariant for the chiral interacting systemwhich adiabatically concurs with the non-interactingsystem: The winding number ν associated with thefictitious Hamiltonian h(k). A similar construction for achiral 1D system has been presented in Ref. 78.

Discussion of the topological Hamiltonian and practicalconsequences

As already mentioned, the above construction cannotbe valid for arbitrary interacting systems. In particularin 1D, the breakdown of the Z classification in thepresence of general interactions has been investigatedin Ref. 62. However, this problem does not pertainto the concept of the topological Hamiltonian itselfbut rather reflects the limited validity of the adiabaticcontinuation of the non-interacting invariants in termsof the single particle Green’s function, i.e., the limitedvalidity of Eq. (63) and Eq. (65). In the validityregime of these equations, one can equivalently use thetopological Hamiltonian h(k) = −G−1(0, k) to classifyan interacting system in any symmetry class. This is ofenormous practical usefulness for at least two reasons.First, we get rid of the ω-integration appearing in Eq.(63) and Eq. (65) which is cumbersome to evaluate.Second, the method of dimensional extension, thoughgenerally valid, is not always the most convenient wayto calculate the topological invariant of a symmetryprotected descendant state. Provided with the formalequivalence between the non-interacting classificationproblem of the topological Hamiltonian and the Green’sfunction topology, we can directly apply all simplifiedschemes that have been introduced to directly calculatenon-interacting invariants of symmetry protected states(see, e.g., Refs. 33, 58, 79–81) to the topological Hamil-tonian. The framework of dimensional extension andEq. (63) or Eq. (65) for the parent state are, with thebenefit of hindsight, only needed to justify the validityof the topological Hamiltonian.

Before closing the section, we would like to discuss therole of the bulk boundary correspondence, in the pres-ence of interactions. In general, interactions can spon-taneously break the protecting symmetry of a symmetryprotected TSM locally at the boundary thus gapping outthe characteristic metallic surface states. Importantly,this spontaneous symmetry breaking at the gapless sur-face will typically happen at a lower critical interactionstrength than in the gapped bulk. This is because the

gapless surface modes offer more phase space for interac-tions. A generally valid bulk boundary correspondenceis hence absent in the interacting case. Within the va-lidity regime of Eq. (63) and Eq. (65) for chiral TSM,respectively, an interacting analogue of the bulk bound-ary correspondence has been reported in Refs. 69 and82. The main difference to the non- interacting case isthat boundary zero-modes, which represent poles of theGreen’s function can be canceled by zeros of the Green’sfunction as far as the calculation of topological invari-ants is concerned. Note that the Green’s function of anon-interacting system does not have zeros.

VII. LIMITATIONS OF THE FRAMEWORK OFTSM

Finally, we would like to point out some limitationsof the concept of TSM. The two main aspects that onecould see critical in the field of TSM are outlined in thefollowing.

First, whereas the topologically quantized Hallconductivity in the integer quantum Hall state, thehistorical role model of all TSM, is a physical observable,the topological invariants of symmetry protected TSMlike the QSH state are not directly physically observablewithout additional unitary symmetries. The quantumHall effect can be understood in terms of the spectralflow associated with the threading of a flux tube83.Along similar lines, the QSH effect can be understoodin terms of a spin charge separation associated withthe threading of a spin flux84. However, this spin flux,as opposed to an ordinary magnetic flux tube, is notimmediately experimentally accessible and the generalobservable consequences of the QSH state have beenshown to be much more subtle85. For several TSM,the directly measurable consequences of the respectivetopological invariants are still under debate or unknown.

Second, the entire construction and classification ofTSM is based on single particle Hamiltonians. In Sec-tion VI D, we discussed how adiabatic interactions can betaken into account and argued that interactions of mod-erate strength are not likely to destroy the phenomenol-ogy of TSM. In order to position the field of TSM in abroader context, we would like to point out that there arealso phenomena of topological origin which emerge onlydue to the presence of interactions. The historically firstphenomenon is the 1

ν FQH effect63,64 which cannot beadiabatically connected to an insulating non-interactingstate. The non-interacting state is in this case a partiallyfilled Landau level which provides an enormous densityof states at the Fermi energy. In a system with periodicboundaries, the 1

ν FQH has a characteristic ν-fold groundstate degeneracy. Interestingly, this simplest FQH statecan still be analyzed in the framework of TBC57. Formore general FQH systems the concept of topological or-

20

der has been introduced by Wen86. A crucial notion inthis framework is the quantum dimension of the topolog-ically ordered system which can be viewed as the groundstate degeneracy of the system on a torus, i.e., with pe-riodic boundary conditions. Most TSM discussed in thisarticle have quantum dimension one, like a trivial insu-lator. From the vantage point of topological order, thesestates are thus trivial. Recently, the concepts of TSMand FQH physics have been combined to the definitionof the fractional Chern insulator87–89 and the fractionaltopological insulator90–93 the first lattice realization ofwhich has been reported in Ref. 94. These states aretranslation-invariant realizations of the FQH effect andits time reversal symmetry protected analogues, respec-tively. A general hierarchy of fractional topological insu-lators has been reported in Ref. 95.

VIII. OUTLOOK

From a conceptual point of view, the entire zoo oftopological states of matter can be seen as conclusivelyunderstood in the framework outlined in this review.However, there are at least two general routes to beconsiderably further explored by future research. First,the precise experimental implications of many topolog-ical states of matter have not been fully unraveled yet.Whereas in the quantum Hall state, the topological in-

variant directly represents a physical observable, namelythe Hall conductivity of the sample, the observability ofthe topological invariants of several topological statesof matter is unknown or still under active debate.This issue is from our point of view closely related tothe rather limited number of promising proposals forconcrete technological applications based on these novelstates of matter. Obviously, successful research in thisdirection will be of decisive importance for the longterm future of the entire field of topological states ofmatter. Second, the influence of interactions and openquantum system effects on topological states of matteris by no means conclusively understood, let alone anexhaustive topological classification of interacting ordissipative systems. As a first step along these lines,a purely dissipation driven topological state has beenreported in Ref. 96 and 97.

ACKNOWLEDGEMENT

We acknowledge interesting discussions with HansHansson, Patrik Recher, Dietrich Rothe, RonnyThomale, Grigori Volovik, and Shou-Cheng Zhang aswell as financial support from the DFG-JST research unit“Topotronics” (BT and JCB) and from the Swedish Sci-ence Research Council (JCB).

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