# Topological Field Theory of Time-Reversal Invariant Insulators

###### Abstract

We show that the fundamental time reversal invariant (TRI) insulator exists in dimensions, where the effective field theory is described by the dimensional Chern-Simons theory and the topological properties of the electronic structure is classified by the second Chern number. These topological properties are the natural generalizations of the time reversal breaking (TRB) quantum Hall insulator in dimensions. The TRI quantum spin Hall insulator in dimensions and the topological insulator in dimension can be obtained as descendants from the fundamental TRI insulator in dimensions through a dimensional reduction procedure. The effective topological field theory, and the topological classification for the TRI insulators in and dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of novel and measurable phenomena, the most striking of which is the topological magneto-electric effect, where an electric field generates a magnetic field in the same direction, with an universal constant of proportionality quantized in odd multiples of the fine structure constant . Finally, we present a general classification of all topological insulators in various dimensions, and describe them in terms of a unified topological Chern-Simons field theory in phase space.

###### Contents

- I Introduction
- II TRB topological insulators in dimensions and its dimensional reduction
- III Second Chern number and its physical consequences
- IV Dimensional reduction to -d TRI insulators
- V Dimensional reduction to -d
- VI Unified theory of topological insulators
- VII Conclusion and discussions
- A Conventions
- B Derivation of Eq. (54)
- C The winding number in the non-linear response of Dirac-type models
- D Stability of edge theories in generic dimensions

## I Introduction

Most states or phases of condensed matter can be described by local order parameters and the associated broken symmetries. However, the quantum Hall (QH) statev. Klitzing et al. (1980); Tsui et al. (1982); Laughlin (1981, 1983) gives the first example of topological states of matter which have topological quantum numbers different from ordinary states of matter, and are described in the low energy limit by topological field theories. Soon after the discovery of the integer QH effect, the quantization of Hall conductance in units of was shown to be a general property of two-dimensional time reversal breaking (TRB) band insulatorsThouless et al. (1982). The integral of the curvature of the Berry’s phase gauge field defined over the magnetic Brillouin zone (BZ) was shown to be a topological invariant called the first Chern number, which is physically measured as the quanta of the Hall conductance. In the presence of many-body interactions and disorder, the Berry curvature and the first Chern number can be defined over the space of twisted boundary conditionsNiu et al. (1985). In the long wave length limit, both the integer and the fractional QH effect can be described by the topological Chern-Simons field theoryZhang et al. (1989) in dimensions. This effective topological field theory captures all physically measurable topological effects, including the quantization of the Hall conductance, the fractional charge, and the statistics of quasi-particlesZhang (1992).

Insulators in dimensions can also have unique topological effects. Solitons in charge density wave insulators can have fractional charge or spin-charge separationSu et al. (1979). The electric polarization of these insulators can be expressed in terms of the integral of the Berry’s phase gauge field in momentum spaceKing-Smith and Vanderbilt (1993); Ortiz and Martin (1994). During an adiabatic pumping cycle, the change of electric polarization, or the net charge pumped across the 1D insulator, is given by the integral of the Berry curvature over the hybrid space of momentum and the adiabatic pumping parameter. This integral is quantized to be a topological integerThouless (1983). Both the charge of the soliton and the adiabatic pumping current can be obtained from the Goldstone-Wilczek formulaGoldstone and Wilczek (1981).

In this paper we shall show that the topological effects in the dimensional insulator can be obtained from the QH effect of the dimensional TRB insulator by a procedure called dimensional reduction. In this procedure one of the momenta is replaced by an adiabatic parameter, or field, and the Goldstone-Wilczek formula, and thus, all topological effects of the dimensional insulators, can be derived from the dimensional QH effect. The procedure of dimensional reduction can be generalized to the higher dimensional TRI insulators and beyond, which is the key result of this paper.

In recent years, the QH effect of the dimensional TRB insulators has been generalized to TRI insulators in various dimensions. The first example of a topologically non-trivial TRI state in condensed matter context was the 4D generalization of the QH effect (4DQH) proposed in Ref. Zhang and Hu, 2001. The effective theory of this model is given by the Chern-Simons topological field theory in dimensionsBernevig et al. (2002). The quantum spin Hall (QSH) effect has been proposed in dimensional TRI quantum modelsC. L. Kane and E. J. Mele (2005a); B.A. Bernevig and S.C. Zhang (2006). The QSH insulator state has a gap for all bulk excitations, but has topologically protected gapless edge states, where opposite spin states counter-propagateC. L. Kane and E. J. Mele (2005a); C. Wu et al. (2006); C. Xu and J. Moore (2006). Recently the QSH state has been theoretically predictedB. A. Bernevig et al. (2006) and experimentally observed in HgTe quantum wellsKönig et al. (2007). TRI topological insulators have also been classified in dimensionsFu et al. (2007); Moore and Balents (2007); Roy (a). These 3D states all carry spin Hall current in the insulating stateMurakami et al. (2004a).

The topological properties of the dimensional TRI insulator can be described by the second Chern number defined over four dimensional momentum space. On the other hand, TRI insulators in and dimensions are described by a topological invariant defined over momentum spaceC. L. Kane and E. J. Mele (2005a, b); Fu and Kane (2006, 2007); Fu et al. (2007); Moore and Balents (2007); Roy (a, b, c). In the presence of interactions and disorder, the momentum space invariant is not well defined, however, one can define a more general topological invariant in terms of spin-charge separation associated with a fluxQi and Zhang ; Ran et al. . One open question in this field concerns the relationship between the classification of the dimensional TRI insulator by the second Chern number and the classification of the and dimensional TRI insulators by the number.

The effective theory of the dimensional TRI insulator is given by the topological Chern-Simons field theoryBernevig et al. (2002); Niemi and Semenoff (1983). While the dimensional Chern-Simons theory describes a linear topological response to an external gauge fieldZhang et al. (1989); Zhang (1992), the dimensional Chern-Simons theory describes a nonlinear topological response to an external gauge field. The key outstanding theoretical problem in this field is the search for the topological field theory describing the TRI insulators in and dimensions, from which all measurable topological effects can be derived.

In this paper, we solve this outstanding problem by constructing topological field theories for the and dimensional TRI insulators using the procedure of dimensional reduction. We show that the dimensional topological insulator is the fundamental state from which all lower dimensional TRI insulators can be derived. This procedure is analogous to the dimensional reduction from the dimensional TRB topological insulator to the dimensional insulators. There is a deep reason why the fundamental TRB topological insulator exists in dimensions, while the fundamental TRI topological insulator exists in dimensions. The reason goes back to the Wigner-von Neumann classificationvon Neumann and Wigner (1929) of level crossings in TRB unitary quantum systems and the TRI symplectic quantum systems. Generically three parameters need to be tuned to obtain a level crossing in a TRB unitary system, while five parameters need to be tuned to obtain a level crossing in a TRI symplectic system. These level crossing singularities give rise to the non-trivial topological curvatures on the 2D and 4D parameter surfaces which enclose the singularities. Fundamental topological insulators are obtained in space dimensions where all these parameters are momentum variables. Once the fundamental TRI topological insulator is identified in dimensions, the lower dimensional versions of TRI topological insulators can be easily obtained by dimensional reduction. In this procedure, one or two momentum variables of the dimensional topological insulator are replaced by adiabatic parameters or fields, and the dimensional Chern-Simons topological field theory is reduced to topological field theories involving both the external gauge field and the adiabatic fields. For the TRI insulators, the topological field theory is given by that of the “axion Lagrangian”, or the dimensional vacuum term, familiar in the context of quantum chromodynamics (QCD), where the adiabatic field plays the role of the axion field or the angle. From these topological field theories, all physically measurable topological effects of the and the dimensional TRI insulators can be derived. We predict a number of novel topological effects in this paper, the most striking of which is the topological magneto-electric (TME) effect, where an electric field induces a magnetic field in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine structure constant . We also present an experimental proposal to measure this novel effect in terms of Faraday rotation. Our dimensional reduction procedure also naturally produces the classification of the and the dimensional TRI topological insulators in terms of the integer second Chern class of the dimensional TRI topological insulators.

The remaining parts of the paper are organized as follows. In Sec. II we review the physical consequences of the first Chern number, namely the -d QH effect and -d fractional charge and topological pumping effects. We begin with the -d time reversal breaking insulators and study the topological transport properties. We then present a dimensional reduction procedure that allows us to consider related topological phenomena in -d and -d. Subsequently, we define a classification of these lower dimensional descendants which relies on the presence of a discrete particle-hole symmetry. This will serve as a review and a warm-up exercise for the more complicated phenomena we consider in the later sections. In Secs. III, IV, and V we discuss consequences of a non-trivial second Chern number beginning with a parent -d topological insulator in Sec. III. In Secs. IV and V we continue studying the consequences of the second Chern number but in the physically realistic -d and -d models which are the descendants of the initial -d system. We present effective actions describing all of the physical systems and their responses to applied electromagnetic fields. This provides the first effective field theory for the TRI topological insulators in -d and -d. For these two descendants of the -d theory, we show that the classification of the decedents are obtained from the 2nd Chern number classification of the parent TRI insulator. Finally, in Sec. VI we unify all of the results into families of topological effective actions defined in a phase space formalism. From this we construct a family tree of all topological insulators, some of which are only defined in higher dimensions, and with topological classifications which repeat every dimensions.

This paper contains many new results on topological insulators, but it can also be read by advanced students as a pedagogical and self-contained introduction of topology applied to condensed matter physics. Physical models are presented in the familiar tight-binding forms, and all topological results can be derived by exact and explicit calculations, using techniques such as response theory already familiar in condensed matter physics. During the course of reading this paper, we suggest the readers to consult Appendix A which covers all of our conventions.

## Ii TRB topological insulators in dimensions and its dimensional reduction

In this section, we review the physics of the TRB topological insulators in dimensions. We shall use the example of a translationally invariant tight-binding modelX.L. Qi et al. (2006) which realizes the QH effect without Landau levels. We discuss the procedure of dimensional reduction, from which all topological effects of the dimensional insulators can be obtained. This section serves as a simple pedagogical example for the more complex case of the TRI insulators presented in Sec. III and IV.

### ii.1 The first Chern number and topological response function in -d

In general, the tight-binding Hamiltonian of a -d band insulator can be expressed as

(1) |

with the lattice sites and the band indices for a -band system. With translation symmetry , the Hamiltonian can be diagonalized in a Bloch wavefunction basis:

(2) |

The minimal coupling to an external electro-magnetic field is given by where is a gauge potential defined on a lattice link with sites at the end. To linear order, the Hamiltonian coupled to the electro-magnetic field is obtained as

with the band indices omitted. The DC response of the system to external field can be obtained by the standard Kubo formula:

(3) | |||||

with the DC current , Green’s function , and the area of the system. When the system is a band insulator with fully-occupied bands, the longitudinal conductance vanishes, i.e. , as expected, while has the form shown in Ref. Thouless et al., 1982:

(4) | |||||

Physically, is the component of the Berry’s phase gauge field (adiabatic connection) in momentum space. The quantization of the first Chern number

(5) |

is satisfied for any continuous states defined on the BZ.

Due to charge conservation, the QH response also induces another response equation:

(7) |

where is the charge density in the ground state. Equations (II.1) and (7) can be combined together in a covariant way:

(8) |

where are temporal and spatial indices. Here and below we will take the units so that .

The response equations (8) can be described by the topological Chern-Simons field theory of the external field :

(9) |

in the sense that recovers the response equations (8). Such an effective action is topologically invariant, in agreement with the topological nature of the first Chern number. All topological responses of the QH state are contained in the Chern-Simons theoryZhang (1992).

### ii.2 Example: two band models

To make the physical picture clearer, the simplest case of a two band model can be studied as an exampleX.L. Qi et al. (2006). The Hamiltonian of a two-band model can be generally written as

(10) |

where is the identity matrix and are the three Pauli matrices. Here we assume that the represent a spin or pseudo-spin degree of freedom. If it is a real spin then the are thus odd under time reversal. If If the are odd in then the Hamiltonian is time-reversal invariant. However, if any of the contain a constant term then the model has explicit time-reversal symmetry breaking. If the are a pseudo-spin then one has to be more careful. Since, in this case, then only is odd under time-reversal (because it is imaginary) while are even. The identity matrix is even under time-reversal and must be even in to preserve time-reversal. The energy spectrum is easily obtained: . When for all in the BZ, the two bands never touch each other. If we also require that , so that the gap is not closed indirectly, then a gap always exists between the two bands of the system. In the single particle Hamiltonian , the vector acts as a “Zeeman field” applied to a “pseudospin” of a two level system. The occupied band satisfies , which thus corresponds to the spinor with spin polarization in the direction. Thus the Berry’s phase gained by during an adiabatic evolution along some path in -space is equal to the Berry’s phase a spin- particle gains during the adiabatic rotation of the magnetic field along the path This is known to be half of the solid angle subtended by , as shown in Fig.1. Consequently, the first Chern number is determined by the winding number of around the originVolovik (2003); X.L. Qi et al. (2006):

(11) |

From the response equations we know that a non-zero implies a quantized Hall response. The Hall effect can only occur in a system with time-reversal symmetry breaking so if then time-reversal symmetry is broken. Historically, the first example of such a two-band model with a non-zero Chern number was a honeycomb lattice model with imaginary next-nearest-neighbor hopping proposed by HaldaneHaldane (1988).

To be concrete, we shall study a particular two band model introduced in Ref. X.L. Qi et al. (2006), which is given by

(12) | |||||

This Hamiltonian corresponds to the form (10) with and . The Chern number of this system is X.L. Qi et al. (2006)

(13) |

In the continuum limit, this model reduces to the dimensional massive Dirac Hamiltonian

In a real space, this model can be expressed in tight-binding form as

(14) | |||||

Physically, such a model describes the quantum anomalous Hall effect realized with both strong spin-orbit coupling ( and terms) and ferromagnetic polarization ( term). Initially this model was introduced for its simplicity in Ref. X.L. Qi et al., 2006, however, recently, it was shown that it can be physically realized in quantum wells with a proper amount of spin polarizationliu .

### ii.3 Dimensional reduction

To see how topological effects of dimensional insulators can be derived from the first Chern number and the QH effect through the procedure of dimensional reduction, we start by studying the QH system on a cylinder. An essential consequence of the nontrivial topology in the QH system is the existence of chiral edge states. For the simplest case with the first Chern number , there is one branch of chiral fermions on each boundary. These edge states can be solved for explicitly by diagonalizing the Hamiltonian (14) in a cylindrical geometry. That is, with periodic boundary conditions in the -direction and open boundary conditions in the -direction, as shown in Fig.2 (a). Note that with this choice is still a good quantum number. By defining the partial Fourier transformation

with the coordinates of square lattice sites, the Hamiltonian can be rewritten as

(15) | |||||

In this way, the 2D system can be treated as independent 1D tight-binding chains, where is the period of the lattice in the -direction. The eigenvalues of the 1D Hamiltonian can be obtained numerically for each , as shown in Fig. 2 (b). An important property of the spectrum is the presence of edge states, which lie in the bulk energy gap, and are spatially localized at the two boundaries: The chiral nature of the edge states can be seen from their energy spectrum. From Fig. 2 (b) we can see that the velocity is always positive for the left edge state and negative for the right one. The QH effect can be easily understood in this edge state picture by Laughlin’s gauge argumentLaughlin (1981). Consider a constant electric field in the -direction, which can be chosen as

The Hamiltonian is written and the current along the -direction is given by

(16) |

with the current of the 1D system. In this way, the Hall response of the 2D system is determined by the current response of the parameterized 1D systems to the temporal change of the parameter . The gauge vector corresponds to a flux threading the cylinder. During a time period , the flux changes from to . The charge that flows through the system during this time is given by

(17) | |||||

with . In the second equality we use the relation between the current and charge polarization of the 1D systems . In the adiabatic limit, the 1D system stays in the ground state of , so that the change of polarization is given by . Thus in the limit can be written as

(18) |

Therefore, the charge flow due to the Hall current generated by the flux through the cylinder equals the charge flow through the -dimensional system , when is cycled adiabatically from to . From the QH response we know is quantized as an integer, which is easy to understand in the 1D picture. During the adiabatic change of from to , the energy and position of the edge states will change, as shown in Fig.2 (c). Since the edge state energy is always increasing(decreasing) with for a state on the left (right) boundary, the charge is always “pumped” to the left for the half-filled system, which leads to for each cycle. This quantization can also be explicitly shown by calculating the polarization , as shown in Fig.2 (d), where the jump of by one leads to . In summary, we have shown that the QH effect in the tight-binding model of Eq. (12) can be mapped to an adiabatic pumping effectThouless (1983) by diagonalizing the system in one direction and mapping the momentum to a parameter.

Such a dimensional reduction procedure is not restricted to specific models, and can be generalized to any 2D insulators. For any insulator with Hamiltonian (2), we can define the corresponding 1D systems

(19) |

in which replaces the -direction momentum and effectively takes the place of When is time-dependent, the current response can be obtained by a similar Kubo formula to Eq. (3), except that the summation over all is replaced by that over only . More explicitly, such a linear response is defined as

(20) | |||||

Similar to Eq. (4) of the 2D case, the response coefficient can be expressed in terms of a Berry’s phase gauge field as

with the sum rule

(22) |

If we choose a proper gauge so that is always single-valued, the expression of can be further simplified to

(23) |

Physically, the loop integral

(24) |

is nothing but the charge polarization of the 1D systemKing-Smith and Vanderbilt (1993); Ortiz and Martin (1994), and the response equation (20) simply becomes . Since the polarization is defined as the shift of the electron center-of-mass position away from the lattice sites, it is only well-defined modulo . Consequently, the change through a period of adiabatic evolution is an integer equal to and corresponds to the charge pumped through the system. Such a relation between quantized pumping and the first Chern number was shown by ThoulessThouless (1983).

Similar to the QH case, the current response can lead to a charge density response, which can be determined by the charge conservation condition. When the parameter has a smooth spatial dependence , the response equation (20) still holds. From the continuity equation we obtain

(25) |

in which is defined with respect to the background charge. Similar to Eq. (8), the density and current response can be written together as

(26) |

where are time and space. It should be noted that only differentiation with respect to appears in Eq. (26). This means, as expected, the current and density response of the system do not depend on the parametrization. In general, when the Hamiltonian has smooth space and time dependence, the single particle Hamiltonian becomes , which has the eigenstates with the band index. Then relabelling as we can define the phase space Berry’s phase gauge field

(27) |

and the phase space current

(28) |

The physical current is obtained by integration over the wavevector manifold:

(29) |

where This recovers Eq. (26). Note that we could have also looked at the component but this current does not have a physical interpretation.

Before moving to the next topic, we would like to apply this formalism to the case of the Dirac model, which reproduces the well-known result of fractional charge in the Su-Schrieffer-Heeger (SSH) modelSu et al. (1979), or equivalently the Jackiw-Rebbi modelJackiw and Rebbi (1976a). To see this, consider the following slightly different version of the tight-binding model (12):

(30) | |||||

with . In the limit , the Hamiltonian has the continuum limit , which is the continuum Dirac model in -d, with a real mass and an imaginary mass . As discussed in Sec. II.2, the polarization is determined by the solid angle subtended by the curve , as shown in Fig. 3. In the limit one can show that the solid angle so that , in which case Eq. (26) reproduces the Goldstone-Wilczek formulaGoldstone and Wilczek (1981) :

(31) |

Specifically, a charge is carried by a domain wall of the field. In particular, for an anti-phase domain wall, , we obtain fractional charge . Our phase space formula (28) is a new result, and it provides a generalization of the Goldstone-Wilczek formula to the most general one-dimensional insulator.

### ii.4 classification of particle-hole symmetric insulators in -d

In the last subsection, we have shown how the first Chern number of a Berry’s phase gauge field appears in an adiabatic pumping effect and the domain wall charge of one-dimensional insulators. In these cases, an adiabatic spatial or temporal variation of the single-particle Hamiltonian, through its parametric dependence on , is required to define the Chern number. In other words, the first Chern number is defined for a parameterized family of Hamiltonians , rather than for a single 1D Hamiltonian . In this subsection, we will show a different application of the first Chern number, in which a topological classification is obtained for particle-hole symmetric insulators in 1D. Such a relation between Chern number and topology can be easily generalized to the more interesting case of second Chern number, where a similar characterization is obtained for TRI insulators, as will be shown in Sec. IV.3 and V.2.

For a one-dimensional tight-binding Hamiltonian , the particle-hole transformation is defined by , where the charge conjugation matrix satisfies and . Under periodic boundary conditions the symmetry requirement is

(32) | |||||

From Eq. (32) it is straightforward to see the symmetry of the energy spectrum: if is an eigenvalue of , so is . Consequently, if the dimension of is odd, there must be at least one zero mode with . Since the chemical potential is constrained to vanish by the traceless condition of , such a particle-hole symmetric system cannot be gapped unless the dimension of is even. Since we are only interested in the classification of insulators, we will focus on the case with bands per lattice site.

Now consider two particle-hole symmetric insulators with Hamiltonians and , respectively. In general, a continuous interpolation between them can be defined so that

(33) |

Moreover, it is always possible to find a proper parametrization so that is gapped for all . In other words, the topological space of all 1D insulating Hamiltonians is connected, which is a consequence of the Wigner-Von Neumann theoremvon Neumann and Wigner (1929).

Suppose is such a “gapped interpolation” between and . In general, for doesn’t necessarily satisfy the particle-hole symmetry. For , define

(34) |

We choose this parameterization so that if we replaced by a momentum wavevector then the corresponding higher dimensional Hamiltonian would be particle-hole symmetric. Due to the particle-hole symmetry of and , is continuous for , and . Consequently, the adiabatic evolution of from to defines a cycle of adiabatic pumping in , and a first Chern number can be defined in the space. As discussed in Sec. II.3, the Chern number can be expressed as a winding number of the polarization

where the summation is carried out over the occupied bands. In general, two different parameterizations and can lead to different Chern numbers . However, the symmetry constraint in Eq. (34) guarantees that the two Chern numbers always differ by an even integer: .

To prove this conclusion, we first study the behavior of under a particle-hole transformation. For an eigenstate of the Hamiltonian with eigenvalue , Eq. (34) leads to

(35) |

in which is the complex conjugate state: where are the position space lattice, and orbital index respectively. Thus is an eigenstate of with energy and momentum . Such a mapping between eigenstates of and is one-to-one. Thus

(36) | |||||

Since is only well-defined modulo , the equality (36) actually means . Consequently, for or we have , so that or . In other words, the polarization is either or for any particle-hole symmetric insulator, which thus defines a classification of particle-hole symmetric insulators. If two systems have different value, they cannot be adiabatically connected without breaking the particle-hole symmetry, because (mod ) is a continuous function during adiabatic deformation, and a value other than and breaks particle-hole symmetry. Though such an argument explains physically why a classification is defined for particle-hole symmetric system, it is not so rigorous. As discussed in the derivation from Eq. (II.3) to Eq. (23), the definition relies on a proper gauge choice. To avoid any gauge dependence, a more rigorous definition of the classification is shown below, which only involves the gauge invariant variable and Chern number .

To begin with, the symmetry (36) leads to

(37) |

which is independent of gauge choice since only the change of is involved. This equation shows that the change of polarization during the first half and the second half of the closed path are always the same.

Now consider two different parameterizations and , satisfying , . Denoting the polarization and corresponding to and , respectively, the Chern number difference between and is given by

(38) |

Define the new interpolations and as

(39) |

and are obtained by recombination of the two paths and , as shown in Fig. 4. From the construction of and , it is straightforward to see that

(40) |

Thus . On the other hand, from Eq. (37) we know , so that . Since , we obtain that is even for any two interpolations and between and . Intuitively, such a conclusion simply comes from the fact that the Chern number and can be different only if there are singularities between these two paths, while the positions of the singularities in the parameter space are always symmetric under particle-hole symmetry, as shown in Fig. 4.

Based on the discussions above, we can define the “relative Chern parity” as

(41) |

which is independent of the choice of interpolation , but only determined by the Hamiltonians . Moreover, for any three particle-hole symmetric Hamiltonians , it is easy to prove that the Chern parity satisfies the following associative law:

Consequently, defines an equivalence relation between any two particle-hole symmetric Hamiltonians, which thus classifies all the particle-hole symmetric insulators into two classes. To define these two classes more explicitly, one can define a “vacuum” Hamiltonian as , where is an arbitrary matrix which does not depend on and which satisfies the particle-hole symmetry constraint . Thus describes a totally local system, in which there is no hopping between different sites. Taking such a trivial system as a reference Hamiltonian, we can define as a topological quantum number of the Hamiltonian . All the Hamiltonians with are classified as trivial, while those with are considered as nontrivial. (Again, this classification doesn’t depend on the choice of “vacuum” , since any two vacua are equivalent.)

Despite its abstract form, such a topological characterization has a direct physical consequence. For a nontrivial Hamiltonian , an interpolation can be defined so that , , and the Chern number is an odd integer. If we study the one-dimensional system with open boundary conditions, the tight binding Hamiltonian can be rewritten in real space as

As discussed in Sec. II.3, there are mid-gap end states in the energy spectrum of as a consequence of the non-zero Chern number. When the Chern number , there are values for which the Hamiltonian has zero energy localized states on the left end of the 1D system, and the same number of values where zero energy states are localized on the right end, as shown in Fig. 5. Due to the particle-hole symmetry between and , zero levels always appear in pairs at and . Consequently, when the Chern number is odd, there must be a zero level at or . Since corresponds to a trivial insulator with flat bands and no end states, the localized zero mode has to appear at . In other words, one zero energy localized state (or an odd number of such states) is confined at each open boundary of a nontrivial particle-hole symmetric insulator.

The existence of a zero level leads to an important physical consequence—a half charge on the boundary of the nontrivial insulator. In a periodic system when the chemical potential vanishes, the average electron density on each site is when there are bands filled. In an open boundary system, define to be the density deviation with respect to on each site. Then particle-hole symmetry leads to