A Eigenstate degeneracy of 2D lattice boson model

Exactly soluble models for fractional topological insulators in 2 and 3 dimensions

Abstract

We construct exactly soluble lattice models for fractionalized, time reversal invariant electronic insulators in 2 and 3 dimensions. The low energy physics of these models is exactly equivalent to a non-interacting topological insulator built out of fractionally charged fermionic quasiparticles. We show that some of our models have protected edge modes (in 2D) and surface modes (in 3D), and are thus fractionalized analogues of topological insulators. We also find that some of the 2D models do not have protected edge modes – that is, the edge modes can be gapped out by appropriate time reversal invariant, charge conserving perturbations. (A similar state of affairs may also exist in 3D). We show that all of our models are topologically ordered, exhibiting fractional statistics as well as ground state degeneracy on a torus. In the 3D case, we find that the models exhibit a fractional magnetoelectric effect.

I Introduction

One of the more surprising discoveries of the past decade has been that time-reversal invariant band insulators come in two kinds: topological insulators and trivial insulators. These two families of insulators exist in both two(1); (2); (3); (4) and three(5); (6); (7); (4) dimensional systems. They are distinguished by the fact that the interface of a topological insulator with the vacuum always carries a gapless edge mode (in two dimensions) or surface mode (in three dimensions), while no such protected boundary modes exist for the trivial insulator.

Though much of our current understanding of topological insulators has focused on non-interacting or weakly interacting systems, it is natural to consider the fate of this physics in the presence of strong interactions. Strongly interacting insulators can be divided into two classes: systems that can be adiabatically connected to (non-interacting) band insulators without closing the bulk gap, and those that cannot. In the former case it has been shown (explicitly in 2D(8), and implicitly in 3D(9)) that the gapless boundary modes of a topological insulator are stable to strong interactions as long as time reversal symmetry and charge conservation are not broken (explicitly or spontaneously). Therefore there is a well-defined notion of interacting topological insulators in systems that are adiabatically connected to band insulators.

Here we will consider the second possibility: strongly interacting, time reversal invariant electron systems whose ground state cannot be adiabatically connected to a band insulator. The same question can be posed: do some of these systems have protected gapless edge modes? This question is particularly interesting in light of the fact that such phases can be fractionalized, leading to a great diversity of possibilities. That is, such phases need not have excitations that resemble electrons; in general, the quasiparticles may have fractional charge and statistics.

Our understanding of these fractionalized insulators is, however, limited. Focusing on the two dimensional case, Ref. (10) analyzed a class of strongly interacting toy models(3) where spin-up and spin-down electrons each form fractional quantum Hall states with opposite chiralities.(11) Ref. (10) concluded that some of these strongly interacting, time reversal invariant insulators have protected edge modes, while some do not. The two kinds of insulators were dubbed “fractional topological insulators” and “fractional trivial insulators” since they are analogous to non-interacting topological and trivial insulators, but they contain quasiparticle excitations with fractional charge and fractional statistics. These models demonstrate that fractionalized analogues of topological insulators are possible in principle, but they do not exhaust all the possibilities for these phases.

In the three dimensional case, even less is known. Refs. (12); (13) used a parton construction to build time reversal invariant insulators with a fractional magnetoelectric effect. However, this work did not prove that these states have protected surface modes (though Ref. (12) did conjecture that this is the case). Also, Refs. (12); (13) did not construct a microscopic Hamiltonian realizing these phases–a standard limitation of parton or slave particle approaches.

In this paper, we address both of these issues. First, we construct a set of exactly soluble lattice electron models – in both two and three dimensions – that realize time reversal invariant insulators with fractionally charged excitations. Second, we prove that some of these fractionalized electronic insulators have protected edge or surface modes (that is, we show that perturbations cannot gap out the boundary modes without breaking time reversal symmetry or charge conservation symmetry, explicitly or spontaneously). In this sense, these models provide concrete examples of “fractional topological insulators” in both two and three dimensions. An added bonus of our analysis is that the argument we use to establish the robustness of the edge or surface modes is not specific to our exactly soluble models, and can be applied equally well to more general fractionalized (or unfractionalized) insulators.

The low energy physics of our models is exactly equivalent to a non-interacting topological insulator built out of fractionally charged fermions. Surprisingly, however, some of the models do not have protected boundary modes. Specifically, we find that in some 2D models (namely those for which our protected-edge argument breaks down) the edge modes can be gapped out by adding an appropriate time reversal invariant, charge conserving perturbation. In the 3D case, our understanding is more limited: we can prove that some of the models have protected surface modes, but we do not know the fate of the surface modes in the remaining models.

The models that we construct and solve are rather far from describing systems that are presently accessible to experiments. Nevertheless, they are of value for two reasons. First, they allow for a study of matters of principle, such as the existence of fractional topological insulators, their properties and their stability. And second, since the phases we consider are robust to arbitrary deformations of the Hamiltonian that do not close the bulk gap (or break time reversal or charge conservation symmetry), these phases may also be realized by models that are significantly different from the ones discussed here.

The paper is organized as follows. In section II we describe the basic physical picture and summarize our results. In section III, we construct models in the 2D case. In section IV, we analyze the physical properties of these models, including the structure of the edge modes and the topological order in the bulk. In section V, we consider the same models in the 3D case. We then analyze the physical properties of the 3D models in section VI, including the structure of the surface modes, the topological order in the bulk, and the nature of the magnetoelectric effect. In the final part of the paper, section VII, we investigate whether the boundary modes in our models are robust to arbitrary time reversal invariant, charge conserving perturbations. The Appendix contains some of the more technical calculations. Table 1 lists the various symbols that we use in the text.

Symbol Description Section
number of sites in the lattice II
number of links in the lattice II
number of plaquettes in the lattice II
integer parameter in boson model III.1
boson creation operator on site III.1
boson creation operator on link III.1
boson occupation number on site III.1
boson occupation number on link III.1
cluster charge on site in boson model III.1
ring exchange term on plaquette III.1
sublattice weighting factor for site III.1
hopping term on link III.1
eigenvalue of III.1.1
eigenvalue of III.1.1
simultaneous eigenstate of III.1.1
simultaneous eigenstate III.1.2
electric charge of charge excitation III.1.2
electron creation operator III.2
electron occupation number III.2
total number of electrons on site III.2
cluster charge in electron model III.2
integer parameter in electron model III.2
eigenvalue of III.2.1
simultaneous eigenstate of III.2.1
electron state with occupation III.2.2
eigenstate III.2.2
electric charge of fermion excitation III.2.2
hopping term for fermion excitations III.3
hopping amplitudes for fermions III.3
creation operator for fermions III.3
Mutual statistics of charges and fluxes IV.2
Statistics of fermions and fluxes IV.2
smallest charged excitation VI.3
Table 1: List of symbols, their description, and the section where they are defined.

Ii Summary of results

This section is aimed at introducing the reader to our exactly soluble models, and to the main results we find by analyzing these models. We will emphasize the physical picture, leaving the detailed calculations to the following sections.

ii.1 Constructing exactly soluble models for fractional topological insulators

To obtain candidate fractional topological insulators, we build models with two important properties: (1) fractionally charged fermionic quasiparticles and (2) a topological insulator band structure for these excitations. Our construction has three steps. In the first step we construct lattice boson models with fractionally charged bosonic excitations. In the second step we add electrons to the lattice, and define an electron-boson interaction that binds each electron to fractionally charged excitations of the bosonic model, thus creating a fractionally charged fermion. In the third step we construct a hopping term on the lattice that allows the fractionally charged fermion to hop between lattice sites without exciting other degrees of freedom. We then choose the hopping terms so that these fermions have a topological insulator band structure.

The boson models we construct are similar in spirit to the “toric code” model and its generalizations(14), but are built out of bosonic charged particles whose total charge is conserved. In this sense, these models are a hybrid between the toric code model (which is exactly soluble but not charge conserving) and the fractionalized bosonic insulators of Refs. (15); (16); (17); (18); (19) (which are charge conserving, but not exactly soluble).

The construction of the models is based on the following recipe. We consider a system of bosons that live on the sites and links of a bipartite square (or in 3D, cubic) lattice. We construct a bosonic Hamiltonian composed of two parts: . Each term has an associated energy scale whose magnitude is of minor significance to our discussion. Both, however, depend on an integer parameter which plays a crucial role, as it determines the fractional charge carried by the quasiparticle excitations.

The first term is the “charging” Hamiltonian. This term depends only on the number of bosons on each site and on each link . Each boson is made of two electrons, of a charge each. The charging Hamiltonian assigns different energies to different charge configurations , by coupling the charge on a site to the charges on the four links neighboring the site. The spectrum of is discrete, as expected from a charging Hamiltonian. The spectrum is also highly degenerate, since many charge configurations have the same energy cost. In fact, the number of degenerate eigenstates of the lowest eigenvalue of is with being the number of links in the lattice and being the number of sites.

The second term is the “hopping” Hamiltonian. This term makes bosons hop between neighboring lattice sites and links. A crucial aspect of our model is that the two parts are mutually commuting: . Thus, the hopping Hamiltonian only has matrix elements between degenerate states of the charging Hamiltonian , and splits the degeneracy for the ground state.

As we want to build an insulator, we need the ground state of to be separated from the excited states by a finite energy gap. Furthermore, because we want fractionally charged excitations, must be topologically ordered(20); (21) (in gapped systems, fractional charge implies the existence of topological order). The presence of topological order means that the degeneracy of the ground state must depend on the topology of the system.(20); (21); (22) More specifically, we need the degeneracy of the ground state to be independent of the system size, and to be different for a system with open and periodic boundary conditions.

The first condition – existence of an energy gap – is guaranteed in our model by having the spectra of the charging Hamiltonian and the hopping Hamiltonian discrete. Note that this is not a common feature to hopping Hamiltonians. The continuous spectrum of the Josephson Hamiltonian is a representative example to the contrary. To make the spectrum discrete, we need to choose a carefully tailored hopping operator. While a conventional hopping Hamiltonian allows a single particle to hop between two neighboring sites, the hopping term we introduce allows only for a simultaneous correlated hopping of several particles around a single plaquette.

The second condition – a ground state degeneracy that depends on the topology of the system – is a consequence of the way that the hopping Hamiltonian splits the degeneracy of the ground state of the charging Hamiltonian. For example, consider the case of the 2D system defined on a torus. Each of the terms in describes hopping around one of the plaquettes of the lattice and only one out of ground states of is also a ground state of . Thus, the ground state degeneracy of the Hamiltonian on a torus is . By Euler’s theorem, this number is exactly . A similar calculation in a 2D open geometry yields a ground state degeneracy of . In the 3D case, the analysis is similar. One finds that the ground state degeneracy in a 3D open geometry is again , while on a 3D torus it is .

This counting agrees with the generalized “toric code” model with gauge group .(14) The quasiparticle excitations of the boson model are also similar to the toric code: there are two types of quasiparticle excitations – “charge” particles and “flux” particles – which are individually bosons but have fractional mutual statistics. Also, like the toric code model, the boson model does not have gapless edge modes. The main difference from the toric code model is that the “charge” quasiparticles carry a fractional electric charge, .

After constructing the bosonic models, we next introduce single electron degrees of freedom that live on the lattice sites. The electrons couple to the bosons through the charging energy, and the electron-boson coupling is characterized by a second integer parameter . We design this coupling so that it energetically binds an electron to a composite of fractionally charged bosonic quasiparticles, each carrying charge . The resulting composite particle then has a fractional charge of , and follows fermionic statistics. We denote the Hamiltonian of this modified lattice model by .

In order for the composite particle to be a stable degree of freedom, it must be able to hop between lattice sites “in one piece”, i.e. without affecting the other types of excitations. In the final step of the construction, we find a hopping term that does just that. We then add to the Hamiltonian , choosing the hopping amplitudes so that the composite particles have a band structure of a topological insulator. The energy gap between the bands is a parameter of , and we assume it to be much smaller than the energy gap of the bosonic excitations.

This construction results in a system of non-interacting fermions of spin- and charge in a topological insulator band structure in either two or three dimensions. The smallest charged excitation in the system carries a charge when is even, and a charge of when is odd. In the former case, this is a bosonic excitation. In the latter, it is a composite of fermionic and bosonic excitations.

ii.2 Properties of the models

The two-dimensional case

In two dimensions our models realize quantized spin Hall states, with a pair of gapless edge modes and a spin-Hall conductivity of . The topological order characterizing the states originates from the bosonic models underlying them. The ground state degeneracy on a torus is . In the bulk there are three types of excitations: the bosonic charge excitation with electric charge , the bosonic flux excitation which is neutral, and the fermion excitation with charge . We find the flux excitation to have a non-trivial mutual statistics with the other two types of excitations. When a bosonic charge excitation of charge winds around a flux excitation, it accumulates a phase of . Consequently, when a fermion, which is a composite of an electron and bosonic charge excitations, winds around a flux particle, it accumulates a phase of .

In certain limits the only active degrees of freedom are those of the fermions at the edge, where a gapless mode exists. The system can then be described as a topological insulator built out of non-interacting fermions of fractional charge . In particular, this description holds when the system is driven at low frequencies and long wave lengths by a weak electromagnetic field or when thermodynamical properties are probed at low temperatures. Under these conditions, the system would show a two-terminal conductance of , the shot noise associated with tunneling between edges would correspond to a charge of , and the heat capacity would be linear in temperature and proportional to the system’s circumference, as expected from a 2D topological insulator of non-interacting charge fermions.

When deviating from these conditions, the bosonic degrees of freedom can become active. Examples include the application of a magnetic flux of the order of a flux quantum, , per plaquette, the application of bias charges of order of to particular sites and the application of an electromagnetic field at frequencies that correspond to the gap to bosonic excitations.

The three dimensional case

In three dimensions our models are strong topological insulators built out of charge fermions, with a gapless Dirac cone on each surface. When time reversal symmetry is broken on the surface, the models exhibit a surface Hall effect with a fractional Hall conductivity of . As in the two dimensional case, the topological order in the 3D model originates from the topological order of the underlying bosonic system. The charged excitations carry electric charges of and and are identical to those in the 2D case, but the flux excitation becomes a flux loop rather than the point particle it is in 2D. The ground state degeneracy on a 3D torus is . Again, in certain limits the bosonic degrees of freedom may be neglected and the only active degrees of freedom are the fermionic ones. The conditions for these limits to hold are similar to those of the two dimensional case.

The bosonic degrees of freedom are active in several cases, one of which is of particular interest. In a 3D topological insulator of non-interacting electrons, a magnetic monopole in the bulk of the insulator binds a half integer electric charge. (23); (24); (25) Hence, a monopole/anti-monopole pair – which may be created by a finite-length solenoid carrying a flux quantum and positioned within the bulk – creates an electric dipole with a half-integer electric charge at its ends. In our model we find that such a solenoid leads to a dipole with a charge which is a half-integer multiple of . Unlike the non-interacting case, however, the energy involved in creating the dipole is proportional to its length – indicating that the two ends of the dipole cannot be effectively separated from one another. The two ends of the dipole can be separated only when the flux carried by the solenoid is flux quanta. Furthermore, because we could presumably trap any number of additional charge quasiparticles near the ends of the solenoid by adding an appropriate local potential, the only quantity which is independent of microscopic details is the monopole charge modulo . Calculating this quantity, we find that the charge at the end of the solenoid is a half-integer multiple of for the models where is odd, and an integer multiple of for the models where is even.

ii.3 The stability of the edge or surface modes

In conventional topological insulators, the edge or surface modes are protected as long as time reversal symmetry and charge conservation are not broken. (4) If either of these symmetries is broken, e.g. by a Zeeman magnetic field that couples to the electron spin or by a proximity-coupling to a superconductor that allows for Cooper pairs to tunnel into and out of the edge or surface modes, these modes may be gapped. The breaking of time reversal symmetry may be spontaneous rather than explicit, induced for example by the Fock term of electron-electron interaction. The stability of the edge or surface modes to perturbations that do not break these symmetries is the distinguishing feature of topological insulators in 2D and strong topological insulators in 3D.

An important question is whether the phases we study here have protected edge or surface modes similar to conventional topological insulators. We find that some of the models do indeed have edge or surface modes protected by time reversal symmetry and charge conservation, while some do not. (Independent of this difference, all the models are topologically ordered, as demonstrated by their topological ground state degeneracy).

The two dimensional case

In the 2D case, we find that our models conform to the general rule derived in Ref. (10): that is, the edge modes are protected if and only if the ratio is odd, where is the spin-Hall conductivity in units of and is the elementary charge in units of . In our models, this criterion is equivalent to the condition that the ratio is odd.

We establish the stability of the edge modes for the models with odd by a general flux insertion argument similar to the used in Ref. (10), and establish the instability in the case of even by explicitly constructing the perturbations whose combination gaps the edge. This combination is rather interesting. As defined, the models have two fermionic edge modes of opposite chiralities – the bosonic excitations are gapped at the edge. In order to gap the fermionic edge modes, we introduce one perturbation whose role is to close the gap of the bosonic excitations at the edge, and then two additional perturbations that couple the bosonic and fermionic modes, gapping them both.

For the closure of the bosonic gap at the edge we apply a perturbation aimed at turning the edge of the bosonic system from an insulator into a superfluid. The natural way of doing that is by introducing a hopping Hamiltonian that allows fractionally charged bosonic excitations at the edge to hop from one site to another. When the hopping term is strong enough it can overcome the charging term described by , thereby closing the gap at the edge. As for the perturbations that couple the bosons and the fermions at the edge, the first such perturbation breaks a spinless boson of charge into two electrons of opposite spin directions on the same lattice site. The second of these perturbations flips the direction of an integer number of electrons’ spins, while simultaneously operating on the flux degrees of freedom on the edge. Both of these perturbations make use of the bosonic degrees of freedom and therefore do not have analogues in non-interacting electron systems.

The three dimensional case

Just as in the 2D case, we find that the 3D models with odd have protected surface modes. We establish this result using a 3D generalization of the flux insertion argument of Ref. (10). We note that this argument is of interest beyond the particular models discussed here, and can be applied to more general fractionalized and conventional insulators. Unlike the 2D case, we are not able to determine the stability of the surface modes for the models with even . Addressing this question requires either the construction of specific perturbations that gap out the surface, or an argument proving that the surface modes are protected.

Iii Lattice models for 2D fractional topological insulators

iii.1 Step 1: 2D lattice boson models with fractional charge

In this section we describe a collection of exactly soluble lattice boson models with fractionally charged excitations—one for each integer . The models can be defined on any bipartite lattice in or higher dimensions. Here, for simplicity, we will focus on the case of the square lattice. Later, when we construct 3D models, we will consider the cubic lattice case.

Figure 1: In the lattice boson model, bosons live on both the sites and links of the square lattice. The Hamiltonian (1) is a sum of a term (2), which acts on four links and one site , and a term (4), which acts on the four sites and links adjacent to a plaquette . The term is a product of four link operators (5) which each act on the sites and the link .

The basic degrees of freedom in these models are charge spinless bosons which live on the sites and links of the square lattice. We denote the boson creation operators on the sites and links by and and the corresponding boson occupation numbers by and . The Hamiltonian can be written as a sum of two terms, one associated with sites , and the other associated with plaquettes of the square lattice (Fig. 1):

(1)

We will take but otherwise arbitrary. The term is a “cluster charge” term which measures the total charge on the site and the four neighboring links with appropriate weighting factors. It is defined as the sum

(2)

where

(3)

and and are the two sublattices of the square lattice. Since is positive, describes a short range repulsive interaction between the bosons. This interaction breaks the sublattice symmetry between the and sublattices, except in the case . The term can be thought of as a ring exchange term. It is defined as the product

(4)

where is a boson hopping term on the link :

(5)

The hopping term describes processes where bosons hop from the sites to the link and vice versa. It is designed so that it has two special properties. First, changes the number of bosons on the site at the center of the link by with the sign when and the sign when . This change is compensated by a corresponding increase or decrease in the number of bosons in the two neighboring sites so that the total number of bosons is conserved. Second, decreases the cluster charge by and increases the cluster charge by and doesn’t affect the charge on any other site:

(6)

An important consequence of this relation is that commutes with the product of around any set of closed loops, and in particular,

(7)

Equation (7) is at the root of why our system is an insulator: the operator has no effect on the cluster charges and hence does not provide for the long-distance transport of electric charge.

The final component of the model is our definition of the boson creation operators . For the site bosons , we use a rotor representation, letting with . The boson occupation number on the sites can therefore be any integer, . On the other hand, we take the link bosons to be a kind of generalized hard-core boson, restricting the boson occupation number to , and defining to be the matrix

(8)

when written in the normalized number basis on the link . We note that while these generalized hard-core bosons are unconventional, they can arise as an effective description of a conventional boson system in an appropriate limit. For example, if the number of bosons on the link is very large but is restricted to a set of contiguous values by appropriate energetics, then the above definition of becomes a good approximation to conventional bosons (up to an overall normalization factor). To summarize, the full Hilbert space for our model is spanned by the occupation number states with and .

Solving the boson model

We will now show that the Hamiltonian (1) is exactly soluble, and compute its exact energy spectrum. We are already part way there, having established that commute with each other (7). Next, we note that

(9)

from which it follows that

(10)

Combining these results with the obvious relation

(11)

we conclude that all commute, and therefore can be diagonalized simultaneously.

The simultaneous eigenstates of these operators can be labeled as , where

(12)

The corresponding energies are

(13)

It is clear from the definition (2) that has integer eigenvalues so is an integer. Similarly, using the fact that changes the occupation number by , we can show that

(14)

so must be a th root of unity. This relation guarantees what we promised in section II: the spectrum of is discrete.

Figure 2: We consider the lattice boson model (1) in two geometries: (a) a rectangular geometry with open boundary conditions, and (b) a periodic (torus) geometry. In the rectangular geometry (a), the operators at the corners act on two links , while those at the edge act on three links.

The only remaining question is to determine the degeneracy of each of the eigenspaces. This degeneracy depends on the geometry we consider. We first consider the case of a rectangular piece of square lattice with open boundary conditions (Fig. 2(a)). We show in Appendix A.1 that in this geometry, there is a unique eigenstate for each collection of satisfying the global constraint

(15)

In other words, are independent and complete quantum numbers, except for this single global constraint. A similar result holds for a (periodic) torus geometry (Fig. 2(b)). In this case, we find that there are states for each collection of satisfying (15) as well as the additional constraint (see Appendix A.2). This degeneracy is a consequence of the fact that the system is topologically ordered (see Section IV.2). Below we will focus exclusively on the open boundary condition geometry, unless otherwise indicated.

Putting this all together, we conclude that the ground state of (1) is the unique state with everywhere. There are two types of elementary excitations: “charge” excitations where for some site , and “flux” excitations where for some plaquette . The total number of charge excitations, in the bulk and edge together, must sum up to modulo . The charge excitations cost an energy of , while the flux loop excitations cost an energy of . In particular, as long as , then the ground state is gapped.

Fractional charge in the boson model

An important property of the boson model (1) is that the charge excitations carry fractional electric charge . We can derive this result by explicitly calculating the electric charge distribution in these states. Consider an eigenstate with some arbitrary configuration of charges , and with no fluxes. It is straightforward to show that the (un-normalized) microscopic wave function for this state in the occupation number basis is given by

(16)

(Again, to be precise, this wave function applies to a system with open boundary conditions). This wave function has a nice property: if we were to measure the occupation number on any link , then we would find each of the possible values, , with equal probability, . This is true on every link , independent of the configuration of charges . It follows that the expectation value of in the state is

(17)

Using this result, together with the constraint , we deduce

(18)

where is the coordination number of lattice ( for the square lattice).

We are now in a position to compute the fractional charge . Consider the case of an isolated charge excitation—that is, a state where at some site and at all nearby sites. Denote this state by . Similarly, consider an eigenstate where both at and at all nearby sites. Then the electric charge carried by the excitation is given by the difference in expectation values

(19)

where is an operator which measures the total number of bosons in some large area containing :

(20)

Using the above results (17-18) we derive

(21)

implying that . Furthermore, we can see from these expressions that this charge is perfectly localized to the site . This perfect localization is specific to the exactly soluble model: in a generic gapped system we expect excitations to have a finite size of order the correlation length.

Alternatively, we can derive the fractional charge using a simple identity: for any set of sites in the square lattice, we have the relation

(22)

Taking expectation values of both sides in the two states , and subtracting gives

(23)

To complete the calculation, we note that the second term on the right hand side vanishes in the limit that becomes large, since in that case, the sum only involves links that are far from , and the excess charge must vanish at large distances from . It follows that , as claimed.

iii.2 Step 2: 2D lattice electron models with fractional charge

We are now ready to construct a model with fractionally charged spin- fermionic quasiparticle excitations. We accomplish this task by modifying the lattice boson model (1) and coupling the bosons to additional (unpaired) electron degrees of freedom.

The Hilbert space for this modified model is very similar to the boson model, except that we introduce an additional spin- electron degree of freedom at each lattice site . In addition, we now think of the lattice bosons as being charge , spin-singlet pairs of electrons. This microscopic picture for the bosons is important conceptually because ultimately we want a model for a fractional topological insulator which is constructed out of electron degrees of freedom.

We will denote the creation and annihilation operator for the (unpaired) electrons by , and their occupation number by . We will use to denote the total number of the (unpaired) electrons on site :

(24)

Later, we will be interested in models with multiple orbitals on each lattice site . In that case, we will let include both the spin and orbital degrees of freedom.

In this notation, the Hamiltonian for the electron model is a sum of three terms:

(25)

where

(26)

and are defined as in Eq. (2-5). We will take , but will consider both positive and negative , and arbitrary integer .

Solving the electron model

The electron model (25) can be solved in the same way as the original boson model. Just as before,

(27)

Also, it is clear that

(28)

Hence, we can simultaneously diagonalize . Let denote the simultaneous eigenstates, where is an integer, is a th root of unity, and . The corresponding energies are:

(29)

By our analysis of the lattice boson model, we know that there is a unique state for each choice of satisfying the global constraint

(30)

(Again, we are assuming a geometry with open boundary conditions).

Putting this all together, and taking for simplicity, we conclude that the ground state is the unique state with . The system has three types of elementary excitations: “charge” excitations with for some site , “flux” excitations where on some plaquette , and spin- “fermion” excitations, where for some site and spin . The charge excitations cost an energy of , the flux excitations cost an energy of , and the fermion excitations cost energy . In particular, as long as and , the ground state is gapped.

Fractional charge in the electron model

Our next task is to show that the fermion excitations carry fractional charge. To do this, we first need to introduce some notation. Let denote the electron occupation basis state

(31)

where is the empty state. Also, let denote the boson occupation state defined in section III.1. A complete basis for the Hilbert space of the electron model is given by tensor product states .

In addition to these general basis states, we will also find it useful to think about the set of eigenstates made up of some arbitrary configuration of fermions with no charge or flux excitations. (To construct these states we must impose the global constraint ). The microscopic wave function for these states is given by

(32)

where are the boson eigenstates defined in (16).

In order to compute the fractional charge carried by the fermion, it suffices to consider a state with an isolated fermion excitation— that is, suppose at some site and vanishes at all nearby sites. Denote this state by . According to the definition (32), this state is a tensor product of an electron state and a boson state:

(33)

We can see that consists of a single spin- electron at site , while corresponds to “charge” excitations at site . Thus, the fermion excitation is a composite particle made of an electron and charge excitations. To compute the total charge of the fermion, we need to add together the contributions coming from these two pieces. By our analysis of the fractional charge in the bosonic model, we know that each charge excitation carries charge . On the other hand, the electron clearly has charge . Adding together these two contributions, we conclude that the fermion excitation has charge

(34)

Time reversal symmetry and the electron model

To construct candidate fractional topological insulators, it will be important to understand how the fermionic excitations in our model transform under time reversal. We use the usual convention for , where the electron creation operators transform according to:

(35)

In this convention, the bosons in (25) transform trivially, since they are spin singlet pairs of electrons:

(36)

Applying these transformation laws, we see that the fermion excitations transform like spin- electrons.

iii.3 Step 3: Building candidate 2D fractional topological insulators

In the last two sections, we have shown that the fermionic excitations of the electron model (25) carry spin- and transform under time reversal just like electrons. In fact, they are virtually indistinguishable from electrons except for the fact that they carry fractional charge (34). Given these properties, it is easy to build a candidate fractional topological insulator: we simply put the fractionally charged fermions into a non-interacting topological insulator band structure. We accomplish this by adding a new term to the electron Hamiltonian (25):

(37)

where

(38)

and are defined as before. The new term gives an amplitude for the fermion excitations to hop from site to site without affecting any of the other degrees of freedom, as we now show. We will assume that so that the bandwidth of the fermion excitations is much smaller than the gap to the bosonic excitations.

We can understand the effect of by computing the matrix elements of this operator between different eigenstates of ,

(39)

This computation is considerably simplified by the fact that

(40)

implying that the matrix elements are only nonzero when , and . In what follows, we specialize to the case, since these are the lowest energy states and this is all we will need to understand the low energy physics. These states contain only fermions and no other excitations. As in (32), we will denote a state with some arbitrary configuration of fermions using the abbreviated notation (Here labels the sites of the lattice, while labels the two possible spin states). To find the matrix elements , we write

(41)

and then analyze each of these two pieces in turn. Using the explicit form of (16), we find

(42)

where

(43)

Similarly, we have

(44)

where

(45)

and the sign depends on the ordering of the electron creation and annihilation operators in Eq. (31). Combining these two results with (41), we derive

(46)

where is defined as in (45). This relation establishes what we promised earlier: gives an amplitude for the fermion excitations to hop from site to site, but does not affect the other types of excitations. (We should not take these properties for granted: for example, if we had not included the operator in the definition of (38) then would have affected the bosonic charge excitations).

Denoting the creation operators for the fermion excitations by , we conclude that the matrix elements of within the low energy subspace are given by the free fermion Hamiltonian

(47)

To complete the construction, we choose the hopping amplitudes and the chemical potential so that is a non-interacting topological insulator. The low energy physics is then described by a topological insulator built out of fractionally charged fermions.

There are many possible choices for , but to be concrete we will focus our discussion on the following tight binding model on the square lattice(26). We consider a model with two orbitals, whose hopping matrix elements are related by time reversal symmetry: letting denote the fermion spin, and denote the orbital index, we have:

(48)

and there is no hopping matrix element connecting the two orbitals. For each spin, the hopping matrix elements for orbital are:

(49)

where are Pauli matrices acting on the spin indices. Here parametrizes the spin-independent hopping terms, and is a spin-orbit coupling. The constant sets the band gap at the momenta ; in the regime , the model is in a topologically insulating phase.

In addition to being a topological insulator, the model (49) conserves the total component of the spin. The ground state consists of a filled spin-up band with Chern number , and a filled spin-down band with Chern number . As a result, the model exhibits a spin-Hall conductivity of

(50)

iii.4 Effect of an electromagnetic field

In this section, we investigate the response of the fractionalized insulator (37) to an applied electromagnetic field. We show that for weak, slowly varying fields, the model behaves like a non-interacting system of fractionally charged fermions. On the other hand, for stronger fields, we find that other, non-fermionic, degrees of freedom contribute to the response.

The first step is to understand how to incorporate a vector potential into the Hamiltonian. As the model (37) contains charged bosons that live on links of the lattice in addition to those that live on the sites, we define a lattice vector potential for each of the two halves of each link (see Fig. 3). The hopping operator is then(27)

(51)

We can simplify this expression with the help of the unitary transformation

A little algebra shows that

(52)

where is the total vector potential on the link . In retrospect, this expression is to be expected as hops a charge from to , and as such, should be multiplied by a phase factor in the presence of an electromagnetic vector potential. Substituting this expression into (37), we find that the Hamiltonian can be written as

(53)