(3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions

-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions

Mauro Cirio Centre for Engineered Quantum Systems, Department of Physics and Astronomy, Macquarie University, North Ryde, NSW 2109, Australia    Giandomenico Palumbo School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom    Jiannis K. Pachos School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom
July 11, 2019

Currently, there is much interest in discovering analytically tractable -dimensional models that describe interacting fermions with emerging topological properties. Towards that end we present a three-dimensional tight-binding model of spinless interacting fermions that reproduces, in the low energy limit, a -dimensional Abelian topological quantum field theory called BF model. By employing a mechanism equivalent to the Haldane’s Chern insulator, we can turn the non-interacting model into a three-dimensional chiral topological insulator. We then isolate energetically one of the two Fermi points of the lattice model. In the presence of suitable fermionic interactions, the system, in the continuum limit, is equivalent to a generalised -dimensional Thirring model. The low energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of -dimensional BF theory at the boundary of the lattice and it provides a way to detect the topological order of the model through fermionic density measurements.

11.15.Yc, 71.10.Fd

I Introduction

The interest in strongly interacting fermionic systems has recently found new applications related to topological phases of matter. In the non-interacting case a complete classification Schnyder et al. (2008); Kitaev () of standard topological insulators Hasan and Kane (2010) of free fermions exists. Unfortunatelly, it is not possible to straightforwardly extend these results to the interacting case. For example, it is not possible to generalise the band theory approach to topological invariants, so more flexible approaches have to be invented Gurarie11 (). The introduction of interactions in a free fermion system can either connect different phases of matter Fidkowski11 () or give access to new ones Wang222 (). Examples of the latter are the two-dimensional topological Mott insulators Raghu11 (), where interactions can open an insulating gap and drive the system to topological phases not accessible in the non-interacting case.

Much progress in the study of interacting fermionic systems has already been made in 1+1 and 2+1 dimensions Fidkowski22 (); Lu111 (). In three spatial dimensions the situation is somehow less clear, though some analysis has been already carried out Wang222 (); Ran11 (). Complications arise already in the effective description, where the Chern-Simons theory Wen (2004) only holds in even spatial dimensions with broken time-reversal symmetry. A natural generalization of Chern-Simons theory is the topological BF theory, which is well defined in any dimensions Blau (). In two spatial dimensions BF theories can be interpreted as double Chern-Simons theories, allowing for the description of time-reversal symmetric topological insulators Bernevig (). BF theories have also been proposed as effective theories for describing topological insulators in any dimension Cho and Moore (2011); Chan et al. (2013); Palumbo2 (); Sodano (); Maciejko (). Nevertheless, very few interacting fermionic models that give rise to BF theory are available.

Here we make another step into the exploration of interactions-driven phases of matter. Our starting point is a cubic lattice of spinless fermions. For particular values of the couplings and in the absence of interactions the system becomes a chiral topological insulator Hosur et al. (2010). Our approach is similar in spirit to Haldane’s Chern insulator Haldane (1988), which gives us the ability to arbitrarily tune the asymmetry in the energy spectrum of the model. This allows us to enter a regime where the dynamics, associated with one of the two Dirac fermions present in the model, is adiabatically eliminated Palumbo and Pachos (2013); Palumbo33 (). Subsequently, we introduce interactions between the tight-binding fermions to obtain a generalization of the -dimensional massive Thirring model Thirring (1958) with a tensorial current. By applying a series of transformations Fradkin and Schaposnik (1994) we show that our system simulates a -dimensional topological massive gauge theory Cremmer and Scherk (1974a); Allen et al. (1991). The short distance behaviour of this theory is dominated by a Maxwell term. The large distance behaviour is characterised by an Abelian BF term which is topological in nature and it gives mass to the gauge field. The connection of the fermionic tight-binding model to the BF theory allows us to directly obtain that the boundary of the lattice is described by the -dimensional BF theory. Finally, we identify analytical expressions for topological invariants associated with the model and relate them to physical local fermionic observables. This method allows us to probe the topological properties of our three-dimensional system and provides a possible platform for simulating -dimensional gauge theories in the laboratory with cold atoms Zohar et al. (2013) in optical lattices Banerjee et al. (2013); Lewenstein et al. (2012); AlbaPachos ().
This article is organized as follows. In Section II a free fermion tight binding model is introduced. We focus on the kinematic sector by analysing the (gapless) energy spectrum, the symmetry properties, and the low energy limit of the model. We also consider the effect of additional mass terms which open a gap in the spectrum and allow us to show the existence of a chiral topological insulating phase. In Section III we leave the free fermion description by introducing 4-bodies interactions in the tight binding model. We then show that in the low energy limit the model is described by bosonic degrees of freedom and we find the corresponding effective theory through a duality operation. Interestingly, the effective theory contains a purely topological term. By proposing opportune bosonization rules we give a map between observables for the effective and microscopic theory. We then explore two features of the theory in its purely topological regime. We find that the boundary of the model is described by a topological theory. Finally, we describe microscopic fermionic observables which can be used to test the topological features of the model.

Ii Free Fermion Model

Let us begin with an overview of the model. We consider spinless fermions, positioned on the vertices of a three-dimensional cubic lattice , as shown in Fig. 1. The tight-binding Hamiltonian is given by


where and and are the creation and annihilation fermion operators at position of the lattice. We define planar unit cells populated by four fermion flavours , as shown in Fig. 1. Let us analyse each term of the Hamiltonian. The first term, which we call kinematic, has coupling and corresponds to nearest-neighbour hopping. The phases are such to create a net flux through each plaquette. The term proportional to describes a staggering between sites along the -direction indicated by . The last term corresponds to tunnelling between the next-next-nearest neighbouring sites, , with coupling . The phase factors , and are defined in Fig. 1. Let us now study this model more explicitely.

Figure 1: The tight-binding model, where spinless fermions (yellow) reside on the vertices of a cubic lattice. The plaquette unit cell has four fermions labelled . The fermions tunnel along the lattice via Hamiltonian (1). Tunnelling takes place along the edges of the cubic lattice (black) with coupling and a phase that is determined by the black arrows, i.e. for positive and for negative direction. A purely imaginary staggering term in the -direction (orange) has coupling and a phase with the same phase convention as . Tunnelling along the diagonals of the cube (green) have coupling and phase with , where the negative (positive) sign is chosen for full (dashed) lines.

The lattice of the unit cells (in blue in Fig. 1) is given by: , with and , , written in units of a fixed reference length. The Hamiltonian in Eq. (1) can be written as


where is a kinematic Hamiltonian (defined through the black links in Fig. 1) which has gapless spectrum. In order to open a gap in the model we introduce which is defined along the red and green links in Fig. 1. Let us now define the two terms of the Hamiltonian one by one.

ii.1 The kinematic model

As can be seen by inspecting Eq. (2) and Fig. 1, the kinematic Hamiltonian of the model can be written as


where is intended and where is an energy scale. This Hamiltonian is known Susskind (1977) to give rise in the continuum limit to two massless Dirac fermions. Let us now calculate the spectrum explicitely.
The reciprocal lattice is defined as where the vectors satisfy and are explicitely defined to be , , . The Brillouin zone (BZ) is defined as the elementary cell in the reciprocal lattice with . A generic vector in the Brillouin zone can be written as . The periodic invariance of the phase space allows us to parametrize the Brillouin zone in a different and somehow more convenient way. We can in fact define it as with , , where the volume of the Brillouin zone is . We now have all the ingredients to define the Fourier transform and analogously for . By introducing the Fourier transformed operators in Eq. (3) we find




and with the kernel given by


where we have defined and . From the explicit expression of and from Fig. 1 we can easily see that the set of vertices and only interacts with the set and . This condition defines a chiral symmetry. In fact, such a symmetry describes the existence of a bipartition of the lattice “broken” by all couplings (see Appendix A for more details). The existence of chiral symmetry allows to cast the Hamiltonian in an off-block diagonal form. In our case this is easily seen: after the definition of a new basis


the Hamiltonian takes the form




This Hamiltonian has eigenvalues (with degeneracy 2) given by


The spectrum has then two double degenerate bands and it becomes gapless at two Fermi points where the two bands touch each other. The two independent Fermi points are given by


In order to study the behaviour around the Fermi points we now define the following matrices


which satisfy the algebra


We now introduce coordinates around the Fermi points for small , and , so that the Hamiltonian around the Fermi points looks like


where . The Hamiltonians in Eq. (14) represent two massless Dirac fermions.

ii.1.1 Symmetries

The symmetries of the kinematic model can be studied by analyzing the Hamiltonian kernel (9). In particular we are interested in checking the behaviour of the model under time-reversal, particle-hole and chiral symmetry. For an introduction to the defintions of these symmetries we refer to Appendix A. In the table below we express the conditions on the Hamiltonian kernel under which these symemtries are satisfied.

Symmetry Condition

Inspection of the Hamiltonian kernel given in (9) shows us that . This condition means that the system breaks time-reversal symmetry and preserves particle-hole symmetry. We also have an explicit chiral symmetry since the Hamiltonian anticommutes with the matrix defined as


which is hermitian and unitary, as expected from the block structure of Hamiltonian (9).

ii.2 Gapped model

Figure 2: Energy bands for the full model described in Eq. (1) as a function of the momentum variable for fixed , in arbitrary units. The parameters and are tuned to open a gap. The model is in fact a description of a chiral topological insulator. The dashed line represents the Fermi energy and highlights the insulating properties of the material.

The kinematic model introduced in the previous section is gapless.We now introduce a gap term. In this way the low energy physics of the model is described by a massive Dirac fermion. Such a mass term has to anticommute with all the matrices (Eq. (12)), square to the identity and we also require it to satisfy chiral symmetry. As can be checked, the mass term has to be proportional to . In the chosen representation, we have


The implementation of such a mass term requires the introduction of additional couplings between the sites and and between and (as can be seen by inspecting the explicit form of in the basis given by Eq. (7)). We introduce a staggering of the and couplings along the axis and a next-next nearest neighbor (NNN) interactions as shown in Fig. 1. The staggering NNN interactions give an equal (opposite) mass term to the two Dirac fermions defined in Eq. (17). Explicitely we define


where we have introduced different energy scales and for the staggering and NNN term respectively. We also introduced a phase associated with the NNN couplings. In momentum space this Hamiltonian becomes


In the basis of Eq. (7) the kernel in momentum space of the interaction Hamiltonian reads


This term is proportional to the matrix so it can be interpreted as a fermion mass as discussed above.
Now, we first notice that, when and the particle-hole symmetry is broken: . Incidentally, it is important to notice that by adding these interactions we did not restore time-reversal symmetry (already broken in the kinematic model) in the full model.
The results of this section imply that the full Hamiltonian breaks time-reversal and particle-hole symmetry while it is symmetric under chiral symmetry. We also note that the joint presence of staggering and NNN interactions allows to arbitrarly tune the fermion masses at the two Fermi points. This is easily seen by evaluating Eq. (19) at the two Fermi points to get two independent masses. More precisely, let us define


which, for the choice becomes


With these definitions we get the expression for the full Hamiltonian around the two Fermi points (to be compared with Eq. (14))


where and . Notice that when or such an arbitrary tuning would not be possible and we would get .
We end up this section with the book-keeping explicit expression for the total Hamiltonian of the model of Eq. (1). From Eq. (9) and (19) and with the definitions (12), (16) and the ones below Eq. (6) the kernel of the total Hamiltonian reads


The spectrum of the total Hamiltonian has two double degenerate bands


where .

ii.3 Chiral Topological Insulator

Symmetry protected phases of matter for models described by a free fermion model are completely classified Schnyder et al. (2008). This classification characterizes phases of matter within 10 different symmetry classes determined by the symmetry properties under time-reversal, particle-hole, and chiral symmetry. More specifically, one starts by continuously deforming the Hamiltonian that describes a free fermion model to a “reference” Hamiltonian RyuArxiv (). This Hamiltonian has all occupied (empty) bands “flatten” with energy (-1) in the whole Brillouin zone. This can be done by defining the operator as




where are the eigenvalues of the occupied bands for the total Hamiltonian and is the number of occupied (empty) bands. The operator is such that , and . This operator has eigenvalues and corresponding to occupied and empty bands. Each of the 10 symmetry classes mentioned above determines a manifold such that . Within each symmetry class (and hence for each manifold ), we want to classify the phases of matter described by the reference Hamiltonians . Two Hamiltonians belong to the same phase if they can be continuously deformed one into the other without encountering a critical point. As shown in Schnyder et al. (2008) one can classify such phases through the th homotopy group of the manifold where is the spatial dimension of the model. For example, for the symmetry class A (all symmetries broken), we have that is isomorphic to the Grassmannian: . In fact, the collection of all energy eigenvectors describes an element of modulo the “gauge” symmetry relabeling the eigenvectors corresponding to occupied and empty bands. Now, for two spatial dimensions we have an infinite number of different phases as implied by (specifying, for example, the number of edge states for the quantum Hall effect, which in fact, being a Chern insulator, belongs to the symmetry class A). In three spatial dimensions we have (where represent the group trivial element) so that only the trivial phase is allowed. Such models can become non-trivial when more symmetries are considered. Specifically, we are interested in the symmetry class AIII where only chiral symmetry is preserved. in this case (positive and energy eigenstates come in pairs, see Appendix A), and one can write in the following block form Schnyder et al. (2008)

Figure 3: Phase diagram of Hamiltonian (1) as a function of the couplings and parameterised by the phase . The gapped regions with non-trivial winding number (yellow) are separated by phase transition (blue lines) from the topologically trivial regions with (white). The winding number, , is correlated with the sign of , in the same way as in Haldane’s model. The critical line for (dashed blue) and for generic value of (solid blue) are depicted.

with . Our model is then described by the function where . Contrary to the class A example, we now find that allowing for non-trivial phases in three spatial dimensions. In fact, we can define Schnyder et al. (2008) a winding number (associated with the map ) labelling all the possible phases as


where .

ii.3.1 Phase Diagram

We now want to see under which conditions on the parameters of the full Hamiltonian of our model (Eq. (1)) we can get non-zero winding number (Eq. (28)).
Let us first resume what we learned so far. First of all, from the results of section II.2 we know that non-trivial topological properties are forbidden for or . In fact, in such a regime, the particle-hole symmetry is not broken leading to the impossibility to define a winding number as shown in Schnyder et al. (2008). Second, we know that the conditions and imply (see Eq. (20)) that either or meaning that the system is critical. Then, in these cases we expect the winding number in Eq. (28) to be not well defined.
Let us now draw the phase diagram of the model in the parameter space (see Fig. 3). A gapless system is described by imposing the equations and (see Eq. (20)). Pictorially, these equations are two straight lines in the plane, parametrized by the phase . They divide the parameter space in four disconnected regions. We studied the behaviour of the winding number in these four regions and we found that two of them are in fact non-trivial with (see Fig. 3). When the parameter tends to the two non-trivial phases disappear as the two critical lines merge together. This result is consistent to the fact that corresponds to a system where particle-hole symmetry is not broken (see the analysis following Eq. (19)) which is a sufficient condition for the absence of topological order Schnyder et al. (2008). The non-triviality of the winding number Eq. (28) (for a certain parameters regime) shows that the system is a chiral topological insulator.
To sumarize, the introduction of NNN neighbour interactions () breaks particle-hole symmetry (provided that ) and gives an opposite contribution to the masses in Eq. (20). On the contrary, the staggered interactions () give an equal contribution to the fermion masses. As a consequence, the simultaneous presence of both interactions allow to arbitrarily tune the masses .
Intuitively, this model presents several formal analogies with the Haldane model Haldane (1988) where spinless electrons hop on the verteces of a honeycomb lattice. Such a model has a kinematic term which preserves time-reversal and inversion symmetries (and breaks particle-hole symmetry) and that gives rise to two gapless Fermi points. In addition, next-nearest neighbour interactions (mimicking a nested magnetic field) and a staggered chemical potential break, rispectively, time-reversal and inversion symmetry. The breaking of each symmetry allows for a non-zero energy gap to appear. More precisely, the fermions at the two Fermi points acquire the same (opposite) mass due to the breaking of time-reversal (inversion) symmetry. In this sense, our NNN neighbour and staggering terms mimic, respectively, the staggered magnetic field and the chemical potential of the Haldane model. In the light of the classification given in Schnyder et al. (2008), the key-feature to build a non-trivial topological phase on top of our (Haldane) kinematic theory is to break the particle-hole (time-reversal) symmetry. Despite these similarities, the Haldane model breaks all symmetries (it describes the physics of the quantum Hall effect without magnetic field) while we have to pay extra attention to preserve chiral symmetry in order to protect the topological phase in dimensions.

Iii Interacting Fermions Model

We now turn to the case of interacting fermions. The starting point is the effective theory described in Eq. (22) with . This model has enough flexibility to allow us to arbitrarily tune the masses around the two Fermi points shown in Fig. 4. Following the approach in Palumbo and Pachos (2013) we can define a hierarchy in the energy scales, given by , and adiabatically eliminate the physics around the second Fermi point, . We now introduce four-body fermionic interactions with coupling that is small compared to the energy scale of , i.e. , and comparable to . These interactions are particularly designed so that they give rise to self-interacting current-current terms in the single Dirac fermion description corresponding to . The resulting effective physics is encoded in the Hamiltonian


where and . There are two types of currents given by and , for with the gamma matrices defined as , and .

Figure 4: Adiabatic elimination of a Fermi point. The energy spectrum (in arbitrary units) of the model described by the Hamiltonian in Eq. (1) (as a function of the momentum variable for fixed ) allows to arbitrary tune the masses around the two Fermi points (see Eq. (20)). In particular, by opportunely choosing the parameters and , it is possible to work in a regime where . If we suppose not to have any perturbation with energy scale bigger than then the low energy physics of the system is completely described around the Fermi point in the Brillouin zone.

A dimensional analysis shows that the four-components Dirac field has dimensions compatible with the units of the Hamiltonian density above. This fixes the dimensions of the current-current interaction terms (to ) which, in fact, implies that . The Hamiltonian in Eq. (29) is the tensorial generalization of the Thirring model Thirring (1958) in dimensions. This generalization of the Thirring model is not renormalizable (at least by means of perturbative methods). It is analogous in spirit to the Nambu-Jona-Lasinio model Nambu1 () and the Fermi effective model Fermi1 () for weak interactions which involves the non renormalizable point-like interaction between two currents. The energy scale associated with the Thirring coupling is given by (which is the only energy scale we can define from , and ). This gives a dimensional analysis justification to the adiabaticity condition given above which allows us to restrict the physics around one Fermi point. From now on we will use units where .

iii.1 Microscopic prescription

We now want to find the microscopic description for the Hamiltonian in Eq. (29). To achieve this, we proceed backwards and substitute in and the expressions for the spinor and the gamma matrices as given by and . After some tedious calcuations one gets




The terms involving only two fermions can be omitted since they give a contribution to the Hamiltonian kernel which is proportional to the identity and can be seen as a constant chemical potential on every site of the lattice. The other terms either interactions between two sites populations or between four sites.
This shows the explicit form of the microscopic interaction needed to simulate the Tensorial Thirring model at low energies given by Eq. (29). We note that some of these interactions are attractive while other are repulsive.

iii.2 Bosonization

Throughout the rest of the section we assume that in Eq. (29). The Thirring model describes relativistic fermions with self-interactions. In order to get a more accessible theory it is possible to linearize the interaction by introducing new degrees of freedom. Following this approach, we now show how to describe the low energy physics of the Tensorial Thirring model with a pure bosonic theory.
As it can be seen from Eq. (29), the effective theory of our model is described, in Euclidean space, by the action


where the Dirac action, , and the action for the currents, , are given in Euclidean space. Clearly, involves products of four spinors. To analytically treat this model we linearise the action in terms of the currents by introducing the Hubbard-Stratonovich transformation Botta Cantcheff and Helayel-Neto (2003). Indeed, we employ the bosonic degrees of freedom (in terms of a vector field and an antisymmetric tensor’ field ) to write


Following Botta Cantcheff and Helayel-Neto (2003) we can integrate out the Dirac fermions to find an effective bosonic theory


where the effective action is defined as


where and . Up to terms of order ( in momentum space) Botta Cantcheff and Helayel-Neto (2003) this effective action can be written as


where is the Levi-Civita symbol. The correction terms are insignificant in the large-wavelength/low-energy regime we are interested in. By neglecting the irrelevant constants, the partition function for the final theory can be written as


We note that the fields and have dimension . We could be tempted to consider the field as a sort of electromagnetic field and as a sort of curvature field. Unfortunately, such an interpretation is not obvious at this stage. In fact, the theory is not invariant under the gauge-like transformation Blau1 ()


where and are a scalar and a vector field respectively. In fact, the kinetic terms and explicitly break invariance under these transformations (as, for example, the vector potential appears explicitely). Hence, the partition function describes a massive spin-1 theory that does not allow easy interpretations. We would like to recast this theory in a more suitable form given in terms of a “vector potential” and a “curvature” field, which naturally leads to the next sections’ topic. This process is analogous to the (2+1)-dimensional one where a duality between a self-dual free massive field theory and a topologically massive theory Deser () has been demonstrated.
As a final note, it is important to stress that it is not possible to apply the bosonization procedure proposed here to free Dirac fermions, i.e. without the presence of the current-current interactions in Eq. (29). This means that we cannot tune the parameter to zero without encountering non-analytical points, which justifies the presence of the factor in Eq. (40). Physically, the naive replacement into the initial (Eq. (29)) and final (Eq. (37)) theories would lead to a mapping between free fermionic degrees of freedom and free bosonic ones, which is clearly forbidden by the statistics of the fields involved. The presented “transmutation” of degrees of freedom holds only for interacting theories (), as for example happens in superconductivity where the interaction between electrons in a metal leads to a physics described by bosonic degrees of freedom in terms of Cooper pairs Cooper1 ().

iii.3 Duality

In order to recast the theory defined in Eq. (37) in a more suitable form, one can employ a BFT quantization procedure Batalin and Fradkin (1987); Batalin and Tyutin (1991) to show the equivalence of the massive spin-1 theory


to one involving an “electromagnetic” field and a so called Kalb-Ramond field Kalb and Ramond (1974). The two theories can in fact be embedded in the same enlarged theory from which they descend as different choices of gauge fixing Harikumar and Sivakumar (2000); Kim et al. (2003). The resulting theory, in the Lorentzian signature, is described by the Cremmer-Scherk Lagrangian Cremmer and Scherk (1974a)


where and . The field is an effective electromagnetic field (with dimension ) while the field (with dimension ) is the so called Kalb-Ramond field. As above, we can define the symmetry transformation


Contrary to the theory described in Eq. (37), this one is explicitly invariant under this “gauge” transformation.
Let us now take some time to analyze the terms appearing in the action in Eq. (40). The first two kinematic terms are geometric (the metric appears explicitly) while the last one has a topological nature and it is the standard BF term. This theory is a topological massive gauge theory in dimensions Cremmer and Scherk (1974a); Allen et al. (1991) and represents the natural abelian generalization of Chern Simons-Maxwell theory in dimensions as the Chern Simons theory cannot exist in dimensions. This way of generating mass for the electromagnetic field (through a topological interaction) is an alternative to the Higgs mechanism and can in fact be connected to superconducting phenomena Hansson (); Balachandran (). The theory is renormalizable Allen et al. (1991) and explicitely gauge invariant in the bulk.

iii.4 Bosonization Rules

The possibility to map the Tensorial Thirring model to a massive gauge theory does not come as a surprise. In fact, in the -dimensional case the massive Thirring model is equivalent to the sine-Gordon massive scalar theory Coleman (1975), while in dimensions is equivalent to the Maxwell-Chern-Simons theory Fradkin and Schaposnik (1994), where, the Maxwell field acquire a mass through a topological mechanism. Motivated by the analogies with the low-dimensional cases we propose the natural generalization of the bosonization rules to the three-dimensional case. These rules connect the degrees of freedom of the equivalent fermionic and bosonic theories (up to multiplicative factors) in the following way

Dimensions Theory Bosonization Rules
1+1 sine-Gordon
2+1 Maxwell-CS
3+1 Cremmer-Scherk

Let us take a little more time to emphasize the analogies with the lower dimensional cases and get some more intuitions on the bosonization procedure. We can note that the Thirring model is always equivalent to some massive theory. In the (1+1)-dimensional case the equivalent theory is a sine-Gordon massive scalar theory. The equivalence with the Thirring model has been shown by Coleman Coleman (1975) (see also Delpine1 () for extension of the proof to the finite temperature case). In (2+1) dimensions the Thirring model has been proven by Fradkin and Schaposnik to be equivalent to a Maxwell-Chern-Simons theory Fradkin and Schaposnik (1994). In this last case, the proof relies on a dualization procedure first showed by Deser and Jackiw Deser () and the equivalent theory corresponds to a massive gauge theory where the mass of the photon is given thanks to the interaction with a topological Chern-Simons term (which, as already mentioned, is an alternative to the Higgs procedure to give mass to a gauge theory). In 3+1 dimensions the Tensorial Thirring model we are studying is going to be equivalent to a so called Cremmer-Scherk model Cremmer and Scherk (1974a) where a Maxwell theory is coupled to a Kalb-Ramond field Kalb and Ramond (1974) thanks to a BF term. This is, in analogy with the (2+1)-dimensional case, a massive gauge theory where the mass comes from the topological interaction with the Kalb-Ramond field. Note that, in this case, we need two fields since we are considering two different types of current-current interactions in the fermionic model. These two fields are necessary to generate mass for the gauge theory in a topological fashion 111Without them, it would be impossible to generate such a mass with just one field. In the abelian case this is very easily seen. In fact, with the presence of one field the only possibilities for a gauge invariant 4-form are and . The first term has a topological nature while the second term is geometric. The topological term can be rewritten as . This allows to conclude that the addition of such a differential to the lagrangian would not change the equation of motion (since it adds a total derivative to the Lagrangian) and for this reason it cannot possibly add a mass to the gauge field.. We also note that similar bozonization rules in dimensions were proposed in Schaposnik1 ().

iii.5 Pure Topological Regime

From now on we work in a regime where the contribution of the topological BF term in (40) is dominant, i.e. we want to work with energy scales much smaller than . Intuitively, this suggests that the Maxwell and Kalb-Ramond field have small kinetic energy compared to their (topologically) acquired mass. In such a regime we are sufficiently close to the ground state and the important contributions to the effective theory come from the topological BF term


where is the spacetime manifold associated with our theory.

iii.5.1 Boundary Behaviour

We now consider the behaviour of our lattice model at its physical boundary. Several approaches are possible based, for example, on the Symanzik method Amoretti () or on gauge invariance analysis Momen (1997); Cho and Moore (2011). Focusing on the latter at the bosonic level a BF theory defined on a non-compact space, , is not manifestly gauge invariant due to contributions from the boundary, . Restoring gauge invariance generates a -dimensional BF theory on the boundary, while leaving the bulk theory unchanged Momen (1997); Cho and Moore (2011). Here, we show that the bosonization rules allow to infer exactly the same theory on the boundary of the -dimensional fermionic lattice model (for more detail we refer to Appendix B).
We start by introducing a minimal coupling between the tight-binding fermions and a pure gauge field parameterised by . This coupling extends (42), in the continuum limit, by


but it leaves the physics of the model unchanged. We can now employ the bosonization rule , together with Stokes’ theorem and an integration by parts to show that


where here (and throughout the rest of the paper for integrations on the boundary) the indices run through the coordinates that parameterise . The field can now be interpreted as a Lagrange multiplier enforcing the condition on . This implies that, locally, , which conveniently implies . This means that the possibility to add to our action is equivalent to the constraint on . We are now ready to find the effective action on the boundary. In fact, we can rewrite the right hand-side of Eq. (42) as


which, by restriction on the boundary, implies the following form for the theory on the boundary


This is indeed a -dimensional abelian BF theory. It is equivalent to a double Chern-Simons theory that describes time-reversal symmetric physics PalumboBFGraphene () on the boundary.

iii.5.2 Physical Observables

We now want to identify physical observables associated with the purely topological part, . Gauge invariant observables of the -dimensional BF theory are given by expectation values of Wilson surface operators Horowitz and Srednicki (1990); Oda and Yahikozawa (1990), which are a generalization of the -dimensional Wilson loop operators. These observables


are defined for any two-dimensional boundary of a three-dimensional volume , where is the Kalb-Ramond field Kalb and Ramond (1974). The corresponding fermionic observables are given by


where is a generic charge of the (string-like) excitations associated with the field . The correspondence is easily proven by an opportunely manipulation of the Noether charge (where ). The joint use of the the bosonization rule (where the constant has been introduced for dimensional reasons) and Stokes’ theorem immediately leads to . This proves that (where any proportionality constant implicit in definition of the bosonization rules is absorbed inside the charge ) or, in other words


For more details of this proof we refer to Appendix C.1. It is a well known fact that