Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional generalizations by Oshikawa and Hastings establish that a translation-invariant lattice model of spin-’s can not have a non-degenerate ground state preserving both spin and translation symmetries. Recently it was shown that LSM theorems can be interpreted in terms of bulk-boundary correspondence of certain weak symmetry-protected topological (SPT) phases. In this work we discuss LSM-type theorems for two-dimensional fermionic systems, which have no bosonic analogs. They follow from a general classification of weak SPT phases of fermions in three dimensions. We further derive constraints on possible gapped symmetry-enriched topological phases in such systems. In particular, we show that lattice translations must permute anyons, thus leading to “symmetry-enforced” non-Abelian dislocations, or “genons”. We also discuss surface states of other weak SPT phases of fermions.
Determining the emergent quantum phase in an interacting quantum many-body system is generally a very difficult question. The celebrated Lieb-Schultz-Mattis-Hastings-Oshikawa (LSMHO) theorem Lieb et al. (1961); Oshikawa (2000); Hastings (2004) points to an exact relation between the microscopic properties and low-energy physics for lattice spin systems: if there is an odd number of spin-’s per unit cell, there can not be a non-degenerate ground state while preserving both and translation symmetries. A symmetric gapped ground state then must be topologically ordered, e.g. a quantum spin liquid Savary and Balents (2017); Zhou et al. (2017). The LSMHO theorem has been quite valuable in the study of frustrated magnets. Recently, LSMHO theorem has been generalized significantly, to more complicated space groups Parameswaran et al. (2013); Watanabe et al. (2015); Po et al. (2017) and other internal symmetry groups (e.g. a Kramers doublet per unit cell in the presence of time-reversal symmetry, applicable to spin-orbit-coupled materials), as well as to itinerant fermions with charge conservation Watanabe et al. (2015). More recently, LSMHO theorems have also found extensions to magnetic translation symmetries, which lead to LSM-type constraints for symmetry-protected topological (SPT) phases Lu (2017); Yang et al. (2017). On the other hand, when the ground state is topologically ordered preserving all symmetries, the LSMHO theorem can be further refined to place stringent constraits on the symmetry-enriched topological order Zaletel and Vishwanath (2015); Cheng et al. (2016); Qi and Cheng (2018). For example, in the “traditional” setup with spin and translation symmetries, one can show that the “background charge” on a lattice must have spin-, i.e. a spinon. These results will be collectively refered to as LSM-type constraints.
Many LSM-type constraints can be unified under the theme of bulk-boundary correspondence Vishwanath and Senthil (2013); Chen et al. (2015) of crystalline SPT phases, as ellaborated in LABEL:ChengPRX2016 (see also LABEL:JianPRB2018,_HuangPRB2017,_ThorngrenPRX2018). Essentially, given a -dimensional lattice system with an internal symmetry group , if there is a projective representation of in each unit cell, the translation-invariant system can be viewed as the boundary of a stack of 1D SPT phases (going into a fictious -th direction), which forms a weak SPT phase protected by . The well-known constraints on the boundary state of such a weak SPT phase become the statement of a LSM-type theorem. This immediately suggests the following generalization: we can construct a bulk as a stack of 1D fermionic SPT (FSPT) phases Fidkowski and Kitaev (2011). End states of 1D FSPT phases are characterized by “fermionic projective representations”, namely symmetry transformations not commuting with the local fermion parity. Unlike other LSM-type theorems for itinerant electrons known in literature Watanabe et al. (2015), LSM-type constraints obtained this way have no “Mott”, or bosonic limits, and do not require charge conservation.
In this work we study fermionic LSM-type constraints, focusing on 2D lattice systems with an internal particle-hole symmetry and translations. After motivating the theorem from the consideration of bulk-boundary correspondence, we present a simple proof of the theorem and then analyze its implications for possible symmetric gapped phases.
Ii Classification of Weak SPT Phases
In this section we present the classification of weak FSPT phases in 3D, whose internal symmetry group is with being the fermion parity symmetry, and the full symmetry group is . The classification is very similar to the bosonic case Cheng et al. (2016), with three “weak invariants”:
There are “strong” SPT phases protected just by alone.
We may define “2D SPT per unit length”. The generators of the bulk states are obtained by stacking 2D SPT layers along the -th direction, where . They will be referred to as “type-I” weak SPT phases.
We define “1D SPT per unit area”. The generators of the bulk states are obtained by packing 1D SPT perpendicular to the -plane. They will be sometimes referred to as “type-II” weak SPT phases.
Lastly, there is also “0D SPT per unit volume”, generated by filling the bulk with 0D SPT states (“charges”).
Since we will consider the physics of the surface, only the type-I and type-II invariants are relevant. The most significant invariant is type-II invariant, i.e. 1D SPT per unit area. Let us review the classification of 1D fermionic SPT phases. It is well-known that 1D topological phases are classified by their end states Pollmann et al. (2012); Chen et al. (2011a, b); Fidkowski and Kitaev (2011); Turner et al. (2011); Schuch et al. (2011). For fermionic systems, we distinguish two cases: in a finite system, if there is a topological ground state degeneracy between states with even and odd parity, then even in the absence of any symmetries the system remains nontrivial, i.e. a class D topological superconductor. Otherwise, we can assume that the (symmetry-protected) degenerate ground states all have even fermion parity. We can think of the two cases as having odd/even number of Majorana zero modes on each end.
Let us consider the latter case, where there exists a well-defined Fock space on each end Bultinck et al. (2017); Kapustin et al. (2016); Turzillo and You (2017). FSPT phases in this case are classified by a pair where is a group homomorphism and . determines whether the local symmetry action on the boundary is fermionic or bosonic and determins the projective phases of the local symmetry action 111When the total symmetry group is a nontrivial extension of by , there is a further obstruction-vanishing condition on and . See e.g. LABEL:TurzilloFSPT1D for details. We will only consider the total symmetry group being .
For a concrete example, we set . Since , there is one nontrivial phase with . A simple physical realization can be found in a system of two identical Kitaev chains, labeled as and . The symmetry is generated by .
On an end there are two Majorana modes, and . Under they transform as
Locally this symmetry action can be implemented by a unitary . Obviously the only mass perturbation breaks the symmetry. We can also define a complex fermion mode , then acts as a particle-hole (or charge-conjugation) transformation: . We will call such a boundary fermionic mode a fermionic projective representation of .
We will also consider 2D SPT per unit length. For with being a unitary group, a complete classification of 2D fermionic SPT phases has been established Gu and Wen (2014); Cheng et al. (2015); Bhardwaj et al. (2017). In this case the classification data is a triple where is again a group homomorphism, is a -cocycle in and now is a -cochain valued in . Physically, determines whether a -defect is Majorana-type or not (i.e. whether it carries a Majorana zero mode). We will only consider trivial in this work. encodes the projective fusion rules of symmetry defects:
We believe the results can also be obtained from a general classification of fermionic SPT phases, such as spin cobordism Kapustin et al. (2015), with an internal symmetry group . As long as the classification is given by a generalized cohomology theory, a Kunneth-type decomposition exists and produces the classification discussed in this section (see LABEL:Xiong). We will come back to this point in Sec. V.
Iii Fermionic LSM Theorems
Motivated by the perspective that views LSM-type theorems as a result of bulk-boundary correspondence, we propose fermionic LSM-type theorems: in a -dimensional lattice, if the degrees of freedom in a unit cell transform as a fermionic projective representation of an internal symmetry group , a corresponding LSM theorem should hold. Namely, if the Hamiltonian preserves all symmetries, the ground state is either gapless, spontaneous symmetry breaking or topologically ordered.
iii.1 A fermionic LSM theorem with unitary symmetry
The main example that we will study is . In this case, the fermionic projective representation of is basically a single fermion mode with a particle-hole symmetry . Thus we are led to consider a lattice model of spinless fermions with exact particle-hole symmetry, and one particle-hole doublet per unit cell. For example, the Hamiltonian may contain purely imaginary hopping terms , as well as interactions such as . Notice that the symmetry fixes the average density of fermions to be , i.e. half filling. So if the fermion number is conserved, the LSMHO theorem already rules out a trivially gapped ground state. However, if the symmetry is broken, without additional symmetries it is certainly possible to have a trivial gapped phase even when the average density is fractional. For example, we can simply form a stack of Kitaev chains.
We now present a proof of the claim. Without loss of generality, we consider a square lattice of size , where fermions obey periodic or anti-periodic boundary conditions in both directions:
The particle-hole symmetry is generated by the following unitary
Notice that , where is the number of sites. In the following we define .
If both and is odd, is fermionic and . This is already an indication that there can not be a fully symmetric SRE state on the torus, since a SRE state should have a unique ground state on any closed manifold and should not know the parity of when .
We may also consider the case when there are even number of sites and is a bosonic operator. The translation symmetry then acts on the fermions as follows:
Generally we find that
Similarly we have
For even and odd , we obtain . Therefore, with PBC , there is again at least two-fold ground state degeneracy. This rules out a completely symmetric SRE ground state. The argument presented here is very similar to the one for translation-invariant Majorana models in LABEL:HsiehPRL2016.
It is also straightforward to show that any quadratic Hamiltonian preserving the symmetries must be gapless. In fact, if we write where and are Majorana operators, the symmetry acts as . Thus at quadratic level, and are decoupled. A translation-invariant Majorana model with one Majorana per site is always gapless since the single-particle dispersion has to be an odd function of the lattice momentum, thus gap closing near zero momentum.
One can easily construct various symmetry-breaking ground states. For example, on a square lattice we can simply form a charge density wave with ordering vector . Alternatively, we can keep the particle-hole symmetry by forming “bond” density waves, pairing with , and with . In the following we present an example of an interacting symmetric gapped phase.
iii.1.1 Coupled-wire construction of a symmetric gapped phase
We construct an example of a symmetric gapped phase on a square lattice, in a highly anisotropic and strongly-interacting limit. First turn on the following couplings along :
The single-particle spectrum is . We obtain a stack of gapless chains indexed by and each of them is a free fermion in . To describe the low-energy physics, we follow the standard bosonization approach Giamarchi (2003) and linearize the spectrum around Fermi points. Within this approximation, each chain has a right-moving mode at and left-moving at :
Under the particle-hole symmetry, the chiral fermion fields transform as
Under translations they transform as
We then bosonize .
To construct a symmetrically gapped phase, the chains must be coupled by interactions. For simplicity of the presentation, we follow an analogous construction in LABEL:MrossPRL2016, inserting plates of gauge theories between neighboring chains. The bulk of a theory can be described by an Abelian Chern-Simons theory with K matrix . Anyonic quasiparticles are generated from the charge and flux , as well as their bound states. Correspondingly, edge modes of the gauge theory are Luttinger liquid with the following Lagrangian:
The bulk-edge correspondence identifies with , and with .
Because of the translation invariance along direction, we can focus on the -th chain and the two sets of edge mode coming from adjacent plates: from the plate between and . The following interactions are turned on Mross et al. (2016):
The gapping terms explicitly preserve and symmetries. To preserve the symmetry, we demand that also acts as charge-conjugation symmetry in the gauge theory:
takes to , to in the topological order. Notice that the interactions also preserve charge, if carries a half electric charge. We assume that are large so the system becomes fully gapped.
Now we have a symmetric, gapped Hamiltonian. We can further check that there is no ground-state degeneracy (except the topological one) from the gapping Hamiltonian at , so there can not be any spontaneous symmetry breaking Wang and Levin (2013). In addition, we also need to make sure that there is no string operator connecting neighboring wires (which is still a local operator) that transforms under the symmetry. The bulk topological order has been analyzed in LABEL:MrossPRL2016 and is just the topological order. However, the symmetry actions on anyons are highly nontrivial, which we determine in the following.
In the gapped phase, the following fields acquire nonzero expectation values:
With our choice of the Hamiltonian, we have , and . Naively, under we find that becomes , and the same to . Because and are not local operators, this can be fixed by applying gauge transformations.
Generally, to be consistent with the topological order, we must have and transform consistently since they come from edge modes bounding the same bulk. To be precise, we assume that under ,
where the phases and satisfy
So that the upper and lower edges of the same plate can be “glued” by and without any symmetry breaking.
Demanding that and are invariant with these gauge transformations, we can easily find and .
Let us analyze how acts on anyons. First, because , the anyons can simply tunnel between different plates. Thus acts trivially on (besides moving it along ). The situation for anyons is quite different. Notice that
Therefore, in terms of the original anyons in each plate, is identified with , where represents the physical fermion. This means that in the 2D phase, the actual anyon has the following identification:
Microscopically, becomes under the translation. Compairing with Eq. (19), we find that acts on anyons as
It is easy to see that the translation does not permute anyons. These results agree with the analysis in LABEL:MrossPRL2016.
So far we have determined how anyon types are permuted under the symmetries. It is also important to understand symmetry fractionalization, encoded in the various additional phases appearing in the transformations of and . Thus we need to check the commutation relations between the symmetries. First, let us check the commutation relation between and . They apparently commute on . For , we will use a heuristic argument here: under , becomes . Since acquires an phase under , acquires an additional phase relative to under ( because we do not know how transforms under ). Therefore, when acting on the anyon, and differ by a phase. Intuitively we can understand the non-commutativity between and translations as having a background charge or 222As we will discuss below, these two values are “gauge-equivalent”., such that when moving around a unit cell one gets a Berry phase. This is well-defined because is invariant under .
A similar calculation shows that and do not commute when acting on anyons locally. We find that and differ by a phase on both and .
iii.2 Fermionic LSM theorems with time-reversal symmetry
We can replace the unitary particle-hole symmetry with an anti-unitary one, i.e. time-reversal symmetry . For one complex fermion per site, there are two possibilities:
. In terms of Majorana operators, they transform as
Or . This kind of transformation forbids all quadratic terms in the Hamiltonian.
. In terms of Majorana operators, they transform as
This is known as a “Majorana doublet”. The complex fermion transforms as . Thus hopping terms like are not allowed, and only real pairing terms can appear at quadratic level:
iii.3 Majorana LSM-type theorem
We can also consider a LSM-type theorem without any internal symmetries (except the fermion parity conservation), in a translation-invariant lattice with an odd number of Majorana modes per site, e.g. a triangular vortex lattice in a superconductor 333One subtlety is that in such a vortex lattice, the natural Majorana hopping model has magnetic translation symmetry.. LABEL:HsiehPRL2016 showed that such a lattice model defined on even by odd torus must have degenerate ground states, protected by the anti-commuting algebra of translation and total fermion parity.
iii.4 LSM-type theorems for other space symmetries
LABEL:PoPRL2017 found the most general LSM-type constraints for 2D magnets, starting from three concrete conditions, known as “Bieberbach” no-go, mirror no-go and rotation no-go.
We conjecture that a general “Bieberbach” no-go holds: if there is a nontrivial fermionic projective representation in the “fundamental domain”, a symmetric SRE state does not exist. Here a “fundamental domain” is a region which tiles the plane under the action of translation and glide symmetries.
However, the generalization of the rotation no-go is more subtle for fermions, as the argument in LABEL:PoPRL2017 is no longer sufficient to exclude SRE states. A systematic study will be presented elsewhere Williamson and Cheng (2018).
Iv Constraints on 2D Symmetry-Enriched Topological Phases
In this section we develop fermionic LSM-type constraints for gapped topological phases, generalizing those for bosonic systems obtained in LABEL:ZaletelPRL2015 and LABEL:ChengPRX2016. We will first develop the necessary formalisms to describe fermionic SET phases and symmetry defects, following the treatment in LABEL:SET. We should note, however, that our analysis does not cover the most general fermionic SET phases. Indications will be given whenever simplifying assumptions are made. The completely general theory will be left for furture works.
Throughout this section we assume that the total symmetry group of the system is .
iv.0.1 Algebraic theory of fermionic topological phases
The mathematical theory of a general two-dimensional topological phase is known as the algebraic theory of anyons (“anyon model”), or unitary braided tensor category (UBTC) Kitaev (2006). In an anyon model , a set of labels represent different anyon types, or topological charges. Among them, there is a unique label “” denoting the trivial bosonic local excitations. The most fundamental property of anyon excitations is their fusion rules:
where the fusion coefficients are integers. Next we can also exchange or braid anyons around another. These information are summarized in the so-called and matrices. is a diagonal matrix, whose diagonal elements are topological twist factors , or the exchange phase between . Elements of matrix are related to mutual braiding statistics between anyons. For bosonic systems, one can further impose the condition of braiding non-degeneracy, or more precisely that the matrix is unitary, which makes it into a unitary modular tensor category (UMTC). Physically, it means that one can always distinguish different topological charges by braiding. There are other finer data for anyon models, notably the and transformations, which play important roles in the discussions of symmetry-enriched topological phases, and we refer the readers to LABEL:kitaev2006 for a more comprehensive review.
Gapped fermionic phases can be modeled as a UBTC, where we include the physical fermion as one of the topological charge types, denoted by in the rest of the paper. The fermion satisfies and , but braids trivially with every other anyon. The UBTC is no longer modular, but only up to the physical fermion. We can thus form “super topological charges”, consist of a doublet
where is a topological charge in the UBTC 444One can show that . Otherwise the ribbon identity if a UBTC implies , contradicting the fact that is a physical fermion. The UBTC that describes fermionic systems is “super-modular” Bruillard et al. (2017), in the sense that braiding is still non-degenerate for supercharges. Equivalently, one can factorize the matrix as
where is unitary. 555However, it is worth emphasizing that the factorization of the matrix does not mean that the UBTC can be factorized into a UMTC and . In general they do not.
Every fermionic topological phase modeled by a UBTC can be embeded into a larger bosonic one , where becomes an emergent fermion. Physically, this can be done by “gauging the fermion parity”, namely coupling the fermionic system to a gauge field sourced by the fermions. This gauging is not unique, but we always consider those with the minimal total quantum dimension equal to , where is the quantum dimension of . Such a bosonic topological phase is called the “modular extension” of . Even with this condition, there are always distinct modular extensions of a given , corresponding to stacking 2D topological superconductors before gauging Kitaev (2006); Bruillard et al. (2017). The modular extension can be written as , where consists of fermion parity fluxes which have mutual braiding phase with .
iv.0.2 Symmetry action on anyons
Let be the UBTC that describes a fermionic topological phase. Following the notations in LABEL:SET, we define a topological symmetry group Aut, consisting of all permutations under which all physical properties remain invariant, e.g.
Notice that here the definitions of symmetries involve the anyon types directly, not just the supercharge types, i.e. and should have identical topological twists. For a given permutation , it may act nontrivially on anyon fusion spaces, in order to preserve and transformations of the UBTC:
Here we assume that fusion multiplicities are or for simplicity, and therefore are phase factors. In general they are unitary transformations. In particular, the identity permutation can still act on fusion spaces in the following way:
Here are phase factors. Such a “trivial” transformations are called natural isomorphisms.
All allowed permutations form the group Aut, which defines the intrinsic symmetry of the emergent topological degrees of freedom. We should however notice that in fermionic systems, there are nontrivial symmetries not captured by Aut, which do not permute any anyons in but permute fermion parity fluxes Cheng and Wang (2018). We will not consider these symmetries for now.
Given a global internal symmetry group , assuming it is unitary for simplicity, we have a group homomorphism . Basically, indicates how a symmetry operation permutes anyons, together with symmetry transformations on fusion spaces. We adopt the following notations from LABEL:SET:
Here represent unitary transformations (and are phases in this case). More precisely, we should actually consider , the equivalent classes of ’s up to natural isomorphisms.
In a gapped phase, let us consider the global symmetry transformation corresponding to a group element , acting on a physical state containing several anyon excitations . Since is on-site, we expect that the action should be “localizable”, meaning that the nontrivial action is localized on anyons (on top of the global actions on splitting spaces when anyons are permuted). Formally, we can write
Here is a local unitary transformation support on the neighborhood of . They form a projective representation of :
Associativity of the local unitaries implies that
Requiring that , we find
In particular, for , we have
We define . The above condition implies
One should note that given a homomorphism (which fixes up to gauge transformations), Eq. (33) and Eq. (34) may not adimit any solutions for . This is captured by a obstruction class identified in LABEL:SET. We will proceed assuming that this obstruction vanishes, and there is no further obstruction realizing the corresponding permutation action . In this case, without loss of generality, we can always choose a representative such that
One should remember that this is merely a choice for convenience. With this choice of , we can easily show that satisfies a similar twisted -cocycle condition.
Eq. (36) implies that we can always write , where is an Abelian anyon. This readily follows from the unitarity of . We denote the set of Abelian anyons by , which naturally forms an Abelian group with multiplication given by fusion. In Appendix A we prove that can always be written as . Notice that unlike the modular case, is only determined up to the transparent fermion, i.e. only the supercharge is fixed by .
A straightforward calculation shows
Therefore we have
In other words, only the supercharge of forms a twisted -cocycle of .
Let us briefly discuss ambiguities in these quantities. is defined up to “1-coboundaries”, i.e.
are physically equivalent to , by redefining . Here are phase factors satisfying for . This coboundary ambiguity in translates into an anyon-valued -coboundary on :
Together, we conclude that equivalence classes of 2-cocycles valued in Abelian supercharges are classified by (the action of on is canonically induced from that of on ).
Each of the class then represents an equivalence class of projective phases characterizing local symmetry actions on anyons. Similar to the bosonic case, we refer to as the symmetry fractionalization class. An important remark is in order: when anyons are permuted, should be understood as torsors, i.e. starting from a SET phase, we can modify the symmetry fractionalization structure by .
iv.0.3 Symmetry defects
Symmetry defects are extrinsic objects carrying symmetry fluxes. For each , we can introduce -defects, going around which a local -action is enacted. There are topologically distinct types of -defects, organized into a -crossed braided category
where contains all -defects . They obey -graded fusion rules:
We will make a simplifying assumption that the symmetry defects do not “absorb” physical fermions, i.e. .
LABEL:SET defines -crossed braiding of defects for bosonic SET phases. While we do not attempt to present the most general theory of -crossed braiding for fermionic SET phases, we will describe in detail an important aspect of -crossed braiding, namely actions on defects.
Consider a pair of group elements and . Suppose in a particular SET phase, -defects transform under the symmetry as
Here is a defect in the sector. Following the convention in LABEL:SET, a counter-clockwise exchange (more precisely, a transformation) of and results in and , since passes through the branch cut of . Since the exchange can be implemented locally, the total topological charge before and after the transformation must remain the same.
In the following we consider what happens when we modify the SET structure by a symmetry fractionalization class . Due to the torsor nature of , we will assume a “reference” SET with -graded fusion rules in Eq. (43) and actions given by (see Eq. (44)), respectively. In the new SET, the fusion rules of defects become
to account for the additional projective phases when braiding defects around anyons. The key point is that although the projective phases are determined by , what appear in the defect fusion rules are .
Let us determine how the symmetry action on defects is modified. Recall that the symmetry action can be implemented by a braid (more precisely, a transformation). We expect that should differ from by an Abelian anyon:
Now compare the defect fusion rules before and after the braid:
In order for them to be equal we find that the “commutator” is given by
Symmetry transformations of defects are subject to the following ambiguities: first, for an Abelian anyon we have
Second, we are free to “relabel” charges in a given defect sector, by where is an Abelian anyon (if is of the form then the relabeling does not do anything). The symmetry transformation becomes
Here one naturally defines . We note that these ambiguities are exactly the coboundary degrees of freedom for .
iv.0.4 Incorporating translation symmetries
Although translations are not internal symmetries, they do preserve locality (i.e. map local operators to local operators), as well as orientation. It is therefore straightforward to formally include translation symmetries into the discussions in Sec. IV.0.2. Similarly, we can also discuss “defects” of translation symmetries, which are lattice dislocations. The mathematical formulation remains basically identical. We will address a subtlety in the physical interpretation of the action of lattice translation on internal symmetry defects below.
iv.1 Derivation of the LSM-type constraint
We now derive a generalized LSM-type constraint in a possible gapped symmetric phase. We follow the argument in LABEL:ChengPRX2016, where it was shown that in a bosonic system, where each unit cell transforms as a projective representation under an internal unitary symmetry , a gapped symmetric topological phase must satisfy
assuming no anyons are permuted. Here is the factor set that defines the projective representation per unit cell. The argument proceeds by considering moving a defect around a unit cell. We can identify two contributions in the expression: the first term from the background anyon charge which transforms as a projective representation of , and the second from the “anyonic spin-orbit coupling” (referring to nontrivial commutation relation between and when acting on anyons). The physical origin of the second term is that the topological charge of a -defect may change under translations. Creation of these additional anyons results in the second phase factor.
In our case, the physical constraint is that the fermion parity in a unit cell changes under the action of the particle-hole symmetry . Therefore, it is clear that anyons have to be permuted by some of the symmetries (, or ) to match the additional fermion that appears under the local symmetry action. Following LABEL:ChengPRX2016, we introduce a -defect into the system, move it around a unit cell, and then apply a local symmetry action to the unit cell in order to restore the original Hamiltonian (removing the branch loop). As the -defect is moved, it may change type and leave behind on its path additional anyon charges. We denote the total (Abelian) charge appearing in this process by . The LSM-type constraint essentially says .
Let us consider adiabatically transporting a -defect where is an internal symmetry. Suppose a unit translation along the -th direction, denoted by , acts on defects as
If is nontrivial, naïvely it might seem that such a translational symmetry action would imply that -defects cannot be adiabatically transported in the -direction, since the topological charge value carried by the defect must change when it moves. This subtlety was already addressed in LABEL:ChengPRX2016. For a -defect carrying the energetically favored topological charge of , adiabatically transporting the defect by one unit length in the -direction involves extending the defect branch line by one unit length and ending with a Hamiltonian that energetically favors topological charge at the new endpoint of the branch line. For example, if where is an Abelian anyon, adiabatically transporting a defect by a unit length in the -direction involves creating a pair, leaving on the new segment of defect branch line, and fusing with the defect to change its topological charge value.
Now we consider the local -action on a unit cell by pair creating a - defect pair, adiabatically transporting the -defect around a path enclosing one unit cell in a counterclockwise fashion, and then pair annihilating the defects. As we have mentioned, this process changes the topological charge in the unit cell enclosed by (which is an anyon, not to be confused with a -defect). Because of the torsoring structure in the SET classification, we will actually calculate the additional Abelian charge accumulated in the unit cell if we modify the SET structure by a fractionalization class .
One subtlety which is only present with nontrivial anyon permutation, is that the charge being created when moving a -defect is position-dependent. We therefore use to denote the charge created by moving a -defect at position by a unit step along the -direction, see Fig. 2 for illustration. To be precise, again due to the torsoring structure, what we actually calculate is the additional Abelian charge associated to the fractionalization class . We will not repeat this point further.
Before we discuss the actual calculation, let us explain two basic rules:
If we move a -defect by a unit length in the -direction, and then move it back by a unit length in the direction so that it returns to the original position, there should be no anyons left. In other words, without loss of generality, we can set
Consider moving a -defect at position by a unit length in the -direction, and the same process but with the -defect initially in . These two processes are related by a lattice translation along , so we should have
Notice that here we write instead of an equality, because one can always absorbs/emits an Abelian anyon of the form where to/from a -defect. These rules are illustrated in Fig. 2(a).
To be more concrete (without any loss of generality), let us choose the path to be , as shown in Fig. 2(b). Keeping track of the topological charge creation and annihilation due to the adiabatic transportation of the -defect creating a loop of defect branch line, we have:
: the topological charge created on the bottom segment of defect branch line is .
: the topological charge created on the right segment of defect branch line is .
: the topological charge created on the upper segment of defect branch line is
: the topological charge created on the left segment of the defect branch line is
The corresponding configuration of topological charges for this -defect branch loop is illustrated in Fig. 2(b).
All together, an Abelian anyon
has been accumulated in the unit cell. We can choose such that
In addition, the action also changes the background charge value:
Note that this result depend on the actual location of the unit cell, since the “background charge” can get permuted as well under translations. However, Eq. (57) can be absorbed by a -defect so we may ignore this contribution.
To summarize, we have found that the total change of topological charge in the unit cell is modified by
We now argue that LSM constraint implies . Let us start from a certain “reference” SET, with the same anyon permutation . We assume that this reference SET can be realized in a lattice model without a fermionic projective representation per unit cell, so after taking a -defect around a unit cell we accumulate no charge: . Now we modify the fractionalization class by and demand that the resulting SET can be realized in a system with the fermionic LSM constraint. This is achieved by requiring .
We can in fact make the expression Eq. (58) more precise. Physically we expect that or is “gauge-invariant” under coboundary transformations of fractionalization classes. However, the right-hand side of the equation, as given, is not. But recall we are allowed to have additional anyons of the form . Demanding gauge invariance, the right-hand side can be uniquely fixed:
An interesting corollary of the LSM-type constraint is that must permute anyons. If we assume that the translations do not permute anyons, we have that
On the other hand the fermionic LSM constraint requries . So we must find an Abelian background charge such that , which is clearly impossible. Therefore, the LSM constraint implies that lattice dislocations must carry non-Abelian zero modes Bombin (2010); You and Wen (2012); Teo et al. (2014), i.e. they are “genons” Barkeshli and Qi (2012); Barkeshli et al. (2013).
iv.2 Surface States of Type-I Weak Fermionic SPT Phases
In this section we study gapped surface topological phases of type-I weak SPT phases.
The coupled wire construction presented in Sec. III.1.1 can also be used as a model for the surface of a type-I weak fermionic SPT phase, with an internal symmetries. The bulk of this SPT is simply a stack of two-dimensional fermionic SPT layers. Each layer consists of two Chern insulators, with and respectively, and the symmetry is the fermion parity of one of the Chern insulators. It is easy to see that the low-energy surface theory is just Eq. (9), and the symmetry acts as