Space-Time Geometry of Topological phases
SSpace-Time Geometry of Topological phases
F. J. Burnell , and Steven H. Simon Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK All Souls College, Oxford, UK (Dated: October 25, 2018)The 2+1 dimensional lattice models of Levin and Wen [PRB 71, 045110 (2005)] provide themost general known microscopic construction of topological phases of matter. Based heavily on themathematical structure of category theory, many of the special properties of these models are notobvious. In the current paper, we present a geometrical space-time picture of the partition functionof the Levin-Wen models which can be described as doubles (two copies with opposite chiralities)of underlying Anyon theories. Our space-time picture describes the partition function as a knotinvariant of a complicated link, where both the lattice variables of the microscopic Levin-Wen modeland the terms of the hamiltonian are represented as labeled strings of this link. This complicatedlink, previously studied in the mathematical literature, and known as Chain-Mail , can be relateddirectly to known topological invariants of 3-manifolds such as the so called Turaev-Viro invariantand the Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle excitations of theLevin-Wen models and we see how they can be understood by adding additional strings to the Chain-Mail link representing quasi-particle world-lines. Our construction gives particularly important newinsight into how a doubled theory arises from these microscopic models.
I. INTRODUCTION
The intersection between topology and the physics ofstrongly correlated systems harbors great theoretical andpractical interest. Motivated by the dream of buildingtopologically protected qubits, numerous approaches tofinding topological phases of matter in physical materialshave been explored. In this work, we focus on perhapsthe most prominent attempt to build theoretical latticemodels that exhibit topological properties – the string netmodels of Levin and Wen . We situate these models inthe context of other mathematics and physics-based ap-proaches to topological theories. In doing so, we seek notonly a new understanding of the models themselves, butalso a methodology which allows us to generalize theseconstructions in potentially fruitful ways.The origins of this field of study extend to the 1980s,when the notion of a “topological quantum field the-ory” (TQFT) was introduced. These TQFTs were firststudied in the context of gravitation with the goal ofconstructing a metric-independent action. Though ulti-mately their success as a theory of gravity was limitedto 2 + 1 spatial dimensions, the exploration forged sev-eral profound connections between topology and physics.Witten’s groundbreaking work showed that there is avery deep and powerful link between Chern-Simons gaugetheory, conformal field theory, and mathematical invari-ants of knots, links, and 3-manifolds. Growing from thisdiscovery, as well as from other developments in the thestudy of conformal field theory , was the realization thatin 2+1 dimensions, point particles can have exotic (“any-onic”) quantum statistics described by a non-abeliantopological field theory. Such statistics are far richer thanthose of bosons and fermions, and even beyond the frac-tional statistics realized by the Jain series of fractionalquantum Hall states.More recently, anyon theories in 2 + 1 dimensions have received renewed attention in the context of topologicalphases of matter — phases of strongly interacting mat-ter whose low energy, long wavelength description is atopological quantum field theory, and whose excitationsconsequently have anyonic statistics. Proposals made byFreedman and Kitaev that such phases of matter couldbe used to build powerful quantum computers provide apractical motive for attempting to realize these in thelab.While there has been some evidence that certainquantum Hall states may be realizations of extremelynontrivial topological phases of matter, suitable for quan-tum computation, experiments in this field are diffi-cult and progress has remained challenging. As a re-sult researchers have begun to consider what other sys-tem might be found which could display similar ex-otic statistics. Earlier work on lattice spin models dis-playing fractional statistics inspired attempts to construct lattice model Hamiltonians exhibiting non-trivial topological phases. Among these constructions,the Levin-Wen models are of particular importance asthey can exhibit the most general range of possible quasi-particle statistics of any known Hamiltonian lattice the-ory. Indeed, Hamiltonians of this kind are thought toexist for a large class of achiral anyon theories knownas quantum doubles. This class encompasses many ofthe previously studied anyon lattice models such as theToric code , many of its generalizations and dou-bled Chern-Simons theories . Further, as the Hamilto-nians are exactly solvable, a complete understanding ofthe spectra and behavior is possible. As such, they serveas a useful prototype for understanding topological be-havior in more complex systems. Because these modelsrealize such a wide range of topological phases, they mayserve as a guide in designing new materials which supportthe anyonic statistics essential for topological quantumcomputation . a r X i v : . [ c ond - m a t . s t r- e l ] A p r In this work, we focus on a large class of Levin-Wen models that are equivalent to doubled anyon the-ories (those in which each type of anyon appears in twoopposite-chirality copies). We will expose the connec-tions between these models and certain topological in-variants developed in the context of knot theory. Specif-ically, we show how a subset of the Levin-Wen modelsare related to the knot-theoretic topological invariants of3-manifolds discovered by Witten and Reshitikhin andTuraev . In particular we show how the vacuum par-tition function of these models can itself be viewed asa knot-invariant, first introduced by Justin Roberts ,known as the Chain-Mail link. Further, we show howquasi-particle excitations can be incorporated into thistopological description. On the one hand, the inclusionof quasi-particles, or defects, into 3-manifold invariantssuch as the Chain-Mail link, is new to the mathemat-ics community (see, however, Refs. 19, 20); whereas onthe other hand, the use of these particular mathemati-cal invariants to describe space-time processes of physicalsystems is new to the physics community.Our approach gives a direct visualization of space-timeprocesses and how they can be interpreted as knots thatrepresent configurations of quantum numbers on a lat-tice. In so doing we obtain a number of new ways oflooking at these systems. First, this re-interpretationof the Levin-Wen partition function renders explicit theconnection between these models and the (continuum)topological field theories, such as Chern-Simons theory,familiar in other areas of physics. Second, we providesa simple proof that the ground state partition functionof these models give exactly the Turaev-Viro invariant ,a well-known mathematical quantity (see also Ref. 22).Third, our approach is not tied to any particular lat-tice structure, and makes obvious how Levin-Wen mod-els may be generalized to arbitrary non-trivalent graphsand lattices. Finally, our work gives a new perspectiveon the nature of “doubling” — that is, the production oftwo copies with opposite chirality– in these models. An-other recent work also gives an interesting descriptionof the connection between Levin-Wen models and dou-bled Chern-Simons theory (which can be generalized todescribe other quantum double theories ), by express-ing the lattice models as models of closed surfaces ontwo sheets of opposite chirality. Though this descriptionis superficially quite different from ours – and indeed,Ref. 23 focuses principally on describing phase transi-tions out of from the purely topological limit by studyingfluctuations in the surface topology – the mathematicalunderpinnings of the two pictures are closely related.In situating the Levin-Wen models in the context ofthe mathematical discourse on topological field theory,we also hope to provide a useful framework from whichto approach topological theories on the lattice. Thisframework is geometrical in character – the model canbe phrased in a ‘pictorial’ manner in which closed loopsof string in 2 + 1 dimensional spacetime represent boththe quantum degrees of freedom of the model and the operators, such as the Hamiltonian, that act on thesedegrees of freedom. While this mapping may sound un-usual, it has some distinct advantages. Notably, once thismapping is made, a large amount of tensor calculus is re-duced to trivial geometric manipulations. We find thatcertain properties of the Hamiltonian – namely, topo-logical invariance and the fact that the Hamiltonian iscomposed of commuting projectors, and hence is exactlysolvable – are reflected in special allowed geometrical re-arrangements of these strings which leave the partitionfunction invariant. In the mathematical context, thesere-arrangements are in both cases related to deformationsof the space-time which preserve its topology. (Similarre-arrangements can also be used to coarse-grain the sys-tem, giving a natural interpretation of the exactness ofthe tensor renormalization group of 25 and 26 for thesestates.) In this sense, our geometrical construction makesapparent certain connections between the Hamiltonian-driven approach of Levin and Wen , and the mathemat-ical approach to topological invariants of 3-manifolds viathe study of knots .This paper is structured as follows: broadly speaking,section II gives a conceptual overview of the general ideasin this paper, sections IV, V, and VI flesh out the techni-cal details and explain the correspondence to the mathe-matical literature, while section VII briefly discusses ap-plications of these results to generalizing the model.Specifically, we begin in section II by focusing onChern-Simons theory as an easy entry point into ourwork. We review the salient features of this theory insection II A. Section II B explains the connection betweenthe lattice model of Levin and Wen and the continuumChern-Simons theory, using loops of Wilson lines to rep-resent quantum numbers on the edges of the lattice aswell as to represent the operators that act upon thesequantum numbers. We describe roughly how this con-struction results in an effective theory for the latticemodel which is the double of the original Chern-Simonstheory. Section IV gives the technical details of our con-struction of the ground state partition function of theLevin-Wen lattice models. We begin in section III with avery brief introduction to the structure of anyon theories(or “Modular Tensor Categories”), and discuss Levin-Wen lattice models built on this structure in sectionIV A. In section IV B we construct the partition func-tion of the Levin-Wen models and show how to view itas a space-time picture of successive projectors, similarto a Trotter decomposition. Section V shows in detailhow this construction of the Levin-Wen partition func-tion gives precisely the Chain-Mail invariant introducedby Roberts . Since the Chain-Mail invariant is easilyshown to be equivalent to another mathematical invari-ant, known as the Turaev-Viro invariant, this makesapparent the explicit connection between the lattice par-tition function and the Turaev-Viro state-sum.In section VI we turn to study quasi-particles in theLevin-Wen models. Section VI B reviews the Levin-Wenconstruction of quasi-particles in their model; sectionVI C shows the analogous construction in our approach,which adds quasi-particle world-lines to the Chain-Maillink. This construction gives an interesting perspectiveon the difference between the right- and left- handed sec-tors of the doubled theory, which we elaborate on in sect.VI D 1 . In Sect. VII, we outline how the pictorial con-struction can be carried out on more general lattice ge-ometries, allowing for Hamiltonians with slightly simplerinteractions. Section VIII summarizes our results anddiscusses interesting open questions.This paper also includes appendices detailing cer-tain diagrammatic calculations, as well as discussions ofsurgery, and categories more general than those discussedin the rest of the paper. We have separated these sectionsfrom the main text for simplicity of presentation. II. CONCEPTUAL OVERVIEW:CHERN-SIMONS THEORY AND TOPOLOGICALLATTICE MODELS
We begin with a qualitative description of our mostinteresting results, using as an example the class of topo-logical quantum field theories most likely to be familiarto physicists – Chern-Simons theories.In this section, we outline our answer to a simple butimportant question: how can a pure Chern-Simons the-ory arise from a lattice Hamiltonian? Though several ex-amples in the literature find an effective long-wavelengthChern-Simons term after integrating out fermions on thelattice , our interest is to construct a lattice model inwhich the local variables can be expressed in terms of theChern-Simons gauge field. As observed by Ref. 16, thecommutation relations of the Chern-Simons gauge fieldobstruct such a description unless the theory is doubled;consistent with this, our construction yields only doubledtheories.The usual prescription for putting a Maxwellian gaugetheory on the lattice places a gauge field on every edgeand recovers the continuum limit by taking the latticeconstant to zero. Though similar prescriptions can becarried out for Abelian Chern-Simons theories , conven-tional lattice formulations valid for general non-Abeliangauge groups have proven elusive. Our objective in thissection is to provide a qualitative description of an al-ternative route to this end — namely, we formulate alattice construction of Chern-Simons theory which is nat-urally ‘topological’, in the sense that it is independent ofthe lattice geometry and captures the topological char-acter of the braiding of Wilson lines. The interestingfeature of these models is that they are independentof the lattice constant a , and thus there is no way inwhich the usual prescription of taking the continuumlimit applies. Rather, the correspondence to a contin-uum theory is achieved by means of a known mathemati-cal equivalence between Chern-Simons theory and knotpolynomials. Hence our model does not encode the con-tinuum theory as a long-wavelength limit, but rather en- codes the continuum theory in a lattice representation.This is possible because the topological model has, forany fixed number of excitations, a finite number of de-grees of freedom. A. Chern-Simons theory
We begin with a few important facts about Chern-Simons theory. Our goal here is to sketch the relation-ship between Chern-Simons theory and knot theory firstdescribed by Witten , which is the cornerstone of ourconstruction. For a more pedagogical overview of Chern-Simons theory, see, for example, Ref. 32. Readers whoare relatively familiar with Chern-Simons theory may beable to skip to section II A 1 where we discuss the Ω string(which is likely to be less familiar, even to many experts).To define a Chern-Simons theory, we pick a Lie group G and let the gauge field A take values in its Lie algebra.The Chern-Simons action is written S CS [ A ] = (1) k π (cid:90) M dx ε lmn (cid:20) A al ∂ m A an + 23 f abc A al A bm A cn (cid:21) where k is an integer known as the level. Here the in-tegral is over the spacetime manifold M and we writecoefficients A aµ where µ is a spatial index and a is theLie algebra index where f abc are the structure constantsof the algebra. We denote this Chern-Simons theory G k (pronounced “ G level k ”).The topological character of Chern-Simons theory re-sults from an unusual feature of the Chern-Simons action:since all indices are contracted using the anti-symmetrictensor ε ijk , the action is defined without a spacetimemetric. Hence the action must be invariant under de-formations of space-time – in other words, it must beinvariant under any continuous change in the geometryof the space-time manifold. As a result the Chern-Simonspartition function Z CS [ M ] = (cid:90) D [ A ] e i S CS [ A ] (2)is a topological invariant of the manifold M . Through-out this work, unless otherwise specified, all partitionfunctions are to be understood as evaluated at T = 0; inthe presence of thermal excitations the partition functionnecessarily scales with the area, and hence is not a purelytopological quantity.One consequence of topological invariance is thatthe gauge-invariant physical observables in pure Chern-Simons theories must also be independent of the space-time metric. This stringent condition leaves us with verylimited possibilities for observables of the theory; in fact,the entire physics of Chern-Simons theory can be de-scribed as a theory of Wilson lines. Explicitly, a Wilsonline operator is defined as W R ( C ) = Tr R P e i (cid:72) C A i dx i (3)where C is a directed closed curve in the 3 D space-time (a knot ), P denotes path ordering, and R is a representationof the gauge group in which we take the trace. Hencewe integrate the component of the gauge field tangentto the curve C , to obtain some element of the gaugegroup. To obtain an observable, we must stipulate arepresentation of the gauge group G in which to computethe trace. For example, in the case of the gauge group SU (2), the representations are labeled by their total spin0 , / , , . . . .There is one important subtlety in this approach, dueto the impact of the constraints that make the actiontopological on the quantized theory. The up-shot of thistechnicality is that after quantizing the theory, only afinite number of representations are allowed — which wewill label 0 , . . . , r with 0 reserved to mean the trivialrepresentation (which is equivalent to the absence of aWilson line) . For example, In the case of SU (2) k wehave r = k nontrivial particles.Thus we may think of Chern-Simons theory, quantizedin this way, as a theory of Wilson lines which are labeledby a finite set of allowed quantum numbers 0 . . . r . Allrelevant information about the theory is then encoded inthe expectation values of products of Wilson lines, whichwe can describe by a set of directed closed curves C j labeled with quantum numbers (representations) i j : (cid:104) W i ( C ) . . . W i n ( C n ) (cid:105) M =1 Z CS ( M ) (cid:90) M D [ A ] W i ( C ) . . . W i n ( C n ) e i S CS (4)If n >
1, we will call the collection of closed curves C j a labeled link , whose components consist of the individ-ual curves C j , and whose labels are determined by therepresentations carried by the corresponding Wilson line.Denoting the resulting link L , we will use the notation (cid:104) L (cid:105) M ≡ (cid:104) W i ( C ) . . . W i n ( C n ) (cid:105) M (5)as shorthand for such expectation values. When the man-ifold M is not specified, we take it to be the 3-sphere S .The expectation values (4) are topological invariants ofthe labeled link – and hence, we may evaluate expecta-tion values of products of Wilson lines in the gauge theoryby computing the relevant link invariants. Indeed, as fa-mously shown by Witten , in the case of SU (2) gaugegroup, one obtains the colored Jones polynomial (withthe “colors” being the different labels of the lines). Formore general gauge group one obtains other topologicalinvariants including many previously known in the math-ematical theory of knots.In a more physical language, we can think of each of thelabels as being a “particle type” and the directed lines asbeing the worldlines traced out by these particles. In fact,this picture can be generalized to include branched loops,where two particles come together at a vertex to form athird in a process known as fusion . Roughly speaking,one can think of binding the incident pair of Wilson linestogether such that the only physically relevant variable is the sum of their labels. In general, the end product of fu-sion will be a superposition of particle types– analogousto combining two spin 1/2 particles to form a superpo-sition of singlet and triplet states. Thus we can think ofChern-Simons theory as more generally being able to as-sign a value to any graph of worldlines that has trivalentvertices as well as over- and under-crossings.
1. Vacuum partition functions and the Ω String
Above, we motivated the claim that an expectation ofan arbitrary product of Wilson lines in the Chern-Simonsgauge theory at zero temperature can be evaluated bycomputing an appropriate link invariant. But there isone key missing ingredient required to make this picturevalid: we must know the correct normalization for Eq.4, which requires evaluating the vacuum partition func-tion. At zero temperature, the topological theory con-tains a finite number of degrees of freedom; consequently,as Witten showed this, too, can be done in terms of linkinvariants– specifically, by evaluating the invariant of alink (whose form is dictated by the space-time manifold)labeled by a special superposition of particle types whichwe call Ω (whose precise definition we defer to Sect. III).In other words, for the correct choice of link components C i , we have Z CS ( M ) = 1 Z CS ( M ) (cid:104) L ( M ) (cid:105) M ≡ Z CS ( M ) (cid:104) W Ω ( C ) . . . W Ω ( C n ) (cid:105) M (6)where M is a ‘reference’ space-time manifold (for ourpurposes, S × S ), whose partition function we willchoose to set to 1. The link thus encodes the topologyof the manifold M . The precise relationship between M and L ( M ) can be understood by a procedure known as Dehn surgery , which we will outline in more detail inSect. V B and Appendix C 3.Because of its special relationship to the ground-statepartition function, the Wilson line labeled by Ω (or Ωstring) plays a pivotal role in our construction. Its priv-ileged status is related to the following useful property:
Killing property
If any particle world line labeled i passes through a non-self-knotted Ω loop, the value ofthe evaluated link invariant Eq. 4 will be zero unlessthat particle is the trivial particle i = 0 (See Fig. 1.a).By adding an appropriate normalization, the Ω can thusbe considered to be a projector that gives one if the vac-uum particle passes through it and gives zero otherwise.We emphasize that if multiple particle world lines passthrough an Ω loop, the projector acts on the combined quantum number of all the particles. Thus the value ofthe link invariant will be nonzero if the quantum numbersof the multiple particles can be combined (or “fused”) to(a) i Ω = δ i, Ω(b) ji k Ω= F i ∗ i jj ∗ k ∗ F kk ∗ kk ∗ ji k Ω ji k FIG. 1. The Ω string. (a) The killing property. (b) A usefulconsequence of (a). The meaning of F will be explained inSect. III. the vacuum quantum number. (See Fig. 1.b) Such fu-sions are described in detail in section III below.Hence we may use Ω to construct projection operators,which will be the constituents of our topological latticeHamiltonian. The importance of Ω in topological invari-ants has been discussed starting with the work of Ref. 3and 17. More recently this type of projector has beenused for analysing anyon models in Refs. 1, 11, and 23. B. Lattice Chern-Simons theory via Wilson lines
Our initial question was how a pure Chern-Simons the-ory can arise from a lattice Hamiltonian. At this point,let us turn the question around and ask how we may con-struct a lattice Hamiltonian based on a Chern-Simonstheory. To this end, let us choose a two dimensional lat-tice and give quantum numbers to the edges of the latticechosen from the set of quantum numbers i ∈ { ...r } of aChern-Simons theory. Although, as we will describe be-low, such a starting point can indeed generate a latticemodel that is equivalent to a continuum Chern-Simonstheory, we will find that it is actually equivalent to the double of the Chern-Simons theory we started with —that is, two copies with opposite chiralities .We will aim to make a direct correspondence between partition functions of our lattice model and those of acontinuum Chern-Simons theory. Hence we work directlywith a three dimensional description, where the third di-rection is time. We discretize time, so that we obtain a3D lattice consisting of multiple copies of our 2D latticeseparated by discrete time intervals δt . We will refer to an edge or plaquette of this 3D lattice as being space-likeif it is at fixed time, and time-like if it extends betweenneighboring time intervals.Our objective is to construct a lattice Hamiltonian H such that by using H to propagate states in time, andtracing over intermediate states, we obtain a partitionfunction of the form (6) – in other words, a partitionfunction which corresponds precisely to that of a topo-logical quantum field theory.Since Chern-Simons theory is essentially a theory ofWilson lines, it is quite reasonable to represent the quan-tum numbers of the 2d lattice edges (the space-like edges)in terms of Wilson lines. We thus associate a quantumnumber on a space-like edge at a given time t with aclosed Wilson loop which runs along this edge at time t ,up the two vertical edges at its endpoints, and back alongthe same edge at time t + δt (Fig. 6(a)). In other wordsan edge with quantum number r i at time t , is describedin the continuum model as a Wilson loop just inside theperimeter of a time-like plaquette corresponding to a par-ticular edge and that particular time.The partition function will require us to sum over allpossible edge variables at every time. Since Ω is actu-ally a sum over all quantum numbers, it will turn outthat labeling the Wilson loop around the perimeter ofthe time-like plaquettes with Ω will precisely effect thedesired sum in the partition function. FIG. 2. Wilson lines used to evaluate the partition function,drawn here on a lattice whose spatial slices are the honey-comb. Green strings, which encircle time-like plaquettes, rep-resent the states at each time step. Blue strings around thespace-like plaquettes represent the plaquette projectors in theHamiltonian; mauve strings encircling the time-like edges rep-resent the vertex projectors. The yellow loops along space-likeedges are necessary to implement the action of the Hamilto-nian, by forcing the plaquette projectors to fuse with the edgevariables at each time step. All loops carry the label Ω.
Having extended the edge variables in 2d to Wilsonloops in 3d, we now attempt to determine an appropri-ate lattice Hamiltonian which will yield a topological the-ory. In existing constructions of topological lattice mod-els such as that of the Levin and Wen , and Kitaev’s ToricCode , the Hamiltonian is a sum of mutually commutingprojectors – one class being applied to every vertex ofthe 2d lattice, and another to every plaquette. As willbe detailed below in section IV A, the vertex projectorsassure that the quantum numbers incident on each ver-tex fuse to zero. The plaquette projectors can be thoughtof similarly as a fusion condition (but in a dual basis, asdiscussed in section V B 2). As mentioned above (Prop-erty 1 of section II A 1), a loop labeled with Ω projectsonto states where all lines passing through the Ω loopfuse to the vacuum particle. Hence, up to normalizationof the projectors, we may use the special Wilson line Ωto construct operators in the Chern-Simons theory thatcarry out the action of the Hamiltonian of a topologicallattice model. Indeed, the vertex projector may be imple-mented by encircling time-like edges with an Ω loop andthe plaquette projectors are implemented by including Ωloops on each space-like plaquette – threading additionalΩ loops which wrap space-like edges.An example of the resulting link of Wilson loops isshown in Fig. II B. This link is referred to as the “Chain-Mail ” link , evoking its resemblance to Chain-Mail ar-mor made of linked rings. Evaluation of the associatedknot invariant, known as the Chain-Mail invariant, isknown to give an invariant of the three manifold —that is, it is independent of the lattice decomposition ofthe manifold.Although all of the strings in the Chain-Mail picture inFig. II B are Ω’s, we remind the reader that they have dif-ferent meanings. The time-like plaquette loops representthe edge variables in the lattice models. The remain-ing loops are part of the Hamiltonian which acts uponthese variables at each time step to effect time propaga-tion of the edge variables. Thus, all of the pieces takentogether, this link is a explicit space-time representationof the partition function of a 2d lattice model.Hence, we associate a link (the Chain-Mail link, L CH ,whose components are all Wilson lines labeled by Ω),with the zero-temperature partition function of our lat-tice model. Specifically, we have used the Chain-Maillink to engineer a lattice Hamiltonian with the interest-ing property that evaluating the link invariant gives, upto a constant, the partition function for the ground statesector of our lattice theory: Z ∼ (cid:104) L CH (cid:105) . (7)(This is nontrivial in topological models, since the groundstate sector is degenerate). As we will explain in detailin Sect. IV, the lattice Hamiltonians we construct in thisway are precisely the set of Hamiltonians described byLevin and Wen (Ref. 1) for which the spectrum of exci-tations corresponds to that of a doubled Chern-Simonstheory.Obtaining the partition function in the presence ofquasi-particles turns out to be similar. Since our formal-ism captures only the topological sector, we must workin the zero-temperature or infinitely-gapped limit, wherethese quasi-particles can appear only as topological de- fects, not as excitations. In the 3d picture, these defectsare also Wilson lines (or external sources in the gauge the-ory) which trace out the quasi-particle worldlines’ space-time paths. Hence the partition function can again beevaluated by computing the appropriate link invariant: Z ( W i , ...W i n ) ∼ (cid:104) L CH ∪ W i ( C ) W i n ( C n ) (cid:105) (8)where the the link now includes both the Chain-Mailand the inserted quasi-particle world-lines. (This is il-lustrated in Fig. 9, and explained in detail in Sect. VI).Restricting this picture to a 2d slice at fixed time, thequasi-particle defects appear as string operators — mean-ing that defects must be created in pairs which appearat the end of unobservable strings.The true power of expressing the partition function inthe form 7 is that it allows us to exploit Eq. 6, whichrelates the link invariant to the Chern-Simons vacuumpartition function. To evaluate the ground state parti-tion function of the lattice model, we need to determinethe Chern-Simons expectation value for the complicated“Chain-Mail ” link that is the space-time representationof the lattice model partition function. However, we mayinvoke Witten’s result to realize that this is equivalent toevaluating the Chern-Simons vacuum partition functionin an appropriately chosen space-time manifold (whosetopology is dictated by the precise linking pattern of thecomponents of the Chain-Mail link). Thus, as promised,we make a connection between the lattice models andthe continuum Chern-Simons theory, not via the usualnotion of taking the lattice spacing to zero, but by usingDehn surgery to see that the two partition functions areequivalent.As we will see in Sect. V, it turns out that the space-time of the continuum theory arrived at in this way is al-ways such that the vacuum partition function is achiral .Specifically, if the (3d) space-time lattice can be ‘filled in’to obtain a manifold M , then the corresponding contin-uum theory is a Chern-Simons theory on M M , that is, M connected to its mirror image. In other words, thoughthe lattice model is constructed entirely in terms of op-erators from a chiral theory, our unusual prescription fortaking the continuum limit results in a continuum the-ory which is, in fact, doubled. Hence we have made anexplicit connection between the topological lattice mod-els of Ref. 1 and the continuum doubled Chern-Simonstheory.In the remainder of this paper, we will present the ideasoutlined here in greater detail, focusing on the more gen-eral case of an arbitrary ‘doubled anyon’ theory, whichmay or may not correspond in practice to a doubledChern-Simons theory. Though this will require a moretechnically involved treatment, the intuitive picture –based on the correspondence between partition functionsof topological theories and link invariants – is essentiallythe same as what we have outlined here.One may also consider more complicated (or even ir-regular) lattice geometries. For example, the so-calledCairo-pentagon lattice tiling and the prismatic pentagon SU (2) SU (2) k particle types: j ∈ { , / , , . . . } ⇐⇒ j ∈ { , / , , . . . k/ } vacuum: j = 0 ⇐⇒ j = 0fusion rules: j × l = (cid:80) j + li = | j − l | i ⇐⇒ j × l = (cid:80) min { j + l,k − j − l } i = | j − l | iF -matrices 6-j symbols of SU (2) ⇐⇒ q-deformed 6-j symbols of SU (2) k TABLE I. Analogy between SU (2) Yang-Mills theory (ie. conventional SU (2) angular momentum addition) and SU (2) k Chern-Simons theory. Note that the former does not constitute an anyon theory in our sense because there are an infinitenumber of particle types. Here each particle j is its own antiparticle, since j can only fuse with j to produce a singlet 0 (thetrivial particle). In the Chern-Simons case, the fusion rules are similar to conventional SU (2) except that they have been deformed so that no fusion ever produces j > k/
2. Note also that there is no analogue of l z in this theory: the particle typesare uniquely specified by the total angular momentum j .Anyon Theory Particle Types Fusion Rules SU (2) k j ∈ { , / , , . . . , k/ } j × l = (cid:80) min { j + l,k − j − l } i = | j − l | i Fibonacci = ( G ) { , } × { , e, m, em } e × m = em ; e × e = m × m = 0 e × em = m ; em × m = e TABLE II. Examples of Well Known Anyon Theories, Their Particle Types, and Fusion Rules. Note that it is always truethat 0 × y = y for any particle type y . lattice tiling both share the feature with the honeycomband square lattices that a LW model based on these lat-tices would couple only 12 edges at a time. III. FORMALISM OF ANYON THEORIES
In Sect. II, we constructed a lattice Chern-Simons the-ory by placing projectors, comprised of a particular su-perposition of Wilson lines, on the edges and plaquettesof a lattice model. To describe the resultant microscopicaction on the states, however, requires rules about howto evaluate the resulting link diagrams. Here we brieflyintroduce the formalism of anyon theories (often called modular tensor categories in the literature) associatedwith these rules. For a more comprehensive introductionto this subject, see Refs. 11,36,37.Generally an anyon theory can be thought of as a gen-eralization of a Chern-Simons theory. Like a Chern-Simons theory (Sect. II A), the anyon theory specifiesa topological invariant of a manifold (analogous to avacuum partition function), and can be used to assigna value (the link invariant ) to any link of world-lineswithin that manifold. As with Chern-Simons theory, us-ing the concept of fusion, the idea of a link invariant canbe generalized to give a value to world line diagrams withbranches as well as knots.The formalism of anyon theories has three importantelements. First of all, fixing an anyon theory specifiesa set of particle types allowed in the theory. Second,it stipulates a set of fusion rules , which determine whathappens when two particles combine at a trivalent vertex to form a third. (Again, physically we can think of bring-ing the two particles so close together that at the lengthscale on which we examine the system they appear to befused into one). These fusion rules will be accompaniedby so-called F -matrices , which describe how the orderof fusing multiple particles together may be changed. Fi-nally, the anyon theory gives a set of braiding rules , whichspecify the statistics of each species of anyons by assign-ing a phase to any exchange process.A familiar example of an anyon theory is the Chern-Simons theory described above. (Note however that notall anyon theories are Chern-Simons theories!) First, ithas a finite set of particle types (the number of which de-pends on the group G and the value of k ). We represent aparticle by its world-line, which carries a label to denotethe particle type. (When G = SU (2), this is the particle’sangular momentum quantum number). Second, if twoparticles are brought close together we can “fuse” themto form a third by combining their labels appropriately.(Again for G = SU (2), this corresponds to performingthe appropriate addition of angular momenta, bearingin mind that for SU (2) k Chern-Simons theory there is amaximum possible angular momentum in the theory, andcombinations which exceed this do not occur.) And third,when two world-lines are braided around each other, theiranyonic nature ensures that the wave function acquiresa complex phase. The close connection between SU (2)angular momentum addition (Yang-Mills) and SU (2) k Chern-Simons theory is shown in Table I. Table II givesseveral examples of fusion rules in well known anyon the-ories.Let us describe the formalism surrounding each ofthese three elements in turn. First, we will graphicallyrepresent a particle as an arrow labeled with the nameof the particle i ∈ { , . . . r } . We may think of this ar-row roughly as the world line of a particle. The iden-tity (“trivial” or “vacuum”) particle (represented eitherby no line, or a dashed line when necessary) is alwayslabeled 0, and particle i has a unique antiparticle type i ∗ ∈ { , . . . r } , which is equivalent to changing the di-rection of the arrow (i.e, the particle i going forward isequivalent to the particle i ∗ going backward). Note thata particle can be its own antiparticle (in which case, linesrepresenting particles need not have arrows).Each of these particle types has an associated quan-tum dimension , denoted ∆ i . This is the value of a cir-cular world-line loop labeled by i when nothing passesthrough the loop. It is useful to define the “total quan-tum dimension” given by the root sum of the squaresof the quantum dimensions of all of the particles in thetheory: D = + (cid:118)(cid:117)(cid:117)(cid:116) r (cid:88) a =0 ∆ a . (9)This quantity will appear in the normalizations of variousquantities in our models.Second, we represent fusion with a tri-valent vertex,which denotes combining the quantum numbers of twoof the incident lines to give the quantum number carriedby the third. The rules for these fusions share certainproperties with their more familiar analogues from thetheory of angular momentum addition: (a) if the quan-tum number of any of the three lines is trivial (vacuum),then the other two lines carry the same quantum num-ber. (I.e, a trivalent vertex may have one incoming par-ticle labeled i , one outgoing particle labeled i and a thirdparticle labeled 0). (b) Generically the result of fusingtwo particles’ lines is not unique – the product may carryone of several possible quantum numbers. This is analo-gous to two spin particles fusing to either a singlet ora triplet (c) Not all trivalent vertices represent allowedcombinations.As with conventional angular momentum addition, onecan change bases (from + basis to singlet and tripletbasis, for example) and a wavfunction will appear differ-ent in the new basis. The relation between two differentbases is given by generalized 6j coefficients, called F ma-trices or recoupling coefficients, and is shown in Fig. 3(a).We can think of this equivalence between two sets of di-agrams as being simply a basis change. As shown in Fig.3(c), the F matrices involving the trivial particle encodethe values of the quantum dimensions of the theory via∆ i = (cid:0) F i ∗ i i ∗ i (cid:1) − . Fig. 3(d) shows a relationship whichwill be useful for evaluating partition functions below. k lmj i = F ijmkln kj n li (a) ij = F i ∗ i jj ∗ n ij n ij (b) i = F i ∗ i i ∗ ia δ a i ai = F i ∗ i i ∗ i ∆ i i (c) ij kln m = F mk ∗ lkm ∗ ∆ m F mi ∗ njlk ∗ i kj (d) FIG. 3. Basic relations of tensor categories. (a) shows thedefining relation for the F matrices. (b) shows how thesedefine fusion between two strings. This prescribes a set ofallowed vertices. (Throughout the figure, the dotted line rep-resents the 0 particle). (c) shows the consequence of applyingthe rule in (b) to a single string, defining the relationship be-tween fusion and the quantum dimension ∆ i . (d) shows asimple consequence of fusion. Third, we must know how to describe braiding (or ex-change statistics) in an anyon theory. This informationis encoded in a matrix R , as shown in Fig. 4. As onemight expect for consistency of anyonic statistics, R is apure phase, and obeys (cid:16) R ijk (cid:17) ∗ = R jik . (10)While the world line diagrams are drawn in two- i jk = R ijk i jk i j = ! k F i ∗ i jj ∗ k j iki j = ! k F i ∗ i jj ∗ k R ijk i jkj i FIG. 4. The un-crossing relations used to define the tensor R which fixes a braiding structure on the modular tensor cate-gory. dimensions, they obviously represent a process in threedimensional spacetime. A crucial part of the structureof anyon theories (and link invariants in general) is thatthe value of the diagram will be independent of whichtwo-dimensional projection is chosen. This independenceof projection is guaranteed by certain consistency condi-tions on the R and F matrices which must be obeyed byany anyon theory (See Refs. 11,36,37).One important consequence of the rules describedabove is that in every anyon theory, we can define anΩ string, via Ω ≡ D r (cid:88) i =0 ∆ i | i (cid:105) (11)which has the Killing property described above in sec-tion II A . (Here we have slightly abused notation andwritten the right hand side as a ket vector to emphasizethat the Ω string is just a linear sum over particle la-bels on that string. ) A string labeled with Ω does notrequire an orientation, as the sum over all particles nec-essarily includes all antiparticles; hence we represent Ωby an un-oriented string.As noted in section II A 1, the evaluation of a link ofΩ’s in a manifold M can be considered as a topologi-cal invariant of a different 3-manifold M (cid:48) , which is ob-tained from M by a process known as surgery. (Thisconnection will be described in detail in sections V B andAppendix C 3 below.) The resulting manifold invariant,known as Z W RT ( M (cid:48) ) (which stands for Witten , Reshi-tikhin and Turaev ), is precisely the Chern-Simons vac-uum partition function Z CS ( M (cid:48) ) in the case of a Chern-Simons theory. It is this connection between links labeledwith the element Ω, and the topological invariant Z W RT ,which lies at the heart of the connection we establishbetween topological lattice models and doubled anyontheories.Note that any anyon theory has a ‘mirror image’, whichis equivalent to the original anyon theory on the mirror image of the manifold or link under consideration. Al-ternately, the mirror image can be obtained by complexconjugating all of the R matrices and leaving the mani-fold and link unchanged. Thus a mirror theory has conju-gates value Z W RT ( M ) = [ Z W RT ( M )] ∗ where M meansthe mirror image of the manifold M . (For Chern-Simonstheories, the mirror image theory can be obtained bytaking taking the level k to − k ). We will refer to thesemirror image theories as left-handed – and hence to theoriginals as right-handed .We emphasize that the fusion rules in Fig. 3 do notdescribe the most general category (although they do de-scribe most simple examples). More generally two parti-cles i and j may fuse to k in multiple possible ways. If onehas such nontrivial “fusion multiplicities”, one needs toinclude an additional index at the fusion vertex indicatingwhich of the possible ways the fusion takes place. (See forexample, the detailed discussion in Refs. 11,36,37). Inthis more general case, each F matrix has four additionalindices. For notational simplicity, and following the leadof Levin and Wen, we do not include these explicitly here.A second issue occurs when the anyon theory containsparticles that are odd under time reversal. In this casethere are minus signs which enter into the translationbetween fusion diagrams and space-time world-line dia-grams. ( These factors are also explained in detail in Refs.11,36,37.) Again, for simplicity, we will not consider suchmodels here. We note, however, that these more generalcases can easily be described by our constructions as well. IV. THE GROUND STATE PARTITIONFUNCTION
In this section we will explain in detail a knot-theorybased construction of the Levin-Wen partition function,which we will relate to the Chain-Mail invariant intro-duced by Roberts in Sect. V. This section will focusonly on obtaining the zero-temperature vacuum parti-tion function of the Levin-Wen models ; quasi-particledefects will be considered in section VI below. Readersfamiliar with tensor categories should note that althoughour approach is most useful in cases where one starts witha valid anyon theory (a.k.a. a modular tensor category)and constructs the lattice model which gives the doubleof this input theory, the discussion of the current sectionis sufficiently general to apply to any Levin-Wen model. A. The Levin-Wen models
The models we will consider are a class of Levin-Wenmodels which describe doubled anyon theories. That is,beginning with a valid anyon theory, we construct a lat-tice model which ends up being equivalent to the achiral double of that theory. (We note that this is not the mostgeneral class of Levin-Wen model; we discuss a more gen-eral form of the approach presented here in Appendix D).0Here we briefly review the key features of these models;we refer the reader to Ref. 1 for more details.Analogous to the doubled Chern-Simons theory de-scribed above in section II B, the ingredients for this con-struction are a Hilbert space consisting of a set of edgelabels 0 . . . r , and two sets of projectors – one acting atvertices, the other at plaquettes– from which to buildthe Hamiltonian. The essence of the Levin-Wen modelsis that 1) all terms in the Hamiltonian commute, but 2)vertex and plaquette projectors are not simultaneouslydiagonal in the basis of edge labels. 1) ensures that themodel is exactly solvable, and 2) yields ground states thatcan be roughly described as a weighted superposition ofall possible edge labelings that satisfy certain “fusion”conditions at each vertex. This is a natural generaliza-tion of the toric code and RVB phase (in which groundstates are a superposition of all possible closed loops onthe lattice). Ref. 1 demonstrates that the ground statedegeneracy of these models depends only on the globaltopology of the 2d lattice (i.e, if the lattice forms a torusor a sphere etc), and that the low-lying excitations aboutthis ground state have anyonic statistics – in other words,they realize topological phases.The detailed construction of these projectors has a nat-ural interpretation in the language of tensor categories.The vertex projector at vertex V will be denoted B ijkV ,where i, j, k are the quantum numbers on the bonds in-cident to that vertex. The projector B V gives one if thevertex is allowed by the fusion rules of the category ,and gives zero otherwise. A configuration of quantumnumbers along the edge which satisfies all of these pro-jectors (except possibly at the position of quasi-particles,if there are any in the system) is known as a “string net”.The plaquette projectors, denoted B P , act on a setof edges surrounding an individual plaquette P to “flip”the quantum numbers of these edges without violatingthe vertex constraints, so that the ground state is indeeda superposition of string nets. This is done by addinga ‘string operator’ (with a particular linear combinationof labels) around the plaquette and fusing it into the ex-isting configuration of edge labels. The effect of such astring operator can be described by a chain of F -matricesdescribing the fusion processes involved. A string oper-ator labeled s which runs around a single plaquette P thus acts on the edge labels i ...i according to: B P ( s ) | i ...i (cid:105) = (cid:32) (cid:89) k =1 F e k i k − i ∗ k si (cid:48)∗ k i (cid:48) k − (cid:33) | i (cid:48) ...i (cid:48) (cid:105) (12)where i ≡ i , and e k is the edge entering vertex k fromoutside the plaquette, as show in Fig. 5. The plaquetteprojectors are a superposition of such string operators: B P = r (cid:88) s =0 a s B P ( s ) (13)where a s are constants. Ref. 1 shows that choosing a s = ∆ s / D (14) ensures that B P is a projector onto states in which theplaquette P contains no external sources, and hence canbe ‘filled in’ without punctures. (Other choices of a s project onto states in which an anyon world-line pierces P . From the perspective of the lattice model, theseanyons are not excitations of the theory, but rather topo-logical defects – specifically, punctures carrying flux – onthe plaquette.) i si i i i i e e e e e e (a) i ! i ! i ! i ! i ! i ! ssss ssi i i i i i i i i i i i e e e e e e (b) = ! k =1 F e k i k − i ∗ k si k i k − i ! i ! i ! i ! i ! i ! e e e e e e (c) FIG. 5. String operator representation of the plaquette pro-jector B P ( s ). (a) The projector is constructed by running alabeled string s (shown in red) around the interior of a plaque-tte. (b) Fusing this string into the edges of the plaquette. (c)Evaluating the bubbles using the relation of Fig. 3(d) givesa map between different labelings consistent with the vertexprojectors. The Levin-Wen Hamiltonian consists of applying theseprojectors to each vertex and plaquette in the honeycomblattice: H = (cid:15) V (cid:88) V (1 − B V ) + (cid:15) P (cid:88) P (1 − B P ) (15)where (cid:15) P and (cid:15) V are the mass gaps for vertex- andplaquette- violatIng excitations, respectively. In the re-mainder of this section, we will show how to evalulatethe partition function for the Hamiltonian, Eq. (15) inthe limit (cid:15) P , (cid:15) V → ∞ , in a way that makes its rela-tionship to the Chern-Simons formulation discussed inSect. II apparent.The technically informed reader will notice that forsimplicity we treat cases where there are no fusion multi-plicities. The more general case is a straightforward ex-tension of this, in which the Hilbert space also includes1variables at the vertices of the honeycomb lattice to in-dicate the relevant fusion channel, as explained in Ap-pendix A of Ref. 1. In this case the F matrices in theHamiltonian have four extra indices to track the four ver-tex variables involved in such a change of basis. All of theresults derived here are equally valid in the presence ofmultiple fusion channels, provided that the lattice modelis constructed from a valid anyon theory. B. Pictorial construction
We now give a graphical construction of the partitionfunction for the Levin-Wen models discussed in the pre-vious section. The essence of this construction is exactlythat described for Chern-Simons theory in Sect. II: werepresent the partition function as a graph – or world-line diagram – consisting of labeled loops encircling theperimeters of plaquettes in the 3D lattice, and projec-tor loops encircling its edges. (The expert reader shouldnote that this construction can also be made for tensorcategories that do not admit a braiding structure. Wediscuss this in more detail in Appendix D). We evaluatethe partition function using a Trotter decomposition ap-proach. That is, we discretize time in short steps andtrace over each time slice separately, such that Z = T r (cid:89) e − H δτ (16)where δτ is a small imaginary time step and the productis over many such small steps. Hence we will visualizethe theory as living on a 3D lattice, with 2D honeycombplanes stacked in the time direction, referring to edgeswithin a honeycomb layer as space-like, and edges join-ing layers as time-like. The reader should note that ourdiagrammatic approach calculates the partition functionin imaginary time; one may view this either as a ther-mal description of a classical model, or as related to thequantum theory by analytic continuation.There are two elements that make this construction ofthe partition function work. First, in Sect. IV B 1, weshow that in the limit that the gap for spontaneously ex-citing quasi-particles is infinite , the operator e − ∆ τH LW isexactly equivalent to acting with a product of vertex andplaquette projectors on the state at time τ . Second, inSect. IV B 2, we establish that the action of the requiredproduct is given by evaluating the Chain-Mail diagram.In particular, along the time-like edges, projecting ontothe 0 string effectively enforces the same constraint as B V : the net quantum number entering each vertex inthe lattice must be 0 at each time step. Similarly pro-jecting onto the 0 string along space-like edges ensuresthat Ω loops around spatial plaquettes give the action of B P .
1. Trotter Decomposition
Before discussing the utility of this geometrical con-struction, let us pause for a moment to understand thedetails of the Trotter decomposition of the partition func-tion of Eq. (15). We may express the Levin-Wen parti-tion function in imaginary time as Z LW = T r N (cid:89) i =1 e − H δτ i (17)= T r N (cid:89) i =1 e − [ (cid:15) V (cid:80) V (1 − B V )+ (cid:15) P (cid:80) P (1 − B P ) ] δτ i (18)= T r (cid:34) N (cid:89) i =1 (cid:89) V e − (1 − B V ) (cid:15) V δτ i (cid:89) P e − (1 − B P ) (cid:15) P δτ i (cid:35) (19)where the third line follows because all projectors in thetheory commute. Here we imagine computing the parti-tion function by discretizing the full imaginary time in-terval τ into N time steps, each of duration δτ i = τN .Further, since B P and B V are projectors, we have (1 − B α ) n = 1 − B α , and hence: e − (1 − B α ) δτ = 1 + ∞ (cid:88) n =1 ( − (cid:15) α δτ ) n n ! (1 − B α ) (20)= e − (cid:15) α δτ − ( e − (cid:15) α δτ − B α (21)Plugging this into the partition function Eq. (17) gives: Z LW = T r (cid:40)(cid:89) V (cid:34) N (cid:89) i =1 ( e − (cid:15) V δτ i − ( e − (cid:15) V δτ i − B V ) (cid:35)(cid:89) P (cid:34) N (cid:89) i =1 ( e − (cid:15) P δτ i − ( e − (cid:15) P δτ i − B P ) (cid:35)(cid:41) (22)= T r (cid:40)(cid:89) V ( e − (cid:15) V τ − ( e − (cid:15) V τ − B V ) (cid:89) P ( e − (cid:15) P τ − ( e − (cid:15) P τ − B P ) (cid:41) where the last equality follows from the fact that( e − (cid:15) α δτ + (1 − e − (cid:15) α δτ ) B α )( e − (cid:15) α δτ + (1 − e − (cid:15) α δτ ) B α )= ( e − (cid:15) α ( δτ + δτ ) + (1 − e − (cid:15) α ( δτ + δτ ) ) B α )(23)In other words, the result is independent of the time slic-ing.Now, if we take the limit (cid:15) α τ → ∞ , effectively restrict-ing the trace to the ground state sector, Eq. (22) reducesto Z LW | T =0 = T r (cid:89) V B V (cid:89) P B P (24)2So, in fact, we may obtain the partition function by sim-ply evaluating this operator over a single time slice. How-ever, since B nV = B V and B nP = B P and further all of the B V and B P are mutually commuting, we are free to ap-ply these operators at every time slice and we will obtainthe same partition function. Thus we may write equallywell Z LW | m α τ →∞ = T r N (cid:89) i =1 (cid:34)(cid:89) V B V ( i ) (cid:89) P B P ( i ) (cid:35) (25)where by B V ( i ) and B P ( i ) we mean to apply B V and B P at time slice i respectively. In fact, this is the form of thepartition function that we will actually use.Our ability to manipulate the partition function in theabove ways relies heavily on the fact the Hamiltonianis made of such simple mutually commuting projectors–or equivalently, that the partition function is topologi-cal, which guarantees that Z LW is independent of thetime slicing. (Indeed, the typical Trotter decompositionis implemented precisely to deal with noncommuting op-erators, which we do not have). Given this, it may seemto be overkill to calculate the partition function by usinga full three-dimensional lattice as in Eq. 25 rather thanonly with a single time slice as in Eq. 24. However, sucha representation will be crucial below. First, we will beable to relate this three dimensional form to previouslystudied mathematical quantities such as the Turaev-Viroinvariant and the Chain-Mail invariant . Second, andmore importantly, when we consider quasi-particles inour theory, we will want to consider quasi-particles fol-lowing world lines in space time; collapsing the systemdown to a single time slice, as in Eq. 25, will be insuffi-cient to describe this physics.
2. Detailed Construction
It remains to show that the expression (Eq. 25) for thepartition function can be obtained by applying the rulesoutlined in Sect. III to the appropriate Chain-Mail dia-gram of anyon world lines. To do so, we first argue thattime-like plaquette loops correctly propagate edge vari-ables between time slices. Second, we will show how thecombination of Ω loops around the edges and plaquettesgives the product of projectors (25). The reader shouldnote that Ω is not strictly a projector; rather D − Ω haseigenvalues 0 and 1. To make the correspondence tothe lattice Hamiltonian explicit here we use projectors,rather than Ω loops, to implement the Hamiltonian. (Wewill show in Sect. V A that these factors of D cancel, anddo not alter the normalization of the partition function.)First, as in Sect. II, we represent states in this pic-ture in terms of oriented closed curves representing anyonworld lines. Thus for each space-like edge labeled i t attime t , we draw a closed loop above this edge and label it i t , as shown in Fig. 6. We may imagine this string as theworld-line of a particle-antiparticle pair, which has beencreated at time t , and annihilated at time t + δt . Wetherefore call such strings particle loops , to distinguishthem from the strings which comprise the Hamiltonian.It is instructive to consider evaluating the diagram con-taining only these world-lines and the space-like edge pro-jectors. This describes particle world-lines propagatingunaltered through time — i.e., a Hamiltonian which iszero. To see this, consider a space-like edge with a par-ticle loop i t − δt below, and i t above (green loops in Fig.6(a)). Applying the projector (yellow rings in Fig. 6(a)along their shared edge at time t , as shown in Fig. 6,fuses the two strings i t − δt , i t in a bi-valent vertex on eachside of the projector. The result can be interpreted as aworld-line for the particle-anti-particle pair i t − δt propa-gating from time t to time t + δt . In this way, addingprojectors along all bonds lying in space-like directionsresults in a picture of an initial set of labels propagatingunaltered through this time slice. In other words, thespace-like edge projectors have the effect of taking an in-ner product between i t − δt and i t (which are orthogonalunless the labels i t − δt and i t are identical).To be precise, we must normalize this inner productcorrectly. To do so, we multiply each edge loop i by ∆ i .This is because fusing i t − δt and i t gives a factor of δ i t − δt ,i t F i ∗ t i t i ∗ t i t = 1∆ i t (26)for each edge. Hence we must give each state a weight of∆ i to cancel these factors and ensure that the time evolu-tion effected by the space-like edge projectors is unitary.To obtain the partition function Eq. 25, we must add the Hamiltonian to this picture. The vertex projectors B V correspond to adding edge projectors to the time-likeedges, as shown in Fig. 6c. Since these projectors forcethe three lines at the vertex to fuse such that their totalquantum number at the vertex is 0 (i.e., such that thethree incident quantum numbers are an allowed fusion),they clearly reproduce the effect of B V . To ensure thatthe correspondence is exact, one must keep track of thevarious coefficients induced by fusion. This calculation isgiven in Appendix A.3 (a) (b)(c) FIG. 6. The pictorial Levin-Wen model, drawn on the hon-eycomb lattice tiled in time. To construct the final picture( the full Chain-Mail link with both vertex and plaquetteprojectors) shown in Fig. 2, a closed loop is drawn aroundevery plaquette in the lattice, and a projector is applied oneach edge. Here we show the intermediate steps in evaluat-ing the diagram. The thick golden lines show the plaquettesin the honeycomb lattice; strings encircling the time-like pla-quettes, which serve to label the states in the model at eachtime slice, are show in green. (a) and (b) Propagation of edgelabels without the Hamiltonian. The yellow rings show theprojector loops on space-like edges. Applying these projectorspropagates the label of each loop forward in time. (c) Apply-ing vertex projectors (purple rings) between space-like layersforces the three edge labels incident at each vertex to fuseto the identity, satisfying the criteria for a string net groundstate.
To add the plaquette string operator B P ( s ), we run astring labeled s around the boundary of the hexagonalplaquette P , passing it through all of the projectors onthe edges of P . The edge projectors force this string tofuse with the edge labels, producing (after some diagram-matic algebra, given in Appendix A) exactly the productof F -matrices in Eq. (12). The action of the plaquetteprojectors B P is thus included by labelling all space-like plaquettes by D − r (cid:88) i =0 ∆ i | i (cid:105) . (27)Fig. 7 shows the effect of applying both Hamiltonianprojectors to a single plaquette at a given time slice.Including edge projectors above each vertex, and pla-quette projectors on every space-like plaquette, we obtain4) (a) (b)(c) i ( t + δt ) e ( t ) i ( t + δt ) i ( t ) i ( t ) s (d) FIG. 7. Applying edge projectors in a single time slice to implement the action of the Hamiltonian on the edge labels of asingle plaquette. (a) The plaquette with edge variables (green) at time t fused beneath each vertex. (b) Fusing the stringcorresponding to the plaquette projector (shown in blue) to the edge variables at time τ (dark green) and τ + δτ (light green–not shown in (a)) by implementing the space-like edge projectors (yellow rings in (a)). Implementing the vertex projectors attime τ + δτ requires fusing the open green strings to each other (not shown). (c) Alternative picture of (b). Here we haveomitted the vertex projectors at τ + δτ , and pulled the (light green) strings carrying these edge labels down into the plane.The bubbles in this diagram can be collapsed to trivalent vertices using the fusion relations, just as in Fig. 5. (d) The diagramwhich remains at each vertex can be evaluated using the identity of Fig. 3(d). The details of this calculation are given inAppendix A. Here e ( t ) = e ( t + δt ). precisely the expression given in Eq. 25 for the partitionfunction. To see this, it suffices to consider a single timeslice. Applying the projectors above each vertex annihi-lates configurations with nonzero total quantum numberentering or leaving a vertex; on the remaining (allowed)configurations, it gives: (cid:89) { P l } (cid:32) r (cid:88) s =0 ∆ s D (cid:89) k =1 F e k i k − i ∗ k si (cid:48)∗ k i (cid:48) k − (cid:33) (28)times the diagram for the remaining time steps.Eq. (28) is precisely the factor we expect from (cid:81) { V i } B V i (cid:81) { P l } B P l at this time step, given a particu-lar set of edge labels. To obtain the zero temperaturepartition function, we then sum over all labelings i of theedges, with appropriate weights ∆ i as described abovein Eq. 26. This is equivalent to labeling each edge loopwith r (cid:88) i =0 ∆ i | i (cid:105) ≡ D| Ω (cid:105) . (29)In summary, the pictorial representation of the fullTrotter decomposition of Eq. 25 is as follows: first wefill the manifold with layers of honeycomb lattice at each (discrete) time step. Draw a labeled loop above eachedge in each honeycomb layer to specify the value of theedge variable in that time slice. To operate with B V ona vertex, apply a projector to the time-like edge on thatvertex. To operate with B P on a plaquette, draw the ap-propriate superposition of strings (cid:80) a s s around that pla-quette. Finally, add a projector on each spacelike edge,to take the inner product between the states representedby time-like loops above and below each time-slice, withthe appropriate string operators sandwiched in between.All projectors, as well as the time-like plaquette loops,are strings of the form (27).Up to normalization, then, the diagram correspondingto the ground state partition function is thus generatedby 1) drawing a closed loop around each plaquette inthe 3 D lattice, labeling each loop with the element Ω,2) applying a projector onto 0 quantum number to eachedge of the 3 D lattice by encircling it with a loop labeledΩ and 3) evaluating the coefficients given by the resultantfusions. The final resulting diagram, the Chain-Mail link,is shown in Fig.1b.Physically, one way to think of the diagrams describedhere is rather complicated pictures detailing the actionof a transfer matrix. Imposing the edge projectors ona time-like edge gives the appropriate transfer matrix5element for B V acting on the 3 incident edges. Simi-larly, imposing the edge projectors on all space-like edgestouching a given plaquette yields the appropriate matrixelements of B P between the initial edge configurations–represented by the labeled strings which emenate fromthe previous time slice, and run ‘backwards’ along theedges of the plaquette– and the final configurations, rep-resented by strings which extend upwards towards thenext time slice, and run ‘forwards’ along the edges of theplaquette.
3. Classical and quantum partition functions
It is worth pausing to clarify one question which somereaders may have at this point: if the lattice models weare interested in are truly quantum creatures, how is itthat we can describe their partition functions in terms ofstate sums, such as the Turaev-Viro invariant, which areessentially classical entities? Viewed as a gauge theory,our system is already quantized – indeed, this quanti-zation is essential to the relation between expectationvalues of Wilson lines and knot polynomials. However,viewed as a theory of labeled strings, the link used toevaluate the partition function is essentially a classicalobject: each component of the link is labeled with thesame element | Ω (cid:105) ; thus quantum superpositions are notrequired to define the Chain-Mail link (or, consequently,the partition function).In fact, quantum mechanics enters only if we wish toexamine the state at a particular instant in time by cut-ting the manifold open at a particular time slice. Thisis characteristic of Chern-Simons theories, in which thequantum mechanical behavior is captured by the appro-priate conformal field theory at the boundary. To exposethis quantum theory in the lattice model, we must firstpinch off the projectors above each vertex, so that theslicing will not cut through any of the strands of theChain-Mail link. In pinching off these vertices, however,we effectively limit the labels incident at each vertex suchthat only the combinations which fuse to 0 flux remain.At this point it is no longer sufficient to think of each edgeas labelled by the single ket | Ω (cid:105) , as this fusion treats eachstring label | i (cid:105) in the superposition differently. (In otherwords, the projectors are diagonal in the basis of stringlabels, rather than a basis in which | Ω (cid:105) is a basis vector).Hence the sum over labels on the time-like loops is trans-formed into a sum over all possible intermediate stringnet configurations. V. DOUBLED ANYON THEORIES ANDRELATION TO THE CHAIN-MAIL INVARIANT
Armed with the diagrammatic representation of theLevin-Wen partition function described in Sect. IV B,we may now return to the main theme of this work –namely, exploring and exploiting the connection of some of the Levin-Wen models to doubled anyon theories (in-cluding doubled Chern-Simons theories). Since such the-ories have been extensively studied both as field theoriesin physics, and as topological invariants in mathematics,this is a useful framework from which to study the latticemodels.To make the desired connection, we will begin byintroducing the Chain-Mail invariant Z CH of Ref. 27described briefly in section II above. Roughly speak-ing, Z CH is obtained by associating a link diagram (theChain-Mail link, L CH ) with a space-time 3-manifold, andevaluating the resulting link invariant (denoted (cid:104) L CH (cid:105) ). (cid:104) L CH (cid:105) is evaluated using the diagrammatical rules ofan anyon theory laid out in Sect. III. Remarkably theChain-Mail invariant Z CH is precisely the same as theinvariant of Turaev-Viro which itself is the same as the square of the Witten-Reshitikhin-Turaev invari-ant Z CH = Z W RT Z W RT . (Here the overline denotes ei-ther complex conjugation of the result, or, equivalently,evaluating the manifold invariant for the mirror imagemanifold.)There are many equivalent links associated with agiven space-time manifold (associated with different lat-tices which can be used to tile space-time – or more tech-nically, with different Heegard splittings of the space-time manifold); we shall see that one of these producesexactly the same link diagram which we associated withthe Levin-Wen partition functions in Sect. IV B. We thenexplore what the other, equivalent, diagrams tell us aboutthe lattice model.
A. The Chain-Mail link and the lattice partitionfunction
A general prescription for constructing the Chain-Mail link L CH is as follows . Given any 3D lattice,wrap a string just inside the perimeter of each plaquette;encircle the plaquette strings that run adjacent to eachedge by another string, and label all strings with theelement Ω. On a cubic lattice, the result is a picture with4 strings running parallel to each edge, linked togetherby an Ω loop. In the case of the 3D lattice which isthe space-time representation of the Levin-Wen model’shoneycomb lattice, the result is exactly the link shown inFig. 6.Using this prescription to construct the Chain-Maillink, the Chain-Mail invariant is given by Z CH = D − n v − n c (cid:104) L CH (cid:105) . (30)(Here D is the total quantum dimension from Eq. 9 and n v is the number of vertices of the lattice and n c is thenumber of 3d cells of the lattice ) Here, (cid:104) L CH (cid:105) meansthat one should evaluate link invariant of the Chain-Maillink using the combinatorial rules of the given anyon the-ory.The Chain-Mail diagram defined here, and the link di-agram presented in Sect. IV B to represent the lattice6partition function, are identical links. Hence in evaluat-ing the two diagrams using the combinatorial rules out-lined in Sect. III, the only difference is in the normaliza-tion: the strings are labeled as specified in Eqs. (29) and(27) for the lattice partition function, and by Ω for theChain-Mail link. To check that the normalizations agree,we must count the factors of D = (cid:112)(cid:80) ri =1 ∆ i in the twodescriptions. Compared to the projectors we used in theconstruction of the Levin-Wen partition function in sec-tion IV B 2, the Chain-Mail link obtains one factor of D for every edge (since we apply projectors along bothspace-like and time-like edges) and for each space-likeplaquette of the lattice. On the other hand, to obtain theLevin-Wen construction as in section IV B 2 we found inEq. 26 that we should normalize the sum over edge vari-ables by multiplying all edge variables with a factor of∆ i . In the Chain-Mail link, the time-like plaquette Ωloops correspond to the edge variables, but are normal-ized to include a factor of ∆ i / D . Thus, the Chain-Maillink obtains a factor of D − for every time-like plaquette,cancelling the factor of D we obtained for the associatedspace-like edge. This leaves one extra factor of D forevery time-like edge and for every space-like plaquette.However, the number of time-like edges is equal to thenumber of vertices n v , and the number of 3d cells n c equals the number of space-like plaquettes. (This count-ing is valid provided we take periodic boundary condi-tions in imaginary time, and the topology of space re-mains fixed during the evolution). Hence the prefactorof the Chain-Mail invariant D − n v − n c precisely cancelsthese factors and we end up with the same normalizationin both cases.In fact, there is some arbitrariness in the normaliza-tion of the topological invariant, which is evaluated rel-ative to a reference manifold whose partition functionwe choose to be 1, as explained in Sect. II A 1. Physi-cally, however, this is not the case: we expect the zero-temperature partition function to count the ground-statedegeneracy. The simplest spatial manifold with ground-state degeneracy 0 is the sphere S (2) ; if we use periodicboundary conditions in imaginary time, we conclude that Z LW ( S (2) × S (1) , T = 0) = 1. The two agree in this casebecause the Chain-Mail invariant is also normalized suchthat Z ( S (2) × S (1) ) = 1. B. Surgery, Handle-Slides, Topological Invariantsand Invariance
In the previous subsection, we showed that the Chain-Mail invariant is none other than the partition functionof a Levin-Wen Hamiltonian constructed from an anyontheory. Here we will discuss how topological invariancecan be understood in the context of these link invariants.This will give us a convenient framework for discussingtopological invariants of quasi-particle world-lines, whichwe will exploit in Sect. VI.We will focus on two observations about the partition function of the lattice model which are apparent fromthe topological properties of the link invariant. First, wecan relate the partition function of the lattice model ona space-time manifold M to that of a theory with no Ωloops on a space-time manifold M M , through a pro-cess called surgery which we will outline below. Thisis the generalization of the connection pointed out inSect. II between these lattice models and doubled contin-uum Chern-Simons theories. Second, the Chain-Mail in-variant has the property, known as the handleslide prop-erty, which implies that certain re-arrangements of thelink components do not alter the partition function. Suchre-arrangements are an interesting tool for visualizing thetopological nature of the partition function.
1. Surgery and Handleslides
We begin with a description of the two propertieswhich we will use in the remainder of this section. (Fora pedagogical introduction to the subject, see Ref. 45).For the purposes of this work, a technical descriptionof surgery will not be necessary; the interested readermay consult Appendix C 3. The essential point is thatsurgery gives a way of establishing an equivalence be-tween partition functions of anyon theories on pairs ofspace-time manifolds M and M (cid:48) with different topolo-gies. The manifolds are related by ‘performing surgery’on one or more closed loops in M to produce M (cid:48) . Infact, surgery is an essential component of the interestingcorrespondence between links in the 3-sphere S and thetopological classification of 3-manifolds. Specifically, itis a well-known, but nontrivial result , that any closed3-manifold can be described by starting with S and per-forming surgery on an appropriate link . This isthe root of the correspondence between topological in-variants of 3-manifolds, and invariants of the correspond-ing links.Another feature of surgery which has interesting con-sequences for the lattice model is that certain re-arrangements can be made to the link in M withoutaffecting the topology of M (cid:48) . Thus, in order fora link invariant to be a topological invariant of a 3-manifold, it must also remain invariant under these samere-arrangements. For our purposes, the most importantof these is the handle-slide property shown in Fig. 8.The handle-slide property is a statement about howloops in a link can ‘slide’ over one another, whilst leavingthe value of the link invariant unchanged. The handle-slide is shown in Fig. 8 a: quite literally, we slide the(red) string on the left over the (blue) loop, irrespectiveof what other strings pass through the loop’s center. Oneway to construct such a link invariant which is unchangedby handle-slides, is to label all strings with Ω. An intu-itive justification for this is shown in Fig. 8 b: Usingthe Killing property of Ω, we may replace any stringsthreaded through Ω by the identity or vacuum string.Since there are now no strings linked with the Ω loop, we7 ji k = ji kji k Ω ∼ ji k Ω ji k FIG. 8. The handle-slide property. A loop (blue) is saidto have the handle-slide property if any string can be de-formed freely around it in the manner shown in (a) withoutchanging the value of the link invariant associated with thediagram. Fig. (b) shows the intuitive reason why Ω loops havethe handle-slide property: Using the Killing property of Ω weforces all strings which pass through the Ω loop to fuse to theidentity, allowing the external string to slide freely across theloop as shown. Once the string has been slid across the loopon the right-hand side, the strings i, j, k may be reattachedthrough the middle of the Ω loop again (exactly cancellingthe coefficient F i ∗ i jj ∗ k ∗ F kk ∗ kk ∗ incurred due to fusion) to givethe right hand side of Fig. (a). may now freely slide any string around this loop with-out obstruction. The fusion of encircled strands onto thevacuum can be un-done once the string has been pulledaround the Ω loop, giving the original diagram with ahandle-slide.
2. Insights on topological invariance
The surgery and handleslide properties give numerousinsights into the nature and properties of the topologicallattice models described with the Chain-Mail link. Herewe describe the most interesting among these.
Lattice theory and continuum gauge theory:
Throughsurgery, we can make an explicit connection between thelattice model and (when a Chern-Simons anyon theoryis used) doubled Chern-Simons theory. The latter canbe obtained not by taking the naive continuum limit,but rather by identifying both the edge variables (appro-priately summed over to give the ground-state partitionfunction) and the terms in the Hamiltonian with anyonworld-line loops, and performing surgery on the resulting link.
Doubling:
Surgery gives one route to understandinghow the lattice theory, which is in general constructedfrom labels in a chiral anyon theory, gets doubled. Ref. 18showed that performing surgery on the Chain-Mail linkof a manifold M produces M M . Here M is the mirrorimage of M , and the Z W RT ( M M ) = Z W RT ( M ) Z W RT ( M ) , (31)this shows that the lattice model, whose partition func-tion is given by the Chain-Mail invariant Z CH = Z W RT ( M M ), is a doubled anyon theory. In appendixC 3 we elaborate further on this connection, examin-ing how the rigorous mapping works even when “quasi-particle” defects are inserted into the Chain-Mail link. Turaev-Viro invariant : Roberts showed that theChain-Mail invariant is rigorously equivalent to theTuraev-Viro invariant, which has been understood todescribe the Levin-Wen ground state partition function.(We give some details of the argument orignially pre-sented in Ref. 18 in appendix C 3). Hence we can bothestablish a rigorous correspondence between the two, andunderstand how the models of Levin and Wen differfrom the Turaev-Viro invariant in the presence of quasi-particle defects (which we will explore below in sectionVI). Independence of Lattice : the Chain-Mail constructioncan be carried out on any lattice, and gives the sameresult – that is, it is a topological invariant of the space-time manifold and does not depend on the particularlattice discretization. We will discuss the implicationsof this for possible alternative Hamiltonians, on latticeswith different geometries, in Sect. VII.Further, once a lattice is selected, it is possible tocoarse-grain the Chain-Mail description without alteringthe partition function. Specifically, as emphasized byRef. 18, in the absence of excitations it is always possibleto eliminate all but a finite number of the plaquettes inthe lattice through a series of handle-slides. (The spe-cific number will depend on the topology of the space-time manifold, and in particular the minimal number of1 and 2-handles required to construct it). This proceduredemonstrates an exact equivalence between the partitionfunction of the lattice model with an arbitrary choice oflattice constant, and a simple product of a finite numberof F -symbols. Indeed, it gives a geometrical understand-ing of the result found by Ref. 25 and 26, who constructan algebraic coarse-graining procedure which is exact forthese topological phases. Commutativity of operators in H : The handle-slide ofΩ also gives a convenient picture of various manifesta-tions of topological invariance in our picture of the Levin-Wen Hamiltonian. For example, with some effort one canshow algebraically that the Hamiltonian is comprised of8commuting projectors. In the Chain-Mail picture, thishas a simple geometrical meaning: Ω-loops on adjacentplaquettes can handle-slide past each other, changing theorder in which they are to be evaluated in time. Similarly,an Ω plaquette loop can handle-slide (along a time-likeplaquette loop) past an edge projector. Hence handle-slide invariance requires that all operators in H commute.Ref. 1 motivates the choice of commuting projectors byarguing that the Hamiltonian of a purely topological the-ory should play the role of imposing an appropriate setof constraints on the wave-functions; the Chain-Mail pic-ture gives an interesting alternative route to understand-ing why commuting projectors are the appropriate build-ing blocks for the Hamiltonian of a topological latticemodel. Independence of time-slicing : At the end of Sect.IV B 1, we argued that as the Hamiltonian is comprisedof commuting projectors, the partition function does notdepend on how many time steps are used in the Trotterdecomposition, or on whether we evaluate all plaquetteoperators simultaneously or at different times. (Here weassume that the topology of space is fixed during the evo-lution). From the point of view of the partition function,this freedom is a consequence of the fact that all oper-ators in H commute. Hence a time step may be ‘sub-divided’ into separate applications of B P – or collapsed,so that all B P ’s act simultaneously. In practice, we cando this in the Chain-Mail picture by handle-sliding anΩ-loop which effectuates an application of B P from onetime slice to the next – in a manner similar to that usedto establish commutativity above. Interestingly, fromthe point of view of the Chain-Mail link, this freedomis required for topological invariance: it corresponds tochanging the lattice by simultaneously adding or remov-ing a plaquette (2-handle) and a 3-cell (3-handle –in thiscase, a solid ball filling in the cells of the lattice). Thistransformation does not alter the topology of the space-time manifold, and hence leaves the partition functionunaltered. Thus again, we find a fundamental connectionbetween commuting projectors and Hamiltonians withtopological ground states.To summarize, by using a pictorial construction of theLevin-Wen partition function at zero temperature, wearrive at a natural correspondence to the Chain-Mail in-variant. This makes explicit the relation of Levin-Wenmodels to doubled Chern-Simons theory. Further, it un-derlines the relationship between topological invarianceand exactly solvable Hamiltonians written in terms ofcommuting projectors. Finally, in this language we arenaturally lead to consider some of the flexibilities of theLevin-Wen models, such as analogous construction on ar-bitrary lattices. VI. QUASI-PARTICLES IN THE PICTORIALMODEL
We now turn to the question of understanding thequasi-particle defects of the theory. We note that,whereas the topological invariants discussed in the abovesections (Chain-Mail, Turaev-Viro), which correspond tothe ground state of the Levin-Wen Hamiltonian, havebeen well studied in the mathematical literature, not allof the situations with violations of terms in the Hamil-tonian which we discuss here have, to the best of ourknowledge, yet been studied (but see Refs. 19 and 20).Since there are two types of terms in the Hamiltonian,the vertex terms and the plaquette terms, we shouldthink about quasi-particle defects that violate one orboth, of these terms. A useful intuition for these twotypes of quasi-particles, which is exact in the case ofAbelian models , is the idea of electric and magneticdefects in analogy to gauge theories: electric defects vi-olate vertex projectors, and magnetic defects violate theplaquette projectors. More generally, though, we shouldrefer to these types of defects as vertex or plaquette de-fects. Here we will explain how to construct each typeof violation in the pictorial model, by inserting extrastrings (quasi-particle world lines) into the Chain-Maillink. This gives geometrical insight into the difference be-tween the two types of quasi-particles, which is reflected,via surgery, in their relationship to Chern-Simons theory. A. Partition Functions with quasi-particles
Before delving into the details of how quasi-particleworld-lines are included in the Chain-Mail link, let usunderstand what the Chain-Mail invariant computes out-side of the ground-state sector – i.e., in the presence ofquasi-particle world lines. As emphasized in Sect. II, thepartition function that we compute is a topological in-variant of the space-time – which is possible only if it isevaluated at T = 0. Thus strictly speaking, the quasi-particle world-lines that we insert are perhaps best con-ceived of as sources created in the system along somespace-time path by an external field, rather than as par-ticles generated by thermal fluctuations. The new groundstate contains quasi-particles following the world-lines ofthese sources; the Chain-Mail invariant then computesthe ground-state partition function in the presence ofthese quasi-particles. Alternatively, we can view theChain-Mail link as capturing only the topological por-tion Z top of the partition function at finite temperature–that is, it captures the physics of the linkings of quasi-particle world lines around non-contractible curves in thespacetime, or around each other, but is insensitive to the e − E g τ contribution of the quasi-particle creation energy E g . This is apparent in the discussion of Sect. IV B 1,where we see that the correspondence between the par-tition function and the evaluation of the pictorial modelis exact only at 0 temperature.9What is this topological contribution from the quasi-particle world-lines to the partition function? In the sim-ple case where world-lines do not enclose non-contractiblecurves on the manifold, we will show presently that theymay be detached completely from the rest of the linkby handle-sliding. In this case Z top factors into the con-tribution of the ground state partition function, timesa contribution from the world line link. More gener-ally world lines may link around non-trivial topology inthe manifold, in which case they can also mix differentground state sectors in the theory. Hence, Z top tracksthe topology of the spacetime manifold, and the linkingof quasi-particle world lines with each other and aroundnon-contractible loops in the space-time. B. Brief Review of Quasi-particles in theLevin-Wen models
Ref. 1 describes all possible excitations by construct-ing operators which act on all edge states i k along somecontinuous path in the lattice. These string operators areconstructed from products of “simple” string operators,which are given by: O s = (cid:89) k F ks ω ks (32) F ks ≡ F e k i ∗ k i k − si (cid:48) k − i (cid:48)∗ k if s turns left at V k F e k i ∗ k − i k si (cid:48) k i (cid:48)∗ k − if s turns right at V k ω ks ≡ ω i k i (cid:48) k s if s turns right then left at V k , V k +1 ω i k i (cid:48) k s if s turns left then right at V k , V k +1 V k are the vertices, i k , i (cid:48) k are the states along thepath of the string before and after fusion with the string s , and e k is the state on the external leg (not traversedby the string) at V k . The F ks are the F -matrices of theconstituent anyon theory that the Levin-Wen model isbuilt on. These are precisely what we obtain by runninga string labeled s along the chosen path, and applying aprojector on each edge to force it to fuse trivially with thetwo states i k , i (cid:48) k . Note that factors ω ks introduce phaseseach time the string crosses from one plaquette to an-other. As we will see below, these ω matrices are closelyrelated to the R matrices of the constituent anyon theory.The operators for closed strings can be shown to com-mute with the Hamiltonian, Eq. (15), whereas operatorsfor open strings connect a particle-anti-particle pair atthe opposing ends of the string. If the phases ω ks aretrivial, then the particle-anti-particle pairs at the ends ofthe strings violate the vertex projectors only. In order tocreate violations of the plaquette projectors at the endsof the strings, it is also necessary to introduce nontrivialvalues of the phases ω ks . C. Quasi-particles in the Chain-Mail picture
We now turn to the task of constructing these quasi-particles in the pictorial model. The basic idea is thatpassing an additional string (labeled s ) through an Ωloop (projector) forces all other strings passing throughthis loop to fuse to s ∗ , rather than to 0 (since s ∗ mustfuse with s to give 0). For example, threading the pro-jector loop above vertex V with a string labeled s fusesthe string labels of the edges entering V to s ∗ , ratherthan 0, creating a defect in the ground state which vio-lates the vertex constraint B V . This suggests that quasi-particles should be added as labeled strings, representingquasi-particle world-lines, linked appropriately throughthe various projectors of the Chain-Mail link. Here wewill first describe how these world lines thread the space-time link, and then explain how they are interpreted inthe Hamiltonian language. We will see in Sect. VI D 2that depending on whether we link this new world linewith vertex projectors or with both plaquette and andvertex projectors, we will get either the right- or left-handed quasi-particle sectors respectively. This gives aninteresting explanation and proof of conclusions aboutquasi-particle statistics reached by Ref. 23. Further, asexpected for the doubled theory, these two sectors willbe completely decoupled. (We refer to the original anyontheory as “right” handed, and its mirror image as “left”handed).
1. Threading quasi-particle strings into the Chain-Mail link
Let us begin by describing how to construct the twotypes of quasi-particles in the Chain-Mail picture. To in-sert vertex violating quasi-particles, we run a string (rep-resenting the quasi-particle’s world line) labeled s alongthe edges of the lattice, linking it through the projectorfor every edge it runs along (Fig. 9(a)). We show shortlythat such strings are associated with right-handed ex-citations, and violate only the vertex terms B V of theHamiltonian. We call these quasi-particles “R-particles”for short.Violations of the plaquette operators (Fig. 9(b)) areconstructed in a similar fashion — by running a labeledquasi-particle world-line along some set of edges in thelattice. However, to create a violation of the plaquetteoperators B P , this string must pass through the loopsencircling the space-like plaquettes. We also stipulatethat such a string links through a time-like plaquette ev-ery time it crosses into a new plaquette in space . Inother words, whenever a string passes between cells of the3D lattice, it must cross through a plaquette (thus be-coming linked with the corresponding plaquette string),rather than an edge. Such strings are associated withleft-handed excitations (landing in M after surgery, aswe will explain shortly) in the doubled theory; we willcall this type of quasi-particle “L-particles”. Note thatbecause they pass through both edge and plaquette loops,0L-particles violate both types of projectors. To constructwhat we would normally associate with a “magnetic” de-fect (violating only plaquette terms) in fact requires acombination of L- and R- particles.To avoid ambiguity, it is possible to formalize the rulesfor choosing trajectories corresponding to the two quasi-particle types. The trajectory for an R particle consists ofa closed curve along the edges of the (3D) lattice. Thisfixes a sequence of edge projectors through which thestring is linked, and hence fully describes its action onthe states in the theory. The trajectory for an L particleconsists similarly of a closed curve on the edges – butin this case we must also specify a continuous trajectoryalong the 3-cells bordering these edges, in order to fixwhich plaquettes the string is linked with. The rules forselecting these 3 cells are simply that for each edge on thepath, there must be one 3 cell which contains this edge,and that only 3-cells containing at east one vertex on thepath of edges can be included. When passing betweenadjacent 3-cells, the string links with the plaquette stringon the face shared by the two 3-cells. This fixes a seriesof plaquettes ‘adjacent’ to the edges on the world-linetrajectory through which the string is to be linked.To understand these strings in the Hamiltonian view-point, we must consider the effect they have on the eval-uated link invariant. As both types of strings threadthrough the edge projectors, applying these projectorsforces them to fuse with the variables on each edge thatthey traverse. If the edge in question lies in a spatialplane, the string is linked only through space-like edgeprojectors and, in the case of L-strings, time-like plaque-tte loops. Since neither of these loops comprise terms inthe Hamiltonian, such a string has no associated ener-getic cost. Rather, they can be thought of as operatorsacting on the quantum state at this time step. (Indeed,evaluating the projector along this edge gives exactly theoperator F sk of Eq. (32), and hence gives precisely thestring operators of Levin and Wen).Excitations in the Hamiltonian picture occur when astring follows an edge that is oriented in the time direc-tion. In this case the string travels through a vertex pro-jector, creating an excitation at this vertex by forcing thequantum numbers of the edges to no longer fuse to thevacuum. In the case of L-particles, the world-line linksalternately with vertex projectors and plaquette projec-tors, creating violations of both.From the point of view of the state of the system inthis time slice, the 2 D string operator terminates at thelast vertex (or vertex and plaquette, for L-particles) thatit enters, leaving a source at this vertex (vertex and pla-quette). This mimics an open string in the Levin-Wenformulation described above. In the 3 D pictorial repre-sentation, which is sensitive only to the topological partof the partition function, this distinction between spaceand time disappears, as the topological partition func-tion is indifferent to the energy gap. The 3 D world-linesare simply closed curves, and Z top depends only on howthey are linked with each other and with the spacetime manifold. (a)(b) FIG. 9. Quasi-particle trajectories. Quasi-particle strings areshown in bright red, and oriented according to the directionof positive imaginary time. All other loops are labeled byΩ. (a) Right-handed quasi-particles run along the centers ofthe thickened edges, passing through the edge projectors (yel-low and purple rings) only. (b) Left-handed quasi-particlesthread alternately through edge (yellow and purple) and pla-quette (blue) projectors, and can be thought of as living onthe plaquettes.
2. Equivalence to quasi-particles in the Levin-Wen models
The Chain-Mail link with quasi-particle world lines in-serted as described above is shown in Fig. 9. To showthat evaluation of the link invariant associated with thismodified Chain-Mail link gives Z top for the Levin-Wenmodels, we must show that strings lying along spatialedges should reproduce the effect of string operators inthe Hamiltonian theory.The link invariant is evaluated by choosing a 2D pro-jection in which to draw the (3D) link, and applying therules (described above in section III) for fusion and un-crossing which are specified by the anyon theory. The1 i i i i i i e e e e e e i i i i i i e e e e e e (a) i ( t + δt ) e ( t + δt ) e ( t ) i ( t + δt ) i ! ( t ) i ( t ) i ( t ) r s (b) FIG. 10. Evaluation of diagrams involving quasi-particleworld-lines. (a) The projection used to reproduce the Levin-Wen string operators. Here we look down at edge variables i t .L particles (at left) link with the plaquette loops which ex-tend downwards in time below the hexagon shown, and hencepass under the strings i t . R strings (right) are not linked,and pass over i t . (b) The result of contracting all projectorsabout a vertex containing a quasi-particle world-line. (This isthe equivalent of Fig. 7(d) in the case where a quasi-particleworld-line enters the vertex). As shown in Appendix A, thediagram evaluates to F i ( t ) e ∗ ( t ) i ( t ) r ∗ i (cid:48) e ∗ ( t + δt ) R ri ( t ) i (cid:48) ( t ) times the result ofthe vertex with no quasi-particle string. If the string turnsin the other direction, passing under the edge i instead of i , this factor is changed to F i ( t ) i ( t ) e ∗ ( t ) re ∗ ( t + δt ) i (cid:48) (cid:16) R ri ( t ) i (cid:48) ( t ) (cid:17) ∗ . Thisagrees with Eq. (32) once we account for the differences inthe string orientations. anyon theory satisfies certain consistency conditions en-suring that the result does not depend on the projectionchosen. It will prove convenient to choose a projectionfrom the positive t direction, looking down into the x − y plane. Let us consider a configuration with a single layer,together with time-like plaquettes below this layer to rep-resent the state of the system at this time slice, as shownin Fig. 10. In the projection that we have chosen, L-strings always cross under the strings from the time-likeplaquettes, since they are linked with these before pro-jection; the R- strings, conversely, always pass over thesetime-like strings. Hence L- and R- strings are assignedopposite phases at each crossing, as one would expect.Finally all edge projectors are applied; Fig. 10(b) showsdiagramatically that the result is simply a factor of R foreach crossing, together with a factor of F s as expectedfor a Levin-Wen string operator. The details of the dia-grammatic evaluation are explained in Appendix A. To see that this gives the same prescription in terms ofleft- and right- turns as Eq. (32), notice that if the R- (L-) string turns right and then left, it crosses over (under)a timelike plaquette string from right to left. Conversely,if it turns left and then right, it crosses over (under) aplaquette string from left to right. Hence by applyingthe un-crossing rules of Fig. 4, we obtain the expectedprescription for assigning phases to right- and left- turns,provided we choose our uncrossing tensor R to be thesame as ω in Eq. (32).An entertaining consequence of the Chain-Mail formu-lation is that the choice of phases in Eq. (32) is notunique. The notion of strings crossing depends on theprojection chosen to evaluate the link invariant – thoughthe end result does not. When the projection is into the x − y plane, as described above, the R - (vertex) particlesappear to cross over, and the L (vertex and plaquette)particles under, the time-like plaquette strings. If theplane of projection is orthogonal to one of the space-likeaxes, however, right-handed strings cross no other stringsafter the edge projectors are applied. From this angle,left-handed strings cross both over and under the time-like plaquette below each edge shared by a pair of plaque-ttes on the string’s trajectory. Fortunately the algebraicstructure of the modular tensor category guarantees thatthe end result will be independent of the angle of view,provided that the relative phases acquired by left- andright- handed string operators j when crossing an edgelabeled i t is ( R i t + δt i t j ) .As noted above, though our R- particles ( or “electric”particles in Abelian gauge theory) violate only vertexprojectors, the L- particles violate both vertex and pla-quette constraints. Thus the L- particles are not strictly“magnetic” excitations (violating plaquettes only) in thelanguage of Abelian gauge theory , but rather a combi-nation of magnetic and electric. A purely ‘magnetic’ ex-citation can only be constructed by taking both a right-and a left- handed string of the same label s (See alsoRef. 23). The left-handed string crosses under, and theright-handed string over, every plaquette loop which theleft-handed string links. We may fuse the two strings to-gether along each edge, as shown in Fig. 11. This fusionwill result in a superposition of labels for the resultingstrings along the edge, as shown in the Figure. As bothstrings carry the same label s , one element of this su-perposition is the 0 string. When the two strings fuseto 0, the result is an isolated loop labeled s encirclingeach plaquette string with which the left-handed quasi-particle was linked. This violates these plaquettes, but novertices, giving a purely magnetic excitation. (In generalthe fusion will generate a superposition of excitations.Only the component of this superposition correspondingto the 0 string corresponds to a purely ‘magnetic’ exci-tation which affects only plaquette variables. ) Interest-ingly, this shows that the purely magnetic quasi-particle(when it exists) is achiral .2 FIG. 11. Combining right- (dark red) and left- (light red)handed quasi-particle strings to create a magnetic excitation.In general fusing the two strings does not produce the 0 stringalong the edges, but rather a linear superposition of the al-lowed fusion products (pale purple and pale blue strings inthe right-hand Figure).
D. Statistics and topological invariance ofquasi-particle strings
Thus far, we have found that we can reproduce theoperators [Eq. (32)] by creating R- strings, which violatevertex projectors and are linked through edge loops, andL- strings, which violate both plaquette and edge pro-jectors and are linked through both edge and plaquetteloops. It is instructive, however, to ask how the topo-logical characteristics expected of quasi-particles in thedoubled theory are manifest in this construction. Thereare several issues to address here. First, the result shouldbe invariant under local geometric deformations of thequasi-particle world lines. Second, we must convince our-selves that the right- and left- handed sectors of the the-ory have the expected mutual statistics — in other words,that they do not interact with each other. Finally, wehave judiciously named the two types of quasi-particlesright- and left- handed — we must show that these namesare indeed justified.It turns out that all three of these are relatively easyto show using the handle-slide property. Handle-slidingallows any string s to slide over a loop labeled by Ω with- out changing the value of the link invariant, irrespectiveof what strings pass through the Ω loop. This allows usto slide quasi-particle strings around on the Chain-Maillink witout affecting the partition function– guarantee-ing that only the topology of the quasi-particle world-lines is relevant. Further, one important consequence ofthe handle-slide is that it changes the linking of s withany string that passes through the Ω loop. This playsan important role in understanding the statistics of thequasi-particles, as we shall see shortly.
1. Handle-slide prescriptions for quasi-particle strings
We will begin by describing the effect of handle-sliding(Sect. V B 1) on both types of quasi-particle strings.Though arbitrary handle-slides of each string type areallowed, we use a convention in which handle-sliding pre-serves the linking conventions of R- and L- particles, asspecified in Sect. VI C 1, respectively.The simplest such prescription for handle-sliding thetwo types of quasi-particles is as follows. R- strings needonly to slide over plaquette loops: since they are neverlinked with the plaquette strings, there is no obstruc-tion to sliding over them and the strings can be maneu-vered freely from plaquette to plaquette in this way (SeeFig. 12). L- strings must slide over both edge loops andplaquette loops in sequence. The first slide unlinks thequasi-particle string from the plaquette loop (in the pro-cess, linking it with another plaquette loops), and thesecond slide moves it across the plaquette (See Fig. 12).These maneuvers have the advantage that they do notintroduce additional twists into the strings, and also donot alter the prescription that L- strings must link witha plaquette loop each time they move between 3-cells,while R strings are linked only through edge projectors.Analogous to the R- strings, handle-sliding L- strings inthis way allows the string to move freely among the pla-quettes in the 3 D lattice. These two types of slides areillustrated in Fig. 12.It is not hard to convince oneself that any deforma-tion of the quasi-particle world-lines that does not changetheir linking with each other, or their winding aroundtopological features of the space-time manifold, can beachieved by an appropriate series of handle-slides. Hence,as promised, invariance of the value of the correspondinglink evaluation under handle-slides guarantees that thepartition function is completely independent of the localgeometry of the quasi-particle strings.
2. Statistics of quasi-particle strings
We may now consider evaluating the partition func-tion in the presence of quasi-particle world lines, withthe goal of understanding their statistics. We will findthat 1) R-strings are “right-handed”, in the sense thatthey obey the same statistics as the original anyon the-ory; 2) L-strings are “left-handed”, having the statisticsof the mirror anyon theory; and 3) R- and L- strings havetrivial mutual statistics. Further, if no quasi-particleworld lines encircle non-contractible loops of the space-time manifold, the partition function factors into sep-3 (a)(b)
FIG. 12. Handle-slide prescriptions for right- and left- handed quasi-particles. Plaquettes to be slid over are highlighted inturquoise. (a) right- handed quasi-particles move around the diagram by handle-sliding over plaquette loops. (b) left-handedquasi-particles slide first across a plaquette loop, and then through edge loops, to un-link from the plaquette. arate ground-state and quasi-particle contributions; wewill show this by un-linking the quasi-particle world-linesfrom the Chain-Mail link entirely.To understand the effect of adding quasi-particles tothe theory without resorting to a brute-force evaluationof the partition function, we may use two powerful tools:surgery and handleslides. First, we may perform surgeryon the Chain-Mail link in the presence of quasi-particles,and track the locus of the quasi-particle world-lines. Sec- ond, we can use handle-slides to visualize more directlywhich quasi-particle world lines can pass through eachother on the lattice, and which cannot. Though less gen-eral than the first approach, the second is more straight-forward, and will be the focus of this section. The moretechnical surgery approach is discussed in Appendix C 3.First, we verify that there are two non-interacting sec-tors. That is, any link of R and L strings can be reduced,via handle-slides, to separate R -particle and L -particle4links. To un-link R and L strings using handle-slides,we may slide the L string over an edge projector throughwhich the R string passes. Since the R string is not linkedto any of the adjacent plaquettes, the L string may nowbe ‘pulled through’ without further affecting the cross-ing, thereby unlinking the two strings (See Figure 13a).This tells us immediately that the right and left handedparticles form independent sectors of the resulting theory(i.e, that the two sectors have trivial mutual statistics).Careful inspection of the handle-slide rules in Fig. 12shows that strings of like chirality cannot be un-linkedby handle-slides. For the R- strings, the reason is clear:these strings slide only over plaquette loops, with whichthey are never linked. Hence the linking of R- stringswith each other is never altered by handle-slides.For the L- strings, the situation is slightly more com-plicated: if two L- strings s and t cross, in attempting toslide s past t we must slide it over both an edge projec-tor and a plaquette projector with which t is linked. Thestrings appear un-linked after the first slide – if t initiallycrossed over s , it now crosses under – but cannot be sep-arated without re-linking, and restoring the over-crossingof the initial configuration. Hence two left-handed stringsalso cannot be un-linked by handle-slides.Thus by handle-sliding, we see that R- and L- stringsconstitute two sectors with mutually trivial statistics.With this knowledge in hand, let us understand thestatistics of the two sectors. To do so, we will factorthe partition function: that is, using handle-slides, wereduce the evaluation of the Chain-Mail invariant withquasi-particle strings (provided these do not encircle non-trivial topology in M ) to the separate evaluations of theChain-Mail link (given by the Chain-Mail invaraint withno sources) and the evaluation of the link invariant of thequasi-particle world lines: Z ( W C , ...W C N ) = (cid:104) L CH (cid:105)(cid:104) L QP (cid:105) (33)where W C i is the Wilson lines – or more generally anyonworld-lines – of the external source on the curve C i . L QP denotes the link of these anyon world lines after theyhave been separated from the Chain-Mail link. Once thepartition function is factored, it is easy to evaluate thequasi-particle statistics.To factor Z , we begin by using handle-slides to un-link all of the L- particles from all of the R- particles(since they can freely slide through each other). We nextperform a series of handle-slides to un-link all remain-ing world-lines from the Ω loops of the Chain-Mail link.The result is illustrated in Fig. 13. In the case of theR- quasi-particles, the resulting link, after being sepa-rated from the Chain-Mail scaffolding, has the same self-linking as when it was attached to the scaffolding (sinceas mentioned above, handle sliding never changes the self-linking of the R- particles). A rather more involved ar-gument (see Appendix B) shows that L-particles can alsoslide entirely off the scaffolding, but that in the processall over-crossings become under-crossings, and vice versa— giving the mirror image of the original world-line link. (a)(b)(c) FIG. 13. Evaluating quasi-particle statistics using handle-slides. Quasi-particle strings are shown in red; all other loopsare labeled with Ω. (a) Passing an L string through an R string can be accomplished by handle-sliding the (verti-cal) L string over a space-like edge projector (shown here inturquoise). (b) An R string link in the lattice model, and theresulting link, seen here from above, after handle-sliding offall Ω loops. (c) As in (b), but for an L string. Note that thedirection of the crossing is reversed in the process of handle-sliding. Hence by examining the world line links after they havebeen separated from the Chain-Mail link in this way, weconclude that the statistics of the R-particles are pre-cisely given by the statistics of the particles from theconsituent anyon theory, and those of the L-particles, byits mirror image.Hence handle-sliding provides a convenient visualiza-5tion of quasi-particle statistics. However, if the quasi-particle world-lines do encircle non-contractible curvesin the spacetime (for example, if they wind around pe-riodic boundaries in space, should our manifold be thetorus cross time), one must resort to the more rigorousapproach of tracking the position of each type of quasi-particle world-line after surgery. Recall that surgery onthe Chain-Mail link gives two connected copies of theoriginal space-time, with opposite orientations. (That is,surgery on the Chain-Mail link in M produces M M ).As show in Appendix C 3, we find that after surgery, Rstrings land in the right-handed copy of the manifold,and L strings land in the left-handed copy. This gives amore general proof that the two types of quasi-particleworld lines must have opposite chirality.In summary, for doubled anyon theories, we find thatviolating vertex projectors or the combination of ver-tex and plaquette projectors creates independent right-and left- handed quasi-particle sectors. By handle-slidingover the Chain-Mail link which describes the ground-state partition function, we see that the picture is invari-ant under geometrical deformations of the world-lines,and that the right and left handed sectors have preciselythe statistics that we expect of the doubled theory. Math-ematically speaking, this follows from the fact that thetwo quasi-particle types land in opposite handed copiesof the manifold after surgery. VII. ALTERNATIVE FORMULATIONS OF THEHAMILTONIAN
From the point of view of topological quantum compu-tation, or more generally condensed matter physics, themost pressing question is how to design experimentallyrealizable systems which are likely to exhibit topologicalphases. One significant obstruction to doing this in theLevin-Wen models is the apparent complexity of the in-teractions. The plaquette projectors act on all 6 edges ofa plaquette in the honeycomb lattice, and the result ingeneral also depends on the states of the 6 external legsentering the vertices at this plaquette. An obvious ques-tion, then, is whether the pictorial description allows usto re-express the model Hamiltonians in a simpler form.
A. Non-trivalent geometries
One is tempted, at first, to try to utilize the fact thatthe correspondence to the Chain-Mail invariant guaran-tees that the partition function is independent of the lat-tice chosen. The Hamiltonian will still be comprised of(local) vertex projectors, and plaquette projectors. How-ever, one might hope that plaquettes with fewer edgesmight lead to simpler, more local, plaquette projectors.There are in fact two complications here. First, wemust understand what happens to B V when the valenceof a vertex is changed. The vertex projector can always be implemented as a single operator, enforcing the con-straint that the net flux entering the vertex is 0. Fromthis perspective, the vertex projector is equally physicalfor any valence of vertex — in the Chain-Mail picturewe simply wrap an Ω around all the strings entering thatvertex. However, since the rules of the fusion categoryspecify only how to fuse three strings at a point, to evalu-ate the link invariant, one ultimately must evaluate sucha projector in terms of string configurations with onlytrivalent vertices . In other words, greater than threestrings can be fused successively to determine their jointquantum number. However, the intermediate quantumnumbers may take multiple values and these values mustbe summed over after the projection. As noted Sect. III,such additional quantum numbers (multiple fusion chan-nels) associated with vertices are actually something thatcan generically occur even at trivalent vertices; for sim-plicity we have so far chosen not to consider this case.However, when we consider vertices with valence greaterthan three, additional quantum numbers at vertices al-ways occur, except for in a very few trivial (Abelian) the-ories. Thus, while the higher valence case is in principlesimilar, in practice it becomes more complicated.The second complication in attempting geometricalsimplification is with the plaquettes themselves. The pla-quette projector always acts on all edges in a plaquette,and is sensitive to the state of all external legs. In gen-eral, if the valency of each vertex is v and there are n V vertices per plaquette, the final state on the n V edges inthe plaquette depends on the initial state of all n V ( v − n V , we decrease the number of states which B P alters – but the total number of edges involved inconstructing B P is still n V ( v − B P appearssignificantly more local, in this sense: for example, onthe hexagonal, square and triangular lattices, we have n V ( v −
1) = 12, 12, and 15, respectively. One may alsoconsider more complicated (or even irregular) lattice ge-ometries. For example, the so-called Cairo-pentagon lat-tice tiling and the prismatic pentagon lattice tiling bothshare the feature with the honeycomb and square latticesthat a LW model based on these lattices would coupleonly 12 edges at a time.
B. Duality transformations
A more promising approach to render the Hamiltonianmore plausible is to attempt to construct models in whichit is natural to impose the constraint of 0 flux throughedges on both the lattice and the dual lattice. As is ap-parent in the Chain-Mail construction, for doubled anyontheories the plaquette projectors serve merely to imple-ment the constraint of 0 flux entering each vertex of thedual lattice; hence these projectors are perfectly local inthe dual lattice description. This yields a picture quitesimilar to that proposed by Ref. 12, in which a quan-6tum loop gas model was constructed with a Hamiltonianconsisting of two sets of vertex projectors – one on thelattice, and one dual lattice. (The difference here is thatthe mapping between lattice and dual-lattice projectorsinvolves all n V ( v −
1) edges required to construct B P ;the construction of Ref. 12 is slightly different and ren-ders the final result somewhat simpler.)The seeming non-locality of B P , then, can be seen asinduced by the change of basis from the dual lattice pic-ture, in which B P is diagonal in the edge variables, tothe lattice picture (in which it is B V that is diagonal).The most obvious way to leverage this fact is to con-struct a model in which B V = 0 is a constraint on theHilbert space in the absence of quasi-particle sources; inthis case the Hamiltonian consists of a single, maximallylocal, term. This would essentially be a generalizationof Yang-Mills theory in the strongly deconfined limit: inthe lattice version of this theory, B V = 0 is the constraintthat the electric field be divergenceless; B P gives the lat-tice action for the magnetic field. In the deconfined limitthe coupling of the E term is 0, and the Hamiltonianmay be expressed in the magnetic basis as a term actingat a single vertex on the dual lattice. Conceptually atleast, this picture is quite natural; indeed, one way tomotivate the Levin-Wen models is as generalizations oflattice gauge theories . VIII. CONCLUSIONS
We have seen how, for Levin-Wen models constructedfrom modular tensor categories, the relationship betweenthe partition function – both for ground state and excitedstate sectors – and the Chain-Mail invariant (decoratedwith quasi-particle world lines, where appropriate) allowsus to connect these models to doubled Chern-Simons the-ory, and more generally to doubled anyon theories. Indoing so, we uncovered several interesting facts aboutthe physical models. First, the particular combinationof terms in the lattice Hamiltonian of Ref. 1 can beunderstood as arising from a link (the ‘picture’ of themodel’s partition function) which in fact corresponds af-ter surgery to a right- and a left- handed copy of theoriginal space-time manifold. There are therefore twonon-interacting sectors in the excitation spectrum – oneassociated with each copy – and the resulting model isachiral. When we translate these excitations back intothe more familiar language of vertex (electric) and pla-quette (magnetic) violations, we find that electric excita-tions are chiral, while magnetic excitations are not. Fur-ther, when the quasi-particle strings do not enclose non-contractible curves, we can easily evaluate the topologicalpart of the partition function, as it factors into ground-state partition function and a piece that represents thequasi-particle excitations only.We would also like to point out that in the contextof the Chain-Mail and Turaev-Viro invariants, the con-struction of the left-handed particle is not, to the best of our knowledge, discussed in the existing literature–though previous authors have considered introducingright-handed and achiral quasi-particle strings. Inthis sense our discussion completes the connection be-tween the work of Roberts , which introduces the Chain-Mail invariant for the ground state, and that of Witten ,and Reshetikhin and Turaev , which describes both theground state partition function and quasi-particles in theun-doubled theory. By constructing both left- and right-handed quasi-particles, we are able to describe the fulldoubled Chern-Simons theory by adding lines of gaugeflux to the Chain-Mail invariant.Besides making the above connections between Levin-Wen models, Chern-Simons theory, and the Chain-Mail(or equivalently, Turaev-Viro) invariant, our work opensseveral interesting new directions. From the point of viewof physics, the main challenge is to utilize the flexibilityinherent in the Chain-Mail construction to connect theserather abstract model Hamiltonians to more physicallymotivated systems. In a future publication, we will alsoaddress the question of whether the above constructioncan be replicated on 3D lattices to generate a topologi-cally non-trivial theory.Another interesting direction is whether this construc-tion can be usefully extended beyond the case of doubledanyon theories. Indeed, the lattice models described byRef. 1 encompass topological phases which are not de-scribed by doubled anyon theories, and hence do not fitinto the framework outlined in this work. (For example,perhaps the simplest topological theory that can be con-structed by the Levin-Wen prescription, the Toric-code ,is not a doubled anyon theory.) By replacing the encir-cling edge loops with generalized projectors, the groundstate partition function can be constructed as describedin Sect. IV B without requiring any information aboutbraiding . However, in this case the formulation is farless instructive, as we know of no analogue of surgery inthis case to relate these models to a continuum descrip-tion.The ultimate goal of understanding how to engineer atopological qubit remains beyond the scope of our presentunderstanding. Nevertheless the search for topologicalphases has inspired a new approach to finding topologi-cal theories in physics – namely, through the constructionof topological lattice models. The deep connections be-tween these lattice constructions, combinatorial topolog-ical invariants as they are studied in mathematics, andthe topological field theories familiar from high-energyphysics, reveal an elegant unity between these seeminglydisparate approaches which, we hope, unveils interestingquestions in all three areas. Appendix A: Fusion coefficients in the ground statepartition function
Here we will track the details of the fusion coefficientsto prove that our partition function is exactly equivalent7to that of the corresponding Levin-Wen Hamiltonian.For many of the results derived here, the following sym-metry relations of the F -symbols will prove useful : F ijmkln = F lkm ∗ jin = F jimlkn ∗ = F klm ∗ ijn ∗ (A1)We begin by adding the vertex projectors B V , whichfuse the 3 strings i t , j t and e t entering each vertical edgeat a trivalent vertex. That is, we first fuse i t and j t togive k t , pushing one of the two ensuing trivalent verticesthrough the projector loop. Next we fuse k t , with e t ,again pushing one of the two trivalent vertices throughthe projector. The line through the projector is now thefusion product of k t with e t , and must be the identitystring. Hence we conclude that k t = e ∗ t , and the fusionincurs a factor of 1∆ e t F j ∗ t j t i t i ∗ t e ∗ t (A2)at each time step.If there are no space-like plaquette strings present, ap-plying projectors along the space-like edge labeled i re-sults in a bivalent vertex between i t and i t + δt , etc, forcing i t = i t + δ t ≡ i along each edge. The coefficient for thisfusion process is F i ∗ t i t i ∗ t i t = it , which precisely cancelsthe factor of ∆ i t included in the partition function. Ap-plying the vertex projector in the next time slice resultsin the diagram shown in Fig. 14. j t i t e t FIG. 14.
This diagram evaluates to F e t i t j t i ∗ t e ∗ t ∆ e t ∆ i t . Combining thetwo factors (fusion below the vertex due to the projector,and evaluation of the diagram) gives: F e t i t j t i ∗ t e ∗ t F j ∗ t j t i t i ∗ t e ∗ t ∆ i t (A3)which is in fact unity, (as justified in Fig. 15) as expected.This holds in general: acting with B V at a vertex to fusethe 3 strings together will induce a coefficient which is i kj = F j ∗ j kk ∗ i ij kj k = F j ∗ j kk ∗ i F ki ∗ jik ∗ ij kk = F j ∗ j kk ∗ i F ki ∗ jik ∗ ∆ k i kj FIG. 15. Graphical justification for the identity F e t i t j t i ∗ t e ∗ t F j ∗ t j t i t i ∗ t e ∗ t ∆ i t = 1. exactly cancelled by the coefficient obtained by evaluat-ing the closed diagram obtained after imposing B V bothabove and below the vertex at the preceding time slice.Next, we must add the plaquette projectors. To seethat these gives the correct form for B P ( s ), we will showthat the net effect of adding a plaqutte string s to thepicture is to induce a factor of F ei t j ∗ t sj ∗ t + δt i t + δt at each ver-tex, and flip the labels from i t , j t to i t + δt , j t + δt . Again,at each vertex V the strings i t , j t , and e t are fused ata trivalent vertex by the projector on the time-like edgeemanating from V . As illustrated in Fig. 7, the space-like edge projectors around the plaquette now fuse thestring s to the strings i t and i t + δt along each edge, toproduce the diagram of Fig. 7(d) at each vertex (with i t ≡ i ( t ) , j t ≡ i ( t ) , e t ≡ e ( t )). This fusion incurs afactor of it + δt F i ∗ t i t ss ∗ i t + δt along each edge. The factor of it + δt cancels the factor of ∆ i t + δt which we have explic-itly included in the partition function. Applying the fu-sion rules in Fig. 3, we may collapse the bubble at eachvertex, leaving a trivalent vertex between the 3 strings i t + δt , j t + δt , and e t . Including the factors incurred by fus-ing s into the edges, this diagram comes with a coefficient: (cid:16) ∆ s F si ∗ t + δt i t i t + δt s ∗ F i ∗ t i t ss ∗ i t + δt (cid:17) F e t i t j ∗ t sj ∗ t + δt i t + δt (A4)where we have used the symmetry relation of Eq. (A1) F sj ∗ t + δt j t e t i t i ∗ t + δt = F e t i t j ∗ t sj ∗ t + δt i t + δt . (A5)The term in parentheses is unity, due to the relationshown in Fig. 15. As noted above, the coefficientsdue to fusion along the time-like edges will always giveunity once all vertex projectors are imposed. Hence, aspromised, the effect of the plaquette string is to inducea coefficient F e t i t j ∗ t sj ∗ t + δt i t + δt at each vertex, and interchangethe labels i t , j t with the labels i t + δt j t + δt . After imposingthe space-like edge projectors, then, the plaquette string s acts by fusion on the labels around a plaquette to giveexactly the product of F -matrices used to define B P ( s ).8 a. Multiple strings and quasi-particles In fact, it is easy to generalize this to the case of sev-eral extra strings acting on the vertex. Again, we mayapply all edge projectors except the one above the ver-tex, to obtain a diagram with 3 external legs. Thesedigarams can always be reduced to the 3-vertex by a se-ries of applications of the identity shown in Fig. 3 d and,for quasi-particle strings, the un-twisting move of Fig. 4a. The extra ∆ coefficients cancel, as do the F factors forfusing the strings together, such that the effect of pass-ing this extra string through is always to multiply by afactor of F (or RF for quasi-particle strings which crossthe edge variables). In general these diagrams will con-tain internal edges, whose labels correspond to those nei-ther of the initial variables i t , j t , e t nor the final variables i t + δt , j t + δt , e t + δt . These can be consistently assigned la-bels i t ..i t n , etc. – in other words, we may equally wellsubdivide the action of all string operators at a vertexsuch that each operates at a different intermediate timestep t k .To clarify the above description, let us consider addinga single quasi-particle string r to the vertex i t , j t , e t , asshown in Fig. 16. r j t si t e t FIG. 16.
The digram shows the 3 strings i t , j t , e t , which havebeen fused to a trivalent vertex by the application of B V at time t , as well as a plaquette string s and thequasi-particle string r . Note that as r crosses under i t , itrepresents an L-particle; R particle strings at time t crossover i t , j t , and e t (but under i t + δt , j t + δt , and e t + δt ).To evaluate this diagram, we first fuse r to the edges i t and e t which it traverses. Next, fuse the string s as-sociated with B P ( s ) to the edges i t , j t on which it acts.Finally, apply the in-plane edge projectors, which fusethe 3 strings i t , j t , e t shown above to i t + δt , j t + δt , e t + δt .This results in the diagram show in Fig. 17, with thecoefficient: F r ∗ r e t e ∗ t e t + δt F i ∗ t i t rr ∗ i (cid:48) t F i (cid:48)∗ t i (cid:48) t ss ∗ i t + δt F j t j ∗ t ss ∗ j ∗ t + δt . (A6)The arrows on e t and i (cid:48) t are taken to have the same ori-entation as those on e t + δt and i t , respectively. e t + δt e t i t + δt j t + δt i t i ! t sr j t FIG. 17.
The diagram is evaluated by first un-twisting r and i t using the identity shown in Fig. 4a, and then usingthe identity of Fig. 3 d twice to collapse the resultingdiagram to a trivalent vertex. The result is a trivalentvertex i t + δt , j t + δt , e t + δt multiplied by the coefficient (cid:16) ∆ r F r ∗ e t + δt e ∗ t e ∗ t + δt r F r ∗ r e t e ∗ t e t + δt (cid:17) (cid:16) ∆ s F si ∗ t + δt i (cid:48) t i t + δt s ∗ F i (cid:48)∗ t i (cid:48) t ss ∗ i t + δt (cid:17)(cid:16) F i ∗ t i t rr ∗ i (cid:48) t F j t j ∗ t ss ∗ j ∗ t + δt (cid:17) R ri t i (cid:48) t F r ∗ i (cid:48) t i ∗ t j t e ∗ t e t + δt F sj t + δt j ∗ t e ∗ t + δt i (cid:48) t i ∗ t + δt (A7)The factors on the first line give unity (see Fig. 15), oncewe account for the fact that the symmetries of the F symbols guarantee that F r ∗ r e t e ∗ t e t + δt = F e t e ∗ t r ∗ re ∗ t + δt F si ∗ t + δt i (cid:48) t i t + δt s ∗ = F i ∗ t + δt si (cid:48) t s ∗ i t + δt . (A8)Similarly, the two F factors in parenthesis in the secondline will cancel corresponding factors at the adjacent ver-tices traversed by the strings r and s , respectively. Thisleaves a net factor of: R ri t i (cid:48) t F r ∗ i (cid:48) t i ∗ t j t e ∗ t e t + δt F sj t + δt j ∗ t e ∗ t + δt i (cid:48) t i ∗ t + δt = R ri t i (cid:48) t F j t e ∗ t i t r ∗ i (cid:48) t e ∗ t + δt F e ∗ t + δt i (cid:48) t j t sj t + δt i t + δt (A9)at each vertex (where we have again used symmetries ofthe F symbols to obtain the equality). Taking ω ijs ≡ R isj for R-particles, and ω ijs = (cid:0) R isj (cid:1) ∗ = R sij for L-particles(which cross over, rather than under, the edge i t ), wesee that the net effect is equivalent to first applying thequasi-particle string operator (c.f. Eq. (32)) runningfrom edge i t to edge e t , which gives the matrix element ω i t i (cid:48) t r F j t e ∗ t i t r ∗ i (cid:48) t e ∗ t + δt , and then applying B P ( s ) to the result-ing state, giving the matrix element F e ∗ t + δt i (cid:48) t j t sj t + δt i t + δt . (Notethat the orientations of the edge labels j and e in Fig.17 differ from those of Fig. 5(c), resulting in different la-bels appearing in their conjugate representations). Henceadding quasi-particle strings to the chain-mail link hasprecisely the same effect on the partition function as act-ing with the quasi-particle string operator of Eq. (32).The derivation here can be extended inductively toshow that any number of strings can be added to thediagram, and will contribute to the partition function9a product of string operators. One useful consequenceof this is that the partition function does not differenti-ate between adding all plaquette projector strings in thesame time slice, and adding each at a separate time. Appendix B: Handle-slides of right- and left- handedquasi-particles
Here we describe in more detail the handle-slide ar-gument for the statistics of the quasi-particles. As ex-plained in the main text, R-particle strings that do notencircle non-contractible curves in the spacetime can bedetached from the scaffolding by a a series of handle-slides over plaquette loops only. Since they are not linkedwith these, it follows that the link has the same chiralityafter removal from the scaffoding as it did when threadedthrough the Chain-Mail link. Hence the statistics of Rparticles are exactly those of the original theory.For the L particles, however, the situation is morecomplicated. To separate any section of a link from theChain-Mail requires a series of handle-slides which nec-essarily include slides over both plaquette and edge Ω-loops. The handle-slide prescription described in Sect. 8ensures that, where possible, these slides occur only overΩ-loops from which the string has already been un-linked.Hence, we first slide over an edge loop to un-link witha particular plaquette, and only then handle-slide overthe plaquette. However, if the L string is non-triviallyknotted, it is not possible to slide it off the lattice with-out handle-sliding over Ω loops with which the stringis linked. This changes the self-linking of the L string,and ultimately as we shall see the chirality of the knot.More generally, if two L strings are linked, they cannotbe separated from the Chain-Mail link using handle-slideswithout handle-sliding one string over Ω loops with whichthe other is linked, thereby changing the chirality of theirlinking.It is helpful to consider the case of a single crossingof L strings. In order to bring this crossing ‘off’ thescaffolding, we must first perform handle-slides until bothstrands are in the same horizontal slice. This processalways involves handle-sliding one strand over a plaquetteloop with which the other is linked, changing the crossingfrom an over-crossing to an under-crossing. As pointedout in the text, if we wish to continue separating the twostrands in this direction, additional slides over edge loopsmust be performed which re-link the two strands with thesame chirality as before, so that the two L stands havenon-trivial mutual statistics. However, if we merely wishto slide the L-string link off the Chain-Mail link, then westop at this intermediate stage.To be more precise, we may slide the L-string off theChain-Mail link by first performing handle-slides so thatthe entirety of the link sits in a single spatial plane S , sothat its strands are linked only through plaquettes, andthrough edge projectors in this plane. This can be donein such a way that the only plaquettes with which the string is linked are time-like plaquettes above S . Oncethe link has been positioned in this way, the handle-slidesrequired to detach it from the Chain-Mail involve slidingover edge projectors sitting above S , and plaquette loopswhich sit in S . As the world-lines comprising the link areunlinked from both of these types of loop, once the linkhas been maneuvered into the plane in this way it slidesoff the lattice without further alteration to its linkingstructure.But, as noted above, when sliding the two strands ata crossing such that the crossing lies in, and linked onlyto time-like plaquettes above, a single spatial plane, allover-crossings and under-crossings are reversed. Hencethe link, when separated from the scaffolding, has theopposite chirality as it does when drawn as quasi-particleworld lines in the lattice model. Appendix C: Surgery considerations
Here we review in more detail the effect of surgery onthe Chain-Mail link itself, and track the location of thetwo types of quasi-particle world-lines. Our ultimate goalis to provide an alternative, more mathematically satis-fying explanation of why right- and left- handed quasi-particle strings are in fact right- and left- handed. Toachieve this, however, requires a digression into the pro-cess of surgery itself.
1. Surgery: a brief description
Let us start with a brief review of the concept ofsurgery. Dehn surgery is a prescription for constructingany closed 3-manifold from a reference closed 3-manifold(usually taken to be the 3-sphere S ) and a (framed) linkwithin that manifold. The prescription for performingsurgery on a link is as follows: Consider a single strandof the link. Topologically, this strand is a circle S , al-though it may be embedded in the manifold nontrivially– i.e, it may be knotted with itself or with other strandsof the link. Now thicken this strand into a solid torus, S × D with S being the strand of the link, and D being a small disk cross section where we thickened thestrand. (If the strand is knotted or linked with otherstrands, the torus is non-trivially embedded in S ). Nowcut this solid torus from the manifold, leaving a boundarythat is the surface of the solid torus, T = S × S wherethe first S is the original strand of the link, and the sec-ond S is the boundary of the thickening disk D . (Againthis may be nontrivially embedded but topologically this T is just a standard torus surface). To obtain a newmanifold without boundary, we can sew onto this result-ing T boundary any other manifold whose boundary isalso T such that the two boundaries meet and “cancel”leaving a new manifold without boundary. While thereare many manifolds that might have such a T surface,the simplest would just be a solid torus. One trivial pos-0sibility is to sew back in the same torus S × D whichwe removed in the first place — in which case we get ex-actly the same manifold we started with. We can thinkof this as “filling in” the second S of the torus surface S × S . The next simplest possibility (and the one wewill be interested in) is to instead sew back in a differentsolid torus D × S which would be “filling in” the first S of the torus surface S × S . (It is hard to visual-ize such a thing since either the initial manifold or finalmanifold cannot be embedded in 3-dimensions).
2. Surgery, Chern-Simons theory, and knotpolynomials
For the interested reader, we will briefly describe howsurgery leads to a correspondence between knot polyno-mials and the partition function of Chern-Simons theory,as first described by Ref. 3. (We note that for a firstreading this section may not be crucial for understand-ing Sect. C 3 below). There are three essential ingredientsto this connection. First, the Hilbert space at a fixed in-stant in time is finite-dimensional; its dimension is fixedby the topology of the space-like surface Σ and the num-ber of Wilson lines piercing this surface. For example, ifthere are no Wilson lines piercing Σ, and Σ has genus 0,then the Hilbert space is one-dimensional. When thereare multiple Wilson lines piercing Σ the Hilbert space istypically multidimensional, as should be expected for atopological (non-Abelian) system with quasi-particle de-fects. Second, certain geometrical transformations on Σresult in a change of basis in the Hilbert space. Third,the effects of such transformations on the Hilbert spacecan equally well be carried out by adding certain Wil-son lines in parts of the spacetime manifold which donot intersect Σ. For example, if Σ is a torus, then lin-ear transformations in the Hilbert space on Σ can becarried out by threading Wilson lines through the solidtorus bounded by Σ. These three crucial properties allhave roots in the connection between Chern-Simons the-ory and rational conformal field theories. We will notattempt to explain their origins here, but merely brieflydescribe their interesting consequences.First, consider the effect of performing a modulartransformation on the surface of the torus. The effect ofthis modular transformation is to interchange the merid-ian and the longitude of the torus, which is carried outvia the action of the modular S matrix: | a (cid:105) → S ab | b (cid:105) .Ref. 3 showed that such a transformation on the surfaceof the torus can be obtained by threading a Wilson lineΩ ≡ (cid:80) ki =0 S i | i (cid:105) around the solid torus enclosed by the T space-like surface upon which the modular transfor-mation is to act. (It turns out that, for the cases of inter-est where the associated CFT is rational and S is unitary,this gives precisely the same definition of Ω as in Eq. (11)). Of course, we could equally well add no Wilson linesto the theory but instead excise the torus, perform themodular transformation, and then glue it back into the space-time manifold. Hence adding a Wilson line labeledby Ω to the theory is equivalent to computing the parti-tion function on a different manifold, related to the firstvia surgery.Incidentally, the connection between the element Ω andthe modular S matrix of a rational CFT furnishes aneasy proof that Ω projects onto 0 flux . The precisecorrespondence which we will need is shown in Fig. 18. S ij j = S j j i FIG. 18. The S matrix can be used to unlink loops fromlines. Using Fig. 18, we see that passing a string labeled j through the Ω loop gives a factor of1 S j k (cid:88) i =0 S i S ij (C1)times the diagram with only the string labeled j . Hencesince the S matrix is unitary and symmetric (and S i is real), the partition function of any Wilson line linkedwith Ω vanishes unless the Wilson line carries trivial flux.Thus a Wilson line L labeled with Ω is an operator thatprojects onto states with 0 flux passing through L .Thus, through its connection to modular transforma-tions in the corresponding rational CFT, we see that aloop labeled with Ω projects onto 0 flux in the latticemodel, and also represents surgery on the space-time.
3. Surgery arguments for the chirality ofquasi-particles
We are now almost ready to track the positions of thequasi-particles after surgery. In order to do so simply,however, we will first present an alternative descriptionof surgery, in which one obtains the 3-manifold as theboundary of an appropriately constructed 4-manifold. Inthis case we can give explicit instructions for constructingthe 4 manifold by gluing handles to the boundary of the4D ball (a.k.a. the 3-sphere) along the strands in a link.The 3 manifold is then given by taking the boundary ofthis 4-manifold.The reason that this can be done at all is that all of theinteresting topology of the boundary of the 4-manifoldcan be captured by attaching 2 handles alone to the S boundary of the 4 ball. A 2-handle is, by definition,something that gets glued onto a manifold M along the1boundary (edge) of a disk. In 2 dimensions, attachinga 2-handle to a manifold M means precisely gluing adisk to M along its boundary. In higher dimensions theprocess is essentially the same, except that the object tobe glued is a disk thickened in the appropriate numberof dimensions. The crucial difference, however, is thatin higher dimensions the (thickened) circle along whichthe disk is glued in can be twisted and knotted– hencethis gluing can, in fact, produce any 3 manifold as theboundary of a 4 manifold .To give the flavor of how this produces the same end-product of surgery as described above, let us consider thesimplest possible link– a single, un-knotted circle. Therecipe, then, is to thicken this circle into a solid tube,and remove this tube from S . Next, glue a 4D 2-handleonto S by attaching it along the excised region. Therelevant feature of the 2-handle is that for every longitudeof the excised tube, there is an associated disk in the 2handle which is attached along this longitude. (This issimply the higher-dimensional analogue of saying thatgluing in a 2-handle amounts to sewing on a disk alongits S boundary – here we sew in a (meridinal) circle’sworth of disks along their (longitudinal) S boundaries).The boundary of the finished product consists of a 3-sphere from which a solid tube has been removed, andglued back in with the meridian and longitude of its -torus boundary interchanged . In fact, this description isvalid for surgery on any link: surgery can be performedby thickening each strand of the link to a hollow tube in S , excising all of the tubes, and then gluing them backin with meridian and longitude interchanged.Armed with this understanding of surgery, we may nowconsider the fate of the Chain-Mail link and its quasi-particles. Let us first introduce some convenient termi-nology. A handle decomposition of a 3-manifold is madeup of a manifold H + of 0 and 1 handles (thickened ver-tices and edges, in our lattice construction), a manifold H − of 2 and 3 handles (thickened plaquettes, togetherwith solid balls filling in the cells on the lattice), anda surface Σ which bounds both H + and H − equippedwith information about how these two are glued together.Thus the 3-manifold is written M = H + ∪ Σ ∪ H − . (Thisis known as a Heegard splitting.) A convenient way todescribe surgery on the Chain-Mail link is given by Bar-rett et. al. , and is based on the following fact: per-forming surgery on the plaquette (resp. edge) strings inΣ × I gives H − h H − (resp. H + h H + ). (The sub-scripts h and h denote a connect sum for each vertexor 3-handle, respectively, in the lattice, as in Ref. 20.)Basically, this is because surgery attaches each of the re-quired 3-dimensional 2-handles, crossed with an intervalin the 4 th dimension, to Σ. Taking the boundary of thisobject, we obtain a manifold which has 2 oppositely ori-ented copies of the 2-handles – but only one copy of eachof the 3-handles which fill these in. This is equivalent totaking 2 oppositely oriented copies of H − , cutting outall of the 3-handles from one copy, and gluing the othercopy in along these excised balls. The same construc- tion holds for surgery on the edge loops, except that inthis case there is one connect sum for each vertex in thelattice.Now, imagine decomposing S in the following way:start with H − , the ensemble of 2 and 3 handles. Glueonto this a copy of Σ × I which contains the edge strings ofthe Chain-Mail link. (Since Σ = ∂H − , we can glue thesetogether on Σ × { } ). Glue in a second copy of Σ × I ,containing the plaquette strings. (Here we join Σ × { } in the first copy to Σ × { } in the second). Finally, gluethis ensemble into the 3 sphere with H − removed (whichwe write as S / Int( H − ) . This gives: S = H − ∪ (Σ × I ) + ∪ (Σ × I ) − ∪ (cid:0) S / Int( H − ) (cid:1) (C2)where the subscripts + and − remind us that we will dosurgery on the plaquette strings in the first copy of Σ × I ,and on the edge strings in the second copy. Here andbelow we use the notation that the order of writing termsshows the order in which pieces are attached together (i.e,each term is connected to the term listed before and afterit).After performing surgery on the two sets of curves, weobtain: H − ∪ H + h H + ∪ H − h H − ∪ (cid:0) S / Int( H − ) (cid:1) = M (3) h M (3) h S (C3)where M (3) ≡ H − ∪ H + is the 3 manifold whose handledecomposition we have used to construct the Chain-Maillink. (As argued by Ref. 20, the multiple connect sums donot change the partition function of the resulting mani-fold, and can be dropped). Note that crucially, the M (3) can be traced back to H − ∪ (Σ × I ) + in Eq. C2 whereasthe M (3) is traced back to (Σ × I ) + ∪ (Σ × I ) − . We referthe reader to Ref. 20 for more details of this construction.Hence surgery on the Chain-Mail link produces twocopies of M , of opposite chirality, joined in a connectedsum. (Or rather, by multiple connect sums, one for eachvertex in the lattice). Where do quasi-particle world-lines land after this surgery? Right-handed quasi-particlestrings are linked only through the edge loops, and hencecan be continuously deformed such that they live in the H − ∪ (Σ × I ) + component of the original decomposition.After surgery, they therefore land in M (3) . Left-handedstrings are linked through both edge and plaquette loops,and hence visit both (Σ × I ) + and (Σ × I ) − portions of thedecomposition. Since the first H − in Eq. (C3) by con-struction contains no quasi-particle strings (and since the0- and 3-cells which are deleted to take the connect sumalso cannot contain quasi-particle strings), these must re-side in the H + ∪ H − portion of the final manifold. Aftersurgery, they are therefore found in M (3) . Thus right-handed strings land in the right-handed copy of M (3) ,and left-handed strings in the left-handed copy, M (3) .This gives them opposite statistics.2 Appendix D: Lattice models in non-braidedcategories
Here we will briefly explain how the construction out-lined in the main text can be applied to lattice modelsbased on tensor categories which are not braided – thatis, to tensor categories which have a consistent set of fu-sion rules, but no matrix R (c.f. Eq. 4) specifying howto un-twist the strands. Thus although these categorieshave a well-defined fusion structure, there is no notionof a braiding structure, ie. of how to evaluate over- andunder- crossings of strings. Much less is known aboutthese types of categories, and no analogue of surgery ex-ists; however we note that a similar construction to thatof Sect. IV can be used to obtain the ground-state par-tition function, and comment on how quasi-particles canbe incorporated into the theory.
1. Ground state partition function
First, we will explain how the ground state partitionfunction of a Levin-Wen Hamiltonian can be evaluatedusing our pictorial representation. The idea is very simi-lar to the case of doubled anyon theories, except that wemust replace the diagram in Fig. 2 with one in which nostrings are linked: in Sect. IV, we expressed both vertexand plaquette projectors in the pictorial model using Ωloops. When the category is not braided this is no longerpossible, as we cannot resolve diagrams in which theseedge loops are linked around the plaquette loops.To circumvent this difficulty, we may replace the edgeloops in the Chain-Mail diagram with an appropriate for-mulation of the projector onto the 0 string . The actionof the vertex projectors is then carried out by the ap-propriate projector in the center of each edge, and theaction of the plaquette projector B P can be implementedwith an Ω loop encircling the plaquette, as before. Inter-estingly, this suggests that the symmetry between latticeand dual lattice descriptions for models constructed fromanyon theories is no longer possible for categories whichhave no braiding structure.To evaluate such diagrams, we must further stipulatethat we first act with all projectors on the edges, thenslide the separated (but un-linked) diagrams apart if nec-essary, before evaluating them using the rules of the fu-sion category. Further, though for the ground state allsuch diagrams can be drawn as planar graphs (i.e. withno strings crossing), in any particular projection of theknot the diagrams may not appear planar. As the cat-egory contains no rules for un-doing crossings, one losesthe general notion of projection independence, and mustevaluate the diagrams in a projection in which they areplanar. We emphasize that both of these choices are ex-ternal to the rules of the category, and must be made suchthat the diagrams to be evaluated contain no crossings.With this caveat in mind, we can follow the steps forevaluating the ground state partition function exactly as in Sect. IV. Again, we find that in the absence of plaque-tte strings, edge variables propagate unaltered upwardsin time, provided that they satisfy the constraint imposedby B V at each vertex. Strings associated to space-likeplaquettes act by fusing with the edge variables (to sat-isfy the constraint of fusing to the 0 string along space-like edges), producing the Levin-Wen action (Eq. (12)) for B P ( s ). As before, in evaluating the partition function,summing over edge variables with appropriate weightsleads us to label the time-like plaquette strings with Ωas well.Since the diagram is no longer a link diagram, onlysome of the interesting features described in Sect. V ap-ply in this case. We note, however, that it is still possibleto establish a connection between the Levin-Wen parti-tion function and the Turaev-Viro invariant. Specifically,if the lattice is chosen such that all vertices in the 3 D lat-tice have valence 4 (in other words, if it is chosen suchthat its dual lattice gives a triangulation of the space-time M ), then acting with the edge projectors reducesthe resulting partition function to the evaluation of atetrahedral diagram at each vertex. Hence in this case, Z LW is clearly the Turaev -Viro invariant of M . Thoughwe expect that these models also should depend only onthe topology of M , and not the choice of lattice, we hencedo not furnish a proof of this fact here– the proofs of in-dependence of handle decomposition (which translates,for our purposes, to independence of the 3D lattice cho-sen) given by Roberts , do not trivially generalize tounbraided categories.
2. Quasi-Particles
A remaining question is whether the pictorial construc-tion can accommodate quasi-particles in the case that thecategory does not describe an anyon theory. In this sec-tion we consider whether quasi-particles can be definedsimilarly in these more general cases. We find that theminimal way in which quasi-particles can be introducedis by assuming that the category has a half-braided struc-ture – that is, that for each quasi-particle type, one typeof crossing (either over or under) may be defined in amanner consistent with the rules of fusion. Interestingly,there are categories of this type which indeed cannot beendowed with a full braiding structure , so that thesemodels are nonetheless more general than those derivedfrom anyon theories.The principle difficulty here is that, as quasi-particlestrings must cross between different 3 cells in the lattice,in general it is not possible to find a projection in whichquasi-particle strings do not cross the plaquette strings.In at least some of the diagrams, after all edge projec-tors are applied, crossings will remain. In principle thisis less of a problem for R quasi-particles, as they are notin fact linked with the plaquette strings. However, in aprojection where R particles do not cross any plaquettestrings, the L quasi-particles cross both under and over3the plaquette loops with which they are linked. To eval-uate such diagrams requires a rule for how to undo bothover and under crossings. By definition, such a structurecannot be consistently assigned unless the category hasa braiding structure, and hence such diagrams can onlybe evaluated if the category is in fact an anyon theory.It is instructive to first understand how this issue isresolved in the Levin-Wen construction. If the theoryis constructed from an anyon theory, then the quasi-particle excitations are right- and left- handed copies ofthe initial category. For example, if strings representworld lines of particles in the of SU (2) k , then the fi-nal model describes doubled Chern-Simons theory witha gauge group SU (2) k,R × SU (2) k,L (or equivalently SU (2) k × SU (2) − k ). Strings representing particles in theright- and left- handed sectors respectively correspondexactly to the right- and left- handed quasi-particles iden-tified in the previous section.If the tensor category has no braiding structure, how-ever, the process of ‘doubling’ is more complex. In prac-tice, the construction of Ref. 1 requires assigning a phaseto a given string type for each crossing – though in the2 D picture, one need not specify whether these cross-ings are over-crossings or under-crossings. Mathemati-cally speaking, then, the tensor category is not braided– but quasi-particles cannot be introduced in these mod-els without specifying some additional information aboutcrossings. We can think of this extra data as specifyinghow to resolve either over- or under- crossings, but notboth.We may use this additional data to specify rules forresolving under-crossings of left-handed quasi-particles,and over-crossings of right-handed particles. In this way,the pictorial construction can reproduce the more general models without full braiding structure – though the al-gorithm for evaluating the partition function apparentlycannot be stated in a projection-independent fashion.However, as noted above, in this more general case onedoes not expect the evaluation of the diagrams to be pro-jection independent, and hence it is not surprising thata specific projection (in this case, the projection whichlooks down at a time slice from the positive τ direction)must be chosen.This construction leaves open many interesting ques-tions about the nature of these more general theories.When the model is not built from an anyon theory,the final quasi-particles consist not of individual right-and left- strings, but rather of specific linear combina-tions which yield quasi-particles for which braiding iswell-defined. Mathematically speaking, the result is the Dreinfeld double of the initial category . It would beinteresting to understand, on grounds other than mathe-matical consistency of the category, how these preferredcombinations arise. Further, one is tempted to askwhether an analogue of the connection to a continuumtopological gauge theory in the anyon theory case is pos-sible here: is there a modified surgery procedure whichallows for such a connection to be made? We will addressthese questions in a future publication. Acknowledgements
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Topological Quantum Computation (AmericanMathematical Society, 2009). In fact, it is always possible to construct a projector ontothe 0-string, which can be used to generalize our construc-tion of the ground-state partition function to the case oftensor categories which do not even have a braiding struc-ture. The ground state is nontrivial in topological models, as itis generically degenerate. The Levin-Wen models require only fusion rules, and notthe matrix R ; such a mathematical structure is known as atensor category and is more general than the anyon theorieswhich we focus on here. See section D. K. Walker, preprint(1991). V. Turaev, preprint(1992). Mathematically inclined readers should note that here we describe how to construct the link for a given lattice; theformal prescription associates a link to any Heegard split-ting of the 3-manifold. Strictly speaking n c is the number of 3-handles of the Hee-gard splitting that defines the manifold . R. E. Gompf and A. I. Stipsicz, (American Mathematical Society, 1999). Amusingly, in Ref. 2, Witten refers to the Lickorish-Wallace theorem as a ‘not too deep result’. W. Lickorish, Ann. of Math. , 531 (1962). W. Lickorish, Proc. Cambridge Philos. Soc. , 307 (1963). A. H. Wallace, Canad. J. Math. , 503 (1960). Such re-arrangements are known as Kirby moves; see e.g.Ref. 45 for a detailed discussion. This move has a natural interpretation in the surgery pic-ture, as one handle (attached during surgery) sliding overanother – hence the name. Strictly speaking handle-slide is a more general propertywhich must also apply to Ω loops that have been self-knotted; a proof that Ω satisfies the general handle-slideproperty can be found in Refs. 61, 37. In this paper, how-ever, we will not need to consider this more general move. V. Oganesyan, T. H. Hansson, and S. L. Sondhi, Annals ofphysics , 497 (2004). The reason for this stems from requiring that the left-handed strings be able to handle-slide freely about thelattice. This type of modification to the Chain-Mail invariant hasalso been discussed by Ref. 20, who provides a proof, viasurgery, of this achirality. We are grateful to P. Bonderson for pointing this out tous. For example, see Ref. 62. C. Kassel,
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