michael a. levin and xiao-gang wen- string-net condensation: a physical mechanism for topological...

8/3/2019 Michael A. Levin and Xiao-Gang Wen- String-net condensation: A physical mechanism for topological phases

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String-net condensation: A physical mechanism for topological phases

Michael A. Levin and Xiao-Gang WenDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Dated: April 2004)

We show that quantum systems of extended objects naturally give rise to a large class of ex-otic phases - namely topological phases. These phases occur when the extended objects, calledstring-nets, become highly fluctuating and condense. We derive exactly soluble Hamiltoniansfor 2D local bosonic models whose ground states are string-net condensed states. Those groundstates correspond to 2D parity invariant topological phases. These models reveal the mathemat-ical framework underlying topological phases: tensor category theory. One of the Hamiltonians -a spin-1/2 system on the honeycomb lattice - is a simple theoretical realization of a fault tolerantquantum computer. The higher dimensional case also yields an interesting result: we find that 3Dstring-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions.Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3and higher dimensions.

PACS numbers: 11.15.-q, 71.10.-w

I. INTRODUCTION

For many years, it was thought that Landaus theoryof symmetry breaking [1] could describe essentially allphases and phase transitions. It appeared that all con-tinuous phase transitions were associated with a brokensymmetry. However, after the discovery of the fractionalquantum Hall (FQH) effect, it was realized that FQHstates contain a new type of order - topological order- that is beyond the scope of Landau theory (for a re-view, see Ref. [2]). Since then the study of topologicalphases in condensed matter systems has been an activearea of research. Topological phases have been investi-gated in a variety of theoretical and experimental sys-

tems, ranging from FQH systems [36], quantum dimermodels [710] , quantum spin models [1119], to quan-tum computing [20, 21], or even superconducting states[22, 23]. This work has revealed a host of interestingtheoretical phenomena and applications, including frac-tionalization, anyonic quasiparticles, and fault tolerantquantum computation. Yet, a general theory of topolog-ical phases is lacking.

One way to reveal the gaps in our understanding isto compare with Landaus theory of symmetry break-ing phases. Landau theory is based on (a) the physicalconcepts of long range order, symmetry breaking, and or-der parameters, and (b) the mathematical framework of

group theory. These tools allow us to solve three impor-tant problems in the study of ordered phases. First, theyprovide low energy effective theories for general orderedphases: Ginzburg-Landau field theories [24]. Second,they lead to a classification of symmetry-breaking states.For example, we know that there are only 230 differentcrystal phases in three dimensions. Finally, they allow usto determine the universal properties of the quasiparticleexcitations (e.g. whether they are gapped or gapless). Inaddition, Landau theory provides a physical picture forthe emergence of ordered phases - namely particle con-

URL: http://dao.mit.edu/~wen

densation.Several components of Landau theory have been suc-

cessfully reproduced in the theory of topological phases.For example, the low energy behavior of topologicalphases is relatively well understood on a formal level:topological phases are gapped and are described by topo-logical quantum field theories (TQFTs).[25] The prob-lem of physically characterizing topological phases hasalso been addressed. Ref. [2] investigated the topologi-cal order (analogous to long range order) that occursin topological phases. The author showed that topo-logical order is characterized by robust ground state de-generacy, nontrivial particle statistics, and gapless edgeexcitations.[3, 13, 26] These properties can be used topartially classify topological phases. Finally, the quasi-

particle excitations of topological phases have been ana-lyzed in particular cases. Unlike the symmetry breakingcase, the emergent particles in topologically ordered (ormore generally, quantum ordered) states include (decon-fined) gauge bosons[27, 28] as well as fermions (in threedimensions) [29, 30] or anyons (in two dimensions) [31].Fermions and anyons can emerge as collective excitationsof purely bosonic models.

Yet, the theory of topological phases is still incomplete.The theory lacks two important components: a physicalpicture (analogous to particle condensation) that clarifieshow topological phases emerge from microscopic degreesof freedom, and a mathematical framework (analogousto group theory) for characterizing and classifying thesephases.

In this paper, we address these two issues for a largeclass of topological phases which we call doubled topo-logical phases. On a formal level, doubled topologicalphases are phases that are described by a sum of twoTQFTs with opposite chiralities. Physically, they arecharacterized by parity and time reversal invariance. Ex-amples include all discrete lattice gauge theories, and alldoubled Chern-Simons theories. It is unclear to what ex-tent our results generalize to chiral topological phases -such as in the FQH effect.

We first address the problem of the physical picturefor doubled topological phases. We argue that in these


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2

String-net condensedNormal

t/U > 1

FIG. 1: A schematic phase diagram for the generic string-netHamiltonian (3). When t/U (the ratio of the kinetic energyto the string tension) is small the system is in the normalphase. The ground state is essentially the vacuum with a fewsmall string-nets. When t/U is large the string-nets condenseand large fluctuating string-nets fill all of space. We expect aphase transition between the two states at some t/U of orderunity. We have omitted string labels and orientations for thesake of clarity.

phases, local energetic constraints cause the microscopicdegrees of freedom to organize into effective extendedobjects called string-nets. At low energies, the mi-croscopic Hamiltonian effectively describes the dynamicsof these extended objects. If the kinetic energy of thestring-nets dominates the string-net tension, the string-nets condense: large string-nets with a typical size onthe same order as the system size fill all of space (seeFig. 1). The result is a doubled topological phase. Thus,just as traditional ordered phases arise via particle con-densation, topological phases originate from string-net

condensation.This physical picture naturally leads to a solution

to the second problem - that of finding a mathemati-cal framework for classifying and characterizing doubledtopological phases. We show that each topological phaseis associated with a mathematical object known as atensor category. [32] Here, we think of a tensor cat-

egory as a 6 index object Fijklmn which satisfies certain

algebraic equations (8). The mathematical object Fijklmncharacterizes different topological phases and determinesthe universal properties of the quasiparticle excitations(e.g. statistics) just as the symmetry group does in Lan-dau theory. We feel that the mathematical frameworkof tensor categories, together with the physical pictureof string-net condensation provides a general theory of(doubled) topological phases.

Our approach has the additional advantage of provid-ing exactly soluble Hamiltonians and ground state wavefunctions for each of these phases. Those exactly solu-ble Hamiltonians describe local bosonic models (or spinmodels). They realize all discrete gauge theories (in anydimension) and all doubled Chern-Simons theories (in(2 + 1) dimensions). One of the Hamiltonians - a spin-1/2 model on the honeycomb lattice - is a simple theo-retical realization of a fault tolerant quantum computer[33]. The higher dimensional models also yield an in-teresting result: we find that (3 + 1)D string-net con-

densation naturally gives rise to both emerging gaugebosons and emerging fermions. Thus, string-net conden-sation provides a mechanism for unifying gauge bosonsand fermions in (3 + 1) and higher dimensions.

We feel that this constructive approach is one of themost important features of this paper. Indeed, in themathematical community it is well known that topolog-ical field theory, tensor category theory and knot theory

are all intimately related [3436]. Thus it is not sur-prising that topological phases are closely connected totensor categories and string-nets. The contribution ofthis paper is our demonstration that these elegant mathe-matical relations have a concrete realization in condensedmatter systems.

The paper is organized as follows. In sections II andIII, we introduce the string-net picture, first in the caseof deconfined gauge theories, and then in the generalcase. We argue that all doubled topological phases aredescribed by string-net condensation.

The rest of the paper is devoted to developing a the-ory of string-net condensation. In section IV, we con-

sider the case of (2 + 1) dimensions. In parts A and B,we construct string-net wave functions and Hamiltoniansfor each (2 + 1)D string-net condensed phase. Then, inpart C, we use this mathematical framework to calculatethe universal properties of the quasiparticle excitationsin each phase. In section V, we discuss the generaliza-tion to 3 and higher dimensions. In the last section, wepresent several examples of string-net condensed states -including a spin-1/2 model theoretically capable of faulttolerant quantum computation. The main mathematicalcalculations can be found in the appendix.

II. STRING-NETS AND GAUGE THEORIES

In this section, we introduce the string-net picture inthe context of gauge theory [27, 37, 38]. We point outthat all deconfined gauge theories can be understood asstring-net condensates where the strings are essentiallyelectric flux lines. We hope that this result provides in-tuition for (and motivates) the string-net picture in thegeneral case.

We begin with the simplest gauge theory - Z2 latticegauge theory [39]. The Hamiltonian is

HZ2 = Ui

x

i + tp

edges ofp

z

j (1)

where x,y,z are the Pauli matrices, and I, i, p label thesites, links, and plaquettes of the lattice. The Hilbertspace is formed by states satisfying

legs of I

xi | = |, (2)

for every site I. For simplicity we will restrict our dis-cussion to trivalent lattices such as the honeycomb lattice(see Fig. 2).

It is well known that Z2 lattice gauge theory is dualto the Ising model in (2 + 1) dimensions [40]. What


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3

x

x x

z

z z

x

xx

x

x

x

z

z

z

p

I

=1

=1 =1

=1

=1

=1

FIG. 2: The constraint termQ

legs ofI

xi

and magnetic termQ

edges ofp

zj

in Z2 lattice gauge theory. In the dual picture,we regard the links with x = 1 as being occupied by astring, and the links with x = +1 as being unoccupied. Theconstraint term then requires the strings to be closed - asshown on the right.

is less well known is that there is a more general dual

description ofZ2 gauge theory that exists in any numberof dimensions [41]. To obtain this dual picture, we viewlinks with x = 1 as being occupied by a string andlinks with x = +1 as being unoccupied. The constraint(2) then implies that only closed strings are allowed inthe Hilbert space (Fig. 2).

In this way, Z2 gauge theory can be reformulated as aclosed string theory, and the Hamiltonian can be viewedas a closed string Hamiltonian. The electric and mag-netic energy terms have a simple interpretation in thisdual picture: the electric energy Ui xi is a stringtension while the magnetic energy t

pedges ofpzj

is a string kinetic energy. The physical picture for theconfining and deconfined phases is also clear. The con-fining phase corresponds to a large electric energy andhence a large string tension U t. The ground stateis therefore the vacuum configuration with a few smallstrings. The deconfined phase corresponds to a largemagnetic energy and hence a large kinetic energy. Theground state is thus a superposition of many large stringconfigurations. In other words, the deconfined phase ofZ2 gauge theory is a quantum liquid of large strings - astring condensate. (Fig. 3a).

A similar, but more complicated, picture exists forother deconfined gauge theories. The next layer of com-plexity is revealed when we consider other Abelian the-ories, such as U(1) gauge theory. As in the case of Z2,U(1) lattice gauge theory can be reformulated as a the-ory of electric flux lines. However, unlike Z2, there ismore then one type of flux line. The electric flux on alink can take any integral value in U(1) lattice gauge the-ory. Therefore, the electric flux lines need to be labeledwith integers to indicate the amount of flux carried bythe line. In addition, the flux lines need to be orientedto indicate the direction of the flux. The final point isthat the flux lines dont necessarily form closed loops.It is possible for three flux lines E1, E2, E3 to meet at apoint, as long as Gauss law is obeyed: E1 + E2 + E3 = 0.Thus, the dual formulation ofU(1) gauge theory involvesnot strings, but more general objects: networks of strings

1

-1

-42

5

-3

2

5

1/2

12

3/2

(a) (b) (c)

57/2

3

FIG. 3: Typical string-net configurations in the dual formu-lation of (a) Z2, (b) U(1), and (c) SU(2) gauge theory. Inthe case of (a) Z2 gauge theory, the string-net configura-tions consist of closed (non-intersecting) loops. In (b) U(1)gauge theory, the string-nets are oriented graphs with edgeslabeled by integers. The string-nets obey the branching rulesE1 + E2 + E3 = 0 for any three edges meeting at a point.In the case of (c) SU(2) gauge theory, the string-nets con-sist of (unoriented) graphs with edges labeled by half-integers1/2, 1, 3/2,.... The branching rules are given by the triangleinequality: {E1, E2, E3} are allowed to meet at a point if andonly ifE1 E2 + E3, E2 E3 + E1, E3 E1 + E2, andE1 + E2 + E3 is an integer.

(or string-nets). The strings in a string-net are labeled,oriented, and obey branching rules, given by Gauss law(Fig. 3b).

This string-net picture exists for general gauge the-ories. In the general case, the strings (electric flux lines)are labeled by representations of the gauge group. Thebranching rules (Gauss law) require that if three stringsE1, E2, E3 meet at a point, then the product of the rep-resentations E1 E2 E3 must contain the trivial rep-resentation. (For example, in the case of SU(2), thestrings are labeled by half-integers E = 1/2, 1, 3/2,...,

and the branching rules are given by the triangle inequal-ity: {E1, E2, E3} are allowed to meet at a point if andonly if E1 E2 + E3, E2 E3 + E1, E3 E1 + E2 andE1 + E2 + E3 is an integer (Fig. 3c)) [37]. These string-nets provide a general dual formulation of gauge theory.As in the case of Z2, the deconfined phase of the gaugetheory always corresponds to highly fluctuating string-nets a string-net condensate.

III. GENERAL STRING-NET PICTURE

Given the large scope of gauge theory, it is naturalto wonder if string-nets can describe more general topo-logical phases. In this section we will discuss this moregeneral string-net picture. (Actually, we will not discussthe most general string-net picture. We will focus on aspecial case for the sake of simplicity. See Appendix Afor a discussion of the most general picture).

We begin with a more detailed definition of string-nets. As the name suggests, string-nets are networks ofstrings. We will focus on trivalent networks where eachnode or branch point is attached to exactly 3 strings. Thestrings in a string-net are oriented and come in varioustypes. Only certain combinations of string types areallowed to meet at a node or branch point. To specifya particular string-net model, one needs to provide the


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i

k

j

FIG. 4: The orientation convention for the branching rules.

i i*

=

FIG. 5: i and i label strings with opposite orientations.

following data:

1. String types: The number of different string typesN. For simplicity, we will label the different stringtypes with the integers i = 1,...,N.

2. Branching rules: The set of all triplets of string-types {{i,j,k}...} that are allowed to meet at apoint. (See Fig. 4).

3. String orientations: The dual string type i as-sociated with each string type i. The duality mustsatisfy (i) = i. The type-i string corresponds tothe type-i string with the opposite orientation. Ifi = i, then the string is unoriented (See Fig. 5).

This data describes the detailed structure of the string-nets. The Hilbert space of the string-net model is thendefined in the natural way. The states in the Hilbertspace are simply linear superpositions of different spatialconfigurations of string-nets.

Once the Hilbert space has been specified, we canimagine writing down a string-net Hamiltonian. The

string-net Hamiltonian can be any local operator whichacts on quantum string-net states. A typical Hamiltonianis a sum of potential and kinetic energy pieces:

H = U HU + tHt (3)

The kinetic energy Ht gives dynamics to the string-nets,while the potential energy HU is typically some kind ofstring tension. When U >> t, the string tension domi-nates and we expect the ground state to be the vacuumstate with a few small string-nets. On the other hand,when t > > U , the kinetic energy dominates, and weexpect the ground state to consist of many large fluc-tuating string-nets. We expect that there is a quantumphase transition between the two states at some t/U onthe order of unity. (See Fig. 1). Because of the analogywith particle condensation, we say that the large t, highlyfluctuating string-net phase is string-net condensed.

This notion of string-net condensation provides a nat-ural physical mechanism for the emergence of topologicalphases in real condensed matter systems. Local energeticconstraints can cause the microscopic degrees of freedomto organize into effective extended objects or string-nets.If the kinetic energy of these string-nets is large, thenthey can condense giving rise to a topological phase. Thetype of topological phase is determined by the structureof the string-nets, and the form of string-net condensa-tion.

But how general is this picture? In the previous sec-tion, we pointed out that all deconfined gauge theoriescan be viewed as string-net condensates. In fact, mathe-matical results suggest that the string-net picture is evenmore general. In (2 + 1) dimensions, all so-called dou-bled topological phases can be described by string-netcondensation (provided that we generalize the string-netpicture as in Appendix A). [34] Physically, this means

that the string-net picture can be applied to essentiallyall parity and time reversal invariant topological phasesin (2+1) dimensions. Examples include all discrete gaugetheories, and all doubled Chern-Simons theories. The sit-uation for dimension d > 2 is less well understood. How-ever, we know that string-net condensation quite gener-ally describes all lattice gauge theories with or withoutemergent Fermi statistics.

IV. STRING-NET CONDENSATION IN (2 + 1)DIMENSIONS

A. Fixed-point wave functions

In this section, we attempt to capture the universalfeatures of string-net condensed phases in (2 + 1) dimen-sions. Our approach, inspired by Ref. [3537, 4244],is based on the string-net wave function. We constructa special fixed-point wave function for each string-netcondensed phase. We believe that these fixed-pointwave functions capture the universal properties of thecorresponding phases. Each fixed-point wave function

is associated with a six index object Fijklmn that satisfiescertain algebraic equations (8). In this way, we derive

a one-to-one correspondence between doubled topologi-cal phases and tensor categories Fijklmn. We would liketo mention that a related result on the classification of(2+1)D topological quantum field theories was obtainedindependently in the mathematical community. [34]

Let us try to visualize the wave function of a string-netcondensed state. Though we havent defined string-netcondensation rigorously, we expect that a string-net con-densed state is a superposition of many different string-net configurations. Each string-net configuration hasa size typically on the same order as the system size.The large size of the string-nets implies that a string-netcondensed wave function has a non-trivial long distance

structure. It is this long distance structure that distin-guishes the condensed state from the normal state.

In general, we expect that the universal features of astring-net condensed phase are contained in the long dis-tance character of the wave functions. Imagine compar-ing two different string-net condensed states that belongto the same quantum phase. The two states will havedifferent wave functions. However, by the standard RGreasoning, we expect that the two wave functions willlook the same at long distances. That is, the two wavefunctions will only differ in short distance details - likethose shown in Fig. 6.

Continuing with this line of thought, we imagine per-forming an RG analysis on ground state functions. All


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FIG. 6: Three pairs of string-net configurations that differonly in their short distance structure. We expect string-netwave functions in the same quantum phase to only differ bythese short distance details.

a

b

c

d

FIG. 7: A schematic RG flow diagram for a string-net modelwith 4 string-net condensed phases a, b, c, and d. All thestates in each phase flow to fixed-points in the long distancelimit. The corresponding fixed-point wave functions a, b,c, and d capture the universal long distance features ofthe associated quantum phases. Our ansatz is that the fixed-point wave functions are described by local constraints ofthe form (4-7).

the states in a string-net condensed phase should flowto some special fixed-point state. We expect that thewave function of this state captures the universal longdistance features of the whole quantum phase. (See Fig.

7).In the following, we will construct these special fixed-

point wave functions. Suppose is some fixed-pointwave function. We know that is the ground state ofsome fixed-point Hamiltonian H. Based on our experi-ence with gauge theories, we expect that H is free. Thatis, H is a sum of local string kinetic energy terms withno string tension terms:

H = tHt = ti

Ht,i

In particular, H is unfrustrated, and the ground statewave function minimizes the expectation values of allthe kinetic energy terms {Hti} simultaneously. Minimiz-ing the expectation value of an individual kinetic energyterm Ht,i is equivalent to imposing a local constrainton the ground state wave function, namely Ht,i| =Ei| (where Ei is the smallest eigenvalue of Ht,i).We conclude that the wave function can be specifieduniquely by local constraint equations. The local con-straints are linear relations between several string-netamplitudes (X1), (X2), (X3)... where the configura-tions X1, X2, X3... only differ by local transformations.

To derive these local constraints from first principlesis difficult, so we will use a more heuristic approach. Wewill first guess the form of the local constraints (ie guessthe form of the fixed-point wave function). Then, in the

next section, we will construct the fixed-point Hamilto-nian and show that its ground state wave function doesindeed satisfy these local relations. Our ansatz is thatthe local constraints can be put in the following graphi-cal form:

i

=

i

(4)

i

=di

(5)

i l

k

j

=ij

i

l

k

i

(6)

m

i

j k

l

=n

Fijmkln

ji

n kl

(7)

Here, i, j, k etc. are arbitrary string types and theshaded regions represent arbitrary string-net configura-tions. The di are complex numbers. The 6 index sym-bol Fijmkln is a complex numerical constant that depends

on 6 string types i, j, m, k, l, and n. If one or moreof the branchings {i,j,m}, {k,l ,m}, {i,n,l}, {j,k,n} isillegal, the value of the symbol Fijmkln is unphysical. How-

ever, for simplicity, we will set Fijmkln = 0 in this case.The local rules (4-7) are written using a new nota-

tional convention. According to this convention, the in-dices i,j,k etc., can take on the value i = 0 in additionto the N physical string types i = 1, ...N. We think ofthe i = 0 string as the empty string or null string.It represents empty space - the vacuum. Thus, we canconvert labeled string-nets to our old convention by sim-ply erasing all the i = 0 strings. The branching rules anddualities associated with i = 0 are defined in the obvious

way: 0 = 0, and {i,j, 0} is allowed if and only if i = j.Our convention serves two purposes: it simplifies nota-tion (each equation in (4- 7) represents several equationswith the old convention), and it reveals the mathematicalframework underlying string-net condensation.

We now briefly motivate these rules. The first rule (4)constrains the wave function to be topologically invari-ant. It requires the quantum mechanical amplitude fora string-net configuration to only depend on the topol-ogy of the configuration: two configurations that can becontinuously deformed into one another must have thesame amplitude. The motivation for this constraint isour expectation that topological string-net phases have

topologically invariant fixed-points.The second rule (5) is motivated by the fundamental

property of RG fixed-points: scale invariance. The wavefunction should look the same at all distance scales.Since a closed string disappears at length scales largerthen the string size, the amplitude of an arbitrary string-net configuration with a closed string should be propor-tional to the amplitude of the string-net configurationalone.

The third rule (6) is similar. Since a bubble is irrel-evant at long length scales, we expect

i l

k

j i j


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6

But ifi = j, the configuration i j is not allowed:

i j

= 0. We conclude that the amplitude

for the bubble configuration vanishes when i = j (6).The last rule is less well-motivated. The main point

is that the first three rules are not complete: anotherconstraint is needed to specify the ground state wavefunction uniquely. The last rule (7) is the simplest lo-cal constraint with this property. An alternative motiva-tion for this rule is the fusion algebra in conformal fieldtheory.[45]

The local rules (4-7) uniquely specify the fixed-pointwave function . The universal features of the string-netcondensed state are captured by these rules. Equiva-lently, they are captured by the six index object Fijmkln ,and the numbers di.

However, not every choice of (Fijmkln , di) corresponds toa string-net condensed phase. In fact, a generic choiceof (Fijmkln , di) will lead to constraints (4-7) that are not

self-consistent. The only (F

ijm

kln , di) that give rise to self-consistent rules and a well-defined wave function are(up to a trivial rescaling) those that satisfy

Fijkji0 =vk

vivjijk

Fijmkln = Flkm

jin = Fjimlkn = F

imjknl

vmvnvjvl

Nn=0

FmlqkpnFjipmnsF

jsnlkr = F

jipqkrF

riq

mls (8)

where vi = vi =

di (and v0 = 1). (See appendix B).

Here, we have introduced a new object ijk defined bythe branching rules:

ijk =

1, if{i,j,k} is allowed,0, otherwise.

(9)

There is a one-to-one correspondence between (2+1)Dstring-net condensed phases and solutions of (8). Thesesolutions correspond to mathematical objects known astensor categories. [32] Tensor category theory is the fun-damental mathematical framework for string-net conden-sation, just as group theory is for particle condensation.We have just shown that it gives a complete classifica-tion of (2 + 1)D string-net condensed phases (or equiva-lently doubled topological phases): each phase is associ-ated with a different solution to (8). We will show laterthat it also provides a convenient framework for derivingthe physical properties of quasiparticles.

It is highly non-trivial to find solutions of (8). How-ever, it turns out each group G provides a solution. Thesolution is obtained by (a) letting the string-type index irun over the irreducible representations of the group, (b)letting the numbers di be the dimensions of the represen-tations and (c) letting the 6 index object Fijmkln be the 6jsymbol of the group. The low energy effective theory ofthe corresponding string-net condensed state turns out tobe a deconfined gauge theory with gauge group G. An-other class of solutions can be obtained from 6j symbols

of quantum groups. It turns out that in these cases, thelow energy effective theories of the corresponding string-net condensed states are doubled Chern-Simons gaugetheories. These two classes of solutions are not necessar-ily exhaustive: Eq. (8) may have solutions other thengauge theories or Chern-Simons theories. Nevertheless,it is clear that gauge bosons and gauge groups emergefrom string-net condensation in a very natural way.

In fact, string-net condensation provides a new per-spective on gauge theory. Traditionally, we think ofgauge theories geometrically. The gauge field A is anal-ogous to an affine connection, and the field strength Fis essentially a curvature tensor. From this point of view,gauge theory describes the dynamics of certain geometricobjects (e.g. fiber bundles). The gauge group determinesthe structure of these objects and is introduced by handas part of the basic definition of the theory. In con-trast, according to the string-net condensation picture,the geometrical character of gauge theory is not funda-mental. Gauge theories are fundamentally theories ofextended objects. The gauge group and the geometricalgauge structure emerge dynamically at low energies andlong distances. A string-net system chooses a particu-lar gauge group, depending on the coupling constants inthe underlying Hamiltonian: these parameters determinea string-net condensed phase which in turn determines asolution to (8). The nature of this solution determinesthe gauge group.

One advantage of this alternative picture is that it uni-fies two seemingly unrelated phenomena: gauge interac-tions and Fermi statistics. Indeed, as we will show insection V, string-net condensation naturally gives rise toboth gauge interactions and Fermi statistics (or fractionalstatistics in (2 + 1)D). In addition, these structures al-ways appear together. [29]

B. Fixed-point Hamiltonians

In this section, we construct exactly soluble lattice spinHamiltonians with the fixed-point wave functions asground states. These Hamiltonians provide an explicitrealization of all (2 + 1)D string-net condensates andtherefore all (2 + 1)D doubled topological phases (pro-vided that we generalize these models as discussed inAppendix A). In the next section, we will use them to

calculate the physical properties of the quasiparticle ex-citations.

For every (Fijmkln , di) satisfying the self-consistency con-ditions (8) and the unitarity condition (14), we can con-struct an exactly soluble Hamiltonian. Let us first de-scribe the Hilbert space of the exactly soluble model. Themodel is a spin system on a (2D) honeycomb lattice, witha spin located on each link of the lattice. Each spin canbe in N+ 1 different states labeled by i = 0, 1,...,N. Weassign each link an arbitrary orientation. When a spinis in state i, we think of the link as being occupied bya type-i string oriented in the appropriate direction. Wethink of the type-0 string or null string as the vacuum(ie no string on the link).


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7

I

p

i*i*

i* ii*

i

kk*

k

k*k

j

j* j

j*j

j*

j*

jj

j*

j

i

i

i

FIG. 8: A picture of the lattice spin model (10). The electriccharge operator Q

I

acts on the three spins adjacent to thevertex I , while the magnetic energy operator B

p

acts on the12 spins adjacent to the hexagonal plaquette p . The term Q

I

constrains the string-nets to obey the branching rules, whileB

p

provides dynamics. A typical state satisfying the low-energy constraints is shown on the right. The empty linkshave spins in the i = 0 state.

The exactly soluble Hamiltonian for our model is givenby

H = I

QIp

Bp, Bp =Ns=0

asBsp (10)

where the sums run over vertices I and plaquettes p ofthe honeycomb lattice. The coefficients as satisfy as =as but are otherwise arbitrary.

Let us explain the terms in (10). We think of thefirst term QI as an electric charge operator. It measures

the electric charge at site I, and favors states with nocharge. It acts on the 3 spins adjacent to the site I:

QI

ki j

= ijk

ki j

(11)

where ijk is the branching rule symbol (9). Clearly, thisterm constrains the strings to obey the branching rulesdescribed by ijk . With this constraint the low energyHilbert space is essentially the set of all allowed string-net configurations on a honeycomb lattice. (See Fig. 8).

We think of the second term Bp as a magnetic flux

operator. It measures the magnetic flux though theplaquette p (or more precisely, the cosine of the magneticflux) and favors states with no flux. This term providesdynamics for the string-net configurations.

The magnetic flux operator Bp is a linear combinationof (N+1) terms Bsp, s = 0, 1,...,N. Each B

sp is an opera-

tor that acts on the 12 links that are adjacent to verticesof the hexagon p. (See Fig. 8). Thus, the Bsp are essen-

tially (N+1)12(N+1)12 matrices. However, the actionofBsp does not change the spin states on the 6 outer linksof p. Therefore the Bsp can be block diagonalized into

(N + 1)6 blocks, each of dimension (N + 1)6 (N + 1)6.Let Bs,g

hijkl

p,ghijkl (abcdef), with a,b,c... = 0, 1,...,N, de-

note the matrix elements of these (N + 1)6 matrices:

Bsp

a

b c

ef

d

h

l

g i

kj

=m,...,r

Bs,ghijkl

p,ghijkl (abcdef)a

b c

ef

d

g

jk

l

ih

(12)

Then the operators Bsp are defined by

Bs,ghijkl

p,ghijkl (abcdef)

= Falg

sglFbghshgF

chisihF

dijsjiF

ejkskjF

fklslk (13)

(See appendix C for a graphical representation of Bsp).One can check that the Hamiltonian (10) is Hermitian ifF satisfies

Fi

j

m

kln = (Fijmkln ) (14)

in addition to (8). Our model is only applicable to topo-logical phases satisfying this additional constraint. Webelieve that this is true much more generally: only topo-logical phases satisfying the unitarity condition (14) arephysically realizable.

The Hamiltonian (10) has a number of interesting

properties, provided that (Fijmkln , di) satisfy the self-consistency conditions (8). It turns out that:

1. The Bsp and QIs all commute with each other.Thus the Hamiltonian (10) is exactly soluble.

2. Depending on the choice of the coefficients as, thesystem can be in N + 1 different quantum phases.

3. The choice as =ds

P

Ni=0 d

2i

corresponds to a topolog-

ical phase with a smooth continuum limit. Theground state wave function for this parameterchoice is topologically invariant, and obeys the lo-cal rules (4-7). It is precisely the wave function ,defined on a honeycomb lattice. Furthermore, QI,Bp are projection operators in this case. Thus, theground state satisfies QI = Bp = 1 for all I, p,while the excited states violate these constraints.

The Hamiltonian (10) with the above choice of as pro-vides an exactly soluble realization of the doubled topo-logical phase described by Fijmkln . We can obtain some in-tuition for the Hamiltonian (10) by considering the case

where Fijmkln is the 6j symbol of some group G. In thiscase, it turns out that QI and Bp are precisely the elec-tric charge and magnetic flux operators in the standardlattice gauge theory with group G. Thus, (10) is theusual Hamiltonian of lattice gauge theory, except withno electric field term. This is nothing more than thewell-known exactly soluble Hamiltonian of lattice gaugetheory. [20, 39] In this way, our construction can beviewed as a natural generalization of lattice gauge the-ory.


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8

In this paper, we will focus on the smooth topolog-ical phase corresponding to the parameter choice as =

dsP

Ni=0 d

2i

(see appendix C). However, we would like to

mention that the other N quantum phases also have non-trivial topological (or quantum) order. However, in thesephases, the ground state wave function does not have asmooth continuum limit. Thus, these are new topologicalphases beyond those described by continuum theories.

C. Quasiparticle excitations

In this section, we find the quasiparticle excitationsof the string-net Hamiltonian (10), and calculate theirstatistics (e.g. the twists and the S matrix s). Wewill only consider the topological phase with smooth con-tinuum limit. That is, we will choose as =

dsP

Ni=0 d

2i

in our

lattice model.Recall that the ground state satisfies QI = Bp = 1 for

all vertices I, and all plaquettes p. The quasiparticle ex-

citations correspond to violations of these constraints forsome local collection of vertices and plaquettes. We areinterested in the topological properties (e.g. statistics) ofthese excitations.

We will focus on topologically nontrivial quasiparticles- that is, particles with nontrivial statistics or mutualstatistics. By the analysis in Ref. [29], we know that thesetypes of particles are always created in pairs, and thattheir pair creation operator has a string-like structure,with the newly created particles appearing at the ends.(See Fig. 9). The position of this string operator isunobservable in the string-net condensed state - only theendpoints of the string are observable. Thus the two ends

of the string behave like independent particles.If the two endpoints of the string coincide so that the

string forms a loop, then the associated closed stringoperator commutes with the Hamiltonian. This followsfrom the fact that the string is truly unobservable; theaction of an open string operator on the ground statedepends only on its endpoints.

Thus, each topologically nontrivial quasiparticle is as-sociated with a (closed) string operator that commuteswith the Hamiltonian. To find the quasiparticles, we needto find these closed string operators.

An important class of string operators are what wewill call simple string operators. The defining propertyof simple string operators is their action on the vacuumstate. If we apply a type-s simple string operator W(P)to the vacuum state, it creates a type-s string along thepath of the string, P. We already have some examples ofthese operators, namely the magnetic flux operators Bsp.When Bsp acts on the vacuum configuration |0, it createsa type-s string along the boundary of the plaquette p.Thus, we can think of Bsp as a short type-s simple stringoperator, W(p).

We would like to construct simple string operatorsW(P) for arbitrary paths P = I1,..., IN on the hon-eycomb lattice. Using the definition ofBsp as a guide, wemake the following ansatz. The string operator W(P)only changes the spin states along the path P. The ma-

trix element of a general type-s simple string operatorW(P) between an initial spin state i1, ...iN and final spinstate i1, ...i

N is of the form

Wi1i

2...i

N

i1i2...iN(e1e2...eN) =

Nk=1

Fsk

Nk=1

k

(15)

where e1,...,eN are the spin states of the N legs of P(see Fig. 9) and

Fsk =

Feki

kik1si

k1ik

, if P turns left at Ik

Feki

k1iksikik1

, if P turns right at Ik(16)

k =

vikvsvik

ikik

, if P turns right, left at Ik, Ik+1vikvsvik

ikik

, if P turns left, right at Ik, Ik+1

1, otherwise(17)

Here, i

j , i

j are two (complex) two index objects thatcharacterize the string W.Note the similarity to the definition of Bsp. The major

difference is the additional factorN

k=1 k. We conjec-

ture that |ji vivsvj | = 1 for a type-s string, soN

k=1 kis simply a phase factor that depends on the initial andfinal spin states i1, i2, ..., iN, i

1, i

2, ..., i

N. This phase

vanishes for paths P that make only left or only rightturns, such as plaquette boundaries p. In that case, thedefinition of W(P) coincides with Bsp.

A straightforward calculation shows that the opera-tor W(P) defined above commutes with the Hamiltonian(10) if ij ,

ij satisfy

mj Fslikjm

li

vjvsvm

=Nn=0

Fjik

snlnkF

jlnksm

ji =Nk=0

kiFiskisj (18)

The solutions to these equations give all the type-s simplestring operators.

For example, consider the case of Abelian gauge the-ory. In this case, the solutions to (18) can be dividedinto three classes. The first class is given by s

= 0,

jivivsvj =

jivivsvj = 1. These string operators create

electric flux lines and the associated quasiparticles areelectric charges. In more traditional nomenclature, theseare known as (Wegner-)Wilson loop operators [39, 46].The second class of solutions is given by s = 0, andji

vivsvj

= (jivivsvj

) = 1. These string operators cre-ate magnetic flux lines and the associated quasiparti-cles are magnetic fluxes. The third class has s = 0 andji

vivsvj

= (jivivsvj

) = 1. These strings create both elec-tric and magnetic flux and the associated quasiparticlesare electric charge/magnetic flux bound states. This ac-counts for all the quasiparticles in (2+1)D Abelian gaugetheory. Therefore, all the string operators are simple inthis case.


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9

However, this is not true for non-Abelian gauge the-ory or other (2 + 1)D topological phases. To computethe quasiparticle spectrum of these more general theo-ries, we need to generalize the expression (15) for W(P)to include string operators that are not simple.

One way to guess the more general expression forW(P) is to consider products of simple string operators.Clearly, if W1(P) and W2(P) commute with the Hamil-

tonian, then W(P) = W1(P)W2(P) also commutes withthe Hamiltonian. Thus, we can obtain other string oper-ators by taking products of simple string operators. Ingeneral, the resulting operators are not simple. IfW1 andW2 are type-s1 and type-s2 simple string operators, thenthe action of the product string on the vacuum state is:

W(P)|0 = W1(P)W2(P)|0 = W1(P)|s2 =s

ss1s2 |s

where |s denotes the string state with a type-s stringalong the path P and the vacuum everywhere else. If wetake products of more then two simple string operators

then the action of the product string on the vacuum isof the form W(P)|0 = s ns|s where ns are some non-negative integers.

We now generalize the expression for W(P) so that itincludes arbitrary products of simple strings. Let W bea product of simple string operators, and let ns be thenon-negative integers characterizing the action of W onthe vacuum: W(P)|0 = s ns|s. Then, one can showthat the matrix elements of W(P) are always of the form

Wi1i

2...i

N

i1i2...iN(e1e2...eN) =

{sk}

N

k=1Fskk

Tr

N

k=1skk

(19)where

skk =

vikvskvik

iksksk+1ik

, if P turns right, left at Ik, Ik+1vikvskvik

iksksk+1ik

, if P turns left, right at Ik, Ik+1

sksk+1 Id, otherwise(20)

and istj,istj are two 4 index objects that characterize

the string operator W. For any quadruple of string typesi,j,s,t, (istj,

istj) are (complex) rectangular matrices

of dimension ns nt. Note that type-s0 simple stringoperators correspond to the special case where ns = s0s.In this case, the matrices istj,

istj reduce to complex

numbers, and we can identify

istj = ijss0ts0 ,

istj =

ijss0ts0 . (21)

As we mentioned above, products of simple string op-erators are always of the form (19). In fact, we be-lieve that all string operators are of this form. Thus,we will use (19) as an ansatz for general string operatorsin (2 + 1)D topological phases. This ansatz is compli-cated algebraically, but like the definition of Bsp, it has asimple graphical interpretation (see appendix D).

A straightforward calculation shows that the closedstring W(P) commutes with the Hamiltonian (10) if

3

W(P)

i23

14

2

1

III

I 123

4

i i

e e

e e

FIG. 9: Open and closed string operators for the latticespin model (10). Open string operators create quasiparti-cles at the two ends, as shown on the left. Closed stringoperators, as shown on the right, commute with the Hamil-tonian. The closed string operator W(P) only acts non-trivially on the spins along the path P = I 1, I 2... (thickline), but its action depends on the spin states on thelegs (thin lines). The matrix element between an initial

state i1, i2,... and a final state i

1, i

2,... is W

i1i

2...

i1i2... (e1e2...) =(F

e2i

2i1si

1i2

Fe3i

3i2si

2i3

...) (vi1vs1vi1

i1i1

vi3vs3vi3

i3i3...) for a type-s sim-

ple string and Wi1i

2...i1i2...

(e1e2...) =P

{sk}(F

e2i

2i1s2i1i2

Fe3i

3i2s3i2i3

...)

Tr(vi1vs1vi1

i1s1s2i1

s2s3Idvi3vs3vi3

i3s3s4i3

...) for a general string.

and satisfy

Ns=0

mrsjFslikjm

lsti

vjvsvm

=Nn=0

Fjik

tnlnrtkF

jlnkrm

jsti =

Nk=0

kstiFit

kisj (22)

The solutions (m, m) to these equations give all thedifferent closed string operators Wm. However, not allof these solutions are really distinct. Notice that twosolutions (1, 1), (2, 2) can be combined to form anew solution (, ):

jsti = j1,sti j2,sti

jsti = j1,sti j2,sti (23)

This is not surprising: the string operator W corre-sponding to (, ) is simply the sum of the two op-erators corresponding to (1,2, 1,2): W

= W1 + W2.Given this additivity property, it is natural to con-

sider the irreducible solutions (, ) that cannot bewritten as a sum of two other solutions. Only the ir-reducible string operators W create quasiparticle-pairsin the usual sense. Reducible string operators W createsuperpositions of different strings - which correspond tosuperpositions of different quasiparticles. [51]

To analyze a topological phase, one only needs to findthe irreducible solutions (, ) to (22). The numberM of such solutions is always finite. In general, each so-lution corresponds to an irreducible representation of analgebraic object. In the case of lattice gauge theory, there


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10

is one solution for every irreducible representation of thequantum double D(G) of the gauge group G. Similarly,in the case of doubled Chern-Simons theories there is onesolution for each irreducible representation of a doubledquantum group.

The structure of these irreducible string operators Wdetermines all the universal features of the topologicalphase. The number M of irreducible string operators is

the number of different kinds of quasiparticles. The fu-sion rules WW =

M=1 h

W determine how bound

states of type- and type- quasiparticles can be viewedas a superposition of other types of quasiparticles.

The topological properties of the quasiparticles are alsoeasy to compute. As an example, we now derive two par-ticularly fundamental objects that characterize the spinsand statistics of quasiparticles: the M twists and theM M S-matrix, s [13, 25, 35, 47].

The twists are defined to be statistical angles ofthe type- quasiparticles. By the spin-statistics theo-rem they are closely connected to the quasiparticle spinss: e

i = e2is . We can calculate by comparing thequantum mechanical amplitude for the following two pro-cesses. In the first process, we create a pair of quasipar-ticles , (from the ground state), exchange them, andthen annihilate the pair. In the second process, we createand then annihilate the pair without any exchange. Theratio of the amplitudes for these two processes is preciselyei .

The amplitude for each process is given by the expec-tation value of the closed string operator W for a par-ticular path P:

A1 = (24)A2 =

(25)Here, | denotes the ground state of the Hamiltonian(10).

Let (, , n) be the irreducible solution corre-sponding to the string operator W. The abovetwo amplitudes can be then be expressed in terms of(, , n) (see appendix D):

A1 =s

d2s Tr(0,sss) (26)

A2 = s

n,sds (27)

Combining these results, we find that the twists are givenby

ei =A1A2 =

s d

2s Tr(0,sss)s n,sds

(28)

Just as the twists are related to the spin and statis-tics of individual particle types , the elements of theS-matrix, s describe the mutual statistics of two parti-cle types , . Consider the following process: We createtwo pairs of quasiparticles , ,, , braid around ,and then annihilate the two pairs. The element s is

FIG. 10: A three dimensional trivalent lattice, obtained bysplitting the sites of the cubic lattice. We replace each vertexof the cubic lattice with 4 other vertices as shown above.

defined to be the quantum mechanical amplitude A ofthis process, divided by a proportionality factor D whereD2 =

(

s n,sds)2. The amplitude A can be cal-

culated from the expectation value of W, W for twolinked paths P:

A =

(29)

Expressing A in terms of (, , n), we find

s =AD

=1

D

ijk

Tr(k,iij) Tr(k

,jji)didj (30)

V. STRING-NET CONDENSATION IN (3 + 1)AND HIGHER DIMENSIONS

In this section, we generalize our results to (3 + 1) andhigher dimensions. We find that there is a one-to-onecorrespondence between (3+1) (and higher) dimensionalstring-net condensates and mathematical objects knownas symmetric tensor categories. [32] The low energyeffective theories for these states are gauge theories cou-pled to bosonic or fermionic charges.

Our approach is based on the exactly soluble latticespin Hamiltonian (10). In that model, the spins live onthe links of the honeycomb lattice. However, the choiceof lattice was somewhat arbitrary: we could equally wellhave chosen any trivalent lattice in two dimensions.

Trivalent lattices can also be constructed in three andhigher dimensions. For example, we can create a space-filling trivalent lattice in three dimensions, by splittingthe sites of the cubic lattice (see Fig. 10). Consider thespin Hamiltonian (10) for this lattice, where I runs overall the vertices of the lattice, and p runs over all theplaquettes (that is, the closed loops that correspondto plaquettes in the original cubic lattice).

This model is a natural candidate for string-net con-densation in three dimensions. Unfortunately, it turnsout that the Hamiltonian (10) is not exactly soluble onthis lattice. The magnetic flux operators Bsp do not com-mute in general.

This lack of commutativity originates from two differ-ences between the plaquettes in the honeycomb latticeand in higher dimensional trivalent lattices. The first


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11

difference is that in the honeycomb lattice, neighboringplaquettes always share precisely two vertices, while inhigher dimensions the boundary between plaquettes cancontain three or more vertices (see Fig. 11). The exis-tence of these interior vertices has the following conse-quence. Imagine we choose orientation conventions foreach vertex, so that we have a notion of left turns andright turns for oriented paths on our lattice (such an

orientation convention can be obtained by projecting the3D lattice onto a 2D plane - as in Fig. 11). Then, nomatter how we assign these orientations the plaquetteboundaries will always make both left and right turns.Thus, we cannot regard the boundaries of the 3D pla-quettes as small closed strings the way we did in twodimensions (since small closed strings always make allleft turns, or all right turns). But the magnetic flux op-erators Bsp only commute if their boundaries are smallclosed strings. It is this inconsistency between the alge-braic definition of Bsp and the topology of the plaquettesthat leads to the lack of commutativity.

To resolve this problem, we need to define a Hamil-

tonian using the general simple string operators W(p)rather then the small closed strings Bsp. Suppose

(isj, isj), s = 0, 1, ...N are type-s solutions of (18). Af-

ter picking some left turn, right turn orientation con-vention at each vertex, we can define the correspondingtype-s simple string operators Ws(P) as in (15). Sup-pose, in addition, that we choose (isj ,

isj) so that the

string operators satisfy Wr Ws =

t rstWt (this prop-erty ensures that Ws(p) are analogous to Bsp). Then, anatural higher dimensional generalization of the Hamil-tonian (10) is

H = I

QIp

Wp, Wp =

Ns=0

asWs(p) (31)

For a two dimensional lattice, the conditions (18) are suf-ficient to guarantee that the Hamiltonian (31) is an ex-actly soluble realization of a doubled topological phase.(This is because the plaquette boundaries p are notlinked and hence the Ws(p) all commute). However,in higher dimensions, one additional constraint is neces-sary.

This constraint stems from the second, and perhapsmore fundamental, difference between 2D and higher di-mensional lattices. In two dimensions, two closed curves

always intersect an even number of times. For higherdimensional lattices, this is not the case. Small closedcurves, in particular plaquette boundaries, can (in asense) intersect exactly once (see Fig. 11). Because ofthis, the objects ijk must satisfy the additional relation:

ijk = ikj (32)

One can show that if this additional constraint is satis-fied, then (a) the higher dimensional Hamiltonian (31)is exactly soluble, and (b) the ground state wave func-tion is defined by local topological rules analogous to(4-7). This means that (31) provides an exactly solublerealization of topological phases in (3 + 1) and higherdimensions.

x

y

z

I

I1

I2

3I4

p

p

p

1

2

3

FIG. 11: Three plaquettes demonstrating the two fundamen-tal differences between higher dimensional trivalent latticesand the honeycomb lattice. The plaquettes p 1, p 2 lie in thexz plane, while p 3 is oriented in the xy direction. Notice thatp 1 and p 2 share three vertices, I 1, I 2, I 3. Also, notice thatthe plaquette boundaries p 1 and p 3 intersect only at theline segment I 3 I 4. The boundary p 1 makes a left turn atI 3, and a right turn at I 4. Thus, viewed from far away, thesetwo plaquette boundaries intersect exactly once (unlike thepair p 1 and p 2).

Each exactly soluble Hamiltonian is associated witha solution (Fijmkln ,

ijk ,

ijk) of (8), (18), (32). By analogy

with the two dimensional case, we conjecture that there isa one-to-one correspondence between topological string-net condensed phases in (3 + 1) or higher dimensions,

and these solutions. The solutions (Fijmkln , ijk ,

ijk) cor-

respond to a special class of tensor categories - symmet-ric tensor categories. [32] Thus, just as tensor categories

are the mathematical objects underlying string conden-sation in (2 + 1) dimensions, symmetric tensor categoriesare fundamental to string condensation in higher dimen-sions.

There are relatively few solutions to (8), (18), (32).Physically, this is a consequence of the restrictions onquasiparticle statistics in 3 or higher dimensions. Un-like in two dimensions, higher dimensional quasiparticlesnecessarily have trivial mutual statistics, and must be ei-ther bosonic or fermionic. From a more mathematicalpoint of view, the scarcity of solutions is a result of thesymmetry condition (32). Doubled topological phases,such as Chern-Simons theories, typically fail to satisfy

this condition.However, gauge theories do satisfy the symmetry con-

dition (32) and therefore do correspond to higher dimen-sional string-net condensates. Recall that the gauge the-ory solution to (8) is obtained by (a) letting the string-type index i run over the irreducible representations ofthe gauge group, (b) letting the numbers di be the dimen-sions of the representations, and (c) letting the 6 index

object Fijmkln be the 6j symbol of the group. One cancheck that this also provides a solution to (18), (32), ifwe set ijk

vjvkvi

= 1 when j = k and the invariant ten-sor in k k i is antisymmetric in the first two indices,and ijk

vjvk

vi= 1 otherwise. This result is to be expected,

since the string-net picture of gauge theory (section II)


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12

is valid in any number of dimensions. Thus, it is notsurprising that gauge theories can emerge from higherdimensional string-net condensation.

There is another class of higher dimensional string-net condensed phases that is more interesting. The lowenergy effective theories for these phases are variants ofgauge theories. Mathematically, they are obtained bytwisting the usual gauge theory solution by

ijk = ijk (1)P(j)P(k) (33)

Here P(i) is some assignment of parity (even, or odd)to each representation i. The assignment must be self-consistent in the sense that the tensor product of tworepresentations with the same (different) parity, decom-poses into purely even (odd) representations. If all therepresentations are assigned an even parity - that is P(i)is trivial - then the twisted gauge theory reduces to stan-dard gauge theory.

The major physical distinction between twisted gaugetheories and standard gauge theories is the quasiparti-

cle spectrum. In standard gauge theory, the fundamen-tal quasiparticles are the electric charges created by theN + 1 string operators Wi. These quasiparticles are allbosonic. In contrast, in twisted gauge theories, all thequasiparticles corresponding to odd representations iare fermionic.

In this way, higher dimensional string-net condensa-tion naturally gives rise to both emerging gauge bosonsand emerging fermions. This feature suggests that gaugeinteractions and Fermi statistics may be intimately con-nected. The string-net picture may be the bridge betweenthese two seemingly unrelated phenomena. [29]

In fact, it appears that gauge theories coupled to

fermionic or bosonic charged particles are the only pos-sibilities for higher dimensional string-net condensates:mathematical work on symmetric tensor categories sug-gests that the only solutions to (8), (18), (32) are thosecorresponding to gauge theories and twisted gauge theo-ries. [48]

We would like to point out that (3 + 1) dimensionalstring-net condensed states also exhibit membrane con-densation. These membrane operators are entirely anal-ogous to the string operators. Just as open string oper-ators create charges at their two ends, open membraneoperators create magnetic flux loops along their bound-aries. Furthermore, just as string condensation makes the

string unobservable, membrane condensation leads to theunobservability of the membrane. Only the boundary ofthe membrane - the magnetic flux loop - is observable.

VI. EXAMPLES

A. N= 1 string model

We begin with the simplest string-net model. In thenotation from section III, this model is given by

1. Number of string types: N = 1

2. Branching rules: (no branching)

3. String orientations: 1 = 1.

In other words, the string-nets in this model contain oneunoriented string type and have no branching. Thus,they are simply closed loops. (See Fig. 3a).

We would like to find the different topological phasesthat can emerge from these closed loops. According tothe discussion in section IV, each phase is captured by

a fixed-point wave function, and each fixed-point wavefunction is specified by local rules (4-7) that satisfy theself-consistency relations (8). It turns out that (8) haveonly two solutions in this case (up to rescaling):

d0 = 1

d1 = F110

110 = 1F000000 = F

101101 = F

011011 = 1

F000111 = F110

001 = F101

010 = F011

100 = 1 (34)

where the other elements of F all vanish. The corre-

sponding local rules (4-7) are:

=

=

(35)

We have omitted those rules that can be derived fromtopological invariance (4).

The fixed-point wave functions satisfying theserules are given by

(X) = (

1)Xc (36)

where Xc is the number of disconnected components inthe string configuration X.

The two fixed-point wave functions correspondto two simple topological phases. As we will see, +corresponds to Z2 gauge theory, while is a U(1) U(1) Chern-Simons theory. (Actually, other topologicalphases can emerge from closed loops - such as in Ref. [4244]. However, we regard these phases as emerging frommore complicated string-nets. The closed loops organizeinto these effective string-nets in the infrared limit).

The exactly soluble models (10) realizing these twophases can be written as spin 1/2 systems with one spin

on each link of the honeycomb lattice (see Fig. 12). Weregard a link with x = 1 as being occupied by a type-1string, and the state x = +1 as being unoccupied (orequivalently, occupied by a type-0 or null string). TheHamiltonians for the two phases are of the form

H = I

QI, p

Bp,

The electric charge term is the same for both phases(since it only depends on the branching rules):

QI,

=1

2(1 +

legs ofI

xi

) (37)


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13

x

x x

z

z

z

z z

z

x x

x

x

x

x

x

xx

x

x

x

p

I

f( )

f( )f( )

f( )

f( )f( )

=1

=1 =1

=1

=1

=1

FIG. 12: The Hamiltonians (39), (40), realizing the twoN= 1string-condensed phases. Each circle denotes a spin-1/2 spin.The links with x = 1 are thought of as being occupied bya type-1 string, while the links with x = +1 are regarded asempty. The electric charge term acts on the three legs of thevertex I with x. The magnetic energy term acts on the 6edges of the plaquette p with z, and acts on the 6 legs of pwith an operator of the form f(x). For the Z2 phase, f= 1,

while for the Chern-Simons phase, f(x) = i(1x)/2.

The magnetic terms for the two phases are

Bp, =1

2(B0p, B1p,) (38)

=1

2

1

edges ofp

zj

legs ofp

(1)

1xj

2

Pp

where Pp is the projection operator Pp =

Ip QI. Theprojection operator Pp can be omitted without affectingthe physics (or the exact solubility of the Hamiltonian).

We have included it only to be consistent with (10). Ifwe omit this term, the Hamiltonian for the first phase(+) reduces to the usual exactly soluble Hamiltonian ofZ2 lattice gauge theory (neglecting numerical factors):

H+ = I

legs of I

xi p

edges ofp

zj (39)

The Hamiltonian for the second phase,

H = I

legs of I

xi +p

(

edges ofp

zj )(

legs ofp

i1x

j

2 )

(40)is less familiar. However, one can check that in bothcases, the Hamiltonians are exactly soluble and the twoground state wave functions are precisely (in the x

basis).Next we find the quasiparticle excitations for the two

phases, and the corresponding S-matrix and twists .In both cases, equation (22) has 4 irreducible solu-

tions (n,s, ij,st,

ij,st), = 1, 2, 3, 4 - corresponding to

4 quasiparticle types. For the first phase (+) these so-lutions are given by:

n1,0 = 1, n1,1 = 0, 01,000 = 1, 11,001 = 1

n2,0 = 0, n2,1 = 1, 12,110 = 1,

02,111 = 1

n3,0 = 1, n3,1 = 0, 0

3,000

= 1, 1

3,001

=

1n4,0 = 0, n4,1 = 1, 14,110 = 1,

04,111 = 1

L-vertex

R-leg

P

j

i

I

FIG. 13: A closed string operator W(P) for the two models(39),(40). The pathP is drawn with a thick line, while the legsare drawn with thin lines. The action of the string operators(41),(44) on the legs is different for legs that branch to theright ofP, R-legs, and legs that branch to the left ofP, L-legs. Similarly, we distinguish between R-vertices and L-vertices which are ends of R-leg and L-leg respectively.

The other elements of vanish. In all cases = .The corresponding string operators for a path P are

W1 = Id

W2 =

edges ofP

zj

W3 =

R-legs

xk

W4 =

edges ofP

zj

R-legs

xk (41)

where the R-legs k are the legs that are to the right

of P. (See Fig. 13). Technically, we should multiplythese string operators by an additional projection opera-tor

IP QI, in order to be consistent with the general

result (19). However, we will neglect this factor since itdoesnt affect the physics.

Once we have the string operators, we can easily cal-culate the twists and the S-matrix. We find:

ei1 = 1, ei2 = 1, ei3 = 1, ei4 = 1 (42)

S =1

2

1 1 1 11 1 1 11 1 1 11 1 1 1

(43)

This is in agreement with the twists and S-matrix forZ2 gauge theory: W1 creates trivial quasiparticles, W2creates magnetic fluxes, W3 creates electric charges, W4creates electric/magnetic bound states.

In the second phase (), we find

n1,0 = 1, n1,1 = 0, 01,000 = 1,

11,001 = 1

n2,0 = 0, n2,1 = 1, 12,110 = 1,

02,111 = i

n3,0 = 0, n3,1 = 1, 13,110 = 1,

03,111 = i

n4,0 = 1, n4,1 = 0, 04,000 = 1,

14,001 = 1

Once again, the other elements of vanish. Also, in allcases, = . The corresponding string operators for a


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14

path P are

W1 = Id

W2 =

edges ofP

zj

R-legs

i1x

j

2

L-vertices

(1)sI

W3 = edges ofPzj R-legs(i)

1xj

2

L-vertices(1)sI

W4 =

R-legs

xj (44)

where the L-vertices I are the vertices ofP adjacent tolegs that are to the left of P. The exponent sI is definedby sI =

14 (1 xi )(1 + xj ), where i, j are the links just

before and just after the vertex I, along the path P. (SeeFig. 13).

We find the twists and S-matrix are

ei1 = 1, ei2 = i, ei3 = i, ei4 = 1 (45)

S = 12

1 1 1 1

1 1 1 11 1 1 11 1 1 1

(46)We see that W1 creates trivial quasiparticles, W2, W3create semions with opposite chiralities and trivial mu-tual statistics, and W4 creates bosonic bound states ofthe semions. These results agree with the U(1) U(1)Chern-Simons theory

L =1

4KIJaIaJ

, I, J = 1, 2 (47)

with K-matrix

K =

2 00 2

(48)

Thus the above U(1)U(1) Chern-Simons theory is thelow energy effective theory of the second exactly solublemodel (with d1 = 1).

Note that the Z2 gauge theory from the first exactlysoluble model (with d1 = 1) can also be viewed as aU(1) U(1) Chern-Simons theory with K-matrix [23]

K =

0 22 0

(49)

B. N= 1 string-net model

The next simplest string-net model also contains onlyone oriented string type - but with branching. Simpleas it is, we will see that this model contains non-Abeliananyons and is theoretically capable of universal fault tol-erant quantum computation [33]. Formally, the model isdefined by


2. Branching rules: {{1, 1, 1}}3. String orientations: 1 = 1.

The string-nets are unoriented trivalent graphs. To findthe topological phases that can emerge from these objects, we solve the self-consistency relations (8). We findtwo sets of self-consistent rules:

=

=1 +

1/2

=

1/2

1

(50)

where = 1

52

. (Once again, we have omitted thoserules that can be derived from topological invariance).Unlike the previous case, there is no closed form expres-sion for the wave function amplitude.

Note that the second solution, d1 = 152 does notsatisfy the unitarity condition (14). Thus, only the firstsolution corresponds to a physical topological phase. Aswe will see, this phase is described by an SO3(3)SO3(3)Chern-Simons theory.

As before, the exactly soluble realization of this phase(10) is a spin-1/2 model with spins on the links of thehoneycomb lattice. We regard a link with x = 1 asbeing occupied by a type-1 string, and a link with x = 1as being unoccupied (or equivalently occupied by a type-0 string). However, in this case we will not explicitlyrewrite (10) in terms of Pauli matrices, since the resultingexpression is quite complicated.

We now find the quasiparticles. These correspond toirreducible solutions of (22). For this model, there are 4such solutions, corresponding to 4 quasiparticles:

1 : n1,0 = 1, n1,1 = 0, 01,000 = 1,

11,001 = 1

2 : n2,0 = 0, n2,1 = 1, 12,110 = 1, (51)

02,111 = 1+ ei/5, 12,111 = 1/2+ e3i/53 : n3,0 = 0, n3,1 = 1,

13,110 = 1,

03,111 = 1+ ei/5, 13,111 = 1/2+ e3i/54 : n4,0 = 1, n4,1 = 1,

04,000 = 1,

14,110 = 1,

14,001 =

2+ ,

04,111 =

1+ ,

14,111 =

5/2

+ ,14,101 = (

14,011)

= 11/4+ (2 e3i/5 + +e3i/5).In all cases, = .

We can calculate the twists and the S-matrix. We find:

ei1 = 1, ei2 = e4i/5, ei3 = e4i/5, ei4 = 1 (52)

S =1

1 + 2

1 2

1 2 2 1

2 1

(53)

We conclude that W1 creates trivial quasiparticles, W2,W3 create (non-Abelian) anyons with opposite chirali-ties, and W4 creates bosonic bound states of the anyons.


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15

These results agree with SO3(3)SO3(3) Chern-Simonstheory, the so-called doubled Yang-Lee theory.

Researchers in the field of quantum computing haveshown that the Yang-Lee theory can function as a univer-sal quantum computer - via manipulation of non-Abeliananyons. [33] Therefore, the spin-1/2 Hamiltonian (10) as-sociated with (50) is a theoretical realization of a univer-sal quantum computer. While this Hamiltonian may be

too complicated to be realized experimentally, the string-net picture suggests that this problem can be overcome.Indeed, the string-net picture suggests that generic spinHamiltonians with a trivalent graph structure will ex-hibit a Yang-Lee phase. Thus, much simpler spin-1/2Hamiltonians may be capable of universal fault tolerantquantum computation.

C. N= 2 string-net models

In this section, we discuss two N = 2 string-net mod-els. The first model contains one oriented string and itsdual. In the notation from section III, it is given by


2. Branching rules: {{1, 1, 1}, {2, 2, 2}}3. String orientations: 1 = 2, 2 = 1.

The string-nets are therefore oriented trivalent graphswith Z3 branching rules. The string-net condensedphases correspond to solutions of (8). Solving these equa-tions, we find two sets of self-consistent local rules:

=

=

=1

(54)

The corresponding fixed-point wave functions aregiven by

(X) = (1)2XcXv/2 (55)

where Xc, Xv, are the number of connected components,

and vertices, respectively in the string-net configurationX. As before, we can construct an exactly soluble Hamil-tonians, find the quasiparticles for the two theories andcompute the twists and S-matrices. We find that the firsttheory + is described by a Z3 gauge theory, while thesecond theory is described by a U(1) U(1) Chern-Simons theory with K-matrix

K =

3 00 3

Both theories have 32 = 9 elementary quasiparticles.In the case of Z3, these quasiparticles are electriccharge/magnetic flux bound states formed from the 3types of electric charges and 3 types of magnetic fluxes.

In the case of the Chern-Simons theory, the quasiparti-cles are bound states of the two fundamental anyons withstatistical angles /3.

The final example we will discuss contains two unori-ented strings. Formally it is given by


2. Branching rules:{{

1, 2, 2}

,{

2, 2, 2}}

3. String orientations: 1 = 1, 2 = 2.

The string-nets are unoriented trivalent graphs, withedges labeled with 1 or 2. We find that there is onlyone set of self-consistent local rules:

i

=di

1

1

1

1

=

1

1

1

1

2

2

1

1 = 1122

2

2

21

21

=

2

1 2

12

2 2

21

2

=

2

1 2

22

m

2 2

2 2

=

2n=0

F22m22n

2

2

2

2n

(56)

where d0 = d1 = 1, d2 = 2, and F22m

22n is the matrix

F22m22n =

12

12

12

12

12

1

21

2 12 0

If we construct the Hamiltonian (10), we find that itis equivalent to the standard exactly lattice gauge theoryHamiltonian [20] with gauge group S3 - the permuta-tion group on 3 objects. One can show that this theorycontains 8 elementary quasiparticles (corresponding tothe 8 irreducible representations of the quantum doubleD(S3)). These quasiparticles are combinations of the 3electric charges and 3 magnetic fluxes.

VII. CONCLUSION

In this paper, we have shown that quantum systemsof extended objects naturally give rise to topologicalphases. These phases occur when the extended objects(e.g. string-nets) become highly fluctuating and con-dense. This physical picture provides a natural mech-anism for the emergence of parity invariant topologicalphases. Microscopic degrees of freedoms (such as spinsor dimers) can organize into effective extended objectswhich can then condense. We hope that this physicalpicture may help direct the search for topological phasesin real condensed matter systems. It would be interest-ing to develop an analogous picture for chiral topologicalphases.


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We have also found the fundamental mathematicalframework for topological phases. We have shown thateach (2 + 1) dimensional doubled topological phase is as-

sociated with a 6 index object Fijmkln and a set of realnumbers di satisfying the algebraic relations (8). Allthe universal properties of the topological phase are con-tained in these mathematical objects (known as tensorcategories). In particular, the tensor category directlydetermines the quasiparticle statistics of the associatedtopological phase (28, 30). This mathematical frame-work may also have applications to phase transitions andcritical phenomena. Tensor categories may characterizetransitions between topological phases just as symmetrygroups characterize transitions between ordered phases.

We have constructed exactly soluble (2 + 1)D latticespin Hamiltonians (10) realizing each of these doubledtopological phases. These models unify (2 + 1)D latticegauge theory and doubled Chern-Simons theory. Oneparticular Hamiltonian - a realization of the doubledYang-Lee theory - is a spin 1/2 model capable of faulttolerant quantum computation.

In higher dimensions, string-nets can also give rise totopological phases. However, the physical and mathe-matical structure of these phases is more restricted. Ona mathematical level, each higher dimensional string-netcondensate is associated with a special kind of tensorcategory - a symmetric tensor category (18), (32). Morephysically, we have shown that higher dimensional string-net condensation naturally gives rise to both gauge inter-actions and Fermi statistics. Viewed from this perspec-tive, string-net condensation provides a mechanism forunifying gauge interactions and Fermi statistics. It mayhave applications to high energy physics [30].

From a more general point of view, all of the phasesdescribed by Landaus symmetry breaking theory canbe understood in terms of particle condensation. Thesephases are classified using group theory and lead to emer-gent gapless scalar bosons [49, 50], such as phonons, spinwaves, etc . In this paper, we have shown that there is amuch richer class of phase - arising from the condensationof extended objects. These phases are classified usingtensor category theory and lead to emergence of Fermistatistics and gauge excitations. Clearly, there is wholenew world beyond the paradigm of symmetry breakingand long range order. It is a virgin land waiting to beexplored.

We would like to thank Pavel Etingof and MichaelFreedman for useful discussions of the mathematical as-pects of topological field theory. This research is sup-ported by NSF Grant No. DMR0123156 and by NSF-MRSEC Grant No. DMR0213282.

APPENDIX A: GENERAL STRING-NET

MODELS

In this section, we discuss the most general string-netmodels. These models can describe all doubled topo-logical phases, including all discrete gauge theories anddoubled Chern-Simons theories.

In these models, there is a spin degree of freedom ateach branch point or node of a string-net, in addition tothe usual string-net degrees of freedom. The dimensionof this spin Hilbert space depends on the string typesof the 3 strings incident on the node.

To specify a particular model one needs to providea 3 index tensor ijk which gives the dimension of thespin Hilbert space associated with

{i,j,k

}(in addition to

the usual information). The string-net models discussedabove correspond to the special case where ijk = 0, 1 forall i,j,k. In the case of gauge theory, ijk is the numberof copies of the trivial representation that appear in thetensor product ij k. Thus we need the more generalstring-net picture to describe gauge theories where thetrivial representation appears multiple times in ijk.

The Hilbert space of the string-net model is definedin the natural way: the states in the string-net Hilbertspace are linear superpositions of different spatial config-urations of string-nets with different spin states at thenodes.

One can analyze string-net condensed phases as before.The universal properties of each phase are captured bya fixed-point ground state wave function . The wavefunction is specified by the local rules (4), (5) andsimple modifications of (6), (7):

i l

k

j

=ij

i

l

k

i

m

i l

j k

=n

(Fijmkln )

i

j kl

n

The complex numerical constant Fijmkln is now a complex

tensor (Fijmkln ) of dimension ijm klm inl jkn .

One can proceed as before, with self-consistency con-ditions, fixed-point Hamiltonians, string operators, andthe generalization to (3 + 1) dimensions. The exactly sol-uble models are similar to (10). The main difference isthe existence of an additional spin degree of freedom ateach site of the honeycomb lattice. These spins accountfor the degrees of freedom at the nodes of the string-nets.

APPENDIX B: SELF-CONSISTENCYCONDITIONS

In this section, we derive the self-consistency condi-tions (8). We begin with the last relation, the so-calledpentagon identity, since it is the most fundamental.To derive this condition, we use the fusion rule (7)

to relate the amplitude

i

k l

m

p q

to the amplitude

mi

j lk

sr

in two distinct ways (see Fig. 14). On the

one hand, we can apply the fusion rule (7) twice to obtain


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i

kj l

m

qp

i m mi

j lk j lk

i m

j lk

j lk

mi

q

sr

p

nn

(a) (b) (c)

(d) (e)

s

r

FIG. 14: The fusion rule (7) can be used to relate the am-plitude of (a) to the amplitude of (c) in two different ways.On the one hand, we can apply the fusion rule (7) twice -along the links denoted by solid arrows - to relate (a) (b) (c). But we can also apply (7) three times - along the linksdenoted by dashed arrows - to relate (a) (d) (e) (c).Self-consistency requires that the two sequences of the opera-tion lead to the same linear relations between the amplitudesof (a) and (c).

the relation

i

k l

m

p q

=r

Fjipqkr

i m

j lk

r

q

=r,s

FjipqkrFriq

mls

mi

j lk

sr

(Here, we neglected to draw a shaded region surround-ing the whole diagram. Just as in the local rules (4-7)the ends of the strings i,j,k,l ,m are connected to somearbitrary string-net configuration). But we can also ap-

ply the fusion rule (7) three times to obtain a differentrelation:

i

k l

m

p q

=n

Fmlqkpn

i m

j lk

n

p

=n,s

FmlqkpnFjipmns

j l

k

mi

ns

=n,r,s

FmlqkpnFjipmnsF

jsnlkr

mi

lk

sr

If the rules are self-consistent, then these two relationsmust agree with each other. Thus, the two coefficients

of

mi

lk

sr

must be the same. This equality implies

the pentagon identity (8).The first two relations in (8) are less fundamental. In

fact, the first relation is not required by self-consistencyat all; it is simply a useful convention. To see this, con-sider the following rescaling transformation on wave func-tions . Given a string-net wave function , we canobtain a new wave function by multiplying the ampli-tude (X) for a string-net configuration X by an arbi-trary factor f(i,j,k) for each vertex

{i,j,k

}in X. As

long as f(i,j,k) is symmetric in i,j,k and f(0, i , i) = 1,

(a) (b) (c) (d)

n ni k l nn l

m m m j

k

i

j

l

l

i

j

j

k

i

k m

FIG. 15: Four string-net configurations related by tetrahe-

dral symmetry. In diagram (a), we show the tetrahedron cor-responding to Gijmkln . In diagrams (b), (c),(d), we show the

tetrahedrons Glkm

jin , Gjimlkn , G

imjknl, obtained by reflecting (a)

in 3 different planes: the plane joining n to the center ofm,the plane joining m to the center ofn, and the plane joiningi to the center ofk. The four tetrahedrons correspond to thefour terms in the second relation of (8).

this operation preserves the topological invariance of .The rescaled wave function satisfies the same set oflocal rules with rescaled Fijmkln :

F

ijm

kln

F

ijm

kln = F

ijm

kln

f(i,j,m)f(k,l ,m)

f(n,l ,i)f(j,k,n) (B1)

Since and describe the same quantum phase, weregard F and F as equivalent local rules. Thus the firstrelation in (8) is simply a normalization convention for For (except when i, j or k vanishes; these cases requirean argument similar to the derivation of the pentagonidentity).

The second relation in (8) has more content. This re-lation can be derived by computing the amplitude for atetrahedral string-net configuration. We have:

n

k

l

j m

i = F

ijm

kln n

n

lkj i

=Fijmkln Fnkj

kn0 dk

n

li

=Fijmkln Fnkj

kn0 Fnil

in0 dkdidn() (B2)=Fijmkln vivjvkvl() (B3)

We define the above combination in the front of ()as:

Gijmkln Fijmkln vivjvkvl (B4)

Imagine that the above string-net configuration lies ona sphere. In that case, topological invariance (together

with parity invariance) requires that Gijmkln be invariantunder all 24 symmetries of a regular tetrahedron. Thesecond relation in (8) is simply a statement of this tetra-

hedral symmetry requirement - written in terms ofFijmkln .(See Fig. 15).

In this section, we have shown that the relations (8)are necessary for self-consistency. It turns out that theserelations are also sufficient. One way of proving this is touse the lattice model (10). A straightforward algebraiccalculation shows that the ground state of 10 obeys thelocal rules (4-7), as long as (8) is satisfied. This estab-lishes that the local rules are self-consistent.


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(a) (b)

FIG. 16: The fattened honeycomb lattice. The strings areforbidden in the shaded region. A string state in the fattenedhoneycomb lattice (a) can be viewed as a superposition ofstring states on the links (b).

APPENDIX C: GRAPHICAL REPRESENTATION

OF THE HAMILTONIAN

In this section, we provide an alternative, graphical,representation of the lattice model (10). This graphicalrepresentation provides a simple visual technique for un-

derstanding properties (a)-(c) of the Hamiltonian (10).We begin with the 2D honeycomb lattice. Imagine wefatten the links of the lattice into stripes of finite width(see Fig. 16). Then, any string-net state in the fat-tened honeycomb lattice (Fig. 16a) can be viewed as asuperposition of string-net states in the original, unfat-tened lattice (Fig. 16b). This mapping is obtained viathe local rules (4-7). Using these rules, we can relate theamplitude (X) for a string-net in the fattened lattice toa linear combination of string-net amplitudes in the orig-inal lattice: (X) =

ai(Xi). This provides a natural

linear relation between the states in the fattened lattice

and those in the unfattened lattice: |X = ai|Xi.This linear relation is independent of the particular wayin which the local rules (4-7) are applied, as long as therules are self-consistent.

In this way, the fattened honeycomb lattice providesan alternative notation for representing the states in theHilbert space of (10). This not

michael a. levin and xiao-gang wen- string-net condensation: a physical mechanism for topological...

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