Xiao-Gang Wen- Quantum order from string-net condensations and origin of light and massless fermions

8/3/2019 Xiao-Gang Wen- Quantum order from string-net condensations and origin of light and massless fermions

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Quantum order from string-net condensations and origin of light and massless

fermions

Xiao-Gang Wen

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Dated: Dec. 2002)

Recently, it was pointed out that quantum orders and the associated projective symmetry groupscan produce and protect massless gauge bosons and massless fermions in local bosonic models. Inthis paper, we demonstrate that a state with such kind of quantum orders can be viewed as astring-net condensed state. The emerging gauge bosons and fermions in local bosonic models canbe regarded as a direct consequence of string-net condensation. The gauge bosons are fluctuationsof large closed string-nets which are condensed in the ground state. The ends of open strings (ornodes of open string-nets) are the charged particles of the corresponding gauge field. For certaintypes of strings, the nodes of string-nets can even be fermions. According to the string-net picture,fermions always carry gauge charges. This suggests the existence of a new discrete gauge field thatcouples to neutrinos and neutrons. We also discuss how chiral symmetry that protects masslessDirac fermions can emerge from the projective symmetry of quantum order.

PACS numbers: 11.15.-q

Contents

I. Introduction 1A. Fundamental questions about light and

fermions 1B. Gapless phonon and symmetry breaking

orders 2C. The existence of light and fermions implies

the existence of new orders 2D. Topological order and quantum order 2E. The quantum orders from string-net

condensations 3F. Organization 5

II. Local bosonic models 5

III. Z2 spin liquid and string-netcondensation on square lattice 6A. Hamiltonians with closed-string-net

condensation 6B. String condensation and low energy effective

theory 7C. Three types of strings and emerging

fermions 7

IV. Classification of different stringcondensations by PSG 8A. Four classes of string-net condensations 8

B. PSG and ends of condensed strings 9C. PSGs classify different string-net

condensations 10D. Different PSGs from the ends of different

condensed strings 11

V. Massless fermion and PSG in string-netcondensed state 11A. Exact soluble spin- 12

12 model 12

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

B. Physical properties of the spin- 1212 model 13

C. Projective symmetry and massless fermions 14

VI. Massless fermions and string-netcondensation on cubic lattice 16

VII. Artificial light and artificial masslesselectron on cubic lattice 16A. 3D rotor model and artificial light 16B. (Quasi-)exact soluble QED on cubic lattice 17C. Emerging chiral symmetry from PSG 18

VIII. QED and QCD from a bosonic model oncubic lattice 20

IX. Conclusion 22

References 23

I. INTRODUCTION

A. Fundamental questions about light and fermions

We have known light and fermions for many years. Butwe still cannot give a satisfactory answer to the followingfundamental questions: What are light and fermions?Where light and fermions come from? Why light and

fermions exist? At moment, the standard answers to theabove fundamental questions appear to be light is theparticle described by a gauge field and fermions arethe particles described by anti-commuting fields. Here,we like to argue that there is another possible answer tothe above questions: our vacuum is filled with string-likeobjects that form network of arbitrary sizes and thosestring-nets form a quantum condensed state. Accord-ing to the string-net picture, the light (and other gaugebosons) is a vibration of the condensed string-nets andfermions are ends of strings (or nodes of string-nets). Thestring-net condensation provides a unified origin of lightand fermions.[62]


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Before discussing the above fundamental questions inmore detail, we would like to clarify what do we mean bylight exists and fermions exist. We know that thereis a natural mass scale in physics the Planck mass.Planck mass is so large that any observed particle have amass at least factor 1016 smaller than the Planck mass.So all the observed particles can be treated as masslesswhen compared with Planck mass. When we ask why

some particles exist, we really ask why those particles aremassless (or nearly massless when compared with Planckmass). So the real issue is to understand what makescertain excitations (such as light and fermions) massless.We have known that symmetry breaking is a way to getgapless bosonic excitations. We will see that string-netcondensation is another way to get gapless excitations.However, string-net condensations can generate masslessgauge bosons and massless fermions.

Second, we would like to clarify what do we mean byorigin of light and fermions. We know that everythinghas to come from something. So when we ask wherelight and fermions come from, we have assumed thatthere are some things simpler and more fundamental thanlight and fermions. In the section II, we define localbosonic models which are simpler than models with gaugefields coupled to fermions. We will regard local bosonicmodels as more fundamental (the locality principle). Wewill show that light and fermions can emerge from a localbosonic model if the model contains a condensation ofnets of string-like object in its ground state.

After the above two clarifications, we can state moreprecisely the meaning of string-net condensation pro-vides another possible answer to the fundamental ques-tions about light and fermions. When we say gaugebosons and fermions originate from string-net condensa-tion, we really mean that (nearly) massless gauge bosons

and fermions originate from string-net condensation in alocal bosonic model.

B. Gapless phonon and symmetry breaking orders

Before considering the origin of massless photon andmassless fermions, let us consider a simpler massless (orgapless) excitation phonon. We can ask three simi-lar questions about phonon: What is phonon? Wherephonon comes from? Why phonon exists? We know thatthose are scientific questions and we know their answers.Phonon is a vibration of a crystal. Phonon comes from

a spontaneous translation symmetry breaking. Phononexists because the translation-symmetry-breaking phaseactually exists in nature. In particular, the gaplessness ofphonon is directly originated from and protected by thespontaneous translation symmetry breaking.[1, 2] Manyother gapless excitations, such as spin wave, superfluidmode etc , also come from condensation of point-like ob-

jects that break certain symmetries.It is quite interesting to see that our understanding of a

gapless excitation - phonon - is rooted in our understand-ing of phases of matter. According to Landaus theory,[3]phases of matter are different because they have differentbroken symmetries. The symmetry description of phases

is very powerful. It allows us to classify all possible crys-tals. It also provides the origin for gapless phonons andmany other gapless excitations. Until a few years ago, itwas believed that the condensations of point-like objects,and the related symmetry breaking and order parame-ters, can describe all the orders (or phases) in nature.

C. The existence of light and fermions implies theexistence of new orders

Knowing light as a massless excitation, one may won-der maybe light, just like phonon, is also a Nambu-Goldstone mode from a broken symmetry. However, ex-periments tell us that a U(1) gauge boson, such as light,is really different from a Nambu-Goldstone mode in 3+1dimensions. Therefore it is impossible to use Landaussymmetry breaking theory and condensation of point-like objects to understand the origin and the massless-ness of light. Also, Nambu-Goldstone modes are alwaysbosonic, thus it is impossible to use symmetry breakingto understand the origin and the (nearly) masslessnessof fermions. It seems that there does not exist any orderthat can give rise to massless light and massless fermions.Because of this, we put light and electron into a differentcategory than phonon. We regarded them as elementaryand introduced them by hand into our theory of nature.

However, if we believe light and electrons, just likephonon, exist for a reason, then such a reason must be acertain order in our vacuum that protect the masslessnessof light and electron. (Here we have assumed that lightand electron are not something that we place in an emptyvacuum. Our vacuum is more like an ocean which isnot empty. Light and electron are collective excitationsthat correspond to certain patterns of water motion.)

Now the question is that what kind of order can give riseto light and electron, and protect their masslessness.

If we really believe in the equality between light, elec-tron and phonon, then the very existence of light andfermions indicates that our understanding of states ofmatter is incomplete. We should deepen and expand ourunderstanding of the states of matter. There should benew states of matter that contain new kind of orders. Thenew orders will produce light and electron, and protecttheir masslessness.

D. Topological order and quantum order

After the discovery of fractional quantum Hall (FQH)effect,[4, 5] it became clear that the Landaus symmetrybreaking theory cannot describe different FQH states,since those states all have the same symmetry. It wasproposed that FQH states contain a new kind of order -topological order.[6] Topological order is new because itcannot be described by symmetry breaking, long rangecorrelation, and local order parameters. Non of the usualtools that we used to characterize phases applies to topo-logical order. Despite of this, topological order is not anempty concept. Topological order can be characterizedby a new set of tools, such as the number of degenerate


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Quantum system Classical system

Gapped

NambuGoldstone mode

"Particle" condensation

Orders

Fermi liquids

Fermi surface topology

Gapless Gauge bosons/Fermions

Projective symmetry group

Conformal algebra, ??

Topological field theory

Nonsymmetry breaking ordersSymmetry breaking orders

Topological orders

Quantum orders

Symmetry group

Stringnet condensation

FIG. 1: A classification of different orders in matter. (Weview our vacuum as one kind of matter.)

ground states, quasiparticle statistics, and edge states. Itwas shown that the ground state degeneracy of a topo-logical ordered state is a universal property since thedegeneracy is robust against any perturbations.[7] Sucha topological degeneracy demonstrates the existence oftopological order. It can also be used to perform faulttolerant quantum computations.[8]

Recently, the concept of topological order was general-ized to quantum order.[9, 10] Quantum order is used todescribe new kinds of orders in gapless quantum states.One way to understand quantum order is to see how itfits into a general classification scheme of orders (see Fig.1). First, different orders can be divided into two classes:symmetry breaking orders and non-symmetry breakingorders. The symmetry breaking orders can be describedby a local order parameter and can be said to containa condensation of point-like objects. All the symmetrybreaking orders can be understood in terms of Landaussymmetry breaking theory. The non-symmetry breakingorders cannot be described by symmetry breaking, nei-ther by the related local order parameters and long range

correlations. Thus they are a new kind of orders. If aquantum system (a state at zero temperature) containsa non-symmetry breaking order, then the system is saidto contain a non-trivial quantum order. We see that aquantum order is simply a non-symmetry breaking orderin a quantum system.

Quantum order can be further divided into many sub-classes. If a quantum state is gapped, then the corre-sponding quantum order will be called topological or-der. The low energy effective theory of a topologicalordered state will be a topological field theory.[11] Thesecond class of quantum orders appear in Fermi liquids(or free fermion systems). The different quantum or-

ders in Fermi liquids are classified by the Fermi surfacetopology.[10, 12]

E. The quantum orders from string-netcondensations

In this paper, we will concentrate on the third class ofquantum orders the quantum orders from condensationof nets of strings, or simply, string-net condensation.[13,14] This class of quantum orders shares some similar-ities with the symmetry breaking orders of particlecondensation. We know that different symmetry break-

ing orders can be classified by symmetry groups. Usinggroup theory, we can classify all the 230 crystal ordersin three dimensions. The symmetry also produces andprotects gapless Nambu-Goldstone bosons. Similarly, aswe will see later in this paper, different string-net con-densations (and the corresponding quantum orders) canbe classified by a mathematical object called projectivesymmetry group (PSG).[9, 10] Using PSG, we can clas-

sify over 100 different 2D spin liquids that all have thesame symmetry.[9] Just like symmetry group, PSG canalso produce and protect gapless excitations. However,unlike symmetry group, PSG produces and protects gap-less gauge bosons and gapless fermions.[9, 15, 16] Becauseof this, we can say light and massless fermions can havea unified origin. They can come from string-net conden-sations.

We used to believe that to have light and fermions inour theory, we have to introduce by hand a fundamen-tal U(1) gauge field and anti-commuting fermion fields,since at that time we did not know any collective modesthat behave like gauge bosons and fermions. Now, we

know that gauge bosons and fermions appear commonlyand naturally in quantum ordered states, as fluctuationsof condensed string-nets and ends of open strings. Thisraises an issue: do light and fermions come from a fun-damental U(1) gauge field and anti-commuting fields asin the 123 standard model or do they come from a par-ticular quantum order in our vacuum? Clearly it is morenatural to assume light and fermions come from a quan-tum order in our vacuum. From the connection betweenstring-net condensation, quantum order, and masslessgauge/fermion excitations, it is very tempting to pro-pose the following answers to the fundamental questionsabout light and (nearly) massless fermions:

What are light and fermions?Light is a fluctuation of condensed string-nets of arbi-trary sizes. Fermions are ends of open strings.

Where light and (nearly) massless fermionscome from?Light and the fermions come from the collective mo-tions of nets of string-like objects that fill our vacuum.

Why light and (nearly) massless fermions exist?Light and the fermions exist because our vacuumchooses to have a string-net condensation.

Had our vacuum chosen to have a particle conden-sation, there would be only Nambu-Goldstone bosonsat low energies. Such a universe would be very bor-ing. String-net condensation and the resulting light and(nearly) massless fermions provide a much more inter-esting universe, at least interesting enough to supportintelligent life to study the origin of light and masslessfermions.

The string-net picture of fermions explains why thereis always an even number of fermions in our universe.The string-net picture for gauge bosons and fermions alsohas an experimental prediction: all fermions must carrycertain gauge charges.[14] At first sight, this prediction


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appears to contradict with the known experimental factthat neutron carry no gauge charges. Thus one may thinkthe string-net picture of gauge bosons and fermions hasalready been falsified by experiments. Here we would liketo point out that the string-net picture of gauge bosonsand fermions can still be correct if we assume the exis-tence of a new discrete gauge field, such as a Z2 gaugefield, in our universe. In this case, neutrons and neutrinos

carry a non-zero charge of the discrete gauge field. There-fore, the string-net picture of gauge bosons and fermionspredict the existence of discrete gauge excitations (suchas gauge flux lines) in our universe.

We would like to remark that, despite the similarity,the above string-net picture of gauge bosons and fermionsis different from the picture of standard superstring the-ory. In standard superstring theory, closed strings corre-spond to gravitons, and open string correspond to gaugebosons. All the elementary particles correspond to dif-ferent vibration modes of small strings in the superstringtheory. Also, the fermions in the standard superstringtheory come from the fermion fields on the world sheet.In our string-net picture, the vacuum is filled with largenets of strings. The massless gauge bosons correspondto the fluctuations of large closed string-nets (ie nets ofclosed strings) and fermions correspond to the ends ofopen strings in string nets. Anti-commuting fields arenot needed to produce (nearly) massless fermions. Mass-less fermions appear as low energy collective modes in apurely bosonic system.

The string-net picture for gauge theories have a longhistory. The closed-string description of gauge fluctua-tions is intimately related to the Wilson loop in gaugetheory.[1719] The relation between dynamical gaugetheory and a dynamical Wilson-loop theory was sug-gested in Ref. [20, 21]. Ref. [22] studied the Hamilto-

nian of a non-local model - lattice gauge theory. It wasfound that the lattice gauge theory contains a string-net structure and the gauge charges can be viewed asends of strings. In Ref. [23, 24] various duality relationsbetween lattice gauge theories and theories of extendedobjects were reviewed. In particular, some statisticallattice gauge models were found to be dual to certainstatistical membrane models.[25] This duality relation isdirectly connected to the relation between gauge theoryand closed-string-net theory[13] in quantum models.

Emerging fermions from local bosonic models also havea complicated history. The first examples of emerg-ing fermions/anyons were the fractional quantum Hall

states,[4, 5] where fermionic/anyonic excitations were ob-tained theoretically from interacting bosons in magneticfield.[26] In 1987, fermion fields and gauge fields wereintroduced to express the spin-1/2 Hamiltonian in theslave-boson approach.[27, 28] However, writing a bosonicHamiltonian in terms of fermion fields does not implythe appearance of well defined fermionic quasiparticles.Emerging fermionic excitations can appear only in decon-fined phases of the gauge field. Ref. [2932] constructedseveral deconfined phases where the fermion fields do de-scribe well defined quasiparticles. However, dependingon the property of deconfined phases, those quasiparti-cles may carry fractional statistics (for the chiral spin

states)[29, 30, 33] or Fermi statistics (for the Z2 decon-fined states).[31, 32]

Also in 1987, in a study of resonating-valence-bond(RVB) states, emerging fermions (the spinons) were pro-posed in a nearest neighbor dimer model on squarelattice.[3436] But, according to the deconfinement pic-ture, the results in Ref. [34, 35] are valid only when theground state of the dimer model is in the Z2 deconfined

phase. It appears that the dimer liquid on square latticewith only nearest neighbor dimers is not a deconfinedstate,[35, 36] and thus it is not clear if the nearest neigh-bor dimer model on square lattice[35] has the fermionicquasiparticles or not.[36] However, on triangular lattice,the dimer liquid is indeed a Z2 deconfined state.[37]Therefore, the results in Ref. [34, 35] are valid for thetriangular-lattice dimer model and fermionic quasiparti-cles do emerge in a dimer liquid on triangular lattice.

All the above models with emerging fermions are 2+1Dmodels, where the emerging fermions can be understoodfrom binding flux to a charged particle.[26] Recently, itwas pointed out in Ref. [14] that the key to emergingfermions is a string structure. Fermions can generallyappear as ends of open strings. The string picture allowsa construction of a 3+1D local bosonic model that hasemerging fermions.

Comparing with those previous results, the new fea-tures discussed in this paper are: (A) Massless gaugebosons and fermions can emerge from local bosonic mod-els as a result of string-net condensation. (B) Mass-less fermions are protected by the string-net condensa-tion (and the associated PSG). (C) String-net condensedstates represent a new kind of phases which cannot bedescribed Landaus symmetry breaking theory. Differentstring-net condensed states are characterized by differ-ent PSGs. (D) QED and QCD can emerge from a local

bosonic model on cubic lattice. The effective QED andQCD has 4N families of leptons and quarks. Each familyhas one lepton and two flavors of quarks.

The bottom line is that, within local bosonic mod-els, massless fermions do not just emerge by themselves.Emerging massless fermions, emerging massless gaugebosons, string-net condensations, and PSG are intimatelyrelated. They are just different sides of same coin - quan-tum order.

According to the picture of quantum order, elemen-tary particles (such as photon and electron) may not beelementary after all. They may be collective excitationsof a local bosonic system below Planck scale. Since we

cannot do experiments close to Planck scale, it is hardto determine if photon and electron are elementary par-ticles or not. In this paper, we would like to show thatthe string-net picture of light and fermions is at least selfconsistent by studying some concrete local boson mod-els which produce massless gauge bosons and masslessfermions through string-net condensations. The local bo-son models studied here are just a few examples among along list of local boson models[8, 28, 29, 3133, 35, 3747]that contain emerging fermions and gauge fields.

Here we would like to stress that the string-net pic-ture for the actual gauge bosons and fermions in ouruniverse is only a proposal at moment. Although string-


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(Z2B,Z2B)

MO

(Z2A,Z2B)

(Z2A,Z2A)(Z2B,Z2A)

Z 2

Z 2Z 2

Z 2

string condense

fluxcharge

string condensestring condense

flux

string condense

charge

+

+

g/J

0

0U/J

FIG. 3: The proposed phase diagram for the H = HU +Hg + HJ model. J is assumed to be positive. The fourstring-net condensed phases are characterized by a pair ofPSGs (PSGcharge ,PSGvortex). MO marks an magnetic or-dered state.

This way, we obtain the Hamiltonian of our spin-1/2

model

H = HU + HJ + Hg (7)

B. String condensation and low energy effectivetheory

When J = 0 in Eq. (7), the model is exactly soluble

since [Fi, Fj ] = 0.[8, 46] All the eigenstates of HU + Hgcan be obtained from the common eigenstates of Fi.Since F2i = 1, the eigenvalues of Fi are simply 1. Thus all the eigenstates of HU + Hg are labeled by 1 oneach plaquette. (Note, this is not true for finite sys-tems where the boundary condition introduce additionalcomplications.[46]) The energies of those eigenstates are

sum of eigenvalues of Fi weighted by U and g.From the results of exact soluble model, we suggest a

phase diagram of our model as sketched in Fig. 3. Wewill show that the phase diagram contains four differentstring-net condensed phases and one phase with no stringcondensation. All the phases have the same symmetryand are distinguished only by their different quantumorders.

Let us first discuss the phase with U,g > 0. We willassume J = 0 and U g. In this limit, all states con-taining open strings will have an energy of order U. Thelow energy states contain only closed strings (or moregenerally closed string-nets) and satisfy

Fi|i=even = 1 (8)For infinite systems, the different low energy states arelabeled by the eigenvalues of Fi on odd plaquettes:

Fi|i=odd = 1 (9)In particular, the ground state is given by

Fi|i=odd = 1. (10)

y

x

x

y

x

y

y

x

x

y

Fi

iF =1

FIG. 4: A hopping of the Z2 charge around four nearest neigh-bor even plaquettes.

All the closed-string-net operators W(Cnet) commutewith HU + Hg. Hence the ground state |0 of HU + Hgsatisfies

0|W(Cnet)|0 = 1. (11)

Thus the U , g > 0 ground state has a closed-stringi-net condensation. The low energy excitations above theground state can be obtained by flipping Fi from 1 to

1

on some odd plaquettes.If we view Fi on odd plaquettes as the flux in Z2 gauge

theory, we find that the low energy sector of model isidentical to a Z2 lattice gauge theory, at least for infi-nite systems. This suggests that the low energy effectivetheory of our model is a Z2 lattice gauge theory.

However, one may object this result by pointing outthat the low energy sector of our model is also identical toan Ising model with one spin on each the odd plaquette.Thus the the low energy effective theory should be theIsing model. We would like to point out that althoughthe low energy sector of our model is identical to an Isingmodel for infinite systems, the low energy sector of our

model is different from an Ising model for finite systems.For example, on a finite even by even lattice with periodicboundary condition, the ground state of our model hasa four-fold degeneracy.[8, 46] The Ising model does nothave such a degeneracy. Also, our model contains anexcitation that can be identified as Z2 charge (see below).Therefore, the low energy effective theory of our model isa Z2 lattice gauge theory instead of an Ising model. TheFi = 1 excitations on odd plaquettes can be viewed asthe Z2 vortex excitations in the Z2 lattice gauge theory.

C. Three types of strings and emerging fermions

What is the Z2 charge excitations? We note that, inthe closed-string-net condensed state, the action of theclosed-string operator Eq. (2) on the ground state is triv-ial. This suggests that the action of the open-string op-erators on the ground state only depend on the ends ofstrings, since two open strings with the same ends onlydiffer by a closed string. Therefore, an open-string op-erator create two particles at its ends when acting onthe string condensed state. Since the strings in Eq. (2)only connect even plaquettes, the particle correspondingto the ends of the open strings always live on the evenplaquettes. We will call such a string T1 string. Form


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For our case, the SG is the translation group SG =

{1, T(2)xy , T(2)xy ,...}. For every element in SG, a(2) SG,there are one or several elements in PSG, a P SG, suchthat a a = a(2). The IGG in our PSG is formed bythe transformations G0 on the singe-particle states thatsatisfy G0 G0 = 1. We find that IGG is generated by

G0|p = |p > (29)G0, TxyGxy and TxyGxy generate the Z2A and Z2BPSGs.

Now we see that the underlying translation symmetrydoes not require the single-particle hopping HamiltonianH(p) to have a translation symmetry. It only requireH(p) to be invariant under the Z2A PSG or the Z2BPSG. When H(p) is invariant under the Z2A PSG, thehopping Hamiltonian has the usual translation symme-try. When H(p) is invariant under the Z2B PSG, thehopping Hamiltonian has a magnetic translation symme-try describing a hopping in a magnetic field with -fluxthrough each odd plaquette.

C. PSGs classify different string-net condensations

After understand the possible PSGs for the hoppingHamiltonian of the ends of strings, now we are ready tocalculate the actual PSGs. Let us consider two groundstates of our model HU+ Hg + Ht. One has Fi|i=odd = 1(for g > 0) and the other has Fi|i=odd = 1 (for g < 0).Both ground states have the same translation symmetryin x+y and xy directions. However, the correspondingsingle-particle hopping Hamiltonian H(p) has different

symmetries. For the Fi|i=odd = 1 state, there is no fluxthrough odd plaquettes and H(p) has the usual transla-tion symmetry. It is invariant under the Z2A PSG. Whilefor the Fi|i=odd = 1 state, there is -flux through oddplaquettes and H(p) has a magnetic translation symme-

try. Its PSG is the Z2B PSG. Thus the Fi|i=odd = 1 stateand the Fi|i=odd = 1 state have different orders despitethey have the same symmetry. The different quantumorders in the two states can be characterized by theirdifferent PSGs.

The above discussion also apply to the Z2 vortex andT2 strings. Thus the quantum orders in our model aredescribed by a pair of PSGs (P SGcharge, P S Gvortex),one for the Z2 charge and one for the Z2 vortex. ThePSG pairs (P SGcharge, P S Gvortex) allows us to distin-

guish four different string-net condensed states of modelH = HU + Hg + Ht.(See Fig. 5.)

Now let us assume U = g in our model:

HU + Hg + Ht = Ht Vi

Fi (30)

The new physical spin model has a larger translationsymmetry generated by i = x and i = y (see Fig.5). Due to the enlarged symmetry group, the quantumorders in the new system should be characterized by anew PSG. In the following, we will calculate the newPSG.

extra translation

symmetry

FM

(Z2B,Z 2A) (Z2A,Z2A)

(Z2A,Z2B)(Z2B,Z2B)

Z 2

Z 2Z 2

Z 2

string condense

fluxcharge

string condensestring condense

flux

string condense

charge

+

+0

0

g/t

U/t

FIG. 5: The proposed phase diagram for the H= HU +Hg +Ht model. t = t

is assumed to be positive. The four string-net condensed phases are characterized by a pair of PSGs(PSGcharge ,PSGvortex). FM marks a ferromagnetic phase.

The single-particle states are given by |p. When p iseven,

|p

corresponds to a Z2 charge and when p is odd,

|p corresponds to a Z2 vortex. We see that a transla-tion by x (or y) will change a Z2 charge to a Z2 vortexor a Z2 vortex to a Z2 charge. Therefore the effectivesingle-particle hopping Hamiltonian H(p) only containhops between even plaquettes or odd plaquettes. Thesingle-particle Hamiltonian H(p) is invariant under thefollowing two transformations G0 and G0:

G0|p = |p >, G0|p = ()p|p > (31)We note that G0G0 = G0G0 = 1. Therefore both G0and G0 correspond to the identity element of the sym-metry group of two-particle states. (G0, G0) generate theIGG of the new PSG. The new IGG is Z2

Z2.

The translations of single-particle states by x and byy are generated by TxGx and TyGy. The translation byx + y and by x y are given by

TxyGxy =TyGyTxGx

TxyGxy =(TyGy)1TxGx (32)

Since TxyGxy and TxyGxy are the translations of the Z2charge and the Z2 vortex discussed above, we find

(TxyGxy)1(TxyGxy)

1TxyGxyTxyGxy = (33)

where = 1 for the (Z2A, Z2A) state with Fi = 1 and

=

1 for the (Z2B, Z2B) state with Fi =

1. Also

TxGx and TyGy must satisfy

(TyGy)1(TxGx)

1TyGyTxGx IGG (34)since on the two-particle states

(T(2)y )1(T(2)x )

1T(2)y T(2)x = 1 (35)

Therefore, (TyGy)1(TxGx)

1TyGyTxGx may take thefollowing possible values 1, 1, ()p, and ()p. Onlychoices p and p are consistent with Eq. (33) and wehave

(TyGy)1(TxGx)

1TyGyTxGx = p (36)


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yx

xy

x

z

z

z

z

y

1

3 2

4

4 3

21

23

4

1

Fi

FIG. 6: Fermion hopping around a plaquette, around asquare, and around a site.

We like to point out that the different choices of =1 do not lead to different PSGs. This is because ifTxGx is a symmetry of the H(p), then TxGx()p is alsoa symmetry of the H(p). However, the change Gx Gx()p will change the sign of . Thus = 1 and = 1 will lead to the same PSG. But the differentsigns of will lead to different PSGs.

(G0, G0) and (TxGx, TyGy) generate the new PSG. Thesingle-particle Hamiltonian H(p) is invariant under such

a PSG. = 1 and = 1 correspond to two differentPSGs that characterize two different quantum orders.The ground state for V > 0 and |V| t (see Eq. (30)) isdescribed by the = 1 PSG. The ground state for V < 0and |V| t is described by the = 1 PSG. The twoground states have different quantum orders and differentstring-net condensations.

D. Different PSGs from the ends of differentcondensed strings

In this section we still assume U = g and consider only

the translation invariant model Eq. (30). In the above wediscussed the PSG for the ends of one type of condensedstrings in different states. In this section, we will concen-trate on only one ground state. We know that the groundstate of our spin-1/2 model contain condensations of sev-eral type of strings. We like to calculate the the differentPSGs for the different condensed strings.

The PSGs for the condensed T1 and T2 strings wereobtained above. Here we will discuss the PSG for theT3 string. Since the ends of the T3 strings live on thelinks, the corresponding single-particle hopping Hamil-tonian Hf(l) describes fermion hopping between links.Clearly, the symmetry group (the PSG) of Hf(l) can be

different from that of H(p).Let us consider fermion hopping around some smallloops. The four hops of a fermion around a site i (see Fig.6) are generated by yi ,

xi ,

yi , and

xi . The total ampli-

tude of a fermion hopping around a site is yi xi

yi

xi =

1. The fermion hopping around a site always sees -flux. The four hops of a fermion around a plaquette p (seeFig. 6) are generated by xi0 ,

yi0+x

, xi0+x+y, and yi0+y

,where i0 is the lower left corner of the plaquette p. Thetotal amplitude of a fermion hopping around a plaquetteis given by yi0+y

xi0+x+y

yi0+xxi0

= Fi0 . When V > 0,

the ground state has Fi = 1. However, since site i0 isnext to the end of T3 string, we have Fi0 = Fi = 1.

In this case, the fermion hopping around a plaquette sees-flux. For V < 0 ground state, we find that fermionhopping around a plaquette sees no flux.

Let us define the fermion hopping l l + x as thecombination of two hops l l + x2 y2 l + x andthe fermion hopping l l + y as the combination ofl l + x2 + y2 l + y (see Fig. 6). Under such adefinition, a fermion hopping around a square l l +x l + x + y l + y l correspond to a fermionhopping around a site and a fermion hopping around aplaquette discussed above (see Fig. 6). Therefore, thetotal amplitude for a fermion hopping around a square isgiven by the sign of V: sgn(V). We find the translationsymmetries (TxGx, TyGy) of the fermion hopping Hf(l)satisfies

(TyGy)1(TxGx)

1TyGyTxGx = sgn(V) (37)

which is different from the translation algebra for H(p)Eq. (36). Hf(l) is also invariant under G0:

G0|l = |l (38)

(G0, TxGx, TyGy) generate the symmetry group - thefermion PSG - of Hf(l). We will call the fermion PSGEq. (37) for sgn(V) = 1 the Z2A PSG and the fermionPSG for sgn(V) = 1 the Z2B PSG. We see that thequantum orders in the ground state can also be charac-terized using the fermion PSG. The quantum order inthe V > 0 ground state is characterized by the Z2A PSGand the quantum order in the V < 0 ground state ischaracterized by the Z2B PSG.

In Ref. [46], the spin-1/2 model Eq. (30) (with t =t = 0) was viewed as a hardcore boson model. Themodel was solved using slave-boson approach by split-ting a boson into two fermions. Then it was shown thefermion hopping Hamiltonian for V > 0 and V < 0 states

have different symmetries, or invariant under differentPSGs. According to the arguments in Ref. [9], the dif-ferent PSGs imply different quantum orders in the V > 0and V < 0 ground state states. The PSGs obtained inRef. [46] for the V > 0 and V < 0 phases agrees exactlywith the fermion PSGs that we obtained above. This ex-ample shows that the PSGs introduced in Ref. [10, 46]are the symmetry groups of the hopping Hamiltonian ofthe ends of condensed strings. The PSG description andthe string-net-condensation description of quantum or-ders are intimately related.

Here we would like to point out that the PSGs intro-duced in Ref. [9, 10] are all fermion PSGs. They are only

one of many different kinds of PSGs that can be used tocharacterize quantum orders. In general, a quantum or-dered state may contain condensations of several types ofstrings. The ends of each type of condensed strings willhave their own PSG.

V. MASSLESS FERMION AND PSG INSTRING-NET CONDENSED STATE

In Ref. [9, 16], it was pointed out that PSG can pro-tect masslessness of the emerging fermions, just like sym-metry can protect the masslessness of Nambu-Goldstone


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13/24


14/24

14

For each fixed configuration sij , there are 2Nsite/2

different states (with even or odd numbers of total fermions). Their energy are given by the fermion hoppingHamiltonian Eq. (43). Let E0({sij}) be the ground stateenergy of Eq. (43). The ground state and the groundstate energy of our spin- 12

12 model Eq. (61) is obtained

by choosing a configuration sij that minimize E0({sij}).We note that E0({sij}) is invariant under the Z2 gaugetransformation Eq. (53).

When g |t|, the ground state of Eq. (41) has Fi =1 which minimize the dominating gi Fi term. Theground state configuration is given by

si,i+x = ()iy , si,i+y = 1. (59)The i fermion hopping Hamiltonian Eq. (43) for theabove configuration describes fermion hopping with -flux per plaquette. The fermion spectrum has a form

Ek = 2

t2 sin2(kx) + t2 sin2(ky). (60)

The low energy excitations of such a hopping Hamilto-nian are described by two two-component massless Diracfermions in 2+1D. We see that the ends of the W stringsare massless Dirac fermions.

Our model also contain Z2 gauge excitations. The Z2vortices are created by flipping Fi = 1 to Fi = 1 insome plaquettes. The Z2 vortex behaves like a -flux tothe gapless fermions. Thus the gapless fermions carry aunit Z2 charge. The low energy effective theory of ourmodel is massless Dirac fermions coupled to a Z2 gaugefield.

C. Projective symmetry and massless fermions

We know that symmetry breaking can produce andprotect gapless Nambu-Goldstone modes. In Ref. [9, 16],it was proposed that, in addition to symmetry break-ing, quantum order can also produce and protect gaplessexcitations. The gapless excitations produced and pro-tected by quantum order can be gapless gauge bosonsand/or gapless fermions. In this paper we show thatthe quantum orders discussed in Ref. [9, 16] are due tostring-net condensations. Therefore, more precisely it isstring-net condensations that produce and protect gap-less gauge bosons and/or gapless fermions. The string-net condensations and gapless excitations are connected

in the following way. Let us consider a Hamiltonian thathas a symmetry described by a symmetry group SG.We assume the ground state has a string-net conden-sation. Then, the hopping Hamiltonian for the ends ofcondensed string will be invariant under a larger group -the projective symmetry group P SG, as discussed in sec-tion IV B. P SG is an extension of the symmetry groupSG, ie P SG contain a normal subgroup IGG such thatPSG/IGG = SG. The relation between P SG and gap-less gauge bosons is simple. Let G be the maximumcontinuous subgroup of IGG. Then the gapless gaugebosons are described by a gauge theory with G as thegauge group.[9, 15] Some times the ends of strings are

fermions. However, the relation between gapless fermionsand P SG is more complicated. Through a case by casestudy of some P SGs[9, 16], we find that certain P SGsindeed guarantee the existence of gapless fermions.

In this section, we are going to study a large familyof exact soluble local bosonic models which depends onmany continuous parameters. The ground states of thelocal bosonic models have a string-net condensation and

do not break any symmetry. We will show that the projective symmetry of the ends of condensed strings pro-tects a massless fermion. As a result, our exact solublemodel always has massless fermion excitations regard-less the value of the continuous parameters (as long asthey are within a certain range). This puts the results ofRef. [9, 16], which were based on mean-field theory, on afirmer ground.

The exact soluble local bosonic models are the spin- 1212

model

H1212

= gi

yxi xyi+x

yxi+x+y

xyi+y

+i

t+,xi ,xi+x + t+,yi ,yi+y + h.c.

(61)

where ab and ,a are given in Eq. (48) and Eq. (47).We will discuss a more general Hamiltonian later.

The Hamiltonian is not invariant under x x par-ity Px. But it has a x x parity symmetry ifPx is followed by a spin rotation

x x. That isPxPxH(PxPx)

1 = H with

Px = 5

x x2

(62)

Similarly for y y parity Py, we havePyPyH(PyPy)

1 = H with

Py = 5

y y2

(63)

In the fermion representation Px and Py generate thefollowing transformations

Px : xi xi , i i ,

Py : yi yi , i i . (64)

Now let us study how the symmetries Tx,y andPx,yPx,y are realized in the hopping HamiltonianEq. (43) for the ends of condensed strings. As discussedin section IV B, the hopping Hamiltonian may not beinvariant under the symmetry transformations Tx,y andPx,yPx,y directly. The hopping Hamiltonian only hasa projective symmetry generated by a symmetry trans-formation followed by a Z2 gauge transformation G(i).Since the -flux configuration does not break any sym-metries, we expect the hopping Hamiltonian for the -flux configuration to be invariant under GxTx, GyTy,GPxPxPx, and GPyPyPy, where Gx,y and GPx,y are thecorresponding gauge transformations. The action of Tx,y


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15

and Px,yPx,y on the fermion are given by

Tx : (ix,iy) (ix+1,iy),Ty : (ix,iy) (ix,iy+1),

PxPx : (ix,iy) (ix,iy),PyPy : (ix,iy) (ix,iy). (65)

For the -flux configuration Eq. (59), we need to choosethe following Gx,y and GPx,y in order for the combinedtransformation Gx,yTx,y and GPx,yPx,yPx,y to be thesymmetries of the hopping Hamiltonian Eq. (43)

Gx = 1, Gy =()ix ,GPx = ()ix , GPy =()iy . (66)

The hopping Hamiltonian is also invariant under a globalZ2 gauge transformation

G0 : i i (67)

The transformations {G0, Gx,yTx,y, GPx,yPx,yPx,y} gen-erate the PSG of the hopping Hamiltonian.

To show that the above PSG protects the masslessnessof the fermions, we consider a more general Hamiltonianby adding

H1212

=Cij

t(Cij)W(Cij) + h.c

(68)

to H1212

, where Cij is an open string connecting site i and

site j and W(Cij) is given in Eq. (56). The new Hamil-tonian is still exactly soluble. We will choose t(Cij) suchthat the new Hamiltonian has the translation symme-

tries and the Px,y parity symmetries. In the following,we would like to show that the new Hamiltonian withthose symmetries always has massless Dirac fermion ex-citations (assuming t(Cij) is not too big comparing tog).

When t(Cij) is not too large, the ground state is stilldescribed by the -flux configuration. The new hoppingHamiltonian for -flux configuration has a more generalform

H =ij

(ijij + h.c.) (69)

The symmetry of the physical spin-1

2

1

2

Hamiltonian re-quires that the above hopping Hamiltonian to be invari-ant under the PSG discussed above. Such an invariancewill guarantee the existence of massless fermions.

The invariance under GxTx and GyTy require that

i,i+m = ()iymxm (70)In the momentum space,

(k1,k2) N1siteij

eik1i+ik2jij

=0(k2)k1k2 + 1(k2)k1k2+Qy (71)

where

0(k) =

mx=even

eikmm,

1(k) =

mx=odd

eikmm. (72)

We note that 0(k) and 1(k) are periodic function in the

Brillouin zone. They also satisfy0(k) = 0(k + Qx), 1(k) = 1(k + Qx). (73)

where Qx = x and Qy = y. In the momentum space,we can rewrite H as

H =k

k(k)k (74)

where Tk = (k, k+Qy). The sum

k is over the re-

duced Brillouin zone: < kx < and /2 < ky

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Xiao-Gang Wen- Quantum order from string-net condensations and origin of light and massless fermions

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