xiao-gang wen- an introduction to quantum order, string-net condensation, and emergence of light and...

8/3/2019 Xiao-Gang Wen- An Introduction to Quantum Order, String-net Condensation, and Emergence of Light and Fermions

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An Introduction to Quantum Order, String-net Condensation,

and Emergence of Light and Fermions∗

Xiao-Gang Wen†

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Dated: Jan. 2004)

We review some recent work on new states of matter. Those states contain a new kind of order –

quantum order – because they cannot be described symmetry breaking. Some quantum orders areshown to be closely related to string-net condensations. Those quantum orders lead to an emergenceof gauge bosons and fermions from pure bosonic models.

I. INTRODUCTION

A. Origin of light/fermion and new orders

The existences of light and fermions are two big mys-teries in nature. The mysteries are so deep that thequestions like, “What are light and fermions?”, “Wheredo light and fermions come from?”, “Why do light and

fermions exist?”, are regarded by many people as philo-sophical or even religious questions.

To appreciate the physical significance those questionslet us ask three simpler questions: “What are phonons?”,“Where do phonons come from?”, “Why do phonons ex-ist?”. We know these three questions to be scientificquestions and we know their answers. Phonons are vi-brations of a crystal. Phonons come from a spontaneoustranslation symmetry breaking. Phonon exists becausethe translation-symmetry-breaking phase actually existsin nature. It is quite interesting to see that our under-standing of a gapless excitation – phonon – is rooted inour understanding of the phases of matter as symmetry

breaking states [1, 2].However, our picture for massless photons and nearly

massless fermions[70] (such as electrons and quarks) isquite different from our picture of gapless phonons. Weregard photons and fermions as elementary particles –the building block of our universe.

But why should we regard photons and fermions aselementary particles? Why don’t we regard photons andfermions as emergent quasiparticles like phonons? Wecan view this question from several different points of view.

First point of view: Before late 1970’s, we felt that we

understood, at least in principle, all the physics aboutphases and phase transitions based on Landau’s sym-metry breaking theory [3, 4]. In such a theory, if westart with a bosonic model, the only way to get gaplessexcitations is via spontaneous breaking of a continuous

∗A more comprehensive description of topological/quantum orderscan be found in Quantum field theory of many-body systems – from

the origin of sound to an origin of light and fermions, Xiao-GangWen, Oxford Univ. Press, 2004.†URL: http://dao.mit.edu/~wen

symmetry [1, 2], which will lead to gapless scalar bosonic

excitations. It seems that there is no way to obtain gap-less gauge bosons and fermions from symmetry breaking.This may be the reason why people think our vacuum(with massless gauge bosons and nearly-gapless fermions)is very different from bosonic many-body systems (whichwere believed to contain only gapless scalar bosonic col-lective excitations, such as phonons). It seems there doesnot exist any order that give rise to massless photons andnearly-massless fermions. This may be the reason whywe regard photons and fermions as elementary particlesand introduce them by hand into our theory of nature.

Second point of view: On the other hand, the resem-blance between the photons and the phonons makes itodd to regard photons as elementary. To appreciate thispoint, let us imagine another universe which contains amassless excitation with three components. This mass-less excitation behaves in every way like the phonon in acrystal. We will not hesitate to declare that the vacuumin that universe is actually a crystal even when no onecan see the particles that form the crystal. Our convic-

tion of the existence of the crystal does not come fromseeing the lattice structure, but from seeing the low en-ergy collective modes of the crystal.

Third point of view: Now back to our universe. Arethe massless photons and nearly massless fermions alsocollective modes of certain order in our vacuum. Notknowing what order can give rise to photons and fermionsmay not imply the photons and fermions to be elemen-tary. More likely, it means that our understanding of order is incomplete. The very existence of light andfermions may indicate that our vacuum contain a newkind of order. The new order will produce light andfermions, and protect its masslessness.

Fourth point of view: If we had a material which isdescribed by bosons (such as a spin system) and if wefound that the low energy excitations in the material aregauge bosons and fermions, we would not hesitate to de-clare that the material contains a new kind of order be-yond the symmetry breaking description. But so far, wehave not find any material that contain emergent gaugebosons and emergent fermions. So we do not know if new order beyond the symmetry breaking exists or not.Other other hand, we may regard our vacuum as a specialmaterial. From this point of view, the light and the elec-trons in the vacuum provided an experimental evidence



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Quantum system Classical system

Gapped

Nambu−Goldstone mode

"Particle" condensation

Orders

Fermi liquidsFermi surface topology

Gapless Gauge bosons/Fermions

Projective symmetry group

Conformal algebra, ??

Topological field theory

Non−symmetry breaking ordersSymmetry breaking orders

Topological orders

Quantum orders

Symmetry group

String−net condensation

FIG. 1: A classification of different orders in matter (and invacuum).

of the existence of new order.

B. Topological order and quantum order

Historically, our understanding of new order beyond

symmetry breaking starts at an unexpected place — frac-tional quantum Hall (FQH) systems. The FQH statesdiscovered in 1982 [5, 6] opened a new chapter of con-densed matter physics. What is really new in FQH statesis that FQH systems contain many different phases atzero temperature which have the same symmetry. Thusthose phases cannot be distinguished by symmetries andcannot be described by Landau’s symmetry breaking the-ory.

Since FQH states cannot be described by Landau’ssymmetry breaking theory, it was proposed that FQHstates contain a new kind of order — topological order[7]. Topological order is new because it cannot be de-

scribed by symmetry breaking, long range correlation, orlocal order parameters. None of the usual tools that weused to characterize a phase applies to topological order.Despite this, topological order is not an empty conceptsince it can be characterized by a new set of tools, suchas the number of degenerate ground states [8, 9], thenon-Abelian Berry’s phase under modular transforma-tions [10], quasiparticle statistics [11], and edge states[12, 13]. Just like Ginzburg-Landau theory is the effec-tive theory of symmetry breaking order, the topologicalfield theory [14] is the effective theory of topological order[7].

It was shown that the ground state degeneracy of a

topologically ordered state is robust against any pertur-bations [9]. Thus the ground state degeneracy is a uni-versal property that can be used to characterize a phase.The existence of topologically degenerate ground statesproves the existence of topological order. The topolog-ically degenerate ground states were found to useful infault tolerant quantum computing [15].

The concept of topological order only applies to statewith finite energy gap. It was recently generalized toquantum order [16] to describe new kind of orders ingapless quantum states. There are two general but vagueunderstanding quantum orders.

In the first understanding, we assume that the orderin a quantum state is encoded in the many-body groundstate wave function. We believe that the symmetry of the ground state wave function cannot characterize allthe possible orders in the many-body state. The extrastructure in the ground state can be viewed as a patternof quantum entanglement in the many-body state. Fromthis point of view, we may say that quantum orders are

patterns quantum entanglement in quantum many-bodystates.

The second way to understand quantum order is tosee how it fits into a general classification scheme of orders (see Fig. 1). First, different orders can be di-vided into two classes: symmetry breaking orders andnon-symmetry breaking orders. The symmetry breakingorders can be described by a local order parameter andcan be said to contain a condensation of point-like objects. The amplitude of condensation corresponds to theorder parameter. All the symmetry breaking orders canbe understood in terms of Landau’s symmetry breakingtheory. The non-symmetry breaking orders cannot be de-scribed by symmetry breaking, nor by the related localorder parameters and long range correlations. Thus theyare a new kind of orders. If a quantum system (a stateat zero temperature) contains a non-symmetry breakingorder, then the system is said to contain a non-trivialquantum order. We see that a quantum order is simplya non-symmetry breaking order in a quantum system.

Quantum orders can be further divided into many sub-classes. If a quantum state is gapped, then the corre-sponding quantum order will be called topological order.The second class of quantum orders appear in Fermi liq-uids (or free fermion systems). The different quantum

orders in Fermi liquids are classified by the Fermi surfacetopology [17]. We will discuss this class of quantum orderbriefly in section II I. The third class of quantum ordersarises from a condensation of nets of strings (string-nets)[18–24]. We will discuss it in sections IV and VI. Thisclass of quantum orders shares some similarities with thesymmetry breaking orders of “particle” condensation.

We know that different symmetry breaking orders canbe classified by symmetry groups. Using group theory,we can classify all the 230 crystal orders in three di-mensions. The phase transitions between different sym-metry breaking orders are described by critical pointwith algebraic correlations. The symmetry also produces

and protects gapless collective excitations – the Nambu-Goldstone bosons – above the symmetry breaking groundstate. Similarly, different string-net condensations (andthe corresponding quantum orders) can be classified bya mathematical object called projective symmetry group[16] (see subsection IV D). Using the projective symme-try group, we can classify over 100 different 2D spin liq-uids that all have the same symmetry. The phase transi-tions between different quantum orders are also describedby critical points. Those phase transitions do not changeany symmetry and cannot be described by order param-eters associated with broken symmetries [25–29]. Just



3

FIG. 2: Our vacuum may be a state filled with string-nets.The fluctuations of the string give rise to gauge bosons. Theends of the strings correspond to electrons, quarks, etc .

like the symmetry group, the projective symmetry groupcan also produce and protect gapless excitations. How-ever, unlike the symmetry group, the projective symme-try group produces and protects gapless gauge bosonsand fermions [16, 30, 31]. Because of this, we can saythat light and massless fermions can have a unified ori-gin. They can emerge from string-net condensations.

C. String-net picture of light and fermions

We used to believe that to have light and fermions inour theory, we have to introduce by hand a fundamentalU (1) gauge field and anti-commuting fermion fields, sinceat that time we did not know any collective modes thatbehave like gauge bosons and fermions. However, due tothe advances of the last 20 years, we now know how toconstruct local bosonic systems that have emergent un-

confined gauge bosons and/or fermions [15, 19, 30, 32–38]. In particular, one can construct ugly bosonic spin

models on a cubic lattice whose low energy effective the-ory is the beautiful QED and QCD with emergent pho-tons, electrons, quarks, and gluons [39].

This raises an issue: do light and fermions in naturecome from a fundamental U (1) gauge field and anti-commuting fields as in the U (1)×SU (2)×SU (3) standardmodel or do they come from a particular quantum orderin our vacuum? Is Coulomb’s law a fundamental law of nature or just an emergent phenomenon? Clearly it ismore natural to assume light and fermions, as well asthe Coulomb’s law, come from a quantum order in ourvacuum. From the connections between string-net con-densation, quantum order, and massless gauge/fermion

excitations, it is very tempting to propose the follow-ing possible answers to the three fundamental questionsabout light and fermions:What are light and fermions?

Light is the fluctuation of condensed strings (of arbitrarysizes) [21, 23, 37]. Fermions are ends of condensed strings[19].Where do light and fermions come from?

Light and fermions come from the collective motions of nets of strings (or string-net) that fill our vacuum (seeFig. 2).Why do light and fermions exist?

P

T

Vapor

Ice

Water

FIG. 3: The phase diagram of water.

Light and fermions exist because our vacuum happen tohave a property called string-net condensation.

Had our vacuum chose to have “particle” condensation,there would be only Nambu-Goldstone bosons at low en-ergies. Such a universe would be very boring. Stringcondensation and the resulting light and fermions pro-vide a much more interesting universe, at least interest-ing enough to support intelligent life to study the originof light and fermions.

Our understanding of quantum/topological orders arebase on many researchs in three main areas: (1) the study

of topological phases in condensed matter systems suchas FQH systems [9, 40–42], quantum dimer models [32,43–46], quantum spin models [10, 16, 33–36, 47–49], oreven superconducting states [50, 51], (2) the study of lattice gauge theory [21–23, 52], and (3) the study of quantum computing by anyons [15, 53, 54]. In this paper,we will use some simple models to introduce the mainpoints of topological/quantum order.

II. STATE OF MATTER AND CONCEPT OFORDER

To start our journey to search new state of matterwith emergent gauge bosons and fermions, we like tofirst discuss the concept of order and review the symme-try breaking description of order. With low temperaturetechnology developed around 1900, physicists discoveredmany new states of matter (such as superconductors andsuperfluids). Those different states have different inter-nal structures, which are called different kinds of orders.The precise definition of order involves phase transition.Two states of many-body systems have the same orderif we can smoothly change one state into the other (bysmoothly changing the Hamiltonian) without encounter

a phase transition (ie without encounter a singularity inthe free energy). If there is no way to change one stateinto the other without a phase transition, than the twostates will have different orders. We note that our def-inition of order is a definition of equivalent class. Twostates that can be connected without a phase transitionare defined to be equivalent. The equivalent class definedthis way is called the universality class. Two states withdifferent orders can be also be said as two states belongto different universality classes. According to our defini-tion, water and ice have different orders while water andvapor have the same order. (See Fig. 3)



4

(a)

A

(b)

A A

BB’

(c)

FFF

φ φ φ

FIG. 4: In the presence of φ → −φ symmetry, switchingof the minima can be continuous and causes a second-orderphase transition. (a) The ground state is symmetric underthe transformation φ → −φ. (c) The ground state breaks theφ → −φ symmetry.

After discovering so many different kinds of orders, ageneral theory is needed to gain a deeper understandingof states of matter. In particular, we like to understandwhat make two orders really different so that we can-not change one order into the other without encountera phase transition. It is a deep insight to connect thesingularity in free energy to a symmetry breaking pic-ture in Fig. 4. Based on the relation between orders andsymmetries, Landau developed a general theory of orders

and phase transitions [3, 55]. According to Landau’s the-ory, the states in the same phase always have the samesymmetry and the states in different phases always havedifferent symmetries. So symmetry is a universal prop-erty that characterized different phases. Landau’s theoryis very successful. Using Landau’s theory and the relatedgroup theory for symmetries, we can classify all the 230different kinds of crystals that can exist in three dimen-sions. By determining how symmetry changes across acontinuous phase transition, we can obtain the criticalproperties of the phase transition. The symmetry break-ing also provides the origin of many gapless excitations,such as phonons, spin waves, etc , which determine the

low energy properties of many systems [1, 2]. A lot of the properties of those excitations, including their gap-lessness, are directly determined by the symmetry.

III. QUANTUM ORDERS AND QUANTUMTRANSITIONS IN FREE FERMION SYSTEMS

However, not all orders are described by symmetry. Infact, free fermion systems are the simplest systems withnon-trivial quantum order. In this section, we will studythe quantum order in a free fermion system to gain some

intuitive understanding of quantum order.To find quantum order, or to even define quantum or-

der, we must find universal properties. The universalproperties are the properties which do not change underany perturbations of the Hamiltonian that do not affectthe symmetry. Once we find those universal properties,we can use them to group many-body wave functionsinto universal classes such that the wave functions ineach class have the same universal properties. Hopefullythose universal classes correspond to quantum phases ina phase diagram. To really show that those universalclasses do correspond to quantum phases, we must show

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

FIG. 5: Two sets of oriented Fermi surfaces in (a) and (b)represent two different quantum orders. The two possible

transition points between the two quantum order (a) and (b)are described by the Fermi surfaces (c) and (d).

that as we deform the Hamiltonian to drive the groundstate from one universality class to another, the groundstate energy always has a singularity at the transitionpoint. (For zero temperature quantum transition, theground state energy play the role of free energy for finitetemperature transition. A singularity in the ground stateenergy signal a quantum phase transition.)

We know that symmetry is a universal property. Theorder determined by such a universal property is our oldfriend – the symmetry-breaking order. So to show theexistence of new quantum order, we must find universalproperties that are different from symmetry.

Let us consider free fermion system with only thetranslation symmetry and the U (1) symmetry from thefermion number conservation. The Hamiltonian has aform

H =ij

c†itijc j + h.c.

(1)

with t∗ij = t ji. The ground state is obtained by fillingevery negative energy level with one fermion. In general,the system contains several pieces of Fermi surfaces.

We note that any small change of tij do not changethe topology of the Fermi surfaces as long as the changedo not break the translation symmetry and do not vio-late the fermion number conservation. So the the Fermisurface topology is a universal properties.

To show that the Fermi surface topology really definesquantum phases, we need to show that any change of the Fermi surface topology will lead to a singularity inthe ground state energy. The Fermi surface topology canchange in two ways as we continuously changing tij : (a)a Fermi surface shrinks to zero (Fig. 5d) and (b) twoFermi surfaces join (Fig. 5c).

When a Fermi surface is about to disappear in a D-dimensional system, the ground state energy density hasa form

ρE =

dDk

(2π)D(k · M · k − µ)Θ(−k · M · k + µ) + ...

where the ... represents non-singular contribution andthe symmetric matrix M is positive (or negative) def-inite. The integral in the above equation simply rep-resents the total energy of the filled states enclosed bythe small Fermi surface. The small Fermi surface isabout to shrink to zero as µ pass zero. We find that



5

the ground state energy density has a singularity atµ = 0: ρE = cµ(2+D)/2Θ(µ) + ..., where Θ(x > 0) = 1,Θ(x < 0) = 0. When two Fermi surfaces are aboutto join, the singularity is still determined by the aboveequation, but now M has both negative and positiveeigenvalues. The ground state energy density has a sin-gularity ρE = cµ(2+D)/2Θ(µ) + ... when D is odd andρE = cµ(2+D)/2 log |µ| + ... when D is even.

We find that the ground state energy density has asingularity at µ = 0 which is exactly the same placewhere the topology of the Fermi surfaces has a change[17]. Therefore the topology of the Fermi surface is auniversal property that define a order. We note that thestates with different Fermi surface topologies all have thesame symmetry. Thus the quantum phase transition thatchange the the topology of the Fermi surface does notchange any symmetry. Therefore the order defined bythe Fermi surface topology is a new kind of order thatcannot be characterized by symmetries. Such an order isan example of quantum order.

IV. QUANTUM ORDER IN BOSON/SPINLIQUIDS

A. Quantum order and new universal properties

After realizing the existence of the quantum order infree fermion systems, we may expect quantum order tobe a general phenomena. In this section we would liketo study the existence of quantum order in interactingboson or spin systems. Instead of looking for univer-sal properties, we would like to first look for boson/spin

states that contain emergent gauge bosons and fermions.The emergence of gauge bosons and fermions indicate theappearance of new quantum order. Then we will studythe universal properties which give a more systematicdescription of quantum orders.

We like to point out that a spin system is a specialcase of boson system since we can regard a site with adown spin as an empty site and a site with a up spin asa occupied site for bosons. In this section we will inter-changeably use both the boson and the spin languages todescribe the same system.

B. Projective construction

In the introduction, we argue that the existence of light and electron implies that our vacuum contains anon-trivial quantum order. However, we do not knowto which system does the quantum order belong. Nowwe look for quantum order in spin systems. So we knowour system. But we do not know what to look for, sincewe have no clue what does a quantum order look likeat microscopic level. So instead of directly searching forquantum order, let us look for something slightly more

familiar: a spin liquid state that does not break any sym-metry.

1. A mean-field theory of spin liquids

To be concrete, let us consider a spin-1/2 system on asquare lattice

H =ij

J ijS i · S j. (2)

In the conventional mean-field theory, we use the ground|Φmi

mean of a free spin Hamiltonian

H mean =i

miS i

to approximate the ground state of the interacting Hamil-tonian H . The mean-field ground state described bym

i =m

i, |Φ

mi

mean, is obtained by minimizing the av-erage energy Φmimean|H |Φmi

mean. However, no matterhow we choose the mean-field ansatz mi, the mean-fieldground state always break spin rotation symmetry andthere is no way to obtain a spin liquid.

We have to use another approach to obtain a spinliquid [56, 57]. We start with a free fermion mean-field Hamiltonian that contains two fermion fields ψi =

ψ1i

ψ2i

:

H mean =

ψ†iuijψi (3)

where uij are two by two complex matrices defined onthe links ij that describe the hopping of the fermions.However, the mean-field ground state of H mean, |Ψ

uijmean,

does not correspond to a spin state. But, we can obtaina spin state from the fermion state by projecting thefermion state |Ψ

uijmean into the subspace where every site

has even numbers of fermion:

|Φuijspin = P|Ψ

uijmean

This is because there are only two states, on each site,that have even number of fermion. One is the emptysite |0 which can be viewed as a spin-down state, and

the other is the state with two fermions ψ†1iψ

†2i|0 which

corresponds to the spin-up state.Although it is not obvious, one can show [16] that iff

uij satisfy

Tr(uij) = imaginary, Tr(uijτ l) = real, l = 1, 2, 3

where τ 1,2,3 are the Pauli matrices, then Φuijspin describes

a spin rotation invariant state. Since the spin state is ob-tained through the projection P , we will call the aboveconstruction projective construction. It is a special caseof the slave-boson construction [56, 57] at zero doping.



6

We see that at least we can use the projective construc-tion to construct a spin liquid state which does not breakthe spin rotation symmetry.

Through the projective construction, we introduced alabel uij that labels a class of spin wave functions. (uijdoes not label all possible spin wave functions.) So we donot have to directly deal with the many-body functionsof of spin liquids which are very hard to visualize. We

only need to deal with uij to understand the propertiesof the spin liquids.

2. The variational ground state

We may view uij as variational parameters. Anapproximate many-body ground state wave function|Φ

uijspin can be obtained by minimizing the average en-

ergy Φuijspin|H |Φ

uijspin where H is the spin Hamiltonian.

Aside from its many variational parameters, there isno reason to expect the projective construction to give a

good approximation of the ground state for a generic spinHamiltonian. So there is no reason to trust any resultsobtained from projective construction. As we will see be-low that the projective construction leads to many unbe-lievable results, such as the emergence of gauge bosons,fermions, or even anyons from purely bosonic systems.So those amazing results may just be the artifacts of aunreliable approach.

However, for certain type of Hamiltonians, the projec-tive construction leads to a good approximation of theground state. In certain large N limits [33, 56], the fluc-tuations around the mean-field ansatz are weak, and theprojective construction gives a good description of the

ground state and the excitations. We can also constructspecial Hamiltonians where the projective constructionleads to an exact ground state (and all the exact ex-cited states. See section V). For those Hamiltonians, theprojective construction does provide a good descriptionof spin liquid states which cannot be provided by otherconventional method. In the following, we will only con-sider those friendly Hamiltonians and trust the results of the projective construction.

I would like to mention that in practice, most of theunbelievable predictions from the projective constructionturn out to be correct. For example, in the research of high T c superconductors, both the d-wave superconduct-

ing state and the pseudo-gap metallic state was predictedby the projective construction prior to experimental ob-servation [56, 58]. This may be the first time in the his-tory of condensed matter physics that a truly new stateof matter – the pseudo-gap metallic state – is predictedbefore the experimental observation [59].

3. Low energy excitations

In the conventional mean-field theory for spin or-dered state, after we obtain the mean-field ground state

|Φmimean that minimize the average energy, we can create

collective excitations above the ground state through thefluctuation of the mean-field ansatz mi = mi + δmi.Those collective excitations correspond to the spin waveexcitations.

In the projective construction, we can create collec-tive excitations in the exactly the same way. The collec-tive excitations above the mean-field ground state Φ

uijspin

correspond to the fluctuations of the mean-field ansatzuij = uij + δuij . The physical spin wave function forsuch type of excitations is obtained via the projection of

the deformed fermion state |Ψuij+δuijmean :

|Φδuijspin = P|Ψ

uij+δuijmean

The ground state obtained from the projective con-struction also contains a second type of excitations. Thistype of excitations corresponds to fermion pair excita-tions. We start with the fermion ground state with a

pair of particle-hole excitations ψ†aiψb j |Ψ

uijmean. After

the projection, we obtain the corresponding physical spinstate

|Φ(a,i,b, j)spin = ψ†

aiψb j |Ψuijmean

that describes a pair of fermions.Clearly the fermions excitations interact with the col-

lective modes δuij. The effective Lagrangian that de-scribes the two types of excitations has a form L(ψ,δuij).It appears that the spin liquid state obtained through theprojective construction always contain fermionic excita-tions described by ψ. The emergent fermions will implythat the spin liquid state is a new state of matter andcontain non-trivial quantum order.

However, the thing is not that easy. It turns out thecollective fluctuations represent gauge fluctuations andthe fermions carry gauge charges. Those fluctuations canmediate an confining interactions between the fermions.As a result, the spin liquid state may not contain anyfermionic excitations and may not represent new state of matter.

To see that the fluctuations of uij represent gauge fluc-tuations, we note that the mean-field Hamiltonian H mean

is invariant under the following SU (2) gauge transforma-tion

ψi →W iψi, W i ∈ SU (2)

uij →W i uij W † j (4)

So the two fermion ground states of the H mean corre-sponding to two ansatz uij and u

ij are related by an

SU (2) gauge transformation if uij = W iuijW † j . Since

the even-fermion states |0 and ψ†1iψ†

2i|0 are invariantunder SU (2) gauge transformation, the projected state|Φ

uijspin = P|Ψ

uijmean is invariant under the SU (2) gauge

transformation:

|Φuijspin = |Φ

W iuijW †j

spin



7

FIG. 6: The fermion dispersion for the ansatz Eq. (6).

As a result, uij is not a one-to-one label of the physi-cal spin state, but a many-to-one label. The mean-fieldenergy E (uij) = Φ

uijspin|H |Φ

uijspin, as a function of real

physical spin state |Φuijspin, is invariant under the SU (2)

gauge transformation E (uij) = E (W iuijW † j ). Similarly,

the effective Lagrangian L(ψ, uij) is also invariant under

the SU (2) gauge transformation:

L(ψ, uij) = L(W iψ, W iuijW † j )

The SU (2) gauge invariance of the effective Lagrangianstrongly affect the dynamics of uij fluctuations. It makesthe uij fluctuations to behave like SU (2) gauge fluctua-

tions. If we write uij = iχeialijτ

l

, then alij play the role

of the SU (2) gauge potential on the lattice.

C. Deconfined phase and new state of matter

We have mentioned that to obtain a new state of mat-ter from the projective construction, the gauge fluctua-tions uij must not mediate a confining interaction. Onemay say that the SU (2) gauge fluctuations always me-diate a confining interaction in 1+2D, so the projectiveconstruction can never produce a spin liquid with emer-gent fermions. Well again thing is not that simple. Itturns out that whether the gauge fluctuations confinethe fermions or not depend on the form of the mean-fieldansatz u

ijthat minimize the average energy H .

To understand how the ansatz uij affect the dynamicsof the gauge fluctuations, it is convenient to introducethe loop variable

P (C i) = uiju jk ...uli (5)

If we write P (C i) as P (C i) = χeiΦl(C i)τ

l

, then Φlτ l isthe SU (2) flux through the loop C i: i → j → k → .. →l → i with base point i. The SU (2) flux correspond tothe gauge field strength in the continuum limit.

1. SU (2) spin liquid

If for a certain spin Hamiltonian H , uij has a form

ui,i+x = −i(−)iyχ,

ui,i+y = −iχ, (6)

the SU (2) flux through any loops is trivial. In thiscase the fluctuations of uij behave like the usual SU (2)gauge fluctuations. The fermion dispersion is deter-mined by mean-field Hamiltonian H mean in Eq. (3):

E k = ±2χ

sin2 kx + sin2 ky. At low energies (E k ∼ 0)

the fermions have a linear dispersion E ∝ |k| (see Fig.6) and can be described by massless Dirac fermions inthe continuum limit. So in the continuum limit, the lowenergy fluctuations are described by

L =N

σ=1

ψσ(∂ µ − ialµτ l)γ µψσ +

1

2gTrf lµνf l,µν (7)

where alµ and f lµν , l = 1, 2, 3 the vector potential and the

field strength of the SU (2) gauge field.It is interesting to see that for the spin liquid described

by the ansatz Eq. (6), the low energy fluctuations is de-scribed by 1+2D QCD! The low energy physical proper-ties of 1+2D QCD is complicated.

But the above construction can be easily generalize tohigher dimensions. The low energy properties of 1+4DQCD is much easier to determine and it contains de-confined phase. As a result, the 1+4D spin liquid con-tains emergent massless fermions and emergent masslessSU (2) gauge bosons. Such a state represent a new stateof matter and contains a non-trivial quantum order. Weget what we are looking for (if we live in four dimensionalspace). Because of the low energy SU (2) gauge fluctua-tions, we will call the ansatz Eq. (6) SU (2) ansatz andthe correspond spin liquid SU (2) spin liquid.

2. U (1) spin liquid

If the ansatz that minimizes the average energy has aform

ui,i+x = −τ 3χ − i(−)i∆,

ui,i+y = −τ 3

χ + i(−)i

∆, (8)

where (−)i ≡ (−)ix+iy , then the low energy uij fluctua-tions are actually described by U (1) gauge fluctuations.This is because the above ansatz contains non-trial SU (2)

flux: P (C i) ∝ eiΦ3τ 3 . Unlike the flux of U (1) gauge field,

the flux of SU (2) gauge field is not invariant under theSU (2) gauge transformations. Instead, the SU (2) fluxtransforms like a Higgs field that carries non-zero SU (2)charge. The non-zero SU (2) flux has a similar effect asthe condensation of a Higgs field, which can give gaugebosons a mass term via the Anderson-Higgs mechanism



8

[60, 61]. For our case, the SU (2) flux in the τ 3 directiongive the a1µ and a2µ components of the gauge field a mass,

but a3µ remains massless. So the SU (2) flux break theSU (2) gauge structure down to a U (1) gauge structure[34]. This is why the spin liquid described by the ansatzEq. (8) contains only massless U (1) gauge fluctuations.The low energy fermions are still described by masslessDirac fermions. So the low energy effective theory of thespin liquid is a 1+2D QED

L =N

σ=1

ψσ(∂ µ − ia3µτ 3)γ µψσ +1

2gf 3µνf 3,µν (9)

where a3µ and f 3µν are the vector potential and the fieldstrength of the U (1) gauge field. Again, the above con-struction can be easily generalize to higher dimensions.Now we do not need to go to 1+4 dimensions. In 1+3

dimensions, the spin liquid [30] already contains emer-gent massless fermions and emergent massless U (1) gaugebosons since 1+3D QED is not confining. Such a 1+3Dspin liquid represents another new state of matter andwill be called U (1) spin liquid.

The close resemblance of the low energy effective the-ory of the spin liquids and the QED/QCD in the standardmodel makes one really wonder: is this how the QED andQCD emerge to be the effective theory that describe ourvacuum [39]?

Even in 1+2D, the U (1) spin liquid was shown to bea stable phase [16, 62, 63] based a combined analysis of

instanton [64] and projective symmetry [16] (see sectionIV D). The existence of the U (1) spin liquid is a strikingphenomenon since the gapless excitations interact downto zero energy, and yet remain to be gapless. The in-teraction is so strong that that are no free fermionic orbosonic quasiparticles ar low energies. Since the U (1)gauge bosons and the fermions are not well defined at anyenergy, the U (1) spin liquid was more correctly called thealgebraic spin liquid [16, 62].

Since there is no spontaneous broken symmetry toprotect the above interacting gapless excitations, thereshould be a “principle” that prevents the gapless excita-

tions from opening an energy gap and makes the alge-braic spin liquids stable. Ref. [16] proposed that quan-tum order is such a principle. To support this idea, itwas shown that just like the symmetry group of sym-metry breaking order protects gapless Nambu-Goldstonemodes, the projective symmetry group (see section IV D)of quantum order protects the interacting gapless excita-tions in the algebraic spin liquid. This result impliesthat the stabilities of algebraic spin liquids are protectedby their projective symmetry groups. The existence of gapless excitations without symmetry breaking is a trulyremarkable feature of quantum ordered states.

3. Z 2 spin liquid

For the ansatz [34]

ui,i+x =ui,i+y = −χτ 3,

ui,i+x+y =ητ 1 + λτ 2,

ui,i−x+y =ητ 1 − λτ 2, (10)

the SU (2) flux Φl(C i)τ l for different loops points indifferent directions in the (τ 1, τ 2, τ 3) space. The non-collinear SU (2) flux break the SU (2) gauge structuredown to a Z 2 gauge structure. Since the Z 2 gauge fluc-tuations only mediate a short ranged interaction, thefermions are not confined even in 1+2D. The spin liq-uid obtained from the ansatz Eq. (10) is a new state of matter that has emergent fermions and Z 2 gauge theory.Non-trivial quantum order can appear in two dimensionalspace. The spin liquid will be called Z 2 spin liquid. Sucha spin liquid corresponds to the short-ranged ResonatingValence Bound state proposed in Ref. [43, 65].

4. Summary

The projective construction is a powerful way con-struct states that represent new state of matter. Thosestates have emergent fermions and gauge bosons, andthus contain a new kind of order that cannot be describedby symmetry. The new order is called quantum orders.Certainly, not all states obtained via the projection con-struction contain non-trivial quantum orders. But manyof them do. The projective construction not only can

produces states with emergent QED and QCD, it canalso produce states with fractional statistics (includingnon-Abelian statistics) [36, 66, 67].

D. Quantum order and projective symmetry group

We know that different symmetry-breaking orders canbe systematically characterize by different symmetrygroup. The group theory description allows us to classify230 different 3D crystals. Knowing the existence of newquantum order in spin liquids, we would like to ask whatmathematical object that we can use to systematicallydescribe different quantum orders? In this subsection, weare going to introduce a mathematical object – projectivesymmetry group and show that the projective symmetrygroup can (partially) characterize different quantum or-ders.

1. The difficulty of seeing quantum order

In subsection IV A, we argue that to describe or to evendefine quantum order, we must find universal properties



9

that are different from the symmetry (such as the topol-ogy of the Fermi surfaces discussed in the section III).However, it is very difficult to find new universal prop-erties of generic many-body wave functions. Let us con-sider the free fermion systems that we discussed before togain some intuitive understanding of the difficulty. Weknow that a free fermion ground state is described by ananti-symmetric wave function of N variables. The anti-

symmetric function has a form of Slater determinant:Ψ(x1,...,xN ) = det(M ) where the matrix elements of M is given by M mn = ψn(xm) and ψn are single-fermionwave functions. The first step to find quantum ordersin free fermion systems is to find a reasonable way togroup the Slater-determinant wave functions into classes.This is very difficult to do if we only know the real spacemany-body function Ψ(x1,...,xN ). However, if we useFourier transformation to transform the real-space wavefunction to momentum-space wave function, then we cangroup different wave functions into classes according totheir Fermi surface topologies. This leads to our under-standing of quantum orders in free fermions systems (see

section III). The Fermi surface topology is the quan-tum number that allows us characterize different quan-tum phases of free fermions. Here we would like to stressthat without the Fourier transformation, it is very diffi-cult to see Fermi surface topologies from the real spacemany-body function Ψ(x1,...,xN ).

For the boson/spin systems, what is missing here is thecorresponding “Fourier” transformation. Just like thetopology of Fermi surface, it is very difficult to see uni-versal properties (if any) directly from the real space wavefunction. At moment there are two ways to understandthe quantum order in boson/spin systems. The first oneis through the projective symmetry group which will bediscussed below. The second one is through string-netcondensation which will be discuss in section VI. Boththe projective symmetry group and the string-net con-densation play the role of the Fourier transformation inthe free fermion system. They allow us the extract theuniversal properties from the very complicated many-body wave functions.

2. Symmetry of the spin liquids

To motivate the projective symmetry group, let us first

consider the symmetry of the spin liquid states obtainedfrom the SU (2), U (1) and Z 2 ansatz Eq. (6), Eq. (8),and Eq. (10). At first sight, those spin liquids appearnot to have all the symmetries. For example, the U (1)ansatz Eq. (8) is not invariant under translation in thex-direction.

However, those ansatz do describe spin states thathave all the symmetries of square lattice, namely the twotranslation symmetries T x: (ix, iy) → (ix + 1, iy) andT y: (ix, iy) → (ix, iy + 1), and three parity symmetries,P x: (ix, iy) → (−ix, iy), P y: (ix, iy) → (ix, −iy), andP xy : (ix, iy) → (iy, ix). This is because the ansatz uij

is a many-to-one label of the physical spin state. Thenon-invariance of the ansatz does not imply the non-invariance of the corresponding physical spin state afterthe projection. We only require the mean-field ansatz tobe invariant up to a SU (2) gauge transformation in orderfor the projected physical spin state to have a symmetry.For example, a T x translation transformation changes theU (1) ansatz Eq. (8) to

U i,i+x = −τ 3χ + i(−)i∆,

U i,i+y = −τ 3χ − i(−)i∆,

The translated ansatz can be transformed into the orig-inal ansatz via a SU (2) gauge transformation W i =(−)iiτ 1. Therefore, after the projection, the ansatzEq. (8) describes a T x translation symmetric spin state.

Using the similar consideration, one can show thatthe SU (2), U (1), and Z 2 ansatz are invariant undertranslation T x,y and parity P x,y,xy symmetry transfor-mations followed by corresponding SU (2) gauge trans-formations GT

x,T

y

and GP x,P

y,P

xy

respectively. Thus thethree ansatz all describe symmetric spin liquids. In thefollowing, we list the corresponding gauge transforma-tions GT x,T y and GP x,P y,P xy for the three ansatz:for the SU (2) ansatz Eq. (6):

GT x(i) =(−)ixGT y (i) = τ 0, GP xy(i) =(−)ixiyτ 0,

(−)ixGP x(i) =(−)iyGP y(i) = τ 0, G0(i) =eiθlτ l

(11)

for the U (1) ansatz Eq. (8):

GT x(i) =GT y(i) = i(−)iτ 1, GP xy (i) =i(−)iτ 1,

GP x(i) =GP y (i) = τ 0, G0(i) =eiθτ 3 (12)

for the Z 2 ansatz Eq. (10):

GT x(i) =GT y(i) = iτ 0, GP xy(i) =τ 0,

GP x(i) =GP y(i) = (−)iτ 1, G0(i) = − τ 0 (13)

In the above we also list the pure gauge transfor-mation G0(i) that leave the ansatz invariant: uij =

G0(i)uijG†0( j).

3. Definition of PSG

The SU (2), U (1) and Z 2 ansatz after the projection,give rise to three spin liquid states. The three stateshave the exactly the same symmetry. The question hereis whether the three spin liquids belong to the same phaseor not. According to Landau’s symmetry breaking the-ory, two states with the same symmetry belong to thesame phase. However, we now know that Landau’s sym-metry breaking theory does not describe all the phases.It is possible that the three spin liquids contain differentorders that cannot be characterized by symmetries. The



10

issue here is to find a new set of quantum numbers thatcharacterize the new orders.

To find a new set of universal quantum numbers thatdistinguish the three spin liquids, we note that althoughthe three spin liquids have the same symmetry, theiransatz are invariant under the symmetry translations fol-lowed by different gauge transformations (see Eq. (11),Eq. (12), and Eq. (13)). So the invariant group of three

ansatz are different. We can use the invariant group of the three ansatz to characterize the new order in the spinliquid. In a sense, the invariant group define a new order– quantum order.

The invariant group of an ansatz is formed byall the combined symmetry transformations and thegauge transformations that leave the ansatz in-variant. Those combined transformations from agroup. Such a group is called the ProjectiveSymmetry Group (PSG). The combined transforma-tions (GT xT x, GT yT y, GP xP x, GP yP y, GP xyP xy) and G0 inEq. (11), Eq. (12), and Eq. (13) generate the three PSG’sfor the three ansatz Eq. (6), Eq. (8), and Eq. (10).

4. Properties of PSG

To understand the properties of the PSG, we wouldlike to point out that a PSG contains a special subgroup,which will be called the invariant gauge group (IGG). AnIGG is formed by pure gauge transformations that leavethe ansatz unchanged

IGG ≡ {G0| uij = G0(i)uijG†0( j)}

For the ansatz Eq. (6), Eq. (8), and Eq. (10), the IGG’s

are SU (2), U (1), and Z 2 respectively. We note thatSU (2), U (1), and Z 2 happen to be the gauge groups thatdescribe the low energy gauge fluctuations in the threespin liquids This relation is not an accident. In generalthe gauge group of the low energy gauge fluctuations fora spin liquid described by an ansatz uij is given by theIGG of the ansatz [16]. This result generalizes the anal-ysis of the low energy gauge group based on the SU (2)flux.

If an ansatz is invariant under the translation T x fol-lowed by a gauge transformation Gx, then it is also invari-ant under the translation T x followed by another gaugetransformation G0Gx, as long as G0 ∈ IGG. So the

gauge transformation associated with a symmetry trans-formation is not unique. The number of the choices of the gauge transformations is the number of the elementsin IGG. We see that as sets, P SG = SG ×IGG where SGis the symmetry group. But as groups, P SG is not thedirect product of SG and IGG. It is a “twisted” product.Using the more rigorous mathematical notation, we have

SG = PSG/IGG

We may also say that the P SG is a projective extensionof SG by IGG.

The SU (2), U (1) and Z 2 ansatz all have the same sym-metry and hence the same symmetry group SG. Theyhave different P SG’s since the same SG is extended bydifferent IGG’s. Here we would like to remark that evenfor a given pair of SG and IGG, there are many differ-ent ways to extent the SG by the IGG, leading to manydifferent P SG’s. For example there are over 100 waysto extend the symmetry group of a square lattice by a

Z 2 IGG. This implies that there are over 100 differentZ 2 spin liquids on a square lattice and those spin liquidsall have the exactly the same symmetry! Finding differ-ent ways of extending a symmetry group SG is a puremathematical problem. Such a calculation will lead toa (partial) classification of the quantum orders (and thespin liquids).

5. PSG is a universal property which protects gaplessexcitations

From the above discussion, we see that a PSG con-tains two parts. The first part is SG which describe thesymmetry of the spin liquid. The second part is IGGwhich describe the gauge “symmetry” of the spin liquid.A generic elements in the PSG is a combination of thesymmetry transformation and the gauge transformation.

We know that symmetry and gauge “symmetry” areuniversal properties, ie perturbative fluctuations cannotbreak the symmetry, nor can they break the gauge “sym-metry”. So both SG and IGG are universal properties.This strongly suggests that the PSG is also a universalproperty.

To directly show a PSG to be a universal property, we

note that the fermion mean-field Hamiltonian H mean inEq. (3) is invariant under the lattice symmetry and theSU (2) gauge transformations Eq. (4). But the mean-field ansatz uij is not invariant under the separate lat-tice symmetry and SU (2) gauge transformations. Sothe mean-field state break the separate lattice symmetryand SU (2) gauge “symmetry” down to a smaller sym-metry. The symmetry group of this smaller symmetryis the PSG. So, the PSG is the symmetry of the mean-field theory with uij ansatz. As a result, the PSG is thesymmetry for the effective Lagrangian Luij (ψ,δuij) thatdescribes the fluctuations around the mean-field ansatz.If the mean-field fluctuations do not have any infrared

divergence, then those fluctuations will be perturbativein nature and cannot change the symmetry – the PSG.What do we mean by “perturbative fluctuations can-

not change the PSG?” We know that a mean-field groundstate is characterized by uij . If we include perturba-tive fluctuations to improve our calculation of the mean-field energy Φ

uijspin|H |Φ

uijspin, then we expect the uij that

minimize the improved mean-field energy to receive per-turbative corrections δuij . The statement “perturbativefluctuations cannot change the PSG?” means that uijand uij + δuij have the same PSG.

As the perturbative fluctuations (by definition) do not



11

change the phase, uij and uij + δuij describe the samephase. In other words, we can group uij into classes(which are called universality classes) such that the uijin each class are connected by the perturbative fluctua-tions. Each universality class describes one phase. Wesee that, if the above argument is true, then the ansatzin a universality class all share the same PSG. In otherwords, the universality classes (or the phases) are classi-

fied by the PSG’s. Thus the PSG is a universal property.We can use the PSG to describe the quantum order inthe spin liquid, as long as the low energy effective theoryLuij (ψ,δuij) does not have any infrared divergence.

In the standard renormalization group analysis of thestability of a phase or a critical point, one needs to in-clude all the counter terms that have the right symme-tries into the effective Lagrangian, since those terms canbe generated by perturbative fluctuations. Then we ex-amine if those allowed counter terms are relevant pertur-bations or not. In our problem, δuij discussed above cor-respond to thoe counter terms. The effective Lagrangianwith the counter term is given by Luij+δuij (ψ,δuij).

The new feature here is that it is incorrect to use thesymmetry group alone to determine the allowed counterterms δuij . We should use PSG to determine the al-lowed counter terms in our analysis of the stability of phases and critical points. That is the counter termsδuij should not change the PSG of uij .

The Z 2 spin liquid Eq. (10) (and other 100 plus Z 2spin liquids) contains no diverging fluctuations. So thePSG description of the quantum order is valid for thiscase. For the SU (2) spin liquid Eq. (6) and the U (1) spinliquid Eq. (8), their low energy effective theory Eq. (7)and Eq. (9) contain log divergence. These are marginalcases where the PSG description of the quantum orderstill apply. In a renormalization group analysis of thestability of the U (1) spin liquid Eq. (8), one can showthat, in a large N limit, the counter terms allowed bythe U (1) PSG Eq. (12) are all irrelevant [62, 63], even if we include the instanton effect [64]. Thus the (large N )U (1) spin liquid is a stable quantum phase. One can alsoshow that non of the allowed counter terms can give thegapless fermions and gapless gauge bosons an energy gap[16, 31]. Thus the gapless excitations in the U (1) spinliquid are protected by by the U (1) PSG despite thosegapless excitations interact down to zero energy.

E. An intuitive understand of quantum order andthe emergent gauge bosons and fermions

The projective construction produces a correlatedmany-body ground state |Φspin = P|Ψ

uijmean. We may

view the complicated correlation in the ground state asa pattern of quantum entanglement. The quantum orderand the associated PSG is a characterization of such apattern of entanglement. The gauge fluctuation abovethe many-body ground state can be viewed as a fluctua-tion of the entanglement. The fermion excitations can be

viewed as topological defects in the entanglement. Fromthis point of view, the theory of quantum order can beregarded as a theory of many-body quantum entangle-ment.

V. AN EXACT SOLUBLE MODEL FROMPROJECTIVE CONSTRUCTION

Usually, the projective construction does not give usexact results. In this section, we are going to construct anexactly soluble model on 2D square lattice [15, 68]. Themodel has a property that the projective constructiongive us exact ground states and all other exact excitedstates.

A. An exact soluble model for the ψ-fermions

First, we would like to construct an exact soluble model

for the ψ-fermions. It is convenient to write the exactsoluble Hamiltonian in terms of four Majorana fermions

2ψ1,i = λxi + iλx

i , 2ψ2,i = λyi + iλy

i (14)

The Majorana fermions satisfy the algebra {λa,i, λb, j} =2δabδij . where a, b = x, x,y, y. The exact soluble fermionHamiltonian is given by

H = −i

gF i, (15)

F i =U i,i+xU i+x,i+x+yU i+x+y,i+yU i+y,i,

U i,i+x =λxi λx

i+x, U i,i+y = λyiλy

i+y, U ij = −U ji.

To see why the above interacting fermion model is ex-actly soluble, we note that U ij commute with each other

and H commute with all the U ij . So we can find theeigenvalues and eigenstates of H by finding the commoneigenstates of U ij:

U ij|{sij} = sij|{sij}

Since (U ij)2 = −1 and U ij = −U ji , sij satisfies sij = ±i

and sij = −s ji. Since H is a function of U ij’s, |{sij} isalso an energy eigenstate of Eq. (15) with energy

E = −i

gF i,

F i =si,i+xsi+x,i+x+ysi+x+y,i+ysi+y,i. (16)

To see if |{sij}’s represent all the exact eigenstatesof H , we need count the states. Let us assume the 2Dsquare lattice to have N site lattice sites and a periodicboundary condition in both directions. In this case thelattice has 2N site links. Since there are total of 22N site

different choices of sij (two choices for each link), thestates |{sij} exhaust all the 4N site states in the (ψ1, ψ2)



12

Hilbert space. Thus the common eigenstates of U ij is notdegenerate and the above approach allows us to obtainall the eigenstates and eigenvalues of the H .

We note that the eigenstate |{sij} is the ground stateof the following free fermion Hamiltonian

H mean =ij

sijU ij (17)

by choosing different sij = ±i, the ground state of theabove mean-field Hamiltonian give rise to all the eigen-states of the interacting fermion Hamiltonian Eq. (15).

B. An exact soluble spin-1/2 model

We note that the Hamiltonian H can only change the

fermion number on each site, ni = ψ†1,iψ1,i + ψ†

2,iψ2,i,by an even number. Thus the H acts within a subspacewhich has an even number of fermions on each site. Wewill call such a subspace physical Hilbert space. The

physical Hilbert space has only two states per site cor-responding to a spin-up and a spin-down state. Whenrestricted within the physical space, H actually describesa spin-1/2 system. To obtain the corresponding spin-1/2Hamiltonian, we note that

σxi = iλy

iλxi , σy

i = iλxi λy

i , σzi = iλx

i λxi (18)

act within the physical Hilbert space and satisfy the al-gebra of Pauli matrices. Thus we can identify σl

i as thespin operator. Using the fact that

(−)ni = λxi λy

iλxi λy

i = 1

within the physical Hilbert space, we can show that the

fermion Hamiltonian Eq. (15) becomes (see Fig. 7)

H spin = −i

gF i, F i = σxi σy

i+xσxi+x+yσy

i+y (19)

within the physical Hilbert space.

C. The projective construction leads to exactresults

All the states in the physical Hilbert space (ie allthe states in the spin-1/2 model) can be obtained fromthe |{sij} states by projecting into the physical Hilbert

space: P|{sij}. The projection operator is given by

P =i

1 + (−)ni

2

Since [P , H ] = 0, the projected state P|{sij}, if non-zero, remain to be an eigenstate of H (or H spin) andremain to have the same eigenvalue. We see that theground state of the mean-field Hamiltonian Eq. (17), af-ter the projection, give rise to all the exact eigenstates of the spin Hamiltonian Eq. (19). The project constructionis exact for Eq. (19)!

D. The Z 2 gauge structure

The physical states (with even numbers of fermions persite) are invariant under local Z 2 transformations gener-ated by

G =i

Gnii

where Gi is an arbitrary function with only two values±1. The Z 2 transformation change ψai to ψai = Giψai

and sij to sij = GisijG j , or more precisely

|{sij} = G|{sij}

Using P G = P , we find that |{sij} and |{sij} giverise to the same physical state after projection (if theirprojection is not zero):

P|{sij} = P|{sij} (20)

Thus, sij is a many-to-one label of the physical spin state.

The above results indicate that we can view isij as aZ 2 gauge potential and the local Z 2 transformation is aZ 2 gauge transformation. Eq. (20) implies that gaugeequivalent gauge potential described the same physicalstate. The fluctuations of sij is described by a Z 2 gaugetheory, which is the low energy effective theory of thespin system Eq. (19).

VI. CLOSED-STRING CONDENSATION

The ground state of the exactly soluble model containsa special property – closed-string condensation. In thissection, we will see that the closed-string condensation isintimately related to the emergence of the gauge struc-ture and the fermions. We have shown that the groundstate of the exactly soluble model can be constructed viathe projective construction. This indicates that the projective construction is probably just a trick to constructstring condensed states.

A. String operators and closed-string condensation

First let us define a string C as a curve that connectthe midpoints of neighboring links (see Fig. 7). Thestring operator has the following form

W (C ) =n

σlnin

(21)

where in are sites on the string. ln = z if the string doesnot turn at site in. ln = x or y if the string makes aturn at site in. ln = x if the turn forms a upper-right orlower-left corner. ln = y if the turn forms a lower-rightor upper-left corner. (See Fig. 7.)

By an explicit calculation, one can show that the closedstring operator defined above commute with the spin



13

F i

σ x

σ x

σ x

σ z

σ z

σ y

σ y

σ y

FIG. 7: The string is formed by a curve connecting the mid-points of the neighboring links. The string operator is formby the product of σx,y,z

ifor sites on the string. The operator

F i is also presented.

Hamiltonian Eq. (19). So the ground state |ground of the spin Hamiltonian is also an eigenstate of the closed-string operator. Since the eigenvalues of the closed stringoperator are ±1. We have ground|W (C closed)|ground =±1. The ground state has a closed-string condensation.

We note that in a symmetry breaking state, the op-erator representing the order parameter has a non-zeroexpectation value Oi = φ. If we define a string oper-

ator as the product of Oi

φ along a loop, the average of

the string operator will be non-zero. However, we do notregard the non-zero average of such a string operator torepresent a string condensation. The real closed-stringcondensation must be “unbreakable”, in the sense thatwe cannot break a closed string operator into several seg-ments and find condensation in each segment. The string

operator

iOi

φ does not satisfy this property. However,

the string operator defined in Eq. (21) is indeed unbreak-able. This is b ecause an open-string operator does not

commute with the spin Hamiltonian and does not con-dense ground|W (C open)|ground = 0. It is such a “un-breakable” closed-string condensation indicates a new or-der in the ground state.

In terms of the Majorana fermions, the closed-stringcan be written as

W (C closed) =ij

iU ij

where ij are the nearest neighbor links that form theclosed string C closed. For a spin state obtained from the

ansatz sij (via the projection, P|{sij), we find that

{sij}|P W (C closed)P|{sij} =ij

isij

(note that for closed strings [W (C closed), P ] = 0). There-fore the closed string operator is nothing but the Wegner-Wilson loop operator [22, 69] for the corresponding Z 2gauge theory. We see an intimate relation between theclosed-string condensation and the emergence of a gaugestructure.

B. Open string operators and Z 2 charges

If the fluctuations of sij represent Z 2 gauge fluctua-tions, what are the Z 2 charges? In a gauge theory, weknow that a Wegner-Wilson operator for an open stringis not gauge invariant and is not a physical operator (ie isnot an operator that acts within the physical Hilbertspace). The open-string operator defined in Eq. (21)act with the physical Hilbert space and is a gauge in-variant physical operator. Thus the Wegner-Wilson op-erator for an open string does not correspond to ouropen string operator. However, in the gauge theory, twocharge operators connected by the Wegner-Wilson oper-

ator, φ†x1

φx2eiR x2x1

dx·a, is gauge invariant. It is such a

operator that correspond to our open string operator.In the gage theory, the Wegner-Wilson loop operator

create a loop of electric flux. The closed-string operatordefined in Eq. (21) has the same physical meaning. In

the gage theory, the operator φ†x1

φx2eiR x2x1

dx·acreates

two charges connected by a line of the electric flux. Our

open string operator defined in Eq. (21) does the samething: it creates two Z 2 charges at its two ends and aelectric flux line connecting the two charges.

Due to the closed-string condensation, the string con-necting the two Z 2 charges is unobservable and costs noenergy. Thus the open string does not create an extendedline-like object, it creates two point-like objects at itsends. Those point-like objects are the Z 2 charges. De-spite the point-like appearance, the Z 2 charges are in-trinsically non-local. There is no way to create a lone Z 2charge. Because of this, the Z 2 charge (and other gaugecharges) usually carry fractional quantum numbers.

C. Statistics of the Z 2 charges

What is the statistics of the Z 2 charges? Usually,bosons are defined as particles described by commutingoperators and fermions as particles described by anti-commuting operators. But this definition is too formaland is hard to apply to our case. The find a new wayto calculate statistics, we need to gain a more physi-cal understanding of the difference between bosons andfermions.

Let us consider the following many-body hopping sys-

tem. The Hilbert space is formed by a zero-particle state|0, one-particle states |i1, two-particle states |i1, i2,etc , where in labels the sites in a lattice. As an iden-tical particle system, the state |i1, i2,... does not de-pend on the order of the indexes i1, i2,.... For example|i1, i2 = |i2, i1. There are no doubly-occupied sites andwe assume |i1, i2,... = 0 if im = in.

A hopping operator tij is defined as follows. When tijacts on state |i1, i2,..., if there is a particle at site j butno particle at site i, then tij moves the particle at site j tosite i and multiplies a complex amplitude t(i, j; i1, i2,...)to the resulting state. Note that the amplitude may de-



14

Exchange No exchange

1

34 51

23

4

5

2

j

k l

i j

k l

i

FIG. 8: (a) The first way to arrange the five hops swaps thetwo particles. (b) The second way to arrange the same fivehops does not swap the two particles.

pend on the locations of all the other particles. TheHamiltonian of our system is given by

H hop =ij

tij

where the sum

ij is over a certain set of pairs ij,

such as nearest-neighbor pairs. In order for the aboveHamiltonian to represent a local system we require that

[tij, tkl] = 0

if i, j, k, and l are all different.What does the hopping Hamiltonian H hop describe? A

hard-core boson system or a fermion system? Whethera many-body hopping system is a boson system or afermion system (or even some other statistical systems)has nothing to do with the Hilbert space. The fact thatthe many-body states are labeled by symmetric indexes(eg |i1, i2 = |i2, i1) does not imply that the many-bodysystem is a boson system. The statistics are determined

by the Hamiltonian H hop.Clearly, when the hopping amplitude t(i, j; i1, i2,...)only depends on i and j, t(i, j; i1, i2,...) = t(i, j), themany-body hopping Hamiltonian will describe a hard-core boson system. The issue is under what conditionthe many-body hopping Hamiltonian describes a fermionsystem.

This problem was solved in Ref. [19]. It was found thatthe many-body hopping Hamiltonian describes a fermionsystem if the hopping operators satisfy

tlk tiltlj = −tlj tiltlk (22)

for any three hopping operators tlj , til, and tlk with i, j,

k, and l all being different. (Note that the algebra has astructure t1t2t3 = −t3t2t1.)

To understand this result, consider the state |i, j,....with two particles at i, j, and possibly other particlesfurther away. We apply a set of five hopping operators{t jl , tlk, til, tlj , tki} to the state |i, j, .... but with differ-ent order (set Fig. 8)

t jl tlktiltlj tki|i, j,.... =C 1|i, j, ....

t jl tlj tiltlk tki|i, j,.... =C 2|i, j, ....

k

l

j

k

i

B

A

i j

l

σ yσ x

σ xσ y

2 1

3

FIG. 9: The Z 2 charge live on the links. The hopping of theZ 2 charges is induced by the open string operator.

where we have assumed that there are no particles atsites k and l. We note that after five hops we get backto the original state |i, j, .... with additional phases C 1,2.However, from Fig. 8, we see that the first way to arrangethe five hops (Fig. 8a) swaps the two particles at i and j, while the second way (Fig. 8b) does not swap the twoparticles. Since the two hopping schemes use the sameset of five hops, the difference between C 1 and C 2 is due

to exchanging the two particles. Thus we require C 1 =−C 2 in order for the many-body hopping Hamiltonianto describe a fermion system. Noting that the first andthe last hops are the same in the two hopping schemes,we find that C 1 = −C 2 if the hopping operators satisfyEq. (22). Eq. (22) serves as an alternative definition of Fermi statistics if we do not want to use anti-commutingalgebra.

For our spin model Eq. (19), the open strings end atthe midpoint of the links. So the Z 2 charges live on thelinks. To apply the above result to the Z 2 charges, wenote that an open string operator connecting midpoints iand j (see Fig. 9) play the role of a hopping operator tij .

Near the site 1 in Fig. 9, the hoping operators betweenthe midpoints i, j, k, and l are given by

tlj = σy1 , til = σz

1 , tlk = σx1 .

The fermion hopping algebra Eq. (22) becomes σy1σz

1σx1 =

−σx1σz

1σy1 which is satisfied. Near the site 2 and 3 in Fig.

9, the hoping operators between the midpoints i, j, k,and l are given by

tlj = σy2 , til = σz

2 , tlk = σx3

The fermion hopping algebra becomes σy2σz

2σx3 =

−σx3σz

2σy2 which is again satisfied. We see that the hop-

ping operators of the Z 2 charges satisfy a fermion hop-ping algebra. So the Z 2 charges are fermions. Fermionscan emerge in a pure bosonic model as ends of condensedstrings.

VII. SUMMARY

Symmetry breaking have dominated our understand-ing of phase and phase transition for over 50 years. Wenow know that symmetry breaking cannot describe allthe possible orders that matter can have. The study of

xiao-gang wen- an introduction to quantum order, string-net condensation, and emergence of light and...

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