xiao-gang wen- quantum orders and symmetric spin liquids

8/3/2019 Xiao-Gang Wen- Quantum Orders and Symmetric Spin Liquids

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Quantum Orders and Symmetric Spin Liquids

Xiao-Gang Wen

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Dated: Aug., 2001)

A concept quantum order is introduced to describe a new kind of orders that generallyappear in quantum states at zero temperature. Quantum orders that characterize universalityclasses of quantum states (described by complex ground state wave-functions) is much richer then

classical orders that characterize universality classes of finite temperature classical states (describedby positive probability distribution functions). The Landaus theory for orders and phase transitionsdoes not apply to quantum orders since they cannot be described by broken symmetries and theassociated order parameters. We introduced a mathematical object projective symmetry group to characterize quantum orders. With the help of quantum orders and projective symmetry groups,we construct hundreds of symmetric spin liquids, which have SU(2), U(1) or Z2 gauge structuresat low energies. We found that various spin liquids can be divided into four classes: (a) Rigidspin liquid spinons (and all other excitations) are fully gaped and may have bosonic, fermionic,or fractional statistics. (b) Fermi spin liquid spinons are gapless and are described by a Fermiliquid theory. (c) Algebraic spin liquid spinons are gapless, but they are not described by freefermionic/bosonic quasiparticles. (d) Bose spin liquid low lying gapless excitations are describedby a free boson theory. The stability of those spin liquids are discussed in details. We find that stable2D spin liquids exist in the first three classes (ac). Those stable spin liquids occupy a finite regionin phase space and represent quantum phases. Remarkably, some of the stable quantum phasessupport gapless excitations even without any spontaneous symmetry breaking. In particular, thegapless excitations in algebraic spin liquids interact down to zero energy and the interaction doesnot open any energy gap. We propose that it is the quantum orders (instead of symmetries) thatprotect the gapless excitations and make algebraic spin liquids and Fermi spin liquids stable. Sincehigh Tc superconductors are likely to be described by a gapless spin liquid, the quantum orders andtheir projective symmetry group descriptions lay the foundation for spin liquid approach to high Tcsuperconductors.

PACS numbers: 73.43. Nq, 74.25.-q, 11.15.Ex

Contents

I. Introduction 2A. Topological orders and quantum orders 2B. Spin-liquid approach to high Tc

superconductors 3C. Spin-charge separation in (doped) spin

liquids 4D. Organization 6

II. Projective construction of 2D spin liquids a review of SU(2) slave-boson approach 6

III. Spin liquids from translationally invariantansatz 9

IV. Quantum orders in symmetric spinliquids 12A. Quantum orders and projective symmetry

groups 12B. Classification of symmetric Z2 spin liquids 14C. Classification of symmetric U(1) and SU(2)

spin liquids 15

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

V. Continuous transitions and spinonspectra in symmetric spin liquids 17A. Continuous phase transitions without

symmetry breaking 17

B. Symmetric spin liquids around theU(1)-linear spin liquid U1Cn01n 17

C. Symmetric spin liquids around theSU(2)-gapless spin liquid SU2An0 20

D. Symmetric spin liquids around theSU(2)-linear spin liquid SU2Bn0 22

VI. Mean-field phase diagram of J1-J2 model 23

VII. Physical measurements of quantumorders 24

VIII. Four classes of spin liquids and their

stability 26A. Rigid spin liquid 26B. Bose spin liquid 26C. Fermi spin liquid 27D. Algebraic spin liquid 27E. Quantum order and the stability of spin

liquids 29

IX. Relation to previously constructed spinliquids 30


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X. Summary of the main results 31

A. General conditions on projectivesymmetry groups 33

References 35

I. INTRODUCTION

Due to its long length, we would like to first outlinethe structure of the paper so readers can choose to readthe parts of interests. The section X summarize the mainresults of the paper, which also serves as a guide of thewhole paper. The concept of quantum order is introducedin section I A. A concrete mathematical description ofquantum order is described in section IV A and sectionIV B. Readers who are interested in the background andmotivation of quantum orders may choose to read sec-tion I A. Readers who are familiar with the slave-bosonapproach and just want a quick introduction to quantum

orders may choose to read sections IV A and IV B. Read-ers who are not familiar with the slave-boson approachmay find the review sections II and III useful. Read-ers who do not care about the slave-boson approach butare interested in application to high Tc superconductorsand experimental measurements of quantum orders maychoose to read sections I A, I B, VII and Fig. 1 - Fig. 15,to gain some intuitive picture of spinon dispersion andneutron scattering behavior of various spin liquids.

A. Topological orders and quantum orders

Matter can have many different states, such as gas,liquid, and solid. Understanding states of matter is thefirst step in understanding matter. Physicists find mattercan have much more different states than just gas, liquid,and solid. Even solids and liquids can appear in manydifferent forms and states. With so many different statesof matter, a general theory is needed to gain a deeperunderstanding of states of matter.

All the states of matter are distinguished by their in-ternal structures or orders. The key step in developingthe general theory for states of matter is the realizationthat all the orders are associated with symmetries (orrather, the breaking of symmetries). Based on the rela-

tion between orders and symmetries, Landau developed ageneral theory of orders and the transitions between dif-ferent orders.[1, 2] Landaus theory is so successful andone starts to have a feeling that we understand, at inprinciple, all kinds of orders that matter can have.

However, nature never stops to surprise us. In 1982,Tsui, Stormer, and Gossard[3] discovered a new kind ofstate Fractional Quantum Hall (FQH) liquid.[4] Quan-tum Hall liquids have many amazing properties. A quan-tum Hall liquid is more rigid than a solid (a crystal),

in the sense that a quantum Hall liquid cannot be com-pressed. Thus a quantum Hall liquid has a fixed and well-defined density. When we measure the electron densityin terms of filling factor , we found that all discoveredquantum Hall states have such densities that the fillingfactors are exactly given by some rational numbers, suchas = 1, 1/3, 2/3, 2/5,.... Knowing that FQH liquidsexist only at certain magical filling factors, one cannot

help to guess that FQH liquids should have some inter-nal orders or patterns. Different magical filling fac-tors should be due to those different internal patterns.However, the hypothesis of internal patterns appearsto have one difficulty FQH states are liquids, and howcan liquids have any internal patterns?

In 1989, it was realized that the internal orders inFQH liquids (as well as the internal orders in chiral spinliquids[5, 6]) are different from any other known ordersand cannot be observed and characterized in any con-ventional ways.[7, 8] What is really new (and strange)about the orders in chiral spin liquids and FQH liquidsis that they are not associated with any symmetries (or

the breaking of symmetries), and cannot be described byLandaus theory using physical order parameters.[9] Thiskind of order is called topological order. Topological orderis a new concept and a whole new theory was developedto describe it.[9, 10]

Knowing FQH liquids contain a new kind of order topological order, we would like to ask why FQH liquidsare so special. What is missed in Landaus theory forstates of matter so that the theory fails to capture thetopological order in FQH liquids?

When we talk about orders in FQH liquids, we arereally talking about the internal structure of FQH liq-uids at zero temperature. In other words, we are talking

about the internal structure of the quantum ground stateof FQH systems. So the topological order is a propertyof ground state wave-function. The Landaus theory isdeveloped for system at finite temperatures where quan-tum effects can be ignored. Thus one should not be sur-prised that the Landaus theory does not apply to statesat zero temperature where quantum effects are impor-tant. The very existence of topological orders suggeststhat finite-temperature orders and zero-temperature or-ders are different, and zero-temperature orders containricher structures. We see that what is missed by Landaustheory is simply the quantum effect. Thus FQH liquidsare not that special. The Landaus theory and symmetrycharacterization can fail for any quantum states at zerotemperature. As a consequence, new kind of orders withno broken symmetries and local order parameters (suchas topological orders) can exist for any quantum states atzero temperature. Because the orders in quantum statesat zero temperature and the orders in classical states atfinite temperatures are very different, here we would liketo introduce two concepts to stress their differences:[11](A) Quantum orders:[93] which describe the universal-ity classes of quantum ground states (ie the universalityclasses of complex ground state wave-functions with in-


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finity variables);(B)Classical orders: which describe the universalityclasses of classical statistical states (ie the universalityclasses of positive probability distribution functions withinfinity variables).From the above definition, it is clear that the quantumorders associated with complex functions are richer thanthe classical orders associated with positive functions.

The Landaus theory is a theory for classical orders,which suggests that classical orders may be characterizedby broken symmetries and local order parameters.[94]The existence of topological order indicates that quan-tum orders cannot be completely characterized by bro-ken symmetries and order parameters. Thus we need todevelop a new theory to describe quantum orders.

In a sense, the classical world described by positiveprobabilities is a world with only black and white. TheLandaus theory and the symmetry principle for classicalorders are color blind which can only describe differentshades of grey in the classical world. The quantumworld described by complex wave functions is a colorful

world. We need to use new theories, such as the theory oftopological order and the theory developed in this paper,to describe the rich color of quantum world.

The quantum orders in FQH liquids have a specialproperty that all excitations above ground state have fi-nite energy gaps. This kind of quantum orders are calledtopological orders. In general, a topological order is de-fined as a quantum order where all the excitations aboveground state have finite energy gapes.

Topological orders and quantum orders are generalproperties of any states at zero temperature. Non trivialtopological orders not only appear in FQH liquids, theyalso appear in spin liquids at zero temperature. In fact,the concept of topological order was first introduced in astudy of spin liquids.[9] FQH liquid is not even the firstexperimentally observed state with non trivial topolog-ical orders. That honor goes to superconducting statediscovered in 1911.[12] In contrast to a common pointof view, a superconducting state cannot be characterizedby broken symmetries. It contains non trivial topologicalorders,[13] and is fundamentally different from a super-fluid state.

After a long introduction, now we can state the mainsubject of this paper. In this paper, we will study a newclass of quantum orders where the excitations above theground state are gapless. We believe that the gaplessquantum orders are important in understanding high Tcsuperconductors. To connect to high Tc superconduc-tors, we will study quantum orders in quantum spin liq-uids on a 2D square lattice. We will concentrate on howto characterize and classify quantum spin liquids withdifferent quantum orders. We introduce projective sym-metry groups to help us to achieve this goal. The projec-tive symmetry group can be viewed as a generalizationof symmetry group that characterize different classicalorders.

B. Spin-liquid approach to high Tc superconductors

There are many different approaches to the high Tcsuperconductors. Different people have different pointsof view on what are the key experimental facts for thehigh Tc superconductors. The different choice of the keyexperimental facts lead to many different approaches andtheories. The spin liquid approach is based on a point of

view that the high Tc superconductors are doped Mottinsulators.[1416] (Here by Mott insulator we mean a in-sulator with an odd number of electron per unit cell.) Webelieve that the most important properties of the high Tcsuperconductors is that the materials are insulators whenthe conduction band is half filled. The charge gap ob-tained by the optical conductance experiments is about2eV, which is much larger than the anti-ferromagnetic(AF) transition temperature TAF 250K, the super-conducting transition temperature Tc 100K, and thespin pseudo-gap scale 40meV.[1719] The insulat-ing property is completely due to the strong correlationspresent in the high Tc materials. Thus the strong cor-

relations are expect to play very important role in un-derstanding high Tc superconductors. Many importantproperties of high Tc superconductors can be directlylinked to the Mott insulator at half filling, such as (a)the low charge density[20] and superfluid density,[21] (b)Tc being proportional to doping Tc x,[2224] (c) thepositive charge carried by the charge carrier,[20] etc .

In the spin liquid approach, the strategy is to try tounderstand the properties of the high Tc superconduc-tors from the low doping limit. We first study the spinliquid state at half filling and try to understand the par-ent Mott insulator. (In this paper, by spin liquid, wemean a spin state with translation and spin rotation

symmetry.) At half filling, the charge excitations canbe ignored due to the huge charge gap. Thus we canuse a pure spin model to describe the half filled system.After understand the spin liquid, we try to understandthe dynamics of a few doped holes in the spin liquidstates and to obtain the properties of the high Tc su-perconductors at low doping. One advantage of the spinliquid approach is that experiments (such as angle re-solved photo-emission,[17, 18, 25, 26] NMR,[27], neutronscattering,[2830] etc ) suggest that underdoped cuper-ates have many striking and qualitatively new propertieswhich are very different from the well known Fermi liq-uids. It is thus easier to approve or disapprove a newtheory in the underdoped regime by studying those qual-

itatively new properties.Since the properties of the doped holes (such as their

statistics, spin, effective mass, etc ) are completely de-termined by the spin correlation in the parent spin liq-uids, thus in the spin liquid approach, each possible spinliquid leads to a possible theory for high Tc supercon-ductors. Using the concept of quantum orders, we cansay that possible theories for high Tc superconductors inthe low doping limits are classified by possible quantumorders in spin liquids on 2D square lattice. Thus one


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way to study high Tc superconductors is to construct allthe possible spin liquids that have the same symmetriesas those observed in high Tc superconductors. Then an-alyze the physical properties of those spin liquids withdopings to see which one actually describes the high Tcsuperconductor. Although we cannot say that we haveconstructed all the symmetric spin liquids, in this paperwe have found a way to construct a large class of symmet-

ric spin liquids. (Here by symmetric spin liquids we meanspin liquids with all the lattice symmetries: translation,rotation, parity, and the time reversal symmetries.) Wealso find a way to characterize the quantum orders inthose spin liquids via projective symmetry groups. Thisgives us a global picture of possible high Tc theories.We would like to mention that a particular spin liquid the staggered-flux/d-wave state[31, 32] may be im-portant for high Tc superconductors. Such a state canexplain[33, 34] the highly unusual pseudo-gap metallicstate found in underdoped cuperates,[17, 18, 25, 26] aswell as the d-wave superconducting state[32].

The spin liquids constructed in this paper can be di-

vided into four class: (a) Rigid spin liquid spinons arefully gaped and may have bosonic, fermionic, or frac-tional statistics, (b) Fermi spin liquid spinons are gap-less and are described by a Fermi liquid theory, (c) Alge-braic spin liquid spinons are gapless, but they are notdescribed by free fermionic/bosonic quasiparticles. (d)Bose spin liquid low lying gapless excitations are de-scribed by a free boson theory. We find some of the con-structed spin liquids are stable and represent stable quan-tum phases, while others are unstable at low energies dueto long range interactions caused by gauge fluctuations.The algebraic spin liquids and Fermi spin liquids are in-teresting since they can be stable despite their gaplessexcitations. Those gapless excitations are not protectedby symmetries. This is particularly striking for algebraicspin liquids since their gapless excitations interact downto zero energy and the states are still stable. We proposethat it is the quantum orders that protect the gaplessexcitations and ensure the stability of the algebraic spinliquids and Fermi spin liquids.

We would like to point out that both stable and unsta-ble spin liquids may be important for understanding highTc superconductors. Although at zero temperature highTc superconductors are always described stable quantumstates, some important states of high Tc superconductors,such as the pseudo-gap metallic state for underdopedsamples, are observed only at finite temperatures. Such

finite temperature states may correspond to (doped) un-stable spin liquids, such as staggered flux state. Thuseven unstable spin liquids can be useful in understand-ing finite temperature metallic states.

There are many different approach to spin liquids.In addition to the slave-boson approach,[6, 15, 16,3133, 3540] spin liquids has been studied usingslave-fermion/-model approach,[4146] quantum dimermodel,[4751] and various numerical methods.[5255] Inparticular, the numerical results and recent experimen-

tal results[56] strongly support the existence of quantumspin liquids in some frustrated systems. A 3D quantumorbital liquid was also proposed to exist in LaTiO3.[57]

However, I must point out that there is no generallyaccepted numerical results yet that prove the existenceof spin liquids with odd number of electron per unit cellfor spin-1/2 systems, despite intensive search in last tenyears. But it is my faith that spin liquids exist in spin-

1/2 systems. For more general systems, spin liquids doexist. Read and Sachdev[43] found stable spin liquids ina Sp(N) model in large N limit. The spin-1/2 modelstudied in this paper can be easily generalized to SU(N)model with N/2 fermions per site.[31, 58] In the large Nlimit, one can easily construct various Hamiltonians[58,59] whose ground states realize various U(1) and Z2 spinliquids constructed in this paper. The quantum ordersin those large-N spin liquids can be described by themethods introduced in this paper.[58] Thus, despite theuncertainly about the existence of spin-1/2 spin liquids,the methods and the results presented in this paper arenot about (possibly) non-existing ghost states. Those

methods and results apply, at least, to certain large-Nsystems. In short, non-trivial quantum orders exist intheory. We just need to find them in nature. (In fact, ourvacuum is likely to be a state with a non-trivial quantumorder, due to the fact that light exists.[58]) Knowing theexistence of spin liquids in large-N systems, it is not sucha big leap to go one step further to speculate spin liquidsexist for spin-1/2 systems.

C. Spin-charge separation in (doped) spin liquids

Spin-charge separation and the associated gauge the-

ory in spin liquids (and in doped spin liquids) are very im-portant concepts in our attempt to understand the prop-erties of high Tc superconductors.[1416, 39, 60] However,the exact meaning of spin-charge separation is differentfor different researchers. The term spin-charge separa-tion has at lease in two different interpretations. In thefirst interpretation, the term means that it is better to in-troduce separate spinons (a neutral spin-1/2 excitation)and holons (a spinless excitation with unit charge) to un-derstand the dynamical properties of high Tc supercon-ductors, instead of using the original electrons. However,there may be long range interaction (possibly, even con-fining interactions at long distance) between the spinonsand holons, and the spinons and holons may not be welldefined quasiparticles. We will call this interpretationpseudo spin-charge separation. The algebraic spin liquidshave the pseudo spin-charge separation. The essence ofthe pseudo spin-charge separation is not that spin andcharge separate. The pseudo spin-charge separation issimply another way to say that the gapless excitationscannot be described by free fermions or bosons. In thesecond interpretation, the term spin-charge separationmeans that there are only short ranged interactions be-tween the spinons and holons. The spinons and holons


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are well defined quasiparticles at least in the dilute limitor at low energies. We will call the second interpretationthe true spin-charge separation. The rigid spin liquidsand the Fermi spin liquids have the true spin-charge sep-aration.

Electron operator is not a good starting point to de-scribe states with pseudo spin-charge separation or truespin-charge separation. To study those states, we usu-

ally rewrite the electron operator as a product of severalother operators. Those operators are called parton oper-ators. (The spinon operator and the holon operator areexamples of parton operators). We then construct mean-field state in the enlarged Hilbert space of partons. Thegauge structure can be determined as the most generaltransformations between the partons that leave the elec-tron operator unchanged.[61] After identifying the gaugestructure, we can project the mean-field state onto thephysical (ie the gauge invariant) Hilbert space and obtaina strongly correlated electron state. This procedure in itsgeneral form is called projective construction. It is a gen-eralization of the slave-boson approach.[15, 16, 33, 36

38, 40] The general projective construction and the re-lated gauge structure has been discussed in detail forquantum Hall states.[61] Now we see a third (but tech-nical) meaning of spin-charge separation: to construct astrongly correlated electron state, we need to use par-tons and projective construction. The resulting effectivetheory naturally contains a gauge structure.

Although, it is not clear which interpretation of spin-charge separation actually applies to high Tc supercon-ductors, the possibility of true spin-charge separation inan electron system is very interesting. The first con-crete example of true spin-charge separation in 2D isgiven by the chiral spin liquid state,[5, 6] where thegauge interaction between the spinons and holons be-comes short-ranged due to a Chern-Simons term. TheChern-Simons term breaks time reversal symmetry andgives the spinons and holons a fractional statistics. Laterin 1991, it was realized that there is another way tomake the gauge interaction short-ranged through theAnderson-Higgs mechanism.[38, 43] This led to a mean-field theory[38, 40] of the short-ranged Resonating Va-lence Bound (RVB) state[47, 48] conjectured earlier. Wewill call such a state Z2 spin liquid state, to stress theunconfinedZ2 gauge field that appears in the low energyeffective theory of those spin liquids. (See remarks at theend of this section. We also note that the Z2 spin liquidsstudied in Ref. [43] all break the 90 rotation symmetry

and are different from the short-ranged RVB state stud-ied Ref. [38, 40, 47, 48].) Since the Z2 gauge fluctuationsare weak and are not confining, the spinons and holonshave only short ranged interactions in the Z2 spin liquidstate. The Z2 spin liquid state also contains a Z2 vortex-like excitation.[38, 62] The spinons and holons can bebosons or fermions depending on if they are bound withthe Z2 vortex.

Recently, the true spin-charge separation, the Z2 gaugestructure and the Z2 vortex excitations were also pro-

posed in a study of quantum disordered superconduct-ing state in a continuum model[63] and in a Z2 slave-boson approach[64]. The resulting liquid state (whichwas named nodal liquid) has all the novel properties ofZ2 spin liquid state such as the Z2 gauge structure andthe Z2 vortex excitation (which was named vison). Fromthe point of view of universality class, the nodal liquid isone kind of Z2 spin liquids. However, the particular Z2

spin liquid studied in Ref. [38, 40] and the nodal liquidare two different Z2 spin liquids, despite they have thesame symmetry. The spinons in the first Z2 spin liquidhave a finite energy gap while the spinons in the nodalliquid are gapless and have a Dirac-like dispersion. In thispaper, we will use the projective construction to obtainmore general spin liquids. We find that one can constructhundreds of different Z2 spin liquids. Some Z2 spin liq-uids have finite energy gaps, while others are gapless.Among those gapless Z2 spin liquids, some have finiteFermi surfaces while others have only Fermi points. Thespinons near the Fermi points can have linear E(k) |k|or quadratic E(k) k2 dispersions. We find there aremore than one Z

2spin liquids whose spinons have a mass-

less Dirac-like dispersion. Those Z2 spin liquids havethe same symmetry but different quantum orders. Theiransatz are give by Eq. (42), Eq. (39), Eq. (106), etc .

Both chiral spin liquid and Z2 spin liquid states areMott insulators with one electron per unit cell if notdoped. Their internal structures are characterized by anew kind of order topological order, if they are gappedor if the gapless sector decouples. Topological order isnot related to any symmetries and has no (local) or-der parameters. Thus, the topological order is robustagainst all perturbations that can break any symmetries(including random perturbations that break translationsymmetry).[9, 10] (This point was also emphasized inRef. [65] recently.) Even though there are no order pa-rameters to characterize them, the topological orders canbe characterized by other measurable quantum numbers,such as ground state degeneracy in compact space as pro-posed in Ref. [9, 10]. Recently, Ref. [65] introduced avery clever experiment to test the ground state degen-eracy associated with the non-trivial topological orders.In addition to ground state degeneracy, there are otherpractical ways to detect topological orders. For example,the excitations on top of a topologically ordered statecan be defects of the under lying topological order, whichusually leads to unusual statistics for those excitations.Measuring the statistics of those excitations also allow us

to measure topological orders.The concept of topological order and quantum order

are very important in understanding quantum spin liq-uids (or any other strongly correlated quantum liquids).In this paper we are going to construct hundreds of dif-ferent spin liquids. Those spin liquids all have the samesymmetry. To understand those spin liquids, we need tofirst learn how to characterize those spin liquids. Thosestates break no symmetries and hence have no order pa-rameters. One would get into a wrong track if trying to


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find an order parameter to characterize the spin liquids.We need to use a completely new way, such as topologicalorders and quantum orders, to characterize those states.

In addition to the above Z2 spin liquids, in this paperwe will also study many other spin liquids with differentlow energy gauge structures, such as U(1) and SU(2)gauge structures. We will use the terms Z2 spin liq-uids, U(1) spin liquids, and SU(2) spin liquids to describe

them. We would like to stress that Z2, U(1), and SU(2)here are gauge groups that appear in the low energy ef-fective theories of those spin liquids. They should not beconfused with the Z2, U(1), and SU(2) gauge group inslave-boson approach or other theories of the projectiveconstruction. The latter are high energy gauge groups.The high energy gauge groups have nothing to do withthe low energy gauge groups. A high energy Z2 gaugetheory (or a Z2 slave-boson approach) can have a lowenergy effective theory that contains SU(2), U(1) or Z2gauge fluctuations. Even the t-J model, which has nogauge structure at lattice scale, can have a low energyeffective theory that contains SU(2), U(1) or Z2 gauge

fluctuations. The spin liquids studied in this paper allcontain some kind of low energy gauge fluctuations. De-spite their different low energy gauge groups, all thosespin liquids can be constructed from any one of SU(2),U(1), or Z2 slave-boson approaches. After all, all thoseslave-boson approaches describe the same t-J model andare equivalent to each other. In short, the high energygauge group is related to the way in which we write downthe Hamiltonian, while the low energy gauge group is aproperty of ground state. Thus we should not regard Z2spin liquids as the spin liquids constructed using Z2 slave-boson approach. A Z2 spin liquid can be constructedfrom the U(1) or SU(2) slave-boson approaches as well.A precise mathematical definition of the low energy gauge

group will be given in section IV A.

D. Organization

In this paper we will use the method outlined inRef. [38, 40] to study gauge structures in various spinliquid states. In section II we review SU(2) mean-fieldtheory of spin liquids. In section III, we construct sim-ple symmetric spin liquids using translationally invari-ant ansatz. In section IV, projective symmetry group isintroduced to characterize quantum orders in spin liq-uids. In section V, we study the transition between dif-ferent symmetric spin liquids, using the results obtainedin Ref. [66], where we find a way to construct all thesymmetric spin liquids in the neighborhood of some wellknown spin liquids. We also study the spinon spectrumto gain some intuitive understanding on the dynamicalproperties of the spin liquids. Using the relation betweentwo-spinon spectrum and quantum order, we propose, insection VII, a practical way to use neutron scatteringto measure quantum orders. We study the stability ofFermi spin liquids and algebraic spin liquids in section

VIII. We find that both Fermi spin liquids and algebraicspin liquids can exist as zero temperature phases. This isparticularly striking for algebraic spin liquids since theirgapless excitations interacts even at lowest energies andthere are no free fermionic/bosonic quasiparticle excita-tions at low energies. We show how quantum order canprotect gapless excitations. Appendix A contains an al-gebraic description of projective symmetry groups, which

can be used to classify projective symmetry groups.[66]Section X summarizes the main results of the paper.

II. PROJECTIVE CONSTRUCTION OF 2DSPIN LIQUIDS A REVIEW OF SU(2)

SLAVE-BOSON APPROACH

In this section, we are going to use projective construc-tion to construct 2D spin liquids. We are going to reviewa particular projective construction, namely the SU(2)slave-boson approach.[15, 16, 33, 3638, 40] The gaugestructure discovered by Baskaran and Anderson[16] inthe slave-boson approach plays a crucial role in our un-derstanding of strongly correlated spin liquids.

We will concentrate on the spin liquid states of a purespin-1/2 model on a 2D square lattice

Hspin =

JijSi Sj + ... (1)

where the summation is over different links (ie ij andji are regarded as the same) and ... represents possi-ble terms which contain three or more spin operators.Those terms are needed in order for many exotic spinliquid states introduced in this paper to become theground state. To obtain the mean-field ground state of

the spin liquids, we introduce fermionic parton operatorsfi, = 1, 2 which carries spin 1/2 and no charge. Thespin operator Si is represented as

Si =1

2fifi (2)

In terms of the fermion operators the Hamiltonian Eq. (1)can be rewritten as

H =ij

12

Jij

fifjf

jfi +

1

2fifif

jfj

(3)

Here we have used = 2 .We also added proper constant terms

i fifi and

ij fifif

jfj to get the above form. Notice that

the Hilbert space of Eq. (3) is generated by the partonoperators f and is larger than that of Eq. (1). Theequivalence between Eq. (1) and Eq. (3) is valid only inthe subspace where there is exactly one fermion per site.Therefore to use Eq. (3) to describe the spin state weneed to impose the constraint[15, 16]

fifi = 1, fifi = 0 (4)


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The second constraint is actually a consequence of thefirst one.

A mean-field ground state at zeroth order is obtainedby making the following approximations. First we replaceconstraint Eq. (4) by its ground-state average

fifi = 1, fifi = 0 (5)Such a constraint can be enforced by including a site

dependent and time independent Lagrangian multiplier:al0(i)(f

ifi 1), l = 1, 2, 3, in the Hamiltonian. At the

zeroth order we ignore the fluctuations (ie the time de-pendence) of al0. If we included the fluctuations of a

l0,

the constraint Eq. (5) would become the original con-straint Eq. (4).[15, 16, 36, 37] Second we replace the

operators fifj and fifi by their ground-state ex-pectations value

ij = 2fi fj, ij =jiij =2fifj, ij =ji (6)

again ignoring their fluctuations. In this way we obtainthe zeroth order mean-field Hamiltonian:

Hmean

=ij

38

Jij

(jif

ifj + ijf

if

j + h.c)

|ij |2 |ij |2

(7)

+i

a30(f

ifi 1) + [(a10 + ia20)fifi + h.c.]

ij and ij in Eq. (7) must satisfy the self consistencycondition Eq. (6) and the site dependent fields al0(i) arechosen such that Eq. (5) is satisfied by the mean-fieldground state. Such ij , ij and a

l0 give us a mean-field

solution. The fluctuations in ij , ij and al

0(i) describethe collective excitations above the mean-field groundstate.

The Hamiltonian Eq. (7) and the constraints Eq. (4)have a local SU(2) symmetry.[36, 37] The local SU(2)symmetry becomes explicit if we introduce doublet

=

12

=

ff

(8)

and matrix

Uij =

ij ijij ij

= Uji (9)

Using Eq. (8) and Eq. (9) we can rewrite Eq. (5) andEq. (7) as:

ili

= 0 (10)

Hmean =ij

3

8Jij

1

2Tr(Uij Uij) (i Uijj + h.c.)

+i

al0i

li (11)

where l, l = 1, 2, 3 are the Pauli matrices. From Eq. (11)we can see clearly that the Hamiltonian is invariant undera local SU(2) transformation W(i):

i W(i) iUij W(i) Uij W(j) (12)

The SU(2) gauge structure is originated from Eq. (2).

The SU(2) is the most general transformation betweenthe partons that leave the physical spin operator un-changed. Thus once we write down the parton expres-sion of the spin operator Eq. (2), the gauge structure ofthe theory is determined.[61] (The SU(2) gauge structurediscussed here is a high energy gauge structure.)

We note that both components of carry spin-up.Thus the spin-rotation symmetry is not explicit in ourformalism and it is hard to tell if Eq. (11) describes aspin-rotation invariant state or not. In fact, for a generalUij satisfying Uij = U

ji , Eq. (11) may not describe a

spin-rotation invariant state. However, if Uij has a form

Uij = iijWij ,ij = real number,

Wij SU(2), (13)

then Eq. (11) will describe a spin-rotation invariant state.This is because the above Uij can be rewritten in a formEq. (9). In this case Eq. (11) can be rewritten as Eq. (7)where the spin-rotation invariance is explicit.

To obtain the mean-field theory, we have enlarged theHilbert space. Because of this, the mean-field theory

is not even qualitatively correct. Let |(Uij)mean be theground state of the Hamiltonian Eq. (11) with energyE(Uij, ali

l). It is clear that the mean-field ground state

is not even a valid wave-function for the spin systemsince it may not have one fermion per site. Thus it isvery important to include fluctuations of al0 to enforceone-fermion-per-site constraint. With this understand-ing, we may obtain a valid wave-function of the spin sys-tem spin({i}) by projecting the mean-field state to thesubspace of one-fermion-per-site:

spin({i}) = 0|i

fii |(Uij)mean. (14)

Now the local SU(2) transformation Eq. (12) can have

a very physical meaning: |(Uij)mean and |(W(i)UijW(j))

mean give rise to the same spin wave-function after projection:

0|i

fii |(Uij)mean = 0|i

fii |(W(i)UijW(j))

mean (15)

Thus Uij and Uij = W(i)UijW(j) are just two differ-

ent labels which label the same physical state. Within themean-field theory, a local SU(2) transformation changes

a mean-field state |(Uij)mean to a different mean-field state|(U

ij)

mean. If the two mean-field states always have the


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8

same physical properties, the system has a local SU(2)symmetry. However, after projection, the physical spinquantum state described by wave-function spin({i}) isinvariant under the local SU(2) transformation. A localSU(2) transformation just transforms one label, Uij , of aphysical spin state to another label, Uij , which labels theexactly the same physical state. Thus after projection,local SU(2) transformations become gauge transforma-

tions. The fact that Uij and Uij label the same physicalspin state creates a interesting situation when we con-

sider the fluctuations ofUij around a mean-field solution some fluctuations of Uij do not change the physicalstate and are unphysical. Those fluctuations are calledthe pure gauge fluctuations.

The above discussion also indicates that in order forthe mean-field theory to make any sense, we must at leastinclude the SU(2) gauge (or other gauge) fluctuationsdescribed by al0 and Wij in Eq. (13), so that the SU(2)gauge structure of the mean-field theory is revealed andthe physical spin state is obtained. We will include thegauge fluctuations to the zeroth-order mean-field theory.The new theory will be called the first order mean-fieldtheory. It is this first order mean-field theory that rep-resents a proper low energy effective theory of the spinliquid.

Here, we would like make a remark about gauge sym-metry and gauge symmetry breaking. We see thattwo ansatz Uij and U

ij = W(i)UijW

(j) have the samephysical properties. This property is usually called thegauge symmetry. However, from the above discussion,we see that the gauge symmetry is not a symmetry. Asymmetry is about two different states having the sameproperties. Uij and U

ij are just two labels that label

the same state, and the same state always have the sameproperties. We do not usually call the same state hav-

ing the same properties a symmetry. Because the samestate alway have the same properties, the gauge symme-try can never be broken. It is very misleading to call theAnderson-Higgs mechanism gauge symmetry breaking.With this understanding, we see that a superconductor isfundamentally different from a superfluid. A superfluidis characterized by U(1) symmetry breaking, while a su-perconductor has no symmetry breaking once we includethe dynamical electromagnetic gauge fluctuations. A su-perconductor is actually the first topologically orderedstate observed in experiments,[13] which has no symme-try breaking, no long range order, and no (local) orderparameter. However, when the speed of light c = ,a superconductor becomes similar to a superfluid and ischaracterized by U(1) symmetry breaking.

The relation between the mean-field state and thephysical spin wave function Eq. (14) allows us to con-struct transformation of the physical spin wave-functionfrom the mean-field ansatz. For example the mean-field

state |(Uij)

mean with Uij = Uil,jl produces a phys-ical spin wave-function which is translated by a dis-tance l from the physical spin wave-function produced

by |(Uij)mean. The physical state is translationally sym-

metric if and only if the translated ansatz Uij and the

original ansatz Uij are gauge equivalent (it does not re-quire Uij = Uij). We see that the gauge structure cancomplicates our analysis of symmetries, since the phys-ical spin wave-function spin({i}) may has more sym-metries than the mean-field state |(Uij)mean before projec-tion.

Let us discuss time reversal symmetry in more detail.

A quantum system described by

it(t) = H(t) (16)

has a time reversal symmetry if (t) satisfying the equa-tion of motion implies that (t) also satisfying theequation of motion. This requires that H = H. Wesee that, for time reversal symmetric system, if is aneigenstate, then will be an eigenstate with the sameenergy.

For our system, the time reversal symmetry means that

if the mean-field wave function (Uij,a

li

l)mean is a mean-field

ground state wave function for ansatz (Uij , ali

l), then(Uij,alil)mean

will be the mean-field ground state wave

function for ansatz (Uij , ali(

l)). That is

(Uij,a

li

l)mean

=

(Uij,ali(

l))mean (17)

For a system with time reversal symmetry, the mean-fieldenergy E(Uij, ali

l) satisfies

E(Uij , ali

l) = E(Uij , ali(

l)) (18)

Thus if an ansatz (Uij , alil) is a mean-field solution, then

(Uij , ali(

l)) is also a mean-field solution with the same

mean-field energy.From the above discussion, we see that under the time

reversal transformation, the ansatz transforms as

Uij Uij = (i2)Uij(i2) = Uij ,ali

l ali l = (i2)(alil)(i2) = alil. (19)

Note here we have included an additional SU(2) gaugetransformation Wi = i2. We also note that under thetime reversal transformation, the loop operator trans-

forms as PC = ei+ill (i2)PC(i2) = ei+i

ll .We see that the U(1) flux changes the sign while theSU(2) flux is not changed.

Before ending this review section, we would like topoint out that the mean-field ansatz of the spin liquidsUij can be divided into two classes: unfrustrated ansatzwhere Uij only link an even lattice site to an odd latticesite and frustrated ansatz where Uij are nonzero betweentwo even sites and/or two odd sites. An unfrustratedansatz has only pure SU(2) flux through each plaquette,while an frustrated ansatz has U(1) flux of multiple of/2 through some plaquettes in addition to the SU(2)flux.


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9

III. SPIN LIQUIDS FROM TRANSLATIONALLYINVARIANT ANSATZ

In this section, we will study many simple examplesof spin liquids and their ansatz. Through those simpleexamples, we gain some understandings on what kind ofspin liquids are possible. Those understandings help usto develop the characterization and classification of spin

liquids using projective symmetry group.Using the above SU(2) projective construction, one

can construct many spin liquid states. To limit ourselves,we will concentrate on spin liquids with translation and90 rotation symmetries. Although a mean-field ansatzwith translation and rotation invariance always generatea spin liquid with translation and rotation symmetries,a mean-field ansatz without those invariances can alsogenerate a spin liquid with those symmetries.[95] Becauseof this, it is quite difficult to construct all the translationand rotation symmetric spin liquids. In this section wewill consider a simpler problem. We will limit ourselvesto spin liquids generated from translationally invariant

ansatz:Ui+l,j+l = Uij, a

l0(i) = a

l0 (20)

In this case, we only need to find the conditions underwhich the above ansatz can give rise to a rotationallysymmetric spin liquid. First let us introduce uij:

3

8JijUij = uij (21)

For translationally invariant ansatz, we can introduce ashort-hand notation:

uij = ui+j

ui+j (22)

where u1,2,3l are real, u0l is imaginary,

0 is the identitymatrix and 1,2,3 are the Pauli matrices. The fermionspectrum is determined by Hamiltonian

H = ij

iujij + h.c.

+i

ial0

li (23)

In k-space we have

H = k

k(u(k) a0 )k (24)

where = 0, 1, 2, 3,

u(k) =l

ul eilk, (25)

a00 = 0, and N is the total number of site. The fermionspectrum has two branches and is given by

E(k) =u0(k) E0(k)

E0(k) =

l

(ul(k) al0)2 (26)

The constraints can be obtained fromEground

al0

= 0 and

have a form

Nili

=

k,E(k)


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10

= 0 in the above ansatz Eq. (28). The spinon in thespin liquid described by = 0 has a large Fermi surface.We will call this state SU(2)-gapless state (This statewas called uniform RVB state in literature). The statewith = has gapless spinons only at isolated k points.We will call such a state SU(2)-linear state to stress thelinear dispersion E |k| near the Fermi points. (Such astate was called the -flux state in literature). The low

energy effective theory for the SU(2)-linear state is de-scribed by massless Dirac fermions (the spinons) coupledto a SU(2) gauge field.

After proper gauge transformations, the SU(2)-gaplessansatz can be rewritten as

ux = i

uy = i (30)

and the SU(2)-linear ansatz as

ui,i+x = i

ui,i+y = i(

)ix (31)

In these form, the SU(2) gauge structure is explicit sinceuij i0. Here we would also like to mention that underthe projective-symmetry-group classification, the SU(2)-gapless ansatz Eq. (30) is labeled by SU2An0 and theSU(2)-linear ansatz Eq. (31) by SU2Bn0 (see Eq. (100)).

When = = 0, The flux PC is non trivial. How-ever, PC commute with PC as long as the two loopsC and C have the same base point. In this case theSU(2) gauge structure is broken down to a U(1) gaugestructure.[38, 40] The gapless spinon still only appearat isolated k points. We will call such a state U(1)-linear state. (This state was called staggered flux state

and/or d-wave pairing state in literature.) After a propergauge transformation, the U(1)-linear state can also bedescribed by the ansatz

ui,i+x = i ()i3ui,i+y = i + ()i3 (32)

where the U(1) gauge structure is explicit. Under theprojective-symmetry-group classification, such a state islabeled by U1Cn01n (see section IV C). The low energyeffective theory is described by massless Dirac fermions(the spinons) coupled to a U(1) gauge field.

The above results are all known before. In the follow-

ing we are going to study a new class of translation androtation symmetric ansatz, which has a form

al0 =0

ux =i0 (3 1)

uy =i0 (3 + 1) (33)

with and non-zero. The above ansatz describes theSU(2)-gapless spin liquid if = 0, and the SU(2)-linearspin liquid if = 0.

After a 90 rotation R90, the above ansatz becomes

ux = i0 (3 + 1)uy = i

0 (3 1) (34)The rotated ansatz is gauge equivalent to the origi-nal ansatz under the gauge transformation GR90(i) =()ix(1 i2)/2. After a parity x x transforma-tion Px, Eq. (33) becomes

ux = i0 (3 1)uy = i

0 (3 + 1) (35)which is gauge equivalent to the original ansatz under thegauge transformation GPx(i) = ()ixi(3 + 1)/

2. Un-

der time reversal transformation T, Eq. (33) is changedto

ux = i0 + (3 1)uy = i0 + (3 + 1) (36)

which is again gauge equivalent to the original ansatz un-der the gauge transformation GT(i) = ()i. (In fact anyansatz which only has links between two non-overlappingsublattices (ie the unfrustrated ansatz) is time reversalsymmetric ifal0 = 0 .) To summarize the ansatz Eq. (33)is invariant under the rotation R90, parity Px, and timereversal transformation T, followed by the following gaugetransformations

GR90(i) =()ix(1 i2)/

2

GPx(i) =()ixi(3 + 1)/

2

GT(i) =()i (37)

Thus the ansatz Eq. (33) describes a spin liquid whichtranslation, rotation, parity and time reversal symme-tries.

Using the time reversal symmetry we can show thatthe vanishing al0 in our ansatz Eq. (33) indeed satisfy theconstraint Eq. (27). This is because al0 al0 under thetime reversal transformation. Thus Emean

al0

= 0 when

al0 = 0 for any time reversal symmetric ansatz, includingthe ansatz Eq. (33).

The spinon spectrum is given by (see Fig. 5a)

E = 2(sin(kx)+sin(ky))2||

2cos2(kx) + 2 cos2(ky)

(38)The spinons have two Fermi points and two small Fermipockets (for small ). The SU(2) flux is non-trivial. Fur-ther more PC1 and PC2 do not commute. Thus the SU(2)gauge structure is broken down to a Z2 gauge structureby the SU(2) flux PC1 and PC2 .[38, 40] We will call thespin liquid described by Eq. (33) Z2-gapless spin liquid.The low energy effective theory is described by mass-less Dirac fermions and fermions with small Fermi sur-faces, coupled to a Z2 gauge field. Since the Z2 gaugeinteraction is irrelevant at low energies, the spinons are


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11

free fermions at low energies and we have a true spin-charge separation in the Z2-gapless spin liquid. The Z2-gapless spin liquid is one of the Z2 spin liquids classifiedin Ref. [66]. Its projective symmetry group is labeled

by Z2A13 13+

30 or equivalently by Z2Ax2(12)n (seesection IV B and Eq. (85)).

Now let us include longer links. First we still limitourselves to unfrustrated ansatz. An interesting ansatz

is given by

al0 = 0

ux = 3 + 1

uy = 3 1

u2x+y = 2

ux+2y = 2u2xy =

2

ux+2y = 2 (39)By definition, the ansatz is invariant under translationand parity x

x. After a 90 rotation, the ansatz is

changed to

ux = 3 1uy = 3 + 1

u2x+y = 2ux+2y = +

2

u2xy = 2ux+2y = +

2 (40)

which is gauge equivalent to Eq. (39) under the gaugetransformation GR90(i) = i3. Thus the ansatz describea spin liquid with translation, rotation, parity and the

time reversal symmetries. The spinon spectrum is givenby (see Fig. 1c)

E =

1(k)2 + 2(k)2 + 3(k)2

1 = 2(cos(kx) + cos(ky))2 = 2(cos(kx) cos(ky))3 = 2[cos(2kx + ky) + cos(2kx ky)

cos(kx 2ky) cos(kx + 2ky)] (41)Thus the spinons are gapless only at four k points(/2, /2). We also find that PC3 and PC4 do notcommute, where the loops C3 = i i + x i + 2x i + 2x + y

i and C4 = i

i + y

i + 2y

i+ 2y x i. Thus the SU(2) flux PC3 and PC4 breakthe SU(2) gauge structure down to a Z2 gauge struc-ture. The spin liquid described by Eq. (39) will be calledthe Z2-linear spin liquid. The low energy effective theoryis described by massless Dirac fermions coupled to a Z2gauge field. Again the Z2 coupling is irrelevant and thespinons are free fermions at low energies. We have a truespin-charge separation. According to the classificationscheme summarized in section IV B, the above Z2-linearspin liquid is labeled by Z2A003n.

Next let us discuss frustrated ansatz. A simple Z2spin liquid can be obtained from the following frustratedansatz

a30 =0, a1,20 = 0ux =

3 + 1

uy =3 1

ux+y =3

ux+y =3 (42)

The ansatz has translation, rotation, parity, and the timereversal symmetries. When a30 = 0, = and = 0,al0

l does not commute with the loop operators. Thus theansatz breaks the SU(2) gauge structure to a Z2 gaugestructure. The spinon spectrum is given by (see Fig. 1a)

E =

2(k) + 2(k)

(k) =2(cos(kx) + cos(ky)) + a30

2(cos(kx + ky) + cos(kx ky))

(k) =2(cos(kx) cos(ky)) + a3

0 (43)which is gapless only at four k points with a linear dis-persion. Thus the spin liquid described by Eq. (42) is aZ2-linear spin liquid, which has a true spin-charge separa-tion. The Z2-linear spin liquid is described by the projec-tive symmetry group Z2A0032 or equivalently Z2A0013.(see section IV B.) From the above two examples of Z2-linear spin liquids, we find that it is possible to obtaintrue spin-charge separation with massless Dirac points(or nodes) within a pure spin model without the chargefluctuations. We also find that there are more than oneway to do it.

A well known frustrated ansatz is the ansatz for the

chiral spin liquid[6]

ux = 3 1uy = 3 + 1

ux+y = 2

ux+y = 2al0 = 0 (44)

The chiral spin liquid breaks the time reversal and paritysymmetries. The SU(2) gauge structure is unbroken.[38]The low energy effective theory is an SU(2) Chern-Simons theory (of level 1). The spinons are gaped and

have a semionic statistics.[5, 6] The third interesting frus-trated ansatz is given in Ref. [38, 40]

ux =uy = 3ux+y =

1 + 2

ux+y =1 2

a2,30 =0, a10 = 0 (45)

This ansatz has translation, rotation, parity and the timereversal symmetries. The spinons are fully gaped and


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12

the SU(2) gauge structure is broken down to Z2 gaugestructure. We may call such a state Z2-gapped spin liquid(it was called sRVB state in Ref. [38, 40]). It is describedby the projective symmetry group Z2Axx0z. Both thechiral spin liquid and the Z2-gapped spin liquid have truespin-charge separation.

IV. QUANTUM ORDERS IN SYMMETRICSPIN LIQUIDS

A. Quantum orders and projective symmetrygroups

We have seen that there can be many different spinliquids with the same symmetries. The stability analysisin section VIII shows that many of those spin liquids oc-cupy a finite region in phase space and represent stablequantum phases. So here we are facing a similar situa-tion as in quantum Hall effect: there are many distinctquantum phases not separated by symmetries and order

parameters. The quantum Hall liquids have finite energygaps and are rigid states. The concept of topologicalorder was introduced to describe the internal order ofthose rigid states. Here we can also use the topologicalorder to describe the internal orders of rigid spin liquids.However, we also have many other stable quantum spinliquids that have gapless excitations.

To describe internal orders in gapless quantum spinliquids (as well as gapped spin liquids), we have intro-duced a new concept quantum order that describesthe internal orders in any quantum phases. The key pointin introducing quantum orders is that quantum phases,in general, cannot be completely characterized by bro-

ken symmetries and local order parameters. This pointis illustrated by quantum Hall states and by the stablespin liquids constructed in this paper. However, to makethe concept of quantum order useful, we need to findconcrete mathematical characterizations the quantum or-ders. Since quantum orders are not described by symme-tries and order parameters, we need to find a completelynew way to characterize them. Here we would like topropose to use Projective Symmetry Group to character-ize quantum (or topological) orders in quantum spin liq-uids. The projective symmetry group is motivated fromthe following observation. Although ansatz for differ-ent symmetric spin liquids all have the same symmetry,the ansatz are invariant under symmetry transformationsfollowed by different gauge transformations. We can usethose different gauge transformations to distinguish dif-ferent spin liquids with the same symmetry. In the fol-lowing, we will introduce projective symmetry group ina general and formal setting.

We know that to find quantum numbers that charac-terize a phase is to find the universal properties of thephase. For classical systems, we know that symmetryis a universal property of a phase and we can use sym-metry to characterize different classical phases. To find

universal properties of quantum phases we need to finduniversal properties of many-body wave functions. Thisis too hard. Here we want to simplify the problem by lim-iting ourselves to a subclass of many-body wave functionswhich can be described by ansatz (uij, a

l0

l) via Eq. (14).Instead of looking for the universal properties of many-body wave functions, we try to find the universal prop-erties of ansatz (uij , a

l0

l). Certainly, one may object

that the universal properties of the ansatz (or the sub-class of wave functions) may not be the universal prop-erties of spin quantum phase. This is indeed the case forsome ansatz. However, if the mean-field state describedby ansatz (uij , a

l0

l) is stable against fluctuations (ie thefluctuations around the mean-field state do not cause anyinfrared divergence), then the mean-field state faithfullydescribes a spin quantum state and the universal proper-ties of the ansatz will be the universal properties of thecorrespond spin quantum phase. This completes the linkbetween the properties of ansatz and properties of phys-ical spin liquids. Motivated by the Landaus theory forclassical orders, here we whould like to propose that theinvariance group (or the symmetry group) of an ansatzis a universal property of the ansatz. Such a group willbe called the projective symmetry group (PSG). We willshow that PSG can be used to characterize quantum or-ders in quantum spin liquids.

Let us give a detailed definition of PSG. A PSG is aproperty of an ansatz. It is formed by all the transfor-mations that keep the ansatz unchanged. Each trans-formation (or each element in the PSG) can be writtenas a combination of a symmetry transformation U (suchas translation) and a gauge transformation GU. The in-variance of the ansatz under its PSG can be expressedas

GUU(uij) =uijU(uij) uU(i),U(j)

GU(uij) GU(i)uijGU(j)GU(i) SU(2) (46)

for each GUU P SG.Every PSG contains a special subgroup, which will be

called invariant gauge group (IGG). IGG (denoted by G)for an ansatz is formed by all the gauge transformationsthat leave the ansatz unchanged:

G = {Wi|WiuijWj = uij, Wi SU(2)} (47)

If we want to relate IGG to a symmetry transformation,then the associated transformation is simply an identitytransformation.

If IGG is non-trivial, then for a fixed symmetry trans-formation U, there are can be many gauge transforma-tions GU that leave the ansatz unchanged. If GUU is inthe PSG ofuij, GGUU will also be in the PSG iff G G.Thus for each symmetry transformation U, the differentchoices ofGU have a one to one correspondence with theelements in IGG. From the above definition, we see that


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13

the PSG, the IGG, and the symmetry group (SG) of anansatz are related:

SG = PSG/IGG (48)

This relation tells us that a PSG is a projective repre-sentation or an extension of the symmetry group.[96] (Inappendix A we will introduce a closely related but differ-

ent definition of PSG. To distinguish the two definitions,we will call the PSG defined above invariant PSG andthe PSG defined in appendix A algebraic PSG.)

Certainly the PSGs for two gauge equivalent ansatzuij and W(i)uijW

(j) are related. From W GUU(uij) =W(uij), where W(uij) W(i)uijW(j), we findW GUU W

1W(uij) = W GUW1U U W(uij) = W(uij),

where WU U W U1 is given by WU(i) = W(U(i)).Thus if GUU is in the PSG of ansatz uij , then(W GUWU)U is in the PSG of gauge transformed ansatzW(i)uijW

(j). We see that the gauge transformationGU associated with the symmetry transformation U ischanged in the following way

GU(i) W(i)GU(i)W(U(i)) (49)

after a gauge transformation W(i).Since PSG is a property of an ansatz, we can group

all the ansatz sharing the same PSG together to form aclass. We claim that such a class is formed by one orseveral universality classes that correspond to quantumphases. (A more detailed discussion of this importantpoint is given in section VIII E.) It is in this sense we saythat quantum orders are characterized by PSGs.

We know that a classical order can be described byits symmetry properties. Mathematically, we say that aclassical order is characterized by its symmetry group.Using projective symmetry group to describe a quantumorder, conceptually, is similar to using symmetry groupto describe a classical order. The symmetry descriptionof a classical order is very useful since it allows us toobtain many universal properties, such as the number ofNambu-Goldstone modes, without knowing the details ofthe system. Similarly, knowing the PSG of a quantumorder also allows us to obtain low energy properties ofa quantum system without knowing its details. As anexample, we will discuss a particular kind of the low en-ergy fluctuations the gauge fluctuations in a quantumstate. We will show that the low energy gauge fluctua-tions can be determined completely from the PSG. In fact

the gauge group of the low energy gauge fluctuations isnothing but the IGG of the ansatz.

To see this, let us assume that, as an example, anIGG G contains a U(1) subgroup which is formed by thefollowing constant gauge transformations

{Wi = ei3 | [0, 2)} G (50)

Now we consider the following type of fluctuations

around the mean-field solution uij : uij = uijeia3ij

3

.

Since uij is invariant under the constant gauge trans-

formation ei3

, a spatial dependent gauge transforma-

tion eii3

will transform the fluctuation a3ij to a3ij =

a3ij + ij . This means that a3ij and a3ij label the samephysical state and a3ij correspond to gauge fluctuations.The energy of the fluctuations has a gauge invarianceE({a3ij}) = E({a3ij}). We see that the mass term of the

gauge field, (a

3

ij)

2

, is not allowed and the U(1) gaugefluctuations described by a3ij will appear at low energies.

If the U(1) subgroup of G is formed by spatial depen-dent gauge transformations

{Wi = eini| [0, 2), |ni| = 1} G, (51)we can always use a SU(2) gauge transformation to ro-tate ni to the z direction on every site and reduce theproblem to the one discussed above. Thus, regardlessif the gauge transformations in IGG have spatial depen-dence or not, the gauge group for low energy gauge fluc-tuations is always given by G.

We would like to remark that some times low energy

gauge fluctuations not only appear near k = 0, butalso appear near some other k points. In this case, wewill have several low energy gauge fields, one for each kpoints. Examples of this phenomenon are given by someansatz of SU(2) slave-boson theory discussed in sectionVI, which have an SU(2) SU(2) gauge structures atlow energies. We see that the low energy gauge struc-ture SU(2) SU(2) can even be larger than the high en-ergy gauge structure SU(2). Even for this complicatedcase where low energy gauge fluctuations appear arounddifferent k points, IGG still correctly describes the lowenergy gauge structure of the corresponding ansatz. IfIGG contains gauge transformations that are indepen-

dent of spatial coordinates, then such transformationscorrespond to the gauge group for gapless gauge fluctua-tions near k = 0. If IGG contains gauge transformationsthat depend on spatial coordinates, then those transfor-mations correspond to the gauge group for gapless gaugefluctuations near non-zero k. Thus IGG gives us a unifiedtreatment of all low energy gauge fluctuations, regardlesstheir momenta.

In this paper, we have used the terms Z2 spin liquids,U(1) spin liquids, SU(2) spin liquids, and SU(2)SU(2)spin liquids in many places. Now we can have a pre-cise definition of those low energy Z2, U(1), SU(2), andSU(2) SU(2) gauge groups. Those low energy gaugegroups are nothing but the IGG of the correspondingansatz. They have nothing to do with the high energygauge groups that appear in the SU(2), U(1), or Z2 slave-boson approaches. We also used the terms Z2 gaugestructure, U(1) gauge structure, and SU(2) gauge struc-ture of a mean-field state. Their precise mathematicalmeaning is again the IGG of the corresponding ansatz.When we say a U(1) gauge structure is broken down toa Z2 gauge structure, we mean that an ansatz is changedin such a way that its IGG is changed from U(1) to Z2group.


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B. Classification of symmetric Z2 spin liquids

As an application of PSG characterization of quantumorders in spin liquids, we would like to classify the PSGsassociated with translation transformations assuming theIGG G = Z2. Such a classification leads to a classificationof translation symmetric Z2 spin liquids.

When

G= Z2, it contains two elements gauge trans-

formations G1 and G2:

G ={G1, G2}G1(i) =

0, G2(i) = 0. (52)Let us assume that a Z2 spin liquid has a translation sym-metry. The PSG associated with the translation groupis generated by four elements GxTx, GyTx where

Tx(uij) = uix,jx, Ty(uij) = uiy,jy. (53)

Due to the translation symmetry of the ansatz, we canchoose a gauge in which all the loop operators of theansatz are translation invariant. That is PC1 = PC2 if

the two loops C1 and C2 are related by a translation.We will call such a gauge uniform gauge.

Under transformation GxTx, a loop operator PCbased at i transforms as PC Gx(i)PTxCGx(i) =Gx(i

)PCGx(i) where i = Txi is the base point of the

translated loop Tx(C). We see that translation invarianceof PC in the uniform gauge requires

Gx(i) = 0, Gy(i) = 0. (54)since different loop operators based at the same basepoint do not commute for Z2 spin liquids. We note thatthe gauge transformations of form W(i) = 0 do notchange the translation invariant property of the loop op-

erators. Thus we can use such gauge transformationsto further simplify Gx,y through Eq. (49). First we canchoose a gauge to make

Gy(i) = 0. (55)

We note that a gauge transformation satisfying W(i) =W(ix) does not change the condition Gy(i) =

0. Wecan use such kind of gauge transformations to make

Gx(ix, iy = 0) = 0. (56)

Since the translations in x- and y-direction commute,Gx,y must satisfy (for any ansatz, Z2 or not Z2)

GxTxGyTy(GxTx)1(GyTy)

1 =

GxTxGyTyT1x G

1x T

1y G

1y G. (57)

That means

Gx(i)Gy(i x)G1x (i y)Gy(i)1 G (58)For Z2 spin liquids, Eq. (58) reduces to

Gx(i)G1x (i y) = +0 (59)

or

Gx(i)G1x (i y) = 0 (60)

When combined with Eq. (55) and Eq. (56), we find thatthere are only two gauge inequivalent extensions of thetranslation group when IGG is G = Z2. The two PSGsare given by

Gx(i

) =

0

, Gy(i

) =

0

(61)and

Gx(i) =()iy0, Gy(i) =0 (62)Thus, under PSG classification, there are only two typesof Z2 spin liquids if they have only the translation sym-metry and no other symmetries. The ansatz that satisfyEq. (61) have a form

ui,i+m =um (63)

and the ones that satisfy Eq. (62) have a form

ui,i+m =()myixum (64)Through the above example, we see that PSG is a very

powerful tool. It can lead to a complete classification of(mean-field) spin liquids with prescribed symmetries andlow energy gauge structures.

In the above, we have studied Z2 spin liquids whichhave only the translation symmetry and no other sym-metries. We find there are only two types of such spinliquids. However, if spin liquids have more symmetries,then they can have much more types. In Ref. [66], wegive a classification of symmetric Z2 spin liquids us-ing PSG. Here we use the term symmetric spin liquidto refer to a spin liquid with the translation symmetryTx,y, the time reversal symmetry T: uij

uij , and

the three parity symmetries Px: (ix, iy) (ix, iy),Py: (ix, iy) (ix, iy), and Pxy: (ix, iy) (iy, ix).The three parity symmetries also imply the 90 rota-tion symmetry. The classification is obtained by notic-ing that the gauge transformations Gx,y, GPx,Py,Pxyand GT must satisfy certain algebraic relations (see ap-pendix A). Solving those algebraic relations and fac-toring out gauge equivalent solutions,[66], we find thatthere are 272 different extensions of the symmetry group{Tx,y, Px,y,xy, T} if IGG G = Z2. Those PSGs are gen-erated by (GxTx, GyTy, GTT, GPxPx, GPyPy, GPxyPxy).The PSGs can be divided into two classes. The firstclass is given by

Gx(i) =0, Gy(i) =0

GPx(i) =ixxpx

iyxpygPx GPy(i) =

ixxpy

iyxpxgPy

GPxy(i) =gPxy GT(i) =itgT (65)

and the second class by

Gx(i) =()iy0, Gy(i) =0GPx(i) =

ixxpx

iyxpygPx GPy(i) =

ixxpy

iyxpxgPy

GPxy(i) =()ixiygPxy GT(i) =itgT (66)


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15

Here the three s can independently take two values1. gs have 17 different choices which are given by (seeRef. [66])

gPxy =0 gPx =

0 gPy =0 gT =

0; (67)

gPxy =0 gPx =i

3 gPy =i3 gT =

0; (68)

gPxy =i3

gPx =0

gPy =0

gT =0

; (69)gPxy =i

3 gPx =i3 gPy =i

3 gT =0; (70)

gPxy =i3 gPx =i

1 gPy =i1 gT =

0; (71)

gPxy =0 gPx =

0 gPy =0 gT =i

3; (72)

gPxy =0 gPx =i

3 gPy =i3 gT =i

3; (73)

gPxy =0 gPx =i

1 gPy =i1 gT =i

3; (74)

gPxy =i3 gPx =

0 gPy =0 gT =i

3; (75)

gPxy =i3 gPx =i

3 gPy =i3 gT =i

3; (76)

gPxy =i3 gPx =i1 gPy =i

1 gT =i3; (77)

gPxy =i1 gPx =

0 gPy =0 gT =i

3; (78)

gPxy =i1 gPx =i

3 gPy =i3 gT =i

3; (79)

gPxy =i1 gPx =i

1 gPy =i1 gT =i

3; (80)

gPxy =i1 gPx =i

2 gPy =i2 gT =i

3; (81)

gPxy =i12 gPx =i

1 gPy =i2 gT =i

0; (82)

gPxy =i12 gPx =i

1 gPy =i2 gT =i

3; (83)

where

ab =a + b

2, ab =

a b2

. (84)

Thus there are 2 17 23 = 272 different PSGs. Theycan potentially lead to 272 different types of symmetricZ2 spin liquids on 2D square lattice.

To label the 272 PSGs, we propose the followingscheme:

Z2A(gpx)xpx(gpy)xpygpxy(gt)t , (85)

Z2B(gpx)xpx(gpy)xpygpxy(gt)t . (86)

The label Z2A... correspond to the case Eq. (65), andthe label Z2B... correspond to the case Eq. (66). A typi-cal label will looks like Z2A1+

2

123. We will also usean abbreviated notation. An abbreviated notation is ob-tained by replacing (0, 1, 2, 3) or (0+,

1+,

2+,

3+) by

(0, 1, 2, 3) and (0, 1,

2,

3) by (n,x,y,z). For exam-

ple, Z2A1+0

123 can be abbreviated as Z2A1n(12)z.Those 272 different Z2 PSGs, strictly speaking, are the

so called algebraic PSGs. The algebraic PSGs are de-fined as extensions of the symmetry group. They can be

calculated through the algebraic relations listed in ap-pendix A. The algebraic PSGs are different from theinvariant PSGs which are defined as a collection of alltransformations that leave an ansatz uij invariant. Al-though an invariant PSG must be an algebraic PSG, analgebraic PSG may not be an invariant PSG. This is be-cause certain algebraic PSGs have the following proper-ties: any ansatz uij that is invariant under an algebraic

PSG may actually be invariant under a larger PSG. Inthis case the original algebraic PSG cannot be an invari-ant PSG of the ansatz. The invariant PSG of the ansatzis really given by the larger PSG. If we limit ourselvesto the spin liquids constructed through the ansatz uij,then we should drop the algebraic PSGs are not invari-ant PSGs. This is because those algebraic PSGs do notcharacterize mean-field spin liquids.

We find that among the 272 algebraic Z2 PSGs, atleast 76 of them are not invariant PSGs. Thus the 272 al-gebraic Z2 PSGs can at most lead to 196 possible Z2 spinliquids. Since some of the mean-field spin liquid statesmay not survive the quantum fluctuations, the number ofphysical Z

2spin liquids is even smaller. However, for the

physical spin liquids that can be obtained through themean-field states, the PSGs do offer a characterizationof the quantum orders in those spin liquids.

C. Classification of symmetric U(1) and SU(2) spinliquids

In addition to the Z2 symmetric spin liquids studiedabove, there can be symmetric spin liquids whose lowenergy gauge structure is U(1) or SU(2). Such U(1) andSU(2) symmetric spin liquids (at mean-field level) areclassified by U(1) and SU(2) symmetric PSGs. The U(1)

and SU(2) symmetric PSGs are calculated in Ref. [66].In the following we just summarize the results.

We find that the PSGs that characterize mean-fieldsymmetric U(1) spin liquids can be divided into fourtypes: U1A, U1B, U1C and U1mn . There are 24 typeU1A PSGs:

Gx =g3(x), Gy = g3(y),

GPx =iyypxg3(px), GPy =

ixypxg3(py)

GPxy =g3(pxy), g3(pxy)i1

GT =itg3(t)|t=1, itg3(t)i1 (87)

and

Gx =g3(x), Gy = g3(y),

GPx =ixxpxg3(px)i

1, GPy = iyxpxg3(py)i

1

GPxy =g3(pxy), g3(pxy)i1

GT =itg3(t)|t=1, itg3(t)i1 (88)

where

ga() eia . (89)


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16

We will use U1Aaxpxbypxcdt to label the 24 PSGs.a, b , c , d are associated with GPx , GPy , GPxy , GT re-

spectively. They are equal to 1 if the corresponding Gcontains a 1 and equal to 0 otherwise. A typical nota-tion looks like U1A1

101 which can be abbreviatedas U1Ax10x.

There are also 24 type U1B PSGs:

Gx =()iyg3(x), Gy = g3(y),GPx =

iyypxg3(px), GPy =

ixypxg3(py)

()ixiyGPxy =g3(pxy), g3(pxy)i1GT =

itg3(t)|t=1, itg3(t)i1 (90)

and

Gx =()iyg3(x), Gy = g3(y),GPx =

ixxpxg3(px)i

1, GPy = iyxpxg3(py)i

1

()ixiyGPxy =g3(pxy), g3(pxy)i1GT =

itg3(t)

|t=1,

itg3(t)i

1 (91)

We will use U1Baxpxbypxcdt to label the 24 PSGs.The 60 type U1C PSGs are given by

Gx =g3(x)i1, Gy = g3(y)i

1,

GPx =ixxpx

iyypxg3(px), GPy =

ixypx

iyxpxg3(py)

GPxy =ixpxyg3(

ipxy

4+ pxy),

GT =itg3(t)|t=1, ixpxyg3(t)i1 (92)

Gx =g3(x)i1, Gy = g3(y)i1,

GPx =ixxpxg3(px)i

1, GPy = iyxpx

ipxyg3(py)i

1

GPxy =ixpxyg3(

ipxy

4+ pxy),

GT =itg3(t)|t=1, ixpxyitg3(t)i1 (93)


1,

GPx =ixxpx

iyypxg3(px), GPy =

ixypx

iyxpxg3(py)

GPxy =g3(pxy)i1

GT =itg3(t)|t=1 (94)


1,

GPx =ixxpx

iyypxg3(px), GPy =

ixypx

iyxpxg3(py)

GPxy =g3(ipxy

4+ pxy)i

1

GT =ixpxy

itg3(t)i

1 (95)


1,

GPx =ixxpxg3(px)i

1, GPy = iyxpx

ipxyg3(py)i

1

GPxy =g3(ipxy

4+ pxy)i

1

GT =itg3(t)|t=1, itixpxyg3(t)i1 (96)

which will be labeled by U1Caxpxbypxcpxydt .The type U1mn PSGs have not been classified. How-

ever, we do know that for each rational number m/n (0, 1), there exist at least one mean-field symmetric spinliquid, which is described by the ansatz

ui,i+x = 3, ui,i+y = g3(

m

nix)

3 (97)

It has m/n flux per plaquette. Thus there are infinitemany type U1mn spin liquids.

We would like to point out that the above 108U1[A,B,C] PSGs are algebraic PSGs. They are only asubset of all possible algebraic U(1) PSGs. However,they do contain all the invariant U(1) PSGs of typeU1A, U1B and U1C. We find 46 of the 108 PSGs arealso invariant PSGs. Thus there are 46 different mean-field U(1) spin liquids of type U1A, U1B and U1C. Theiransatz and labels are given in Ref. [66].

To classify symmetric SU(2) spin liquids, we find 8different SU(2) PSGs which are given by

Gx(i) =gx, Gy(i) = gy

GPx(i) =ixxpx

iyxpygPx , GPy(i) =

ixxpy

iyxpxgPy

GPxy(i) =gPxy , GT(i) = ()igT (98)and

Gx(i) =()iy

gx, Gy(i) = gyGPx(i) =

ixxpx

iyxpygPx , GPy(i) =

ixxpy

iyxpxgPy

GPxy(i) =()ixiygPxy , GT(i) = ()igT (99)where gs are in SU(2). We would like to use the followingtwo notations

SU2A0xpx0xpy

SU2B0xpx0xpy (100)

to denote the above 8 PSGs. SU2A0xpx0xpy is for

Eq. (98) and SU2B0xpx0xpy for Eq. (99). We find only

4 of the 8 SU(2) PSGs, SU2A[n0, 0n] and SU2B[n0, 0n],

leads to SU(2) symmetric spin liquids. The SU2An0state is the uniform RVB state and the SU2Bn0 stateis the -flux state. The other two SU(2) spin liquids aregiven by SU2A0n:

ui,i+2x+y = + i0

ui,i2x+y = i0ui,i+x+2y = + i

0

ui,ix+2y = + i0 (101)


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17

and SU2B0n:

ui,i+2x+y = + i()ix0ui,i2x+y = i()ix0ui,i+x+2y = + i

0

ui,ix+2y = + i0 (102)

The above results give us a classification of symmet-ric U(1) and SU(2) spin liquids at mean-field level. Ifa mean-field state is stable against fluctuations, it willcorrespond to a physical U(1) or SU(2) symmetric spinliquids. In this way the U(1) and the SU(2) PSGs alsoprovide an description of some physical spin liquids.

V. CONTINUOUS TRANSITIONS ANDSPINON SPECTRA IN SYMMETRIC SPIN

LIQUIDS

A. Continuous phase transitions without symmetrybreaking

After classifying mean-field symmetric spin liquids, wewould like to know how those symmetric spin liquidsare related to each other. In particular, we would liketo know which spin liquids can change into each otherthrough a continuous phase transition. This problem isstudied in detail in Ref. [66], where the symmetric spinliquids in the neighborhood of some important symmet-ric spin liquids were obtained. After lengthy calculations,we found all the mean-field symmetric spin liquids aroundthe Z2-linear state Z2A001n in Eq. (39), the U(1)-linearstate U1Cn01n in Eq. (32), the SU(2)-gapless stateSU2An0 in Eq. (30), and the SU(2)-linear state SU2Bn0in Eq. (31). We find that, at the mean-field level, theU(1)-linear spin liquid U1Cn01n can continuously changeinto 8 different Z2 spin liquids, the SU(2)-gapless spinliquid SU2An0 can continuously change into 12 U(1) spinliquids and 52 Z2 spin liquids, and the SU(2)-linear spinliquid SU2Bn0 can continuously change into 12 U(1) spinliquids and 58 Z2 spin liquids.

We would like to stress that the above results on thecontinuous transitions are valid only at mean-field level.Some of the mean-field results survive the quantum fluc-tuations while others do not. One need to do a case by

case study to see which mean-field results can be validbeyond the mean-field theory. In Ref. [40], a mean-fieldtransition between a SU(2) SU(2)-linear spin liquidand a Z2-gapped spin liquid was studied. In particularthe effects of quantum fluctuations were discussed.

We would also like to point out that all the abovespin liquids have the same symmetry. Thus the contin-uous transitions between them, if exist, represent a newclass of continuous transitions which do not change anysymmetries.[67]

B. Symmetric spin liquids around the U(1)-linearspin liquid U1Cn01n

The SU(2)-linear state SU2Bn0 (the -flux state), theU(1)-linear state U1Cn01n (the staggered-flux/d-wavestate), and the SU(2)-gapless state SU2An0 (the uniformRVB state), are closely related to high Tc superconduc-tors. They reproduce the observed electron spectra func-

tion for undoped, underdoped, and overdoped samplesrespectively. However, theoretically, those spin liquidsare unstable at low energies due to the U(1) or SU(2)gauge fluctuations. Those states may change into morestable spin liquids in their neighborhood. In the next afew subsections, we are going to study those more sta-ble spin liquids. Since there are still many different spinliquids involved, we will only present some simplified re-sults by limiting the length of non-zero links. Those spinliquids with short links should be more stable for simplespin Hamiltonians. The length of a link between i and jis defined as |ix jx| + |iy jy|. By studying the spinondispersion in those mean-field states, we can understand

some basic physical properties of those spin liquids, suchas their stability against the gauge fluctuations and thequalitative behaviors of spin correlations which can bemeasured by neutron scattering. Those results allow usto identify them, if those spin liquids exist in certain sam-ples or appear in numerical calculations. We would liketo point out that we will only study symmetric spin liq-uids here. The above three unstable spin liquids may alsochange into some other states that break certain symme-tries. Such symmetry breaking transitions actually havebeen observed in high Tc superconductors (such as thetransitions to antiferromagnetic state, d-wave supercon-ducting state, and stripe state).

First, let us consider the spin liquids around theU(1)-linear state U1Cn01n. In the neighborhood of theU1Cn01n ansatz Eq. (32), there are 8 different spin liq-uids that break the U(1) gauge structure down to a Z2gauge structure. Those 8 spin liquids are labeled by dif-ferent PSGs despite they all have the same symmetry.In the following, we will study those 8 Z2 spin liquids inmore detail. In particular, we would like to find out thespinon spectra in them.

The first one is labeled by Z2A0013 and takes the fol-lowing form

ui,i+x =1 2

ui,i+y =1 + 2

ui,i+x+y = + 11

ui,ix+y = + 11

ui,i+2x =21 + 2

2

ui,i+2y =21 22

a10 =0, a2,30 = 0 (103)

It has the same quantum order as that in the ansatzEq. (42). The label Z2A0013 tells us the PSG that char-


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18

acterizes the spin liquid. The second ansatz is labeled byZ2Azz13:

ui,i+x =1 2

ui,i+y =1 + 2

ui,i+x+y = 11ui,ix+y = + 1

1

ui,i+2x =ui,i+2y = 0a1,2,30 = 0 (104)

The third one is labeled by Z2A001n (or equivalentlyZ2A003n):

al0 =0

ui,i+x =1 + 2

ui,i+y =1 2

ui,i+2x+y =3

ui,ix+2y = 3ui,i+2

xy =

3

ui,i+x+2y = 3 (105)Such a spin liquid has the same quantum order asEq. (39). The fourth one is labeled by Z2Azz1n:

al0 =0

ui,i+x =1 + 2

ui,i+y =1 2

ui,i+2x+y =11 + 1

2 + 3

ui,ix+2y =11 12 + 3

ui,i+2xy =11 + 1

2

3

ui,i+x+2y =11 12 3 (106)The above four ansatz have translation invariance. The

next four Z2 ansatz do not have translation invariance.(But they still describe translation symmetric spin liquidsafter the projection.) Those Z2 spin liquids areZ2B0013:

ui,i+x =1 2

ui,i+y =()ix(1 + 2)ui,i+2x = 21 + 22ui,i+2y = 21 22

a10 =0, a2,30 = 0, (107)Z2Bzz13:

ui,i+x =1 2

ui,i+y =()ix(1 + 2)ui,i+2x+2y = 11ui,i2x+2y =1

1

a1,2,30 = 0, (108)

(a) (b)

(c) (d)

FIG. 1: Contour plot of the spinon dispersion E+(k ) as a

function of (kx/2, ky/2) for the Z2-linear spin liquids. (a)is for the Z2A0013 state in Eq. (103), (b) for the Z2Azz13state in Eq. (104), (c) for the Z2A001n state in Eq. (105), (d)for the Z2Azz1n state in Eq. (106).

Z2B001n:

u i,i+x =1 + 2

u i,i+y =()ix(1 2)u i,i+2x+y =()ix3u i,ix+2y = 3

u i,i+2xy =()ix3u i,i+x+2y = 3

al0 =0, (109)

and Z2Bzz1n:

ux =1 + 2

uy =()ix(1 2)u 2x+y =()ix(11 + 12 + 3)

u x+2y =11 12 + 3

u2xy

=(

)ix(1

1 + 1

2

3)

u x+2y =11 12 3

al0 =0. (110)

The spinons are gapless at four isolated points witha linear dispersion for the first four Z2 spin liquidsEq. (103), Eq. (104), Eq. (105), and Eq. (106). (SeeFig. 1) Therefore the four ansatz describe symmetricZ2-linear spin liquids. The single spinon dispersion forthe second Z2 spin liquid Z2Azz13 is quite interesting. It


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19

(a) (b)

(c) (d)

FIG. 2: Contour plot of the spinon disp ersion

min(E1(k ), E2( k )) as a function of (kx/2, ky/2) forthe Z2-linear states. (a) is for the Z2B0013 state inEq. (107), (b) for the Z2Bzz13 state in Eq. (108), (c) for theZ2B001n state in Eq. (109), (d) for the Z2Bzz1n state inEq. (110).

has the 90 rotation symmetry around k = (0, ) and theparity symmetry about k = (0, 0). One very importantthing to notice is that the spinon dispersions for the fourZ2-linear spin liquids, Eq. (103), Eq. (104), Eq. (105),and Eq. (106) have some qualitative differences betweenthem. Those differences can be used to physically mea-sure quantum orders (see section VII).

Next let us consider the ansatz Z2B0013 in Eq. (107).The spinon spectrum for ansatz Eq. (107) is determinedby

H = 2 cos(kx)0 2 cos(kx)2 2 cos(ky)1 + 2 cos(ky)3 + 4 (111)

where kx (0, ), ky (, ) and0 =

1 3, 1 =1 1,2 =

2 3, 3 =2 1,4 =

1 0. (112)assuming 1,2 = 2 = 0. The four bands of spinondispersion have a form E1(k), E2(k). We find thespinon spectrum vanishes at 8 isolated points near k =(/2, /2). (See Fig. 2a.) Thus the state Z2B0013 is aZ2-linear spin liquid.

Knowing the translation symmetry of the above Z2-linear spin liquid, it seems strange to find that the spinonspectrum is defined only on half of the lattice Brillouinzone. However, this is not inconsistent with translationsymmetry since the single spinon excitation is not phys-ical. Only two-spinon excitations correspond to physical

excitations and their spectrum should be defined on thefull Brillouin zone. Now the problem is that how to ob-tain two-spinon spectrum defined on the full Brillouinzone from the single-spinon spectrum defined on half ofthe Brillouin zone. Let |k, 1 and |k, 2 be the two eigen-states of single spinon with positive energies E1(k) andE2(k) (here kx (/2, /2) and ky (, )). Thetranslation by x (followed by a gauge transformation)

change |k, 1 and |k, 2 to the other two eigenstates withthe same energies:|k, 1 |k + y, 1|k, 2 |k + y, 2 (113)

Now we see that momentum and the energy of two-spinonstates |k1, 1|k2, 2 |k1 + y, 1|k2 + y, 2 aregiven by

E2spinon =E1(k1) + E2(k2)

k =k1 + k2, k1 + k2 + x (114)

Eq. (114) allows us to construct two-spinon spectrumfrom single-spinon spectrum.

Now let us consider the ansatz Z2Bzz13 in Eq. (108).The spinon spectrum for ansatz Eq. (108) is determinedby

H = 2 cos(kx)0 2 cos(kx)2 2 cos(ky)1 + 2 cos(ky)3 (115) 21 cos(2kx + 2ky)4 + 21 cos(2kx 2ky)4

where kx (0, ), ky (, ) and

0 =1 3, 1 =1 1,

2

=2

3,

3=2

1,

4 =1 0. (116)

We find the spinon spectrum to vanish at 2 isolated pointsk = (/2, /2). (See Fig. 2b.) The state Z2Bzz13 is aZ2-linear spin liquid.

The spinon spectrum for the ansatz Z2B001n inEq. (109) is determined by

H = 2 cos(kx)0 2 cos(kx)2 2 cos(ky)1 + 2 cos(ky)3+ 2(cos(kx + 2ky) + cos(kx + 2ky))4

2(cos(2kx + ky) + cos(2kx

ky))5 (117)

where kx (0, ), ky (, ) and

0 =1 3, 1 =1 1,

2 =2 3, 3 =2 1,

4 =3 3, 5 =3 1. (118)

The spinon spectrum vanishes at 2 isolated points k =(/2, /2). (See F

xiao-gang wen- quantum orders and symmetric spin liquids

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