Quantum Orders and Symmetric Spin Liquids

59 Pages·1.5 MB·

by  Xiao-Gang Wen

Quantum Orders and Symmetric Spin Liquids
(the original version)
Xiao-Gang Wen∗
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
(Dated: June3, 2001)
A concept – quantum order – is introduced to describe a new kind of orders that generally
2 appear in quantum states at zero temperature. Quantum orders that characterize universality
0 classes of quantum states (described by complex ground state wave-functions) is much richer then
0 classicalordersthatcharacterizeuniversalityclassesoffinitetemperatureclassicalstates(described
2 bypositiveprobabilitydistributionfunctions). TheLandau’stheoryforordersandphasetransitions
does not apply to quantum orders since they cannot be described by broken symmetries and the
n
associated orderparameters. Weintroduced amathematical object – projective symmetry group –
a
tocharacterizequantumorders. Withthehelpofquantumordersandprojectivesymmetrygroups,
J
we construct hundreds of symmetric spin liquids, which have SU(2), U(1) or Z2 gauge structures
2
at low energies. We found that various spin liquids can be divided into four classes: (a) Rigid

spin liquid – spinons (and all other excitations) are fully gaped and may have bosonic, fermionic,
]
l or fractional statistics. (b) Fermi spin liquid – spinons are gapless and are described by a Fermi
e
liquid theory. (c) Algebraic spin liquid – spinons are gapless, but they are not described by free
-
r fermionic/bosonic quasiparticles. (d) Bose spin liquid – low lying gapless excitations are described
t byafreebosontheory. Thestabilityofthosespinliquidsarediscussedindetails. Wefindthatstable
s
. 2Dspinliquidsexistin thefirstthreeclasses (a–c). Those stablespinliquidsoccupyafiniteregion
t
a in phase space and represent quantum phases. Remarkably, some of the stable quantum phases
m support gapless excitations even without any spontaneous symmetry breaking. In particular, the
gapless excitations in algebraic spin liquids interact down to zero energy and the interaction does
-
d not open any energy gap. We propose that it is the quantum orders (instead of symmetries) that
n protect thegapless excitationsand makealgebraic spin liquidsand Fermispin liquidsstable. Since
o highTc superconductorsarelikelytobedescribedbyagaplessspinliquid,thequantumordersand
c theirprojectivesymmetrygroupdescriptionslaythefoundationforspinliquidapproachtohighTc
[ superconductors.

2
PACSnumbers: 73.43.Nq,74.25.-q,11.15.Ex
v
1
7
Contents C. Classification of symmetric U(1) and SU(2)
0
7 spin liquids 15
0
I. Introduction 2
1 V. Continuous transitions and spinon
A. Topological orders and quantum orders 2
0 spectra in symmetric spin liquids 17
t/ B. Spin-liquid approach to high Tc A. Continuous phase transitions without
a superconductors 3 symmetry breaking 17
m
C. Spin-charge separation in (doped) spin B. Symmetric spin liquids around the
- liquids 5 U(1)-linear spin liquid U1Cn01n 17
d
D. Organization 6 C. Symmetric spin liquids around the
n
o SU(2)-gapless spin liquid SU2An0 20
c II. Projective construction of 2D spin liquids D. Symmetric spin liquids around the
v: – a review of SU(2) slave-boson approach6 SU(2)-linear spin liquid SU2Bn0 22
i
X VI. Mean-field phase diagram of J -J model 23
III. Spin liquids from translationally invariant 1 2
r
a ansatz 9 VII. Physical measurements of quantum
orders 23
IV. Quantum orders in symmetric spin
liquids 12 VIII. Four classes of spin liquids and their
A. Quantum orders and projective symmetry stability 26
groups 12 A. Rigid spin liquid 26
B. Classification of symmetric Z spin liquids 14 B. Bose spin liquid 26
2
C. Fermi spin liquid 26
D. Algebraic spin liquid 27
E. Quantum order and the stability of spin
∗URL:http://dao.mit.edu/~wen liquids 29
2
IX. Relation to previously constructed spin A. Topological orders and quantum orders
liquids 30
Matter can have many different states, such as gas,
X. Summary of the main results 30 liquid, and solid. Understanding states of matter is the
firststepinunderstandingmatter. Physicistsfindmatter
canhavemuchmoredifferentstatesthanjustgas,liquid,
A. Classification of projective symmetry
and solid. Even solids and liquids can appear in many
groups 33
differentformsandstates. Withsomanydifferentstates
1. General conditions on projective symmetry
of matter, a general theory is needed to gain a deeper
groups 33
understanding of states of matter.
2. Classification of Z2 projective symmetry All the states of matter are distinguished by their in-
groups 34 ternal structures or orders. The key step in developing
3. Classification of U(1) projective symmetry the general theory for states of matter is the realization
groups 36 that all the orders are associated with symmetries (or
rather, the breaking of symmetries). Based on the rela-
4. Classification of SU(2) projective symmetry
tionbetweenordersandsymmetries,Landaudevelopeda
groups 43
generaltheory of orders and the transitions between dif-
ferent orders.[1, 2] Landau’s theory is so successful and
B. Symmetric perturbations around one starts to have a feeling that we understand, at in
symmetric spin liquids 44 principle, all kinds of orders that matter can have.
1. Construction of symmetric perturbations 44 However, nature never stops to surprise us. In 1982,
Tsui, Stormer, and Gossard[3] discovered a new kind of
2. Symmetric perturbations around the
state – FractionalQuantum Hall(FQH) liquid.[4] Quan-
Z -linear state Z2A003z 45
2
tumHallliquidshavemanyamazingproperties. Aquan-
3. Symmetric perturbations around the
tum Hall liquid is more “rigid” than a solid (a crystal),
U(1)-linear state U1Cn01n 45
in the sense that a quantum Hall liquid cannot be com-
4. Symmetric perturbations around the pressed. ThusaquantumHallliquidhasafixedandwell-
SU(2)-gapless state SU2An0 47 defined density. When we measure the electron density
5. Symmetric perturbations around the in terms of filling factor ν, we found that all discovered
SU(2)-linear state SU2Bn0 52 quantum Hall states have such densities that the filling
factorsareexactlygivenby some rationalnumbers, such
as ν = 1,1/3,2/3,2/5,.... Knowing that FQH liquids
References 57
exist 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-
torsshouldbe due to those different internal“patterns”.
I. INTRODUCTION However, the hypothesis of internal “patterns” appears
to have one difficulty – FQH states are liquids, and how
Due to its long length, we would like to first outline can liquids have any internal “patterns”?
the structure of the paper so readers can choose to read In 1989, it was realized that the internal orders in
thepartsofinterests. ThesectionXsummarizethemain FQH liquids (as well as the internal orders in chiral spin
results of the paper, which also serves as a guide of the liquids[5, 6]) are different from any other known orders
wholepaper. Theconceptofquantumorderisintroduced and cannot be observed and characterized in any con-
in section IA. A concrete mathematical description of ventional ways.[7, 8] What is really new (and strange)
quantum order is described in section IVA and section about the orders in chiral spin liquids and FQH liquids
IVB. Readers who are interestedin the backgroundand is that they are not associated with any symmetries (or
motivation of quantum orders may choose to read sec- thebreakingofsymmetries),andcannotbe describedby
tion IA. Readers who are familiar with the slave-boson Landau’stheoryusingphysicalorderparameters.[9]This
approachandjustwantaquickintroductiontoquantum kindoforderiscalledtopologicalorder. Topologicalorder
ordersmaychoosetoreadsectionsIVAandIVB. Read- is a new concept and a whole new theory was developed
ers who are not familiar with the slave-boson approach to describe it.[9, 10]
may find the review sections II and III useful. Read- Knowing FQH liquids contain a new kind of order –
ers who do not care about the slave-bosonapproach but topologicalorder, we would like to ask why FQH liquids
are interested in application to high T superconductors are so special. What is missed in Landau’s theory for
c
and experimental measurements of quantum orders may states of matter so that the theory fails to capture the
choosetoreadsectionsIA, IB,VII andFig. 1-Fig. 15, topological order in FQH liquids?
to gain some intuitive picture of spinon dispersion and When we talk about orders in FQH liquids, we are
neutron scattering behavior of various spin liquids. really talking about the internal structure of FQH liq-
3
uids at zero temperature. In other words, we are talking also appear in spin liquids at zero temperature. In fact,
abouttheinternalstructureofthequantumgroundstate the conceptoftopologicalorderwasfirstintroducedin a
of FQH systems. So the topological order is a property study of spin liquids.[9] FQH liquid is not even the first
of ground state wave-function. The Landau’s theory is experimentally observed state with non trivial topolog-
developed for system at finite temperatures where quan- ical orders. That honor goes to superconducting state
tum effects can be ignored. Thus one should not be sur- discovered in 1911.[12] In contrast to a common point
prisedthat the Landau’stheory does not apply to states ofview,asuperconductingstatecannotbecharacterized
at zero temperature where quantum effects are impor- bybrokensymmetries. Itcontainsnontrivialtopological
tant. The very existence of topological orders suggests orders,[13] and is fundamentally different from a super-
that finite-temperature orders and zero-temperature or- fluid state.
ders are different, and zero-temperature orders contain After a long introduction, now we can state the main
richerstructures. WeseethatwhatismissedbyLandau’s subject of this paper. In this paper, we will study a new
theory is simply the quantum effect. Thus FQH liquids class of quantum orders where the excitations above the
arenotthatspecial. TheLandau’stheoryandsymmetry ground state are gapless. We believe that the gapless
characterization can fail for any quantum states at zero quantum orders are important in understanding high T
c
temperature. As a consequence, new kind of orders with superconductors. To connect to high T superconduc-
c
no broken symmetries and local order parameters (such tors, we will study quantum orders in quantum spin liq-
astopologicalorders)canexistforanyquantumstatesat uids on a 2D square lattice. We will concentrate on how
zero temperature. Because the orders in quantum states to characterize and classify quantum spin liquids with
at zero temperature and the orders in classical states at different quantum orders. We introduce projective sym-
finite temperatures arevery different, here we wouldlike metrygroupstohelpustoachievethisgoal. Theprojec-
to introduce two concepts to stress their differences:[11] tive symmetry group can be viewed as a generalization
(A) Quantum orders:[92] which describe the universal- of symmetry group that characterize different classical
ity classes of quantum ground states (ie the universality orders.
classes of complex ground state wave-functions with in-
finity variables);
(B)Classical orders: which describe the universality
B. Spin-liquid approach to high Tc superconductors
classes of classical statistical states (ie the universality
classes ofpositive probability distribution functions with
There are many different approaches to the high T
infinity variables). c
superconductors. Different people have different points
From the above definition, it is clear that the quantum
of view on what are the key experimental facts for the
ordersassociatedwith complex functions are richerthan
high T superconductors. The different choice of the key
the classical orders associated with positive functions. c
experimentalfactsleadtomanydifferentapproachesand
The Landau’s theory is a theory for classical orders,
theories. The spinliquid approachis basedona pointof
whichsuggeststhatclassicalordersmaybecharacterized
view that the high T superconductors are doped Mott
by broken symmetries and local order parameters.[93] c
insulators.[14,15,16](HerebyMottinsulatorwemeana
The existence of topological order indicates that quan-
insulator with an odd number of electron per unit cell.)
tum orders cannot be completely characterized by bro-
We believe that the most important properties of the
ken symmetries and order parameters. Thus we need to
high T superconductors is that the materials are in-
develop a new theory to describe quantum orders. c
sulators when the conduction band is half filled. The
In a sense, the classical world described by positive
charge gap obtained by the optical conductance experi-
probabilitiesisaworldwithonly“blackandwhite”. The
ments is about 2eV, which is much larger than the anti-
Landau’stheoryandthesymmetryprincipleforclassical
ferromagnetic(AF) transitiontemperatureT 250K,
AF
orders are color blind which can only describe different ∼
the superconducting transition temperature T 100K,
c
“shades of grey” in the classical world. The quantum ∼
and the spin pseudo-gap scale ∆ 40meV.[17, 18, 19]
worlddescribedbycomplexwavefunctionsisa“colorful” ∼
The insulating property is completely due to the strong
world. Weneedtousenewtheories,suchasthetheoryof
correlations present in the high T materials. Thus the
c
topologicalorderandthetheorydevelopedinthispaper,
strongcorrelationsareexpecttoplayveryimportantrole
to describe the rich “color” of quantum world.
inunderstandinghighT superconductors. Manyimpor-
c
The quantum orders in FQH liquids have a special tantpropertiesofhighT superconductorscanbedirectly
c
property that all excitations above ground state have fi- linkedtotheMottinsulatorathalffilling,suchas(a)the
nite energygaps. Thiskindofquantumordersarecalled low charge density[20] and superfluid density,[21] (b) T
c
topological orders. In general, a topological order is de- being proportional to doping T x,[22, 23, 24] (c) the
c
finedas aquantumorderwhereallthe excitations above positive charge carried by the cha∝rge carrier,[20] etc .
ground state have finite energy gapes. In the spin liquid approach, the strategy is to try to
Topological orders and quantum orders are general understand the properties of the high T superconduc-
c
properties ofany states atzero temperature. Nontrivial tors from the low doping limit. We first study the spin
topological orders not only appear in FQH liquids, they liquid state at half filling and try to understand the par-
4
ent Mott insulator. (In this paper, by spin liquid, we The algebraic spin liquids and Fermi spin liquids are in-
mean a spin state with translation and spin rotation teresting since they can be stable despite their gapless
symmetry.) At half filling, the charge excitations can excitations. Those gapless excitations are not protected
be ignored due to the huge charge gap. Thus we can bysymmetries. Thisis particularlystrikingforalgebraic
use a pure spin model to describe the half filled system. spin liquids since their gapless excitations interact down
After understand the spin liquid, we try to understand tozeroenergyandthestatesarestillstable. Wepropose
the dynamics of a few doped holes in the spin liquid that it is the quantum orders that protect the gapless
states and to obtain the properties of the high T su- excitations and ensure the stability of the algebraic spin
c
perconductors at low doping. One advantage of the spin liquids and Fermi spin liquids.
liquid approach is that experiments (such as angle re-
Wewouldliketopointoutthatbothstableandunsta-
solvedphoto-emission,[17, 18,25, 26]NMR,[27], neutron
blespinliquidsmaybeimportantforunderstandinghigh
scattering,[28, 29, 30] etc ) suggest that underdoped cu-
T superconductors. Although at zero temperature high
perates have many striking and qualitatively new prop- c
T superconductorsarealwaysdescribedstablequantum
ertieswhichareverydifferentfromthewellknownFermi c
states,someimportantstatesofhighT superconductors,
liquids. It is thus easier to approve or disapprove a new c
such as the pseudo-gap metallic state for underdoped
theoryintheunderdopedregimebystudyingthosequal-
samples, are observed only at finite temperatures. Such
itatively new properties.
finite temperature states may correspondto (doped) un-
Since the properties of the doped holes (such as their stable spin liquids, such as staggered flux state. Thus
statistics, spin, effective mass, etc ) are completely de- even unstable spin liquids can be useful in understand-
termined by the spin correlation in the parent spin liq- ing finite temperature metallic states.
uids, thus in the spin liquid approach,eachpossible spin
liquid leads to a possible theory for high T supercon- There are many different approach to spin liquids. In
c
ductors. Using the concept of quantum orders, we can addition to the slave-boson approach,[6, 15, 16, 31, 32,
say that possible theories for high T superconductors in 33, 35, 36, 37, 38, 39, 40] spin liquids has been stud-
c
the low doping limits are classified by possible quantum iedusingslave-fermion/σ-modelapproach,[41,42,43,44,
orders in spin liquids on 2D square lattice. Thus one 45, 46] quantum dimer model,[47, 48, 49, 50, 51] and
way to study high T superconductors is to construct all various numerical methods.[52, 53, 54, 55] In particular,
c
the possible spin liquids that have the same symmetries the numericalresults andrecentexperimentalresults[56]
as those observed in high T superconductors. Then an- stronglysupporttheexistenceofquantumspinliquidsin
c
alyze the physical properties of those spin liquids with some frustrated systems. A 3D quantum orbital liquid
dopings to see which one actually describes the high Tc was also proposed to exist in LaTiO3.[57]
superconductor. Although we cannot say that we have
However, I must point out that there is no generally
constructed all the symmetric spin liquids, in this paper
accepted numerical results yet that prove the existence
wehavefoundawaytoconstructalargeclassofsymmet-
of spin liquids with odd number of electron per unit cell
ricspinliquids. (Herebysymmetricspinliquidswemean
for spin-1/2 systems, despite intensive search in last ten
spin liquids with all the lattice symmetries: translation,
years. But it is my faith that spin liquids (with odd
rotation, parity, and the time reversal symmetries.) We
number of electron per unit cell) exist in spin-1/2 sys-
also find a way to characterize the quantum orders in
tems. For more general systems, spin liquids do ex-
those spin liquids via projective symmetry groups. This
ist. Read and Sachdev[43] found stable spin liquids in
gives us a global picture of possible high T theories.
c a Sp(N) model in large N limit. The spin-1/2 model
We would like to mention that a particular spin liquid
studiedinthis papercanbe easilygeneralizedtoSU(N)
– the staggered-flux/d-wave state[31, 32] – may be im-
model with N/2 fermions per site.[31, 58] In the large
portant for high T superconductors. Such a state can
c N limit, one can easily construct various Hamiltonians
explain[33, 34] the highly unusual pseudo-gap metallic
whose ground states realize various U(1) and Z spin
2
state found in underdoped cuperates,[17, 18, 25, 26] as
liquids constructed in this paper.[59] The quantum or-
well as the d-wave superconducting state[32].
ders in those large-N spin liquids can be described by
The spin liquids constructed in this paper can be di- the methods introduced in this paper. Thus, despite the
vided into four class: (a) Rigid spin liquid – spinons are uncertainty about the existence of spin-1/2 spin liquids,
fully gaped and may have bosonic, fermionic, or frac- the methods and the results presented in this paper are
tional statistics, (b) Fermi spin liquid – spinons are gap- not about (possibly) non-existing “ghost states”. Those
lessandaredescribedby aFermiliquidtheory,(c)Alge- methods and results apply, at least, to certain large-N
braic spin liquid – spinons are gapless, but they are not systems. In short, non-trivial quantum orders exist in
described by free fermionic/bosonic quasiparticles. (d) theory. We just need to find them in nature. (In fact,
Bose spin liquid – low lying gapless excitations are de- ourvacuumislikelytobeastatewithanon-trivialquan-
scribed by a free bosontheory. We find some of the con- tumorder,duetothefactthatlightexists.[58])Knowing
structedspinliquidsarestableandrepresentstablequan- the existence of spin liquids in large-N systems, it is not
tumphases,whileothersareunstableatlowenergiesdue such a big leap to go one step further to speculate that
to long range interactions caused by gauge fluctuations. spin liquids exist for spin-1/2 systems.
5
C. Spin-charge separation in (doped) spin liquids ductors, the possibility of true spin-charge separation in
an electron system is very interesting. The first con-
crete example of true spin-charge separation in 2D is
Spin-charge separation and the associated gauge the-
given by the chiral spin liquid state,[5, 6] where the
ory in spin liquids (and in doped spin liquids) are very
gauge interaction between the spinons and holons be-
important concepts in our attempt to understand the
comes short-ranged due to a Chern-Simons term. The
properties ofhigh T superconductors.[14, 15, 16, 39, 60]
c
Chern-Simons term breaks time reversal symmetry and
However, the exact meaning of spin-charge separation is
givesthespinonsandholonsafractionalstatistics. Later
differentfordifferentresearchers. Theterm“spin-charge
in 1991, it was realized that there is another way to
separation” has at lease in two different interpretations.
make the gauge interaction short-ranged through the
Inthefirstinterpretation,thetermmeansthatitisbetter
Anderson-Higgs mechanism.[38, 43] This led to a mean-
to introduce separatespinons (a neutral spin-1/2excita-
field theory[38, 40] of the short-ranged Resonating Va-
tion) and holons (a spinless excitation with unit charge)
lence Bound(RVB) state[47, 48] conjectured earlier. We
to understand the dynamical properties of high T su-
c
will call such a state Z spin liquid state, to stress the
perconductors, instead of using the original electrons. 2
unconfinedZ gauge field that appears in the low energy
However, there may be long range interaction (possibly, 2
effectivetheoryofthosespinliquids. (Seeremarksatthe
evenconfininginteractionsatlongdistance)betweenthe
endofthis section. We alsonote thattheZ spinliquids
spinons and holons,andthe spinons and holons maynot 2
studied in Ref. [43] all break the 90◦ rotation symmetry
be well defined quasiparticles. We will call this inter-
and are different from the short-ranged RVB state stud-
pretation pseudo spin-charge separation. The algebraic
iedRef.[38,40,47,48].) Since theZ gaugefluctuations
spinliquidshavethepseudospin-chargeseparation. The 2
are weak and are not confining, the spinons and holons
essence of the pseudo spin-charge separation is not that
have only short rangedinteractions in the Z spin liquid
spin and charge separate. The pseudo spin-charge sepa- 2
state. TheZ spinliquidstatealsocontainsaZ vortex-
ration is simply another way to say that the gapless exci- 2 2
like excitation.[38, 62] The spinons and holons can be
tationscannotbedescribedbyfreefermions orbosons. In
bosons or fermions depending on if they are bound with
the second interpretation, the term “spin-charge separa-
the Z vortex.
tion”meansthatthereareonlyshort rangedinteractions 2
betweenthespinonsandholons. Thespinonsandholons Recently,thetruespin-chargeseparation,theZ2gauge
are well defined quasiparticles at least in the dilute limit structure and the Z2 vortex excitations were also pro-
oratlowenergies. We willcallthe secondinterpretation posed in a study of quantum disordered superconduct-
the true spin-charge separation. The rigid spin liquids ing state in a continuum model[63] and in a Z2 slave-
andthe Fermispinliquids havethe truespin-chargesep- boson approach[64]. The resulting liquid state (which
aration. was named nodal liquid) has all the novel properties of
Electron operator is not a good starting point to de- Z2 spin liquid state such as the Z2 gauge structure and
scribe states with pseudo spin-charge separation or true theZ2 vortexexcitation(whichwasnamedvison). From
spin-charge separation. To study those states, we usu- the point ofview ofuniversalityclass,the nodalliquidis
ally rewrite the electron operator as a product of sev- one kind of Z2 spin liquids. However, the particular Z2
eral other operators. Those operators are called parton spin liquid studied in Ref. [38, 40] and the nodal liquid
operators. (The spinon operator and the holon opera- are two different Z2 spin liquids, despite they have the
tor are examples of parton operators). We then con- same symmetry. The spinons in the first Z2 spin liquid
struct mean-field state in the enlarged Hilbert space of have a finite energy gap while the spinons in the nodal
partons. The gauge structure can be determined as the liquidaregaplessandhaveaDirac-likedispersion. Inthis
most general transformations between the partons that paper, we will use the projective construction to obtain
leave the electron operator unchanged.[61] After identi- moregeneralspinliquids. Wefindthatonecanconstruct
fying the gauge structure, we can project the mean-field hundreds of different Z2 spin liquids. Some Z2 spin liq-
state onto the physical (ie the gauge invariant) Hilbert uids have finite energy gaps, while others are gapless.
space and obtain a strongly correlated electron state. Among those gapless Z2 spin liquids, some have finite
This procedure in its general form is called projective Fermi surfaces while others have only Fermi points. The
construction. It is a generalization of the slave-boson spinonsneartheFermipointscanhavelinearE(k) k
approach.[15, 16, 33, 36, 37, 38, 40] The general pro- or quadratic E(k) k2 dispersions. We find ther∝e a|re|
∝
jective construction and the related gauge structure has morethanoneZ2spinliquidswhosespinonshaveamass-
beendiscussedindetailforquantumHallstates.[61]Now less Dirac-like dispersion. Those Z2 spin liquids have
we see a third (but technical) meaning of spin-charge the same symmetry but different quantum orders. Their
separation: to construct a strongly correlated electron ansatz are give by Eq. (42), Eq. (39), Eq. (88), etc .
state, we need to use partons and projective construc- Both chiral spin liquid and Z spin liquid states are
2
tion. The resulting effective theory naturally contains a Mott insulators with one electron per unit cell if not
gauge structure. doped. Their internal structures are characterized by a
Although, it is not clear which interpretation of spin- new kind of order– topologicalorder,if they are gapped
charge separation actually applies to high T supercon- or if the gapless sector decouples. Topological order is
c
6
not related to any symmetries and has no (local) or- property of ground state. Thus we should not regardZ
2
der parameters. Thus, the topological order is robust spinliquidsasthespinliquidsconstructedusingZ slave-
2
against all perturbations that can break any symmetries boson approach. A Z spin liquid can be constructed
2
(including random perturbations that break translation from the U(1) or SU(2) slave-boson approaches as well.
symmetry).[9, 10] (This point was also emphasized in Aprecisemathematicaldefinitionofthelowenergygauge
Ref. [65] recently.) Even though there are no order pa- group will be given in section IVA.
rameterstocharacterizethem,thetopologicalorderscan
becharacterizedbyothermeasurablequantumnumbers,
suchasgroundstatedegeneracyincompactspaceaspro- D. Organization
posed in Ref. [9, 10]. Recently, Ref. [65] introduced a
very clever experiment to test the ground state degen- In this paper we will use the method outlined in
eracy associated with the non-trivial topological orders. Ref. [38, 40] to study gauge structures in various spin
In addition to ground state degeneracy, there are other liquid states. In section II we review SU(2) mean-field
practicalwaystodetecttopologicalorders. Forexample, theory of spin liquids. In section III, we construct sim-
the excitations on top of a topologically ordered state ple symmetric spin liquids using translationally invari-
canbedefectsoftheunderlyingtopologicalorder,which ant ansatz. In section IV, projective symmetry group is
usually leads to unusual statistics for those excitations. introduced to characterize quantum orders in spin liq-
Measuringthestatisticsofthoseexcitationsalsoallowus uids. In section V, we study the transition between dif-
to measure topological orders. ferent symmetric spin liquids, using the results obtained
The concept of topological order and quantum order in appendix B, where we find a way to construct all the
are very important in understanding quantum spin liq- symmetric spin liquids in the neighborhood of some well
uids (or any other strongly correlated quantum liquids). known spin liquids. We also study the spinon spectrum
In this paper we are going to construct hundreds of dif- togainsomeintuitiveunderstandingonthepropertiesof
ferent spin liquids. Those spin liquids all have the same the spin liquids. Using the relation between two-spinon
symmetry. To understand those spin liquids, we need to spectrum and quantum order, we propose, in section
first learn how to characterize those spin liquids. Those VII, a practical way to use neutron scattering to mea-
states break no symmetries and hence have no order pa- sure quantum orders. We study the stability of Fermi
rameters. One would get into a wrong track if trying to spin liquids and algebraic spin liquids in section VIII.
find an order parameter to characterize the spin liquids. We find that both Fermi spin liquids and algebraic spin
Weneedtouseacompletelynewway,suchastopological liquids can exist as zero temperature phases. This is
orders and quantum orders, to characterize those states. particularly striking for algebraic spin liquids since their
gapless excitations interacts even at lowest energies and
In addition to the above Z spin liquids, in this paper
2
there are no free fermionic/bosonic quasiparticle excita-
we will also study many other spin liquids with different
tions at low energies. We show how quantum order can
low energy gauge structures, such as U(1) and SU(2)
protectgaplessexcitations. Appendix Acontainsamore
gauge structures. We will use the terms Z spin liq-
2
detailed discussionon projective symmetry group, and a
uids,U(1)spinliquids,andSU(2)spinliquidstodescribe
classification of Z , U(1) and SU(2) spin liquids using
them. We would like to stress that Z , U(1), and SU(2) 2
2
the projective symmetry group. Section X summarizes
here are gauge groups that appear in the low energy ef-
the main results of the paper.
fective theoriesofthosespinliquids. They shouldnotbe
confused with the Z , U(1), and SU(2) gauge group in
2
slave-boson approach or other theories of the projective
II. PROJECTIVE CONSTRUCTION OF 2D
construction. The latter are high energy gauge groups.
SPIN LIQUIDS – A REVIEW OF SU(2)
The high energy gauge groups have nothing to do with
SLAVE-BOSON APPROACH
the low energy gauge groups. A high energy Z gauge
2
theory (or a Z slave-boson approach) can have a low
2
In this section, we are going to use projective con-
energy effective theory that contains SU(2), U(1) or Z
2
struction to construct 2D spin liquids. We are going
gauge fluctuations. Even the t-J model, which has no
to review a particular projective construction, namely
gauge structure at lattice scale, can have a low energy
the SU(2) slave-boson approach.[15, 16, 33, 36, 37, 38,
effective theory that contains SU(2), U(1) or Z gauge
2
40] The gauge structure discovered by Baskaran and
fluctuations. The spin liquids studied in this paper all
Anderson[16] in the slave-boson approach plays a cru-
contain some kind of low energy gauge fluctuations. De-
cialrole in our understanding of strongly correlatedspin
spite their different low energy gauge groups, all those
liquids.
spin liquids can be constructed from any one of SU(2),
We will concentrate on the spin liquid states of a pure
U(1), or Z slave-boson approaches. After all, all those
2
spin-1/2 model on a 2D square lattice
slave-bosonapproachesdescribe the same t-J model and
are equivalent to each other. In short, the high energy
gaugegroupisrelatedtothewayinwhichwewritedown Hspin = JijSi Sj +... (1)
·
the Hamiltonian, while the low energy gauge group is a <ij>
X
7
where the summation is over different links (ie ij and χ and η in Eq. (7) must satisfy the self consistency
ij ij
ji are regarded as the same) and ... represenhts piossi- condition Eq. (6) and the site dependent fields al(i) are
h i 0
ble terms which contain three or more spin operators. chosen such that Eq. (5) is satisfied by the mean-field
Those terms are needed in order for many exotic spin ground state. Such χ , η and al give us a mean-field
ij ij 0
liquid states introduced in this paper to become the solution. The fluctuations in χ , η and al(i) describe
ij ij 0
ground state. To obtain the mean-field ground state of the collective excitations above the mean-field ground
thespinliquids,weintroducefermionicpartonoperators state.
f , α=1,2 which carries spin 1/2 and no charge. The The Hamiltonian Eq. (7) and the constraints Eq. (4)
iα
spin operator S is represented as have a local SU(2) symmetry.[36, 37] The local SU(2)
i
symmetry becomes explicit if we introduce doublet
1
S = f† σ f (2)
i 2 iα αβ iβ
IntermsofthefermionoperatorstheHamiltonianEq.(1) ψ = ψψ1 = ff↑† (8)
can be rewritten as (cid:18) 2(cid:19) (cid:18) ↓(cid:19)
1 1 and matrix
H = J f† f f† f + f† f f† f (3)
−2 ij iα jα jβ iβ 2 iα iα jβ jβ
Xhiji (cid:18) (cid:19) U = χ†ij ηij =U† (9)
HWeereawlsoe haadvdeeduspedroσpeαrβc·oσnαs′tβa′nt=t2erδmαβs′δα′β f−†δfαβδαa′nβ′d. ij ηi†j −χij! ji
i iα iα
f† f f† f to get the above form. Notice that Using Eq. (8) and Eq. (9) we can rewrite Eq. (5) and
hiji iα iα jβ jβ P
Eq. (7) as:
the Hilbert space of Eq. (3) is generated by the parton
P
operators f and is larger than that of Eq. (1). The
α
equivalence between Eq. (1) and Eq. (3) is valid only in ψi†τlψi =0 (10)
the subspace where there is exactly one fermion per site. D E
Therefore to use Eq. (3) to describe the spin state we
need to impose the constraint[15, 16] H = 3J 1Tr(U† U ) (ψ†U ψ + h.c.)
mean 8 ij 2 ij ij − i ij j
fi†αfiα =1, fiαfiβǫαβ =0 (4) Xhiji (cid:20) (cid:21)
The second constraint is actually a consequence of the + alψ†τlψ (11)
0 i i
first one. i
X
Amean-fieldgroundstateat“zeroth”orderisobtained
bymakingthefollowingapproximations. Firstwereplace whereτl, l =1,2,3arethePaulimatrices. FromEq.(11)
constraint Eq. (4) by its ground-state average wecanseeclearlythattheHamiltonianisinvariantunder
a local SU(2) transformation W(i):
f† f =1, f f ǫ =0 (5)
h iα iαi h iα iβ αβi
ψ W(i) ψ
Such a constraint can be enforced by including a site i → i
dependent and time independent Lagrangian multiplier: Uij W(i) Uij W†(j) (12)
→
al(i)(f† f 1), l =1,2,3, in the Hamiltonian. At the
0 iα iα− The SU(2) gauge structure is originated from Eq. (2).
zeroth order we ignore the fluctuations (ie the time de-
pendence) of al. If we included the fluctuations of al, The SU(2) is the most general transformation between
0 0 the partons that leave the physical spin operator un-
the constraint Eq. (5) would become the original con-
changed. Thus once we write down the parton expres-
straint Eq. (4).[15, 16, 36, 37] Second we replace the
sion of the spin operator Eq. (2), the gauge structure of
operators f† f and f f by their ground-state ex-
iα jβ iα iβ thetheoryisdetermined.[61](TheSU(2)gaugestructure
pectations value
discussed here is a high energy gauge structure.)
ηijǫαβ = 2 fiα fjβ , ηij =ηji We note that both components of ψ carry spin-up.
− h i
χ δ =2 f† f , χ =χ† (6) Thus the spin-rotation symmetry is not explicit in our
ij αβ h iα jβi ij ji formalism and it is hard to tell if Eq. (11) describes a
again ignoring their fluctuations. In this way we obtain spin-rotationinvariantstateornot. Infact,forageneral
the zeroth order mean-field Hamiltonian: U satisfying U = U† , Eq. (11) may not describe a
ij ij ji
H spin-rotationinvariant state. However, if Uij has a form
mean
3
= −8Jij (χjifi†αfjα+ηijfi†αfj†β ǫαβ +h.c) Uij = iρijWij,
Xhiji h ρij = real number,
−|χij|2−|ηij|2 (7) Wij ∈ SU(2), (13)
+ a30(fi†αfiα−1)(cid:3)+[(a10+ia20)fiαfiβǫαβ +h.c.] thenEq.(11)willdescribeaspin-rotationinvariantstate.
Xi h i This is because the aboveUij canbe rewrittenin a form
8
Eq.(9). Inthis caseEq.(11)canbe rewrittenas Eq.(7) physical properties. This property is usually called the
where the spin-rotation invariance is explicit. “gaugesymmetry”. However,fromthe abovediscussion,
To obtain the mean-field theory, we have enlarged the we see that the “gaugesymmetry” is not a symmetry. A
Hilbert space. Because of this, the mean-field theory symmetry is about two different states having the same
is not even qualitatively correct. Let Ψm(Ueiajn) be the properties. Uij and Ui′j are just two labels that label
ground state of the Hamiltonian Eq. (1|1) withi energy the samestate,andthesamestatealwayshavethe same
E(U ,alτl). It is clear that the mean-field ground state properties. We do not usually call the same state hav-
ij i
is not even a valid wave-function for the spin system ing the same properties a symmetry. Because the same
since it may not have one fermion per site. Thus it is statealwayhavethesameproperties,the“gaugesymme-
very important to include fluctuations of al to enforce try”canneverbebroken. Itisverymisleadingtocallthe
0
one-fermion-per-site constraint. With this understand- Anderson-Higgsmechanism“gaugesymmetrybreaking”.
ing,we mayobtainavalidwave-functionofthe spinsys- Withthisunderstanding,weseethatasuperconductoris
temΨ ( α )byprojectingthemean-fieldstatetothe fundamentally different from a superfluid. A superfluid
spin i
subspace of{on}e-fermion-per-site: is characterizedby U(1) symmetry breaking, while a su-
perconductorhasnosymmetrybreakingonceweinclude
Ψ ( α )= 0 f Ψ(Uij) . (14) the dynamicalelectromagnetic gaugefluctuations. A su-
spin { i} h | iαi| meani perconductor is actually the first topologically ordered
i
Y
state observed in experiments,[13] which has no symme-
NowthelocalSU(2)transformationEq.(12)canhave try breaking, no long range order, and no (local) order
a very physical meaning: Ψ(Uij) and Ψ(W(i)UijW†(j)) parameter. However, when the speed of light c = ,
giverisetothe samespinw| amvee-afnuinction|afmteeranprojectioni: a superconductor becomes similar to a superfluid and∞is
characterizedby U(1) symmetry breaking.
0 f Ψ(Uij) = 0 f Ψ(W(i)UijW†(j)) (15) The relation between the mean-field state and the
h | iαi| meani h | iαi| mean i physical spin wave function Eq. (14) allows us to con-
i i
Y Y
struct transformation of the physical spin wave-function
Thus Uij and Ui′j = W(i)UijW†(j) are just two differ- from the mean-field ansatz. For example the mean-field
entlabelswhichlabelthesamephysicalstate. Withinthe state Ψ(Ui′j) with U′ = U produces a phys-
mean-field theory, a local SU(2) transformation changes | meani ij i−l,j−l
ical spin wave-function which is translated by a dis-
amean-fieldstate Ψm(Ueiajn) toadifferentmean-fieldstate tance l from the physical spin wave-function produced
Ψm(Uei′ajn) . If the tw| o meain-field states always have the by Ψm(Ueiajn) . The physical state is translationally sym-
s|ame phiysical properties, the system has a local SU(2) met|ric if aind only if the translated ansatz U′ and the
ij
symmetry. However, after projection, the physical spin original ansatz U are gauge equivalent (it does not re-
ij
quantumstatedescribedbywave-functionΨspin({αi})is quire Ui′j = Uij). We see that the gauge structure can
invariant under the local SU(2) transformation. A local complicates our analysis of symmetries, since the phys-
SU(2)transformationjusttransformsonelabel,Uij,ofa ical spin wave-function Ψspin( αi ) may has more sym-
physicalspinstatetoanotherlabel,Ui′j,whichlabelsthe metries than the mean-field sta{te}Ψ(Uij) before projec-
mean
exactly the same physical state. Thus after projection, | i
tion.
local SU(2) transformations become gauge transforma-
Let us discuss time reversal symmetry in more detail.
tions. The fact that U and U′ label the same physical
ij ij A quantum system described by
spin state creates a interesting situation when we con-
siderthefluctuationsofU aroundamean-fieldsolution
ij i~∂ Ψ(t)=HΨ(t) (16)
– some fluctuations of U do not change the physical t
ij
state and are unphysical. Those fluctuations are called
hasatimereversalsymmetryifΨ(t)satisfyingthe equa-
the pure gauge fluctuations.
tion of motion implies that Ψ∗( t) also satisfying the
The above discussion also indicates that in order for equation of motion. This requir−es that H = H∗. We
themean-fieldtheorytomakeanysense,wemustatleast
see that, for time reversal symmetric system, if Ψ is an
include the SU(2) gauge (or other gauge) fluctuations
eigenstate, then Ψ∗ will be an eigenstate with the same
described by al and W in Eq. (13), so that the SU(2)
0 ij energy.
gauge structure of the mean-field theory is revealed and
Foroursystem,thetimereversalsymmetrymeansthat
the physical spin state is obtained. We will include the
gaugefluctuations to the zeroth-ordermean-fieldtheory. ifthe mean-fieldwavefunctionΨ(Uij,aliτl) isamean-field
mean
The new theory will be called the first order mean-field ground state wave function for ansatz (U ,alτl), then
ij i
theory. It is this first order mean-field theory that rep- Ψ(Uij,aliτl) ∗ will be the mean-field ground state wave
mean
resents a proper low energy effective theory of the spin
liquid. f(cid:16)unction for(cid:17)ansatz (Ui∗j,ali(τl)∗). That is
Here,wewouldlikemakearemarkabout“gaugesym-
metry” and “gauge symmetry breaking”. We see that Ψ(Uij,aliτl) ∗ =Ψ(Ui∗j,ali(τl)∗) (17)
two ansatz U and U′ =W(i)U W†(j) have the same mean mean
ij ij ij
(cid:16) (cid:17)
9
Forasystemwithtimereversalsymmetry,themean-field symmetric spin liquid. First let us introduce u :
ij
energy E(U ,alτl) satisfies
ij i 3
J U =u (21)
E(U ,alτl)=E(U∗,al(τl)∗) (18) 8 ij ij ij
ij i ij i
For translationally invariant ansatz, we can introduce a
Thusifanansatz(U ,alτl)isamean-fieldsolution,then
ij i short-hand notation:
(U∗,al(τl)∗) is also a mean-field solution with the same
ij i
mean-field energy. u =uµ τµ u (22)
Fromthe abovediscussion,we seethatunder the time ij −i+j ≡ −i+j
reversaltransformation, the ansatz transforms as where u1,2,3 are real, u0 is imaginary, τ0 is the identity
l l
matrix and τ1,2,3 are the Pauli matrices. The fermion
U U′ =( iτ2)U∗(iτ2)= U ,
ij → ij − ij − ij spectrum is determined by Hamiltonian
alτl a′lτl =( iτ2)(alτl)∗(iτ2)= alτl. (19)
i → i − i − i H = ψ†u ψ +h.c. + ψ†alτlψ (23)
Note here we have included an additional SU(2) gauge − i j−i j i 0 i
transformationW = iτ2. We also note that under the Xhiji(cid:16) (cid:17) Xi
i
time reversal transfor−mation, the loop operator trans- In k-space we have
forms as P = eiθ+iθlτl ( iτ2)P∗(iτ2) = e−iθ+iθlτl.
C → − C H = ψ†(uµ(k) aµ)τµψ (24)
We see that the U(1) flux changes the sign while the − k − 0 k
SU(2) flux is not changed. Xk
Before ending this review section, we would like to where µ=0,1,2,3,
point out that the mean-field ansatz of the spin liquids
Uij can be divided into two classes: unfrustrated ansatz uµ(k)= uµleil·k, (25)
where U only link an even lattice site to an odd lattice
ij l
X
siteandfrustratedansatzwhereU arenonzerobetween
ij
two even sites and/or two odd sites. An unfrustrated a00 = 0, and N is the total number of site. The fermion
ansatz hasonly pure SU(2)flux througheachplaquette, spectrum has two branches and is given by
while an frustrated ansatz has U(1) flux of multiple of
E (k)=u0(k) E (k)
π/2 through some plaquettes in addition to the SU(2) ± ± 0
flux.
E (k)= (ul(k) al)2 (26)
0 − 0
s l
X
III. SPIN LIQUIDS FROM TRANSLATIONALLY
The constraints can be obtained from ∂Eground = 0 and
INVARIANT ANSATZ ∂al
0
have a form
In this section, we will study many simple examples N ψ†τlψ
of spin liquids and their ansatz. Through those simple h i ii
examples, we gain some understandings on what kind of = ul(k)−al0 ul(k)−al0 =0 (27)
spin liquids are possible. Those understandings help us E (k) − E (k)
0 0
to develop the characterizationand classification of spin k,EX−(k)<0 k,EX+(k)<0
liquids using projective symmetry group. which allow us to determine al, l = 1,2,3. It is inter-
0
Using the above SU(2) projective construction, one esting to see that if u0 = 0 and the ansatz is unfrus-
i
canconstructmanyspinliquidstates. Tolimitourselves, trated, then we can simply choose al = 0 to satisfy the
0
we will concentrate on spin liquids with translation and mean-fieldconstraints(sinceuµ(k)= uµ(k+(π,π))for
90◦ rotation symmetries. Although a mean-field ansatz −
unfrustrated ansatz). Such ansatz always have time re-
with translationand rotationinvariance alwaysgenerate
versalsymmetry. ThisisbecauseU and U aregauge
ij ij
a spin liquid with translation and rotation symmetries, −
equivalent for unfrustrated ansatz.
a mean-field ansatz without those invariances can also
Now let us study some simple examples. First let us
generateaspinliquidwiththosesymmetries.[94]Because
assume that only the nearest neighbor coupling u and
xˆ
ofthis,itisquitedifficulttoconstructallthe translation
u are non-zero. In order for the ansatz to describe a
yˆ
and rotation symmetric spin liquids. In this section we
rotationallysymmetric state,the rotatedansatzmustbe
will consider a simpler problem. We will limit ourselves
gauge equivalent to the original ansatz. One can easily
to spin liquids generated from translationally invariant
check that the following ansatz has the rotation symme-
ansatz:
try
Ui+l,j+l =Uij, al0(i)=al0 (20) al = 0
0
In this case, we only need to find the conditions under uxˆ = χτ3+ητ1
which the above ansatz can give rise to a rotationally u = χτ3 ητ1 (28)
yˆ
−
10
since the 90◦ rotation followed by a gauge transforma- theprojective-symmetry-groupclassification,theSU(2)-
tion W = iτ3 leave the ansatz unchanged. The above gapless ansatz Eq. (30) is labeled by SU2An0 and the
i
ansatz also has the time reversal symmetry, since time SU(2)-linear ansatz Eq. (31) by SU2Bn0 (see Eq. (82)).
reversal transformation u u followed by a gauge When χ = η = 0, The flux P is non trivial. How-
ij ij C
→− 6 6
transformation W =iτ2 leave the ansatz unchanged. ever, P commute with P as long as the two loops
i C C′
Tounderstandthegaugefluctuationsaroundtheabove C and C′ have the same base point. In this case the
mean-field state, we note that the mean-field ansatz SU(2) gauge structure is broken down to a U(1) gauge
may generate non-trivial SU(2) flux through plaquettes. structure.[38, 40] The gapless spinon still only appear
Those flux may break SU(2) gauge structure down to at isolated k points. We will call such a state U(1)-
U(1)orZ gaugestructuresas discussedin Ref.[38, 40]. linear state. (This state was called staggered flux state
2
In particular, the dynamics of the gauge fluctuations in and/ord-wavepairingstateinliterature.) Afteraproper
the break down from SU(2) to Z has been discussed in gauge transformation, the U(1)-linear state can also be
2
detail in Ref. [40]. According to Ref. [38, 40], the SU(2) described by the ansatz
flux playsa roleof Higgsfields. A non-trivialSU(2)flux
correspond to a condensation of Higgs fields which can ui,i+xˆ =iχ ( )iητ3
− −
break the gauge structure and give SU(2) and/or U(1) u =iχ+( )iητ3 (32)
i,i+yˆ
gauge boson a mass. Thus to understand the dynamics −
ofthegaugefluctuations,weneedtofindtheSU(2)flux. where the U(1) gauge structure is explicit. Under the
The SU(2) flux is defined for loops with a base point. projective-symmetry-group classification, such a state is
Theloopstartsandendsatthebasepoint. Forexample, labeled by U1Cn01n (see Eq. (B4) and IVC). The low
we can consider the following two loops C with the energy effective theory is described by massless Dirac
1,2
samebasepointi: C =i i+xˆ i+xˆ+yˆ i+yˆ i fermions (the spinons) coupled to a U(1) gauge field.
1
and C is the 90◦ rotatio→n of C →: C = i →i+yˆ→ The above results are all known before. In the follow-
2 1 2
→ →
i xˆ+yˆ i xˆ i. The SU(2)flux forthe twoloops ing we are going to study a new class of translation and
− → − →
is defined as rotation symmetric ansatz, which has a form
PC1 ≡ui,i+yˆui+yˆ,i+xˆ+yˆui+xˆ+yˆ,i+xˆui+xˆ,i =uy†ˆu†xˆuyˆuxˆ al0 =0
P u u u u =u u†u†u uxˆ =iητ0 χ(τ3 τ1)
C2 ≡ i,i−xˆ i−xˆ,i−xˆ+yˆ i−xˆ+yˆ,i+yˆ i+yˆ,i xˆ yˆ xˆ yˆ − −
(29) uyˆ =iητ0 χ(τ3+τ1) (33)
−
with χ and η non-zero. The above ansatz describes the
AsdiscussedinRef.[38,40],iftheSU(2)fluxP forall
C
loops are trivial: P τ0, then the SU(2) gauge struc- SU(2)-gapless spin liquid if χ=0, and the SU(2)-linear
C
∝ spin liquid if η =0.
ture is unbroken. This is the case when χ = η or when
After a 90◦ rotation R , the above ansatz becomes
η = 0 in the above ansatz Eq. (28). The spinon in the 90
spin liquid described by η =0 has a large Fermi surface.
u = iητ0 χ(τ3+τ1)
We will call this state SU(2)-gapless state (This state xˆ − −
was called uniform RVB state in literature). The state uyˆ = iητ0 χ(τ3 τ1) (34)
− −
with χ=η has gaplessspinons only atisolatedk points.
The rotated ansatz is gauge equivalent to the origi-
We will call such a state SU(2)-linear state to stress the
nal ansatz under the gauge transformation G (i) =
linear dispersion E k near the Fermi points. (Such a R90
state was called the∝π|-fl|ux state in literature). The low ( )ix(1 iτ2)/√2. After a parity x x transforma-
− − → −
tion P , Eq. (33) becomes
energy effective theory for the SU(2)-linear state is de- x
scribedbymasslessDiracfermions(thespinons)coupled
u = iητ0 χ(τ3 τ1)
to a SU(2) gauge field. xˆ − − −
Afterpropergaugetransformations,theSU(2)-gapless uyˆ =iητ0 χ(τ3+τ1) (35)
−
ansatz can be rewritten as
whichisgaugeequivalenttotheoriginalansatzunderthe
uxˆ = iχ gaugetransformationGPx(i)=(−)ixi(τ3+τ1)/√2. Un-
der time reversal transformation T, Eq. (33) is changed
u = iχ (30)
yˆ to
and the SU(2)-linear ansatz as u = iητ0+χ(τ3 τ1)
xˆ
− −
u = iητ0+χ(τ3+τ1) (36)
ui,i+xˆ = iχ yˆ −
u = i( )ixχ (31) whichisagaingaugeequivalenttotheoriginalansatzun-
i,i+yˆ
− derthegaugetransformationG (i)=( )i. (Infactany
T
−
Intheseform,theSU(2)gaugestructureisexplicitsince ansatzwhichonlyhaslinksbetweentwonon-overlapping
u iτ0. Herewewouldalsoliketomentionthatunder sublattices (ie the unfrustrated ansatz) is time reversal
ij
∝