Quantum frustration in organic Mott insulators: from spin liquids to unconventional superconductors

9.3 MB·English

by  B. J. Powell

Quantum frustration in organic Mott insulators: from spin liquids to
unconventional superconductors
B. J. Powell∗ and Ross H. McKenzie†
Department of Physics, University of Queensland, Brisbane, 4072, Australia
(Dated: January 17, 2011)
Wereviewtheinterplayoffrustrationandstrongelectroniccorrelationsinquasi-two-dimensional
organic charge transfer salts, such as (BEDT-TTF)2X and EtnMe4−nPn[Pd(dmit)2]2. These
two forces drive a range of exotic phases including spin liquids, valence bond crystals, pseudo-
1 gapped metals, and unconventional superconductivity. Of particular interest is that in several
1 materials with increasing pressure there is a first-order transition from a spin liquid Mott in-
0 sulating state to a superconducting state. Experiments on these materials raise a number of
2 profoundquestionsaboutthequantumbehaviouroffrustratedsystems,particularlytheintimate
connection between spin liquids and superconductivity. Insights into these questions have come
n
from a wide range of theoretical techniques including first principles electronic structure, quan-
a
tummany-bodytheoryandquantumfieldtheory. Inthisreviewweintroducesomeofthebasic
J
ideas of the field by discussing a simple frustrated Heisenberg model with four spins. We then

4 describethekeyexperimentalresults,emphasizingthatfortwomaterials,κ-(BEDT-TTF)2Cu2-
1 (CN)3 and EtMe3Sb[Pd(dmit)2]2, there is strong evidence for a spin liquid ground state, and
for another, EtMe3P[Pd(dmit)2]2, there is evidence of a valence bond crystal ground state. We
] reviewtheoreticalattemptstoexplainthesephenomena,arguingthattheycanbecapturedbya
l Hubbard model on the anisotropic triangular lattice at half filling, and that Resonating Valence
e
Bond (RVB) wavefunctions capture most of the essential physics. We review evidence that this
-
r Hubbardmodelcanhaveaspinliquidgroundstateforarangeofparametersthatarerealisticfor
t therelevantmaterials. Inparticular,spatialanisotropyandringexchangearekeytodestabilising
s
. magneticorder. Weconcludebysummarisingtheprogressmadethusfarandidentifyingsomeof
t
a thekeyquestionsstilltobeanswered.
m
-
d
Contents 1. Dimermodelofthebandstructureof
n
κ-(BEDT-TTF)2X 12
o
I. Introduction 2 2. TheHubbardU 14
c
[ A. Motivation: frustration,spinliquids,andspinons 2 3. The(BEDT-TTF)2 dimer 15
1. Keyquestions 2 B. Insulatingphases 16

2 2. Ahierarchyoftheories: fromquantumchemistry 1. Antiferromagneticandspinliquidphases 16
v tofieldtheory 3 2. Isthespinliquidinκ-(BEDT-TTF)2Cu2(CN)3
1 3. Organicchargetransfersaltsareanimportant gapped? 18
8 classofmaterials 3 3. The6Kanomaly 20
3 4. Whatarespinliquids? 4 C. Mottmetal-insulatortransition 20
5 5. Whatarespinons? 6 1. CriticalexponentsoftheMotttransition 21
. 6. Antiferromagneticfluctuations 6 2. Opticalconductivity 22
7
7. Quantumcriticalpoints 6 3. Thespinliquidtometaltransition 23
0
4. ReentranceoftheMotttransition-explanation
B. Keyconsequencesoffrustration 7
0
fromundergraduatethermodynamics 23
1 1. Reductionofthecorrelationlength 7
D. Magneticfrustrationinthenormalstate 24
: 2. Competingphases 7
v 3. Alternativemeasuresoffrustration 8 1. Dynamicalmean-fieldtheory(DMFT) 24
Xi 4. Geometricfrustrationofkineticenergy 9 23.. FNeMrmRialinqduitdherepgsimeuedogap 2266
r II. Toy models to illustrate the interplay of 4. Thereisnopseudogapin
a frustration and quantum fluctuations 9 κ-(BEDT-TTF)2Cu2(CN)3 28
5. Otherevidenceforapseudogapintheweakly
A. FoursiteHeisenbergmodel 9
frustratedmaterials 28
1. Effectofaringexchangeiteraction 10
6. Testsofthepseudogaphypothesis 29
B. FoursiteHubbardmodel 10
7. TheNernsteffectandvortexfluctuationsabove
III. κ-(BEDT-TTF)2X 11 E. TheTcsuperconductingstate 3310
A. Crystalandelectronicstructure 11
1. κ-(BEDT-TTF)2Cu2(CN)3 31
2. Weaklyfrustratedmaterials 32
IV. β(cid:48)-Z[Pd(dmit) ] 33
2 2
∗Electronicaddress: [email protected] A. Crystalandelectronicstructure 34
†Electronicaddress: [email protected] B. Frustratedantiferromagnetism 36
2
C. Spinliquidbehaviourinβ(cid:48)-Me3EtSb-[Pd(dmit)2]2 References 62
(Sb-1) 36
D. IsthereavalencebondcrystalorspinPeierlsstate
inβ(cid:48)-Me3EtP-[Pd(dmit)2]2 (P-1)? 38 I. INTRODUCTION
E. Paramagnetictonon-magnetictransitionin
Et2Me2Sb[Pd(dmit)2]2 (Sb-2)andCs[Pd(dmit)2]2
(Cs-00) 40 In the early 1970’s, Anderson and Fazekas (Anderson,
1. Et3MeSbimpuritiesinEt2Me2Sb[Pd(dmit)2]2 1973; Fazekas and Anderson, 1974) proposed that the
(Sb-2) 41 ground state of the antiferromagnetic Heisenberg spin-
F. Motttransitionunderhydrostaticpressureand 1/2 model on the triangular lattice did not break spin
uniaxialstress 41
rotational symmetry, i.e., had no net magnetic moment.
V. Nuclear magnetic resonance as a probe of spin A state of matter characterised by well defined local mo-
fluctuations 43 ments and the absence of long range order has become
1. Long-rangeantiferromagneticspinfluctuation knownasaspinliquid(Normand,2009). Suchstates,are
model 43
known in one-dimensional (1d) systems, but 1d systems
2. Quantumcriticalspinfluctuationmodel 44
have some very special properties that are not germane
3. Localspinfluctuationmodel 44
to higher dimensions. Until very recently there has been
VI. Quantum many-body lattice Hamiltonians 44 a drought of experimental evidence for spins liquids in
A. HeisenbergmodelfortheMottinsulatingphase 44 higher dimensions (Lee, 2008).
1. RVBstates 44
In1987Anderson(Anderson,1987),stimulatedbythe
2. Isotropictriangularlattice 45
discovery of high-T superconductivity in layered copper
3. Roleofspatialanisotropy(J(cid:48)(cid:54)=J) 45 c
4. Ringexchange 46 oxides, made a radical proposal that has given rise to
5. Dzyaloshinski-Moriyainteraction 48 lively debate ever since. We summarise Anderson’s pro-
6. Theeffectofdisorder 48 posal as:
B. Hubbardmodelontheanisotropictriangularlattice 49
1. Phasediagram 49 The fluctuating spin singlet pairs produced
2. Laddermodels 49 by the exchange interaction in the Mott in-
sulating state become charged superconduct-
VII. Emergence of gauge fields and fractionalised
ing pairs when the insulating state is de-
quasi-particles 50
A. Spinonsdeconfinewhenincommensuratephasesare stroyedbydoping,frustrationorreducedcor-
quantumdisordered 51 relations.
B. sp(N)theory 51
C. Experimentalsignaturesofdeconfinedspinons 52 These fluctuations are enhanced by spin frustration and
D. Non-linearsigmamodelsformagnons 52 low dimensionality. Furthermore, partly inspired by res-
E. Fieldtheorieswithdeconfinedspinons 53 onating valence bond (RVB) ideas from chemical bond-
F. Fieldtheorieswithbosonicspinonsandvisons 53
ing(Anderson,2008;ShaikandHiberty,2008),Anderson
G. Fieldtheorieswithfermionicspinonsandgaugefields53
proposedavariationalwavefunctionfortheMottinsula-
H. Effectivefieldtheoriesforquasi-particlesinthe
metallicphase 53 tor: a BCS superconducting state from which all doubly
occupied sites are projected out.
VIII. Relation to other frustrated systems 54
In the decades since, there has been an enormous out-
A. β-(BDA-TTP)2X 54 growth of ideas about spin liquids and frustrated quan-
B. λ-(BETS)2X 54
tumsystems,whichwewillreview. Wewillalsoconsider
C. Sodiumcobaltates 54
D. Cs2CuCl4 55 the extent to which several families of organic charge
E. Monolayersofsolid3He 55 transfer salts can be used as tuneable systems to test
F. Pyrochlores 55 suchideasabouttheinterplayofsuperconductivity,Mott
G. Kagomematerials 55
insulation, quantum fluctuations, and spin frustration.
H. Spin-1materials 56
Agoalofthisreviewisnottobeexhaustivebutrather
I. Cuprates 56
J. J1−J2 model 56 to be pedagogical, critical, and constructive. We will
K. Shastry-Sutherlandlattice 56 attempt to follow the goals for such reviews proposed
L. Surfaceof1T-TaSe2 57 long ago (Herring, 1968).
M. Honeycomblattice 57
IX. Alternative models of organic charge transfer
A. Motivation: frustration, spin liquids, and spinons
salts 57
A. Quarterfilledmodels 57
B. Theroleofphonons 58 1. Key questions
C. Weak-coupling,spinfluctuations,andtheFermi
surface 58 A major goal for this review will be to address the
following questions:
X. Conclusions 59
A. Someopenquestions 60
1. Is there a clear relationship between superconduc-
Acknowledgements 61 tivity in organic charge transfer salts and in other
3
strongly correlated electron systems?
Continuum
 =ield
 theory

Quasi-­‐particles
  Gauge
 =ields

2. Are there materials for which the ground state of
the Mott insulating phase is a spin liquid?
Lattice
 model
 Hamiltonian

3. What is the relationship between spin liquids and
superconductivity? In particular, does the same Localised
 spins
  Fermions

fermionic pairing occur in both?
Schrodinger's
 equation
 &
 Coulomb's
 law

4. What are the quantum numbers (charge, spin,
statistics) of the quasiparticles in each phase? Electrons
 in
 atomic
 orbitals
 &
 molecular
 orbitals

5. Are there deconfined spinons in the insulating FIG. 1 The hierarchy of objects and descriptions associated
phase of any of these materials? with theories of organic charge transfer salts. The arrows
point in the direction of decreasing length scales, increasing
energy scales, and increasing numbers of degrees of freedom.
6. Can spin-charge separation occur in the metallic
At the level of quantum chemistry (Schro¨dinger’s equation
phase?
and Coulomb’s law) one can describe the electronic states
of single (or pairs of) molecules in terms of molecular or-
7. In the metallic phase close to the Mott insulating bitals (which can be approximately viewed as superpositions
phase is there an anisotropic pseudogap, as in the ofatomicorbitals). Justafewofthesemolecularorbitalsin-
cuprates? teractsignificantlywiththoseofneigbouringmoleculesinthe
solid. Low-lyingelectronicstatesofthesolidcanbedescribed
8. What is the simplest low-energy effective quantum in terms of itinerant fermions on a lattice and an effective
HamiltoniansuchasaHubbardmodel(seeSectionVI.B).In
many-body Hamiltonian on a lattice that can de-
theMottinsulatingphasetheelectronsarelocalisedonsingle
scribeallpossiblegroundstatesofthesematerials?
lattice sites and can described by a Heisenberg spin model
(see Section VI.A). The low-lying excitations of these lattice
9. Can a RVB variational wave function give an ap-
Hamiltonians and long-wavelength properties of the system
propriatetheoreticaldescriptionofthecompetition may have a natural description in terms of quasi-particles
betweentheMottinsulatingandthesuperconduct- whichcanbedescribedbyacontinuumfieldtheorysuchasa
ing phase? non-linearsigmamodel. Atthislevelunexpectedobjectsmay
emergesuchasgaugefieldsandquasi-particleswithfractional
10. Is there any significant difference between destroy- statistics (see Section VII).
ing the Mott insulator by hole doping and by in-
creasing the bandwidth?
2. A hierarchy of theories: from quantum chemistry to field
theory
11. Forsystemsclosetotheisotropictriangularlattice,
does the superconducting state have broken time-
The quantum many-body physics of condensed mat-
reversal symmetry?
ter provides many striking examples of emergent phe-
nomena at different energy and length scales (Anderson,
12. How can we quantify the extent of frustration?
1972;Coleman,2003;LaughlinandPines,2000;McKen-
Are there differences between classical and quan-
zie, 2007; Wen, 2004). Figure 1 illustrates how this is
tum frustration? If so what are the differences?
played out in the molecular crystals, which form the fo-
cus of this article, showing the stratification of differ-
13. What is the relative importance of frustration and
ent theoretical treatments and the associated objects.
static disorder due to impurities?
It needs to be emphasized that when it comes to the-
oretical descriptions going up the hierarchy is extremely
14. Is the “chemical pressure” hypothesis valid?
difficult, particularly determining the quantum numbers
of quasi-particles and the effective interactions between
15. Istherequantumcriticalbehaviourassociatedwith
them, starting from a lattice Hamiltonian.
quantum phase transitions in these materials?
16. Do these materials illustrate specific “organising
3. Organic charge transfer salts are an important class of
principles” that are useful for understanding other materials
frustrated materials?
Organic charge transfer salts have a number of fea-
At the end of the review we consider some possible turesthatmakethemaplaygroundforthestudyofquan-
answers to these questions. tummany-bodyphysics. Theyhaveofseveralproperties
Imada,Fujimori,and4Tokura: Metal-insulatortransitions 1041
that are distinctly different from other strongly corre-
Mott,1990),inhisoriginalformulationMottarguedthat
lated electron materials, such as transition metal oxides
theexistenceoftheinsulatordidnotdependonwhether
and intermetallics. These properties include:
the system was magnetic or not.
They are available in ultra-pure single crystals, Slater(1951),ontheotherhand,ascribedtheoriginof
•
which allow observation of quantum magnetic os- the insulating behavior to magnetic ordering such as the
cillations such as the de Haas van Alphen effect. antiferromagnetic long-range order. Because most Mott
insulators have magnetic ordering at least at zero tem-
The superconducting transition temperature and
• uppercriticalfieldarelowenoughthatonecande- perature, the insulator may appear due to a band gap
stroy the superconductivity and probe the metallic generatedbyasuperlatticestructureofthemagneticpe-
state in steady magnetic fields less than 20 Tesla. riodicity. In contrast, we have several examples in which
As a result, one can observe rich physics in exper- spin excitation has a gap in the Mott insulator without
imentally accessible magnetic fields and pressure magnetic order. One might argue that this is not com-
ranges
patible with Slater’s band picture. However, in this case,
both charge and spin gaps exist similarly to the band
Chemicalsubstitutionprovidesameanstotunethe FIG. 1. Metal-insulator phase diagram based on the Hubbard
• ground state. mFIoGd.el2inStchheempalatnicepohfasUe/dtiaagnrdamfillainssgocnia.tTedhewsithhadtheedMaroetat-is in insulator. This could give an adiabatic continuity be-
Hubbard metal-insulator transition. [Figure after Reference
tween the Mott insulator and the band insulator, which
p(rIminacdipaleetmael.t,al1li9c98b)u].t uTnhdeerMtohtet sintrsounlagtiinngflupehnacseeoocfctuhres matetal-
Chemical doping (and the associated disorder) is
• ihnasluflafitlolirngtraannsditiwohne,ninthwehoicnh-sictaerrCieorusloamreb eraepsiulylsioloncaUlizeisd by we discuss in Sec. II.B.
not necessary to induce transitions between differ-
emxutrcihnsliacrgfoerrctehsansutchheahsoprapnindgomenneersgsyatnadnedletchteroans-sloactitaicteedcou- In addition to the Mott insulating phase itself, a more
ent phases.
pblainngd.wTiwdtohr.ouAtetsrafnosrittihoenMtoITa m(meteatlalilc-inpshualsaetoorccturarsnseiittihoenr) are difficult and challenging subject has been to describe
by doping away from half filling [FC-MIT= filling controlled
Thesematerialsarecompressibleenoughthatpres- shown: the FC-MIT (filling-control MIT) and the BC-MIT and understand metallic phases near the Mott insulator.
• metal-insulatortransition]orbydecreasingtheratioU/t[BC-
sures of the order of kbars can induce transitions (bandwidth-control MIT). In this regime fluctuations of spin, charge, and orbital
MIT = bandwidth controlled metal-insulator transition]. In
between different ground states. correlations are strong and sometimes critically en-
the cuprates a FC-MIT occurs whereas in the organic charge
transfer salts considered in this review one might argue that hanced toward the MIT, if the transition is continuous
Consequently, over the past decade it has been possi- BC-fiMnIdTtohcceurresf.oOrentahlel otthheerohtahnedr,paetrohmapssooncecushpoiueldd,coann-d in or weakly first order. The metallic phase with such
ble to observe several unique effects due to strongly cor- sideorradethrirtdocgoe-otrtdhinraotue,ghthtehferulsattrtaitcioenh,ainveadtdoitsiopnentodtahelong strong fluctuations near the Mott insulator is now often
related electrons, sometimes phenomena that have not fillintigmaendinbaniodnwsidatlhr.eaTdhyisowcocuulpdieledadbtyo tohtehenrotieonlecotfarons.
called the anomalous metallic phase. A typical anoma-
been seen in inorganic materials. These significant ob- frustration controlled transition (FrC-MIT). In the Hubbard
This needs a considerable addition of energy and so
lous fluctuation is responsible for mass enhancement in
servations include: model on the anisotropic lattice at half-filling for fixed U/t
is extremely improbable at low temperatures.’’
increasing the hopping t(cid:48)/t can drive an insulator to metal V O , where the specific-heat coefficient￿and the Pauli
2 3
Magnetic field induced superconductivity. Ttrhaenssietioonb(sceormvaptairoenFsigluaruen3c9h)e.d[Cthopeyrliognhtg(a19n9d8)cboynttihneuing paramagnetic susceptibility ￿ near the MIT show sub-
•
hAimsteorricyanofPthhyesicfiaellSdocoieftys.t]rongly correlated electrons, par- stantial enhancement from what would be expected
A first-order transition between a Mott insulator
• ticularly the effort to understand how partially filled fromthenoninteractingbandtheory.Tounderstandthis
and superconductor induced with deuterium sub-
bands could be insulators and, as the history developed, mass enhancement, the earlier pioneering work on the
stitution, anionsubstitution, pressure, ormagnetic
Figure2illustratesschematicallytwopossibledifferent
field. how an insulator could become a metal as controllable MIT by Hubbard (1964a, 1964b) known as the Hubbard
routestodestroyingtheMottinsulatingphase, eitherby
parameters were varied. This transition illustrated in approximation was reexamined and treated with the
varying the band filling or by varying the bandwidth.
A valence bond solid in a frustrated antiferromag- Fig. 1 is called the metal-insulator transition (MIT). The Gutzwiller approximation by Brinkmann and Rice
• Another possible route is by varying the amount of frus-
net.
itnrsautiloantinogf pthheassepianndinittesraflcuticotnusa.tioAnnsiinmmpoerttaalnstacroenisned-eed (1970).
A spin liquid in a frustrated antiferromagnet. tqhueenmcoesotfoAuntdsetarsnodninRgVaBn’sdtphreoormyionfetnhteffielalitnugrecsonotfrosltlreodngly Fermi-liquid theory asserts that the ground state and
• cmoertraell-aintesudlaetloercttrroannssitaionnd(FhCav-Me IlTon)gisbtheeatntcheen“tprareletxo- re- low-energy excitations can be described by an adiabatic
Novel critical exponents near the critical point of siestairncghminagtnheitsicfiesilndg.let pairs of the insulating state be- switching on of the electron-electron interaction. Then,
•
Mott metal-insulator transition. coImnetchhearpgaesdtssuixpteyrcyoenadrusc,tminugcphaiprsrowghreenssthhaesinbseuelantomrade naively, the carrier number does not change in the adia-
Collapse of the Drude peak in the optical con- firsodmopbeodthsutffihceioernetltyicastlraonngdlye”xp(Aernidmeersnotna,l1s9id8e7)s.inItuinsder- batic process of introducing the electron correlation, as
• ductivity (a signature of the destruction of quasi- stthaenrdefionrgesimtroponrgtlayntcotroreulnadteerdstealnedctwrohnetshaenrdthMisIeTxste.nIdnsthe- is celebrated as the Luttinger theorem. Because the
to the bandwidth controlled metal-insulator transition
particles) above temperatures of order of tens of oretical approaches, Mott (1949, 1956, 1961, 1990) took Mott insulator is realized for a partially filled band, this
(BC-MIT) were one has equal numbers of “holons” and
Kelvin in the metallic phase. the first important step towards understanding how adiabatic continuation forces the carrier density to re-
“doublons”. Moregenerally,animportantquestion,that
electron-electron correlations could explain the insulat- main nonzero when one approaches the MIT point in
BulkmeasurementoftheFermisurfaceusingangle- has not yet received adequate attention, is what are the
• dependent magnetoresistance. isnimgislataritteie,saannddwdieffecraenllcethsibsetswtaeteentthhee FMCo-MttIiTnsaunldattohre. He the framework of Fermi-liquid theory. Then the only
cBoCn-sMidIeTr?ed a lattice model with a single electronic or- way to approach the MIT in a continuous fashion is the
Low superfluid density in a weakly correlated bital on each site. Without electron-electron interac- divergence of the single-quasiparticle mass m* (or more
•
metal. tions,asinglebandwouldbeformedfromtheoverlapof strictly speaking the vanishing of the renormalization
t4h.eWahtaotmariecsopirnbliitqaulidss?in this system, where the band be- factor Z) at the MIT point. Therefore mass enhance-
Multi-ferroic states.
• comes full when two electrons, one with spin-up and the ment as a typical property of metals near the Mott insu-
Superconductivity near a charge ordering transi- othTehriswqiuthestsiponinh-dasowrence,notlcycubpeeynereavcihewseitdei.nHdeotwaielv(eBra,-two lator is a natural consequence of Fermi-liquid theory.
•
tion. elelenctst,ro2n01s0s;itNtionrgmoanndt,h2e0s0a9m; Seascihtdeewv,o2u0ld09fae)e.lTahlearregaerCe ou- If the symmetries of spin and orbital degrees of free-
lomb repulsion, which Mott argued would split the band dom are broken (either spontaneously as in the mag-
in two: The lower band is formed from electrons that netic long-range ordered phase or externally as in the
occupied an empty site and the upper one from elec- case of crystal-field splitting), the adiabatic continuity
trons that occupied a site already taken by another elec- assumed in the Fermi-liquid theory is not satisfied any
tron. With one electron per site, the lower band would more and there may be no observable mass enhance-
be full, and the system an insulator. Although he dis- ment.Infact,aMITwithsymmetrybreakingofspinand
cussed the magnetic state afterwards (see, for example, orbitaldegreesoffreedomisrealizedbythevanishingof
Rev.Mod.Phys.,Vol.70,No.4,October1998
5
several alternative definitions. The definition that we Sachdev (Sachdev, 2009b) pointed out that such
thinkisthemostilluminating,becauseitbringsouttheir Heisenberg models have possible ground states in four
truly exotic nature, is the following. classes: Neel order, spiral order, a valence bond crystal,
or a spin liquid. Examples of the first two occur on the
A spin liquid has a ground state in which square and the triangular lattices respectively. For both
there is no long-range magnetic order and casesspinrotationalsymmetryandlatticesymmetryare
no breaking of spatial symmetries (rotation broken. Foravalencebondcrystal,onlythespatialsym-
or translation) and which is not adiabatically metry is broken. It may be that valence bond crystal
connected to the band (Bloch) insulator. ground state occurs on the anisotropic triangular lattice
(cf. Section VI.A).
One can write down many such quantum states. Indeed,
Normand (Normand, 2009) considered three different
Wen classified hundreds of them for the square lattice
classesofspinliquids,eachbeingdefinedbytheirexcita-
(Wen, 2002). But the key question is whether such a
tion spectrum. If we denote the energy gap between the
state can be the ground state of a physically realistic
singlet ground state and the lowest-lying triplet state by
Hamiltonian. A concrete example is the ground state of
∆ and the gap to the first excited singlet state by ∆ .
T S
theone-dimensionalantiferromagneticHeisenbergmodel
The three possible cases are:
withnearest-neighbourinteractions. However,despitean
exhaustive search since Anderson’s 1987 Science paper, 1. ∆S =0 and ∆T =0.
(cid:54) (cid:54)
(Anderson, 1987) it seems extremely difficult to find a
2. ∆ =0 and ∆ =0.
physicallyrealisticHamiltonianintwodimensionswhich S T (cid:54)
has such a ground state. 3. ∆ =∆ =0.
S T
As far as we are aware there is still no definitive
counter-example to the following conjecture: Normand refers to the first two as Type I and Type II
respectively. ThethirdcaseisreferredtoasanAlgebraic
Consider a family of spin-1/2 Heisenberg spinliquid. Thecase∆T =0and∆S =0isnotanoption
(cid:54)
models on a two-dimensional lattice with because, by Goldstone’s theorem, it would be associated
shortrangeantiferromagneticexchangeinter- with broken spin-rotational symmetry.
actions (pairwise, ring exchange and higher An important question is how to distinguish these dif-
ordertermsareallowed). TheHamiltonianis ferent states experimentally. It can be shown that for a
invariantunderSU(2) L,whereLisaspace singlet ground state at zero temperature singlet excited
×
group and there is a non-integer total spin states do not contribute to the dynamical spin suscep-
in the repeat unit of the lattice Hamiltonian. tibility. If the susceptibility is written in the spectral
Let γ be a parameter which can be used to representation, 1
distinguishdifferentHamiltoniansinthefam-
nS+( q)0 2
ily (e.g., it could be the relative magnitude χ (q,ω)= exp( β(E E ))|(cid:104) | − | (cid:105)| ,
−+ n 0
of different interaction terms in the Hamilto- − − En E0 ω
n − −
nian). Thenanon-degenerategroundstateis (cid:88) (1)
onlypossiblefordiscretevaluesofγ(e.g.,ata it is clear that the matrix elements of the spin operators
quantum critical point). In other words, the between the singlet ground state and any singlet excited
ground state spontaneously breaks at least state must be zero. This means that at low tempera-
one of the two symmetries SU(2) and L over tures, only triplet excitations contribute to the uniform
all continuous ranges of γ. magneticsusceptibility,theNMRrelaxationrate,Knight
shift, and inelastic neutron scattering cross section. In
Therequirementofnon-integerspinintherepeatunit contrast, both singlet and triplet excitations contribute
ensuresthatthegeneralisationoftheLieb-Schultz-Mattis tothespecificheatcapacityandthethermalconductivity
theoremtodimensionsgreaterthanone(Aletetal.,2006; at low temperatures. Hence, comparing the temperature
Hastings, 2004) does not apply. The theorem states that dependence of thermal and magnetic properties should
forspin-1/2systemswithonespinperunitcellonatwo- allowonetodistinguishTypeIspinliquidsfromTypeII
dimensionallattice,ifthegroundstateisnon-degenerate spin liquids. Furthermore, the singlet spectrum will not
and there is no symmetry breaking, one cannot have a shiftinamagneticfieldbutthetripletswillsplitandthe
non-zero energy gap to the lowest excited state. Note corresponding spectral weight be redistributed.
that,thetriangular,kagome,andpyrochlorelatticescon- Oneimportantreasonforwantingtounderstandthese
tain one, three, and four spins per unit cell respectively details of the spin liquid states is that the spin excita-
(Normand, 2009). Hence, Hasting’s theorem cannot be tion spectrum may well be important for understanding
used to rule out a spin liquid for the pyrochlore lattice.
One of the best candidate counter examples to the
aboveconjectureistheHeisenbergmodelonthetriangu-
larlatticewithringexchange(LiMingetal.,2000)which 1 Here β =1/kBT is the inverse temperature and S±(q) are the
will be discussed in more detail in Section VI.A. spinraising/loweringoperators.
6
unconventional superconductivity. This has led to a lot are fully gapped and may have either bosonic,
of attention being paid to a magnetic resonance seen by fermionic, or fractional statistics.
inelastic neutron scattering the cuprates. It is still an
Fermi spin liquid: spinons are gapless and are de-
openquestionastowhetherthistripletexcitationiscor-
scribedbyaFermiliquidtheory(thespinon-spinon
related with superconductivity (Chubukov et al., 2006;
interactions vanish as the Fermi energy is ap-
Cuketal.,2004;HaoandChubukov,2009;Hwangetal.,
proached).
2004). Strong coupling RVB-type theories focus on sin-
gletexcitationswhereasweak-couplingantiferromagnetic Bose spin liquid: low-lying gapless excitations are de-
spinfluctuationtheoriesfocusontripletexcitations. This scribed by a free-boson theory.
importantdifferenceisemphasizedanddiscussedinare-
view on the cuprates (Norman, 2006). Algebraic spin liquid: spinons are gapless, but they
arenotdescribedbyfreefermionicandfreebosonic
quasiparticles.
5. What are spinons?
A key question is what are the quantum numbers and 6. Antiferromagnetic fluctuations
statistics of the lowest lying excitations. In a Neel or-
dered antiferromagnet these excitations are “magnons” Ithasbeenproposedthataninstabilitytoad-wavesu-
or“spinwaves”whichhavetotalspinoneandobeyBose- perconducting state can occur in a metallic phase which
Einstein statistics (Auerbach, 1994). Magnons can be is close to an antiferromagnetic instability (Scalapino
viewedasaspinflippropagatingthroughthebackground et al., 1986). This has been described theoretically by
of Neel ordered spins. They can also be viewed as the an Eliashberg-type theory in which the effective pairing
Goldstonemodesassociatedwiththespontaneouslybro- interaction is proportional to the dynamical spin suscep-
ken symmetry of the ground state. tibility, χ(ω,(cid:126)q) (Moriya and Ueda, 2003). If this quan-
In contrast, in a one-dimensional antiferromagnetic tity has a significant peak near some wavevector then
spin chain (which has a spin liquid ground state) the that will significantly enhance the superconducting Tc in
lowest lying excitations are gapless spinons which have aspecificpairingchannel. NMRrelaxationratesarealso
total spin-1/2 and obey “semion” statistics, which are determined by χ(ω,(cid:126)q) and so NMR can provide useful
intermediate between fermion and boson statistics (i.e. information about the magnetic fluctuations. For exam-
thereisaphasefactorofπ/2associatedwithparticleex- ple, a signature of large antiferromagnetic fluctuations is
change) (Haldane, 1991). The spinons are “deconfined” thedimensionlessKorringaratiothatismuchlargerthan
in the sense that if a pair of them are created (for exam- one.
ple, in an inelastic neutron scattering experiment) with From a local picture one would like to know the
different momentum then they will eventually move in- strengthoftheantiferromagneticexchangeJ betweenlo-
finitely far apart. Definitive experimental signatures of calised spins in the Mott insulating and the bad metallic
this deconfinement are seen in the dynamical structure phase. In RVB theory J sets the scale for the supercon-
factor S(ω,(cid:126)q) which shows a continuum of low-lying ex- ductingtransitiontemperature. Itisimportanttorealise
citations rather than the sharp features associated with that this is very different from picture of a “glue” in the
spinwaves. ThisisclearlyseeninthecompoundKCuF , Eliashberg-type theories where superconductivity arises
3
whichiscomposedoflinearchainsofspin-1/2copperions due to the formation of Cooper pairs between Fermi liq-
(Tennant et al., 1995). The most definitive evidence for uid quasi-particles (Anderson, 2007; Maier et al., 2008)
suchexcitationsinarealtwo-dimensionalmaterialcomes (also see section IX.C).
from Cs CuCl (Coldea et al., 2003; Kohno et al., 2007)
2 4
above the Neel ordering temperature. Below the Neel
7. Quantum critical points
temperature these excitations become confined into con-
ventionalmagnons(Fjærestadetal.,2007;Starykhetal.,
Figure 3 shows a schematic phase diagram associated
2010). It is an open theoretical question as to whether
with a quantum critical point (Coleman and Schofield,
thereisanytwo-dimensionalHeisenbergmodelwithsuch
2005; Sachdev, 1999). We will discuss the relevance of
excitationsatzerotemperature,otherthanataquantum
suchdiagramstotheorganicchargetransfersaltsbelow.
critical point (Singh, 2010).
We will see that some of the theoretical models (such as
What type of spinon statistics might be possible in
the Heisenberg model on an anisotropic triangular lat-
two dimensions? Wen used quantum orders and projec-
tice)doundergoaquantumphasetransitionfromamag-
tivesymmetrygroups,toconstructhundredsofsymmet-
neticallyorderedtoaquantumdisorderedphasewithan
ric spin liquids, having either SU(2), U(1), or Z gauge
2
energy gap to the lowest lying triplet excitation.
structures at low energies (Wen, 2002). He divided the
A particularly important question is whether any sig-
spin liquids into four classes, based on the statistics of
naturesofquantumcriticalbehaviorhavebeenseeninor-
the quasi-particles and whether they were gapless:
ganicchargetransfersites. Mosttransitionsatzerotem-
Rigid spin liquid: spinons (and all other excitations) peraturearefirst-order. Perhaps,theclearestevidenceof
7
Curie-Weiss theory and dynamical mean-field the-
ory (Section III.D).
In Heisenberg models frustrated spin interactions
•
produce incommensurate correlations. These can
also change the symmetry of the superconducting
pairing (Powell and McKenzie, 2007), lead to new
tripletexcitations(phasons)(Chandraetal.,1990),
and the emergence of new gauge fields which are
deconfining (Section VII).
Frustration of kinetic energy (such as in non-
•
bipartite lattices or by next-nearest-neighbor hop-
ping) reduces nesting of the Fermi surface and sta-
bilises the metallic state (Section VI.B).
FIG. 3 Schematic phase diagram associated with a quan-
tum critical point. The vertical axis is temperature and the 1. Reduction of the correlation length
horizontal axis represents a coupling constant, g. Quantum
fluctuationsincreasewithincreasingg andforacriticalvalue The temperature dependence of the correlation length
gc thereisaquantumphasetransitionfromanorderedphase ξ(T) and the static structure factor S(Q(cid:126)), associated
(with broken symmetry) to a disordered phase, usually as-
sociated with an energy gap, ∆ ∼ (g−g )zν where z is the with the classical ordering wavevector Q(cid:126) has been cal-
c
dynamicalcriticalexponentandν isthecriticalexponentas- culated for both the triangular lattice and square lattice
sociated with the correlation length ξ ∼ |g−g |−ν. In the Heisenberg models using high-temperature series expan-
c
quantum critical region the only energy scale is the temper- sions (Elstner et al., 1993, 1994). For the triangular lat-
ature and the correlation length ξ ∼ 1/T1/z. In this region tice the correlation length has values of about 0.5 and 2
therearealsonoquasi-particles(i.e.,anysingularitiesinspec- lattice constants, at temperatures T = J and T = 0.2J,
tral functions are not isolated poles but rather branch cuts). respectively. In contrast, the model on the square lat-
tice has correlation lengths of about 1 and 200 lattice
constants, at T = J and T = 0.2J, respectively. At
quantum critical fluctuations come from the NMR spin T = 0.2J the static structure factor has values of about
relaxationrateinκ-(BEDT-TTF)2Cu2(CN)3,whichwill 1and3000forthetriangularandsquarelattices, respec-
be discussed in Section VII.D. tively. Hence, frustration leads to a significant reduction
of the spin correlation length. These distinct differences
in temperature dependence can be understood in terms
B. Key consequences of frustration of frustration producing a ‘roton’ like minimum in the
triplet excitation spectra of the triangular lattice model
We briefly list some key consequences of frustration. (Zheng et al., 2006).
Many of these are discussed in more detail later in the We discuss later how the temperature dependence of
review. the uniform magnetic susceptibility of several frustrated
charge transfer salts can be fit to that of the Heisen-
Frustration enhances the number of low energy ex-
berg model on the triangular lattice with J = 250 K
•
citations. This increases the entropy at low tem-
(Shimizu et al., 2003; Tamura and Kato, 2002; Zheng,
peratures (Ramirez, 1994). The temperature de-
Singh, McKenzie and Coldea, 2005). This implies that
pendence of the magnetic susceptibility is flatter
ξ 2a at 50 K. This is consistent with estimates of
and the peak occurs at a lower temperature (Sec- (cid:39)
thespin-spincorrelationlengthinorganicchargetransfer
tion I.B.3).
salts from low temperture NMR relaxation rates (Yusuf
et al., 2007).
Quantum fluctuations in the ground state are en-
•
hanced due to the larger density of states at low
energies. These fluctuations can destroy magneti-
2. Competing phases
cially ordered phases (Section VI.A).
Singlet excitations are stabilised and singlet pair- One characteristic feature of strongly correlated elec-
• ing correlations are enhanced. Resonating valence tron systems that, we believe, should be discussed more
bond states have a larger overlap with the true is how sensitive they are to small perturbations. This is
ground state of the system (Section VI.A). particularly true in frustrated systems. A related issue
is that there are often several competing phases which
Intersite correlations are reduced which enhances are very close in energy. This can make variational wave
•
the accuracy of single site approximations such as functions unreliable. Getting a good variational energy
8
may not be a good indication that the wave function
captures the key physics. Below we give two concrete
examples to illustrate this point.
Firstly, consider the spin1/2Heisenberg model onthe
isotropic triangular on a lattice of 36 sites, and with
exchange interaction J. Exact diagonalisation (Sindz-
ingre et al., 1994) gives a ground state energy per site
of 0.5604J and a net magnetic moment (with 120 de-
−
gree order as in the classical model) of 0.4, compared
to the classical value of 1/2. In contrast, a variational
short-range RVB wavefunction has zero magnetic mo-
ment and a ground state energy of 0.5579J. Yet, it
−
is qualitatively incorrect because it predicts no magnetic
order (and thus no spontaneous symmetry breaking) in
the thermodynamic limit. Note, however, that the en-
ergy difference is only J/400. [For details and references
see Table III in (Zheng et al., 2006)].
The second example concerns the spin 1/2 Heisenberg
model on the anisotropic triangular lattice, viewed as
chains with exchange J(cid:48) and frustrated interchain cou-
pling J. For J(cid:48) 3J this describes the compound
Cs CuCl . The trip∼let excitation spectrum of the model FIG. 4 Effect of frustration on the temperature dependence
2 4
of the magnetic susceptibility χ(T) for the Heisenberg model
has been calculated both with a small Dzyaloshinski-
onananisotropictriangularlattice(Zheng, Singh, McKenzie
MoriyainteractionD,andwithout(D =0). Itisstriking
andColdea,2005). Thevariationofkeyparametersisshown
that even when D J(cid:48)/20 it induces energy changes in
the spectrum of ene∼rgies as large as J(cid:48)/3, including new as a function of the ratio J(cid:48)/(J +J(cid:48)). Tp is the tempera-
ture at which the susceptibility is a maximum, with a value
energy gaps (Fjærestad et al., 2007). For J(cid:48) J the χ ≡ χ(T ). T is the Curie-Weiss temperature which can
(cid:29) p p cw
ground state turns out to be “exquisitely sensitive” to be extracted from the high-temperature dependence of the
other residual interactions as well (Starykh et al., 2010). susceptibility. B is the magnetic saturation field and Ag2
sat
is the Curie-Weiss constant. All quantities were calculated
by a high-temperature series expansion. All of the quantities
plotted have extreme values for the isotropic triangular lat-
3. Alternative measures of frustration
tice, suggesting that in some sense it is the most frustrated.
[Modified from (Zheng, Singh, McKenzie and Coldea, 2005).
Balents recently considered how to quantify the
Copyright (2005) by the American Physical Society.]
amount of frustration in an antiferromagnetic material
(or model) and its tendency to have a spin liquid ground
state(Balents,2010). Heusedameasure(Ramirez,1994)
alternative measures of frustration. The sensitivity of
f = T /T , the ratio of the Curie-Weiss tempera-
CW N
the temperature dependence of the susceptibility to the
ture T to the Neel temperature, T at which three-
CW N
ratioJ(cid:48)/J hasbeenusedtoestimatethisratioforspecific
dimensional magnetic ordering occurs.
materials (Zheng, Singh, McKenzie and Coldea, 2005).
One limitation of this measure is that it does not sep-
arate out the effects of fluctuations (both quantum and In some sense then, the temperature Tp at which the
thermal), dimensionality, and frustration. For strictly susceptibilityhasamaximumandthemagnitudeofthat
one or two dimensional systems, T is zero. For quasi- susceptibility is a measure of the amount of frustration.
N
two-dimensional systems the interlayer coupling deter- Thisisconsistentwithsomeintuition(orisitjustpreju-
mines T . Thus, f would be larger for a set of weakly dice?) that for the anistropic triangular lattice the frus-
N
coupled unfrustrated chains than for a layered triangu- tration is largest for the isotropic case. These measures
lar lattice in which the layers are moderately coupled offrustrationarenotdependentondimensionalityandso
together. do not have the same problems discussed above that the
Section II of (Zheng, Singh, McKenzie and Coldea, ratio f does. On the other hand, these measures reflect
2005)containsadetaileddiscussionoftwodifferentmea- short-rangeinteractionsratherthanthetendencyforthe
suresoffrustrationformodelHamiltonians: (1)thenum- system to fail to magnetically order.
ber of degenerate ground states, and (2) the ratio of Another issue that needs to be clarified is how one
the ground state energy to the base energy [the sum of might distinguish quantum and classical frustration. In
all bond energies if they are independently fully satis- general the nearest neighbour spin correlation f
s
fied.] This measure was introduced previously for classi- Sˆ Sˆ will be reduced by frustration. Entanglemen≡t
i j
(cid:104) · (cid:105)
cal models (Lacorre, 1987). measures from quantum information theory can be used
Figure4 showsresults thatmight be thebasis ofsome to distinguish truly quantum from classical correlations.
9
For a spin rotationally invariant state (i.e. a total spin exactly cancelled by the Aharonov-Bohm phase associ-
singlet state) f is related to a measure of entanglement ated with hopping around the cluster for a particular
s
between two spins in a mixed state, known as the con- choiceofappliedmagneticfield(BarfordandKim,1991).
currence C by (Cho and McKenzie, 2006) Thus a magnetic field may be used to lift the effects of
kinetic energy frustration. The quantum mechanical na-
C =max 0, 2f 1/2 . (2) ture of kinetic energy frustration is in distinct contrast
s
{ − − }
to geometrical frustration in antiferromagnets which can
Hence, there is maximal entanglement (C =1) when the occur for purely classical spins.
twospinsareinasingletstateandarenotentangledwith
the rest of the spins in the system. Once the spin cor-
relations decrease to f = 1/4 there is no entanglement
s
between the two spins. II. TOY MODELS TO ILLUSTRATE THE INTERPLAY
OF FRUSTRATION AND QUANTUM FLUCTUATIONS
4. Geometric frustration of kinetic energy WenowconsidersomemodelHamiltoniansonjustfour
lattice sites. The same Hamiltonians on an infinite lat-
In a non-interacting electron model we are aware of tice are relevant to the organic charge transfer salts and
only two proposed quantitative measures of the geomet- will be discussed in Sections VI.A and VI.B. Although
rical frustration of the kinetic energy. Both are based on suchsmallsystemsarefarfromthethermodynamiclimit,
the observation that, for frustrated lattices with t > 0, these models can illustrate some of the essential physics
an electron at the bottom of the band does not gain the associated with the interplay of strong electronic corre-
full lattice kinetic energy, while a hole at the top of the lations, frustration, and quantum fluctuations. These
band does. Barford and Kim (Barford and Kim, 1991) toy models illustrate the quantum numbers of impor-
suggested that for tight binding models a measure of the tantlow-lyingquantumstates,thedominantshort-range
frustration is then ∆ = (cid:15)max (cid:15)min , where (cid:15)max and correlations, and how frustration changes the competi-
(cid:15)min aretheenergies(rel|atkive|to−t|heken|ergyoftheksystem tion between these states. Furthermore, understanding
k
with no electrons) of the top and bottom of the band re- these small clusters is a pre-requisite for cluster exten-
spectively. Thisfrustrationincreasesthedensityofstates sionsofdynamicalmean-fieldtheory(Ferreroetal.,2009)
forpositiveenergiesfort>0(negativeenergiesfort<0) and rotationally invariant slave boson mean-field theory
which represents an increased degeneracy and enhances (Lechermannetal.,2007)whichdescribesbandselective
the many-body effects when the Fermi energy is in this and momentum space selective Mott transitions. Insight
regime. can also be gained by considering two, three, and four
coupledAndersonimpurities(Ferreroetal.,2007). Small
Together with Merino, we previously argued (Merino
clusters are also the basis of the contractor renormalisa-
etal.,2006)thatasimplermeasureofthekineticenergy
tion (CORE) method which has been used to study the
frustration is W/2z t, where W is the bandwidth and z
| | dopedHubbardmodel(AltmanandAuerbach,2002)and
is the coordination number of the lattice. The smaller
frustrated spin models (Berg et al., 2003).
this ratio, the stronger the frustration is, while for an
unfrustrated lattice W/2z t = 1. But, for example, on A similar approach of just considering four sites has
| |
the triangular lattice kinetic energy frustration leads to been taken before when considering the ground state of
a bandwidth, W = 9t, instead of 12t as one might aHeisenbergmodelonadepletedlatticewhichisamodel
| | | |
na¨ıvely predict from W =2z t since z =6. forCaV4O9 (Uedaetal.,1996). Theauthorsfirstconsid-
| |
We argued that geometrical frustration of the kinetic eredasingleplaquettewithfrustration,albeitalongboth
energy is a key concept for understanding the properties diagonals(seealsoSection3in(Valkovetal.,2006)). Dai
oftheHubbardmodelonthetriangularlattice. Inpartic- and Whangbo (Dai and Whangbo, 2004) considered the
ular, it leads to particle-hole asymmetry which enhances Heisenberg model on a triangle and a tetrahedra. Simi-
many-bodyeffectsforelectron(hole)dopedt>0(t<0) larfoursiteHeisenbergHamiltonianshavealsobeendis-
lattices. cussed in the context of mixed valence metallic clusters
It should be noted that geometrical frustration of the ofparticularinteresttochemists(Augustyniak-Jablokow
kinetic energy is a strictly quantum mechanical effect et al., 2005).
arising from quantum interference. This interference
arises from hopping around triangular plaquettes which
will have an amplitude proportional to t3, which clearly
changessignwhentchangessign. Incontrastonthe,un- A. Four site Heisenberg model
frustrated, square lattice the smallest possible plaquette
is the square and the associated amplitude for hopping ThefoursiteHeisenbergmodelillustratesthatfrustra-
aroundasquareisindependentofthesignoftasitispro- tion can lead to energy level crossings and consequently
portional to t4. Barford and Kim noted that the phase to changes in the quantum numbers of the ground state
collected by hopping around a frustrated cluster may be and lowest lying excited state.
10
The Hamiltonian is (see Figure 5(a)) a) J b)
1 2 H + V = H + V =
ˆ = J Sˆ1 Sˆ2+Sˆ2 Sˆ3+Sˆ3 Sˆ4+Sˆ4 Sˆ1 J J’ J ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿
H · · · · ￿ ￿ ￿ ￿
+J(cid:16)(cid:48)Sˆ1 Sˆ3. (cid:17) (3) 4 J 3 H − V = H − V =
·
c) (2,1,1), (1,1￿￿,1)￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿
It is helpful to introduce the total spin along each of the
0
diagonals, Sˆ = Sˆ +Sˆ and Sˆ = Sˆ +Sˆ , and note (0,1,1)
13 1 3 24 2 4
that these operators commute with each other and with
J-2 (1,1,0)
the Hamiltonian. The total spin of all four sites can be E/
writtenintermsoftheseoperators: Sˆ =Sˆ13+Sˆ24. Thus, -4 (S,S13,S24)=(0,0,0),
the total spin S, and the total spin along each of the (1,0,1)

two diagonals, S and S are good quantum numbers.
13 24 -6
0 2 4 6 8
The term in (3) associated with J can be rewritten as
J'/J
J/2(Sˆ2 Sˆ2 Sˆ2 ). Hence, the energy eigenvalues are
− 13− 24
FIG. 5 Eigenstates and eigenvalues of a frustrated Heisen-
1 1 berg model on a single plaquette. (a) The exchange inter-
E(S,S ,S ) = JS(S+1)+ (J(cid:48) J)S (S +1)
13 24 13 13 actions in the model. (b) The two resonating valence bond
2 2 −
stateswhichspanallthesingletstates(compareequations(5)
1 3
JS (S +1) J(cid:48). (4) and(6)). (c)Dependenceoftheenergyeigenvaluesasafunc-
24 24
−2 − 4 tionofthediagonalinteractionJ(cid:48)/J. Notethatthequantum
numbersofthelowestlyingexcitedstatechangewhenJ(cid:48) =J
Figure 5 (c) shows a plot of these energy eigenvalues as and the ground state changes when J(cid:48) = 2J. Furthermore,
a function of J(cid:48)/J. We note that the quantum numbers thetworesonatingvalencebondstatesbecomedegenerateat
of the lowest lying excited state change when J(cid:48) = J J(cid:48) =2J.
and J(cid:48) = 4J, and that the ground state changes when
J(cid:48) =2J.
The two singlet states can also be written as lin- 1. Effect of a ring exchange iteraction
ear combinations of two orthogonal valence bond states,
Consider adding to Hamiltonian (3) the term
denote H and V , which descibe a pair of singlets
| (cid:105) | (cid:105)
along the horizontal and vertical directions, respectively
(see Figure 5 (b)). The state with quantum numbers ˆ(cid:3) = J(cid:3)(Pˆ1234+Pˆ4321) (7)
H
(S,S13,S24)=(0,0,0) is = J(cid:3)(Pˆ12Pˆ34+Pˆ14Pˆ23 Pˆ13Pˆ24+Pˆ13+Pˆ24 1)
− −
1 whereJ(cid:3) describesthering-exchangeinteractionaround
|0,0,0(cid:105)= √2(|H(cid:105)−|V(cid:105)) (5) a single plaquette, the operator Pˆ12 = 2Sˆ1 Sˆ2 + 1/2
·
permutes spins 1 and 2, and Pˆ is the permutation
1234
and the state with (S,S ,S )=(0,1,1) state is operator around the plaquette (Misguich and L’huillier,
13 24
2005; Thouless, 1965).
1 Intuitively,
0,1,1 = (H + V ). (6)
| (cid:105) √2 | (cid:105) | (cid:105) ˆ(cid:3) H =2J(cid:3) V ˆ(cid:3) V =2J(cid:3) H (8)
H | (cid:105) | (cid:105) H | (cid:105) | (cid:105)
Both of these singlet states are resonating valence bond
Hence, the RVB states (5) and (6) are eigenstates of the
states (see Figure 5 (b)).
ring-exchange Hamiltonian with eigenvalues 2J(cid:3) and
The Hamiltonian has C2v with the C2 axes along each 2J(cid:3), respectively. Hence, ring exchange has−a similar
diagonal (and out of the plane). The two singlet states effect to the diagonal interaction in that it stabilises the
above have A1 and A2 symmetry, respectively. However, state 0,0,0 .
if J(cid:48) = 0 there is C symmetry and the (0,0,0) and | (cid:105)
4v
(0,1,1) states have A and B symmetry, respectively.
1 1
The latter, which is the ground state, connects naturally B. Four site Hubbard model
to the B symmetry of a d superconducting order
1 x2−y2
parameter on the square lattice. Acomprehensivestudyof thet(cid:48) =0model(whichhas
It is possible to relate the two singlet states to the C symmetry)hasbeengivenbySchumann(Schumann,
4v
physical states of a Z gauge field on a single plaquette 2002). TheanalysisissimplifiedbyexploitingthisSU(2)
2
(see Section 3.2 of (Alet et al., 2006)). The gauge flux symmetry associated with particle-hole symmetry (Noce
operator on the plaquette F flips the bonds between andCuoco,1996). Inparticular,theHamiltonianmatrix
p
horizontal and vertical. The RVB states (5) and (6) are thendecomposesintoblocksofdimension3orless. Schu-
eigenstates of F with eigenvalues 1. mann has also solved the model on a tetrahedron and a
p
±