b. j. powell and ross h. mckenzie- quantum frustration in organic mott insulators: from spin liquids...

8/3/2019 B. J. Powell and Ross H. McKenzie- Quantum frustration in organic Mott insulators: from spin liquids to unconventio

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Quantum frustration in organic Mott insulators: from spin liquids tounconventional superconductors

B. J. Powell and Ross H. McKenzie

Department of Physics, University of Queensland, Brisbane, 4072, Australia

(Dated: August 2, 2010)

We review the interplay of frustration and strong electronic correlations in quasi-two-dimensionalorganic charge transfer salts, such as BEDT-TTF2X and EtnMe4nP n[Pd(dmit)2]2. These twoforces drive a range of exotic phases including spin liquids, valence bond crystals, pseudogappedmetals, and unconventional superconductivity. Of particular interest is that in several materialsthere is a direct transition as a function of pressure from a spin liquid Mott insulating state toa superconducting state. Experiments on these materials raise a number of profound questionsabout the quantum behaviour of frustrated systems, particularly the intimate connection betweenspin liquids and superconductivity. Insights into these questions have come from a wide range oftheoretical techniques including first principles electronic structure, quantum many-body theoryand quantum field theory. In this review we introduce the basic ideas of the field by discussing asimple frustrated Heisenberg model with four spins. We then describe the key experimental results,emphasizing that for two materials, -(BEDT-TTF)2Cu2(CN)3 and Et1Me3Sb[Pd(dmit)2]2, thereis strong evidence for a spin liquid ground state, and for Et1Me3P[Pd(dmit)2]2, a valence bondsolid ground state. We review theoretical attempts to explain these phenomena, arguing that

this can be captured by a Hubbard model on the anisotropic triangular lattice at half filling, andthat resonating valence bond wavefunctions can capture most of the essential physics. We reviewevidence that this model can have a spin liquid ground state for a range of parameters that arerealistic for the relevant materials. We conclude by summarising the progress made thus far andidentifying some of the key questions still to be answered.

Contents

I. Introduction 2A. Motivation: frustration, spin liquids, and spinons 2

1. Key questions 22. A hierarchy of theories: from quantum chemistry

to field theory 33. Organic charge transfer salts are an important

class of materials 34. What is a spin liquid? 45. What are spinons? 56. Antiferromagnetic fluctuations 67. Quantum critical points 6

B. Key consequences of frustration 61. Estimates of the correlation length 72. Competing phases 73. Measures of frustration 74. Geometric frustration of kinetic energy 8

II. Toy models to illustrate the interplay offrustration and quantum fluctuations 9A. Four site Heisenberg model 9

1. Effect of a ring exchange iteraction 9

B. Four site Hubbard model 10

III. -(BEDT-TTF)2X 10A. Crystal and electronic structure 11

1. Dimer model of the band structure of-(BEDT-TTF)2X 11

2. The Hubbard U 133. The (BEDT-TTF)2 dimer 15

Electronic address: [email protected]

B. Insulating phases 16

1. Antiferromagnetic and spin liquid phases 16

2. Is the spin liquid gapped? 18

3. The 6 K anomaly 19

C. Mott metal insulator transition 19

1. Critical exponents of the Mott transition 20

2. Optical conductivity 20

3. The spin liquid to metal transition 22

4. Reentrance of the Mott transition - explanationfrom undergraduate thermodynamics 22

D. Magnetic frustration in the normal state 23

1. Dynamical mean-field theory (DMFT) 23

2. Fermi liquid regime 24

3. NMR and the pseudogap 25

4. There is no pseudogap in-(BEDT-TTF)2Cu2(CN)3 27

5. Other evidence for a pseudogap in the weaklyfrustrated materials 27

6. Tests of the pseudogap hypothesis 27

7. The Nernst effect and vortex fluctuations aboveTc 28

E. The superconducting state 29

1. -(BEDT-TTF)2Cu2(CN)3 29

2. Weakly frustrated materials 30

IV. -Z[Pd(dmit)2

]2

31

A. Crystal and electronic structure 32

B. Frustrated antiferromagnetism 34

C. Spin liquid behaviour in -Me3EtSb-[Pd(dmit)2]2(Sb-1) 34

D. Is there a valence bond crystal or spin Peierls state in-Me3EtP-[Pd(dmit)2]2 (P-1)? 36

E. Paramagnetic to non-magnetic transition inEt2Me2Sb[Pd(dmit)2]2 (Sb-2) and Cs[Pd(dmit)2]2(Cs-00) 38

arXiv:1007.5381v1

[cond-mat.str-el]

30Jul2010
mailto:[email protected]:[email protected]


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1. Et3MeSb impurities in Et2Me2Sb[Pd(dmit)2]2(Sb-2) 39

F. Mott transition under hydrostatic pressure anduniaxial stress 40

V. Nuclear magnetic resonance as a probe of spinfluctuations 41

1. Long-range antiferromagnetic spin fluctuationmodel 41

2. Quantum critical spin fluctuation model 42

3. Local spin fluctuation model 42

VI. Quantum many-body lattice Hamiltonians 42A. Heisenberg model for the Mott insulating phase 42

1. RVB states 432. Isotropic triangular lattice 433. Role of spatial anisotropy (J = J) 444. Ring exchange 455. Dzyaloshinski-Moriya interaction 466. The effect of a magnetic field 467. The effect of disorder 46

B. Hubbard model on the anisotropic triangular lattice 471. Phase diagram at zero temperature 472. Phase diagram at non-zero temperature 47

VII. Emergence of gauge fields and fractionalised

quasi-particles 47A. Spinons deconfine when incommensurate phases are

quantum disordered 48B. sp(N) theory 49C. Experimental signatures of deconfined spinons 50D. Non-linear sigma models for magnons 50E. Field theories with deconfined spinons 50F. Field theories with bosonic spinons and visons 50G. Field theories with fermionic spinons and gauge fields51H. Effective field theories for quasi-particles in the

metallic phase 51

VIII. Relation to other frustrated materials 51A. -(BDA-TTP)2X 52B. Cobaltates 52C. Cs2CuCl4 52

D. Monolayers of solid 3He 52E. Pyrochlores 52F. Kagome materials 52G. Spin-1 materials 53H. Cuprates 53

I. Shastry-Sutherland lattice 53J. Surface of 1T-TaSe2 53

K. Honeycomb lattice 53

IX. Conclusions 54A. Some open questions 55

Acknowledgements 56

References 56

I. INTRODUCTION

In the early 1970s, Anderson and Fazekas (Anderson,1973; Fazekas and Anderson, 1974) proposed that theground state of the Heisenberg spin-1/2 model on thetriangular lattice did not break spin rotational symme-try, i.e., had no net magnetic moment. A state of mattercharacterised by we defined local moments and the ab-sence of long range order has become known as a spin

liquid (Normand, 2009). Such states, are known in one-dimensional (1d) systems, but 1d systems have some veryspecial properties that are not germane to higher dimen-sions. Until very recently there has been a drought ofexperimental evidence for spins liquids in higher dimen-sions (Lee, 2008).

In 1987 Anderson (Anderson, 1987), stimulated by thediscovery of high-Tc superconductivity in layered copper

oxides, made a radical proposal that has given rise to heatdebate ever since. We summarise Andersons proposal as:

The fluctuating spin singlet pairs producedby the exchange interaction in the Mott in-sulating state become charged superconduct-ing pairs when the insulating state is de-stroyed by doping, frustration or reduced cor-relations.

These fluctuations are enhanced by spin frustration andlow dimensionality. Furthermore, partly inspired by res-onating valence bond (RVB) ideas from chemical bonding(Shaik and Hiberty, 2008), Anderson proposed a varia-

tional wave function for the Mott insulator: a BCS su-perconducting state from which all doubly occupied sitesare projected out.

In the decades since, there has been an enormous out-growth of ideas about spin liquids and frustrated quan-tum systems, which we will review. We will also considerthe extent to which several families of organic chargetransfer salts can be used as tuneable systems to testsuch ideas about the interplay of superconductivity, Mottinsulation, quantum fluctuations, and spin frustration.

A goal of this review is not to be exhaustive but ratherto be pedagogical, critical, and constructive. We willattempt to follow the goals for such reviews set long agoby Herring (Herring, 1968).

A. Motivation: frustration, spin liquids, and spinons

1. Key questions

A major goal for this review will be to address thefollowing questions:

1. Is there a clear relationship between superconduc-tivity in organic charge transfer salts and in otherstrongly correlated electron systems?

2. Are there materials for which the ground state ofthe Mott insulating phase is a spin liquid?

3. What is the relationship between spin liquids andsuperconductivity? In particular, does the samefermionic pairing occur in both?

4. What are the quantum numbers (charge, spin,statistics) of the quasiparticles in each phase?

5. Are there deconfined spinons in the insulatingphase?


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6. Can spin-charge separation occur in the metallicphase?

7. In the metallic phase close to the Mott insulatingphase is there an anisotropic pseudogap, as in thecuprates?

8. What is the simplest low-energy effective quantum

many-body Hamiltonian on a lattice that can de-scribe all possible ground states of these materials?

9. Can a RVB variational wave function give an ap-propriate theoretical description of the competitionbetween the Mott insulating and the superconduct-ing phase?

10. Is there any significant difference between destroy-ing the Mott insulator by hole doping and by in-creasing the bandwidth?

11. For systems close to the isotropic triangular lattice,does the superconducting state have broken time-

reversal symmetry?

12. How can we quantify the extent of frustration?What is the difference between classical and quan-tum frustration?

13. What is the relative importance of frustration andstatic disorder due to impurities?

14. Is the chemical pressure hypothesis valid?

15. Is there quantum critical behaviour associated withquantum phase transitions in these materials?

16. Do these materials illustrate specific organisingprinciples that are useful for understanding otherfrustrated materials?

At the end of the review we consider some possible an-swers to these questions.

2. A hierarchy of theories: from quantum chemistry to fieldtheory

The quantum many-body physics of condensed mat-ter provides many striking examples of emergent phe-nomena at different energy and length scales (Anderson,1972; Coleman, 2003; Laughlin and Pines, 2000; McKen-zie, 2007; Wen, 2004). Figure 1 illustrates how this isplayed out in the subject of this article showing the strat-ification of different theoretical treatments and the asso-ciated objects. It needs to be emphasized that when itcomes to theoretical descriptions going up the hierarchyis extremely difficult, particularly determining the quan-tum numbers of quasi-particles and the effective interac-tions between them, starting from a lattice Hamiltonian.

Schrodingerequation&Coulomb'slaw

ElectronsinAtomicorbitals&MolecularOrbitals

LatticemodelHamiltonian

Localisedspins fermions

Continuum@ieldtheory

Quasi-particles Gauge@ields

FIG. 1 The hierarchy of ob jects and descriptions associatedwith theories of organic charge transfer salts. The arrowspoint in the direction of decreasing length scales and increas-ing energy scales. At the level of quantum chemistry one candescribe the electronic states of single (and pairs of) moleculesin terms of molecular orbitals (which can be approximatelyviewed as superpositions of atomic orbitals). Just a few of the-semolecular orbitals interact significantly with those of neig-bouring molecules in the solid. Low-lying electronic states ofthe solid can be described in terms of itinerant fermions on alattice and an effective Hamiltonian such as a Hubbard model(Section VI.B). In the Mott insulating phase the electrons arelocalised on single lattice sites and can described by a Heisen-

berg spin model (Section VI.A). The low-lying excitations ofthese lattice Hamiltonians and long-wavelength properties ofthe system may have a natural description in terms of quasi-particles which can be described by a continuum field theorysuch as a non-linear sigma model. At this level unexpectedobjects may emerge such as gauge fields and quasi-particleswith fractional statistics (Section VII).

3. Organic charge transfer salts are an important class ofmaterials

Organic charge transfer salts have a number of featuresthat make them a playground for quantum many-body

physics. As a result of several properties distinctly differ-ent from other strongly correlated electron materials suchas transition metal oxides and intermetallics one can ob-serve rich physics in experimentally accessible magneticfields and pressure ranges. These properties include:

They are available in ultra-pure single crystalswhich allow observation of quantum magnetic os-cillations such as the de Haas van Alphen effect.

The superconducting transition temperature andupper critical field are low enough that one can de-stroy the superconductivity and probe the metallicstate in steady magnetic fields less than 20 tesla.

Chemical substitution provides a means to tune theground state

Chemical doping (and the associated disorder) isnot necessary to induce transitions between differ-ent phases

These materials are compressible enough that pres-sures of the order of kbars can induce transitionsbetween different ground states


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Consequently, over the past decade it has been possi-ble to observe several unique properties of strongly cor-related electron materials, sometimes ones that have notbeen seen in inorganic materials such as transition metaloxides. These significant observations include:

Magnetic field induced superconductivity.

A first-order transition between a Mott insulatorand superconductor induced with deuterium sub-stitution, anion substitution, pressure, or magneticfield.

Valence bond solid in a frustrated antiferromagnet.

A spin liquid in a frustrated antiferromagnet.

A new universality class near the Mott transition.

In the metallic phase collapse of the Drude peak inthe optical conductivity (and thus quasi-particles)above temperatures of order of tens of Kelvin.

Bulk measurement of the Fermi surface using angle-dependent magnetoresistance.

Low superfluid density in a weak correlated metal.

Multiferroic states.

Superconductivity near a charge ordering transi-tion.

Figure 2 illustrates schematically the two differentroutes to destroying the Mott insulating phase. An im-portant consequence of Anderson RVB theory of the FC-MIT is that the preexisting magnetic singlet pairs of theinsulating state become charged superconducting pairswhen the insulator is doped sufficiently strongly (An-derson, 1987). It is therefore important to understandwherther this extends to the BC-MIT were one has equalnumbers of holons and doublons. More generally, animportant question, that has not yet received adequateattention, is what are the similarities and differences be-tween the FC-MIT and the BC-MIT?

4. What is a spin liquid?

This question has recently been reviewed in detail (Ba-lents, 2010; Normand, 2009). There are several alterna-tive definitions. The definition that we think is the mostilluminating, because it brings out their truly exotic na-ture, is the following.

A spin liquid is a quantum ground state inwhich there is no long-range magnetic orderand no breaking of spatial symmetries (rota-tion or translation).

FIG. 2 Schematic phase diagram associated with the Mott-Hubbard metal-insulator transition. [Figure after Reference(Imada et al., 1998)]. The Mott insulating phase occurs athalf filling and when the on-site Coulomb repulsion U ismuch larger than the hopping energy t and the associatedband width. A transition to a metallic phase occurs either

by doping away from half filling [FC-MIT= filling controlledmetal-insulator transition] or by decreasing the ratio U/t [BC-MIT = bandwidth controlled metal-insulator transition]. Inthe cuprates a FC-MIT occurs whereas in the organic chargetransfer salts considered in this review one might argue thatBC-MIT occurs. On the other hand, perhaps one should con-sider a third co-ordinate, the frustration, in addition to thefilling and band width. This would lead to the notion of anFrC-MIT, a frustration controlled transition. In the Hubbardmodel on the anisotropic lattice at half-filling for fixed U/tan increasing t/t can drive an insulator to metal transition(compare Figure 37).

One can write down many such quantum states. Indeed,Wen classified hundreds of them for the square lattice(Wen, 2002). But the key question is whether such astate can be the ground state of a physically realisticHamiltonian. A concrete example is the ground state ofthe one-dimensional antiferromagnetic Heisenberg modelwith nearest-neighbour interactions. However, despite anexhaustive search since Andersons 1987 Science paper,(Anderson, 1987) it seems extremely difficult to find aphysically realistic Hamiltonian in two dimensions whichhas such a ground state.

As far as we are aware there is still no definitivecounter-example to the following conjecture:

Consider a family of spin-1/2 Heisenbergmodels on a two-dimensional lattice withshort range antiferromagnetic exchange inter-actions (both pairwise and ring exchange areallowed). The Hamiltonian is invariant underSU(2)L, where L is a space group and thereis a non-integer total spin in the repeat unitof the lattice Hamiltonian. Let be a param-eter which can be used to distinguish differ-ent Hamiltonians in the family (e.g., it could


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be the relative magnitude of different interac-tion terms in the Hamiltonian). Then a non-degenerate ground state is only possible fordiscrete values of (e.g., at a quantum crit-ical point). In other words, the ground statespontaneously breaks at least one of the twosymmetries SU(2) and L over all continuousranges of .

The requirement of non-integer spin in the repeat unitis so the Hastings generalisation of the Lieb-Schultz-Mattis theorem to dimmensions greater than one doesnot apply (Alet et al., 2006; Hastings, 2004). Note that,for the triangular, kagome, and pyrochlore lattices theseunits contain one, three, and four spins respectively (Nor-mand, 2009). Hence, Hastings theorem cannot be usedto rule out a spin liquid for the pyrochlore lattice.

One of the best candidate counter examples to theabove conjecture is the Heisenberg model on the triangu-lar lattice with ring exchange (LiMing et al., 2000) whichwill be discussed in more detail in Section VI.A.

Sachdev (Sachdev, 2009) pointed out that such Heisen-

berg models have possible ground states in four classes:Neel order, spiral order, a valence bond solid, or a spinliquid. Examples of the first three occur on the square,triangular, and Kagome lattices (Singh and Huse, 2007)respectively. For the first two cases spin rotational sym-metry and lattice symmetry are broken. For the latter,only the spatial symmetry is broken.

Normand (Normand, 2009) considered three differentclasses of spin liquids, each being defined by their excita-tion spectrum. If we denote the energy gap between thesinglet ground state and the lowest-lying triplet state byT and the gap to the first excited singlet state by S.The three possible cases are:

1. S = 0 and T = 0.2. S = 0 and T = 0.3. S = T = 0.

He refers to the first two as Type I and Type II respec-tively. The latter case are referred to as Algebraic spinliquids. The case T = 0 and S = 0 is not an optionbecause, by Goldstones theorem, it would be associatedwith broken spin-rotational symmetry.

An important question is how to distinguish these dif-ferent states experimentally. It can be shown that for asinglet ground state at zero temperature singlet excitedstates do not contribute to the dynamical spin suscep-

tibility. If the susceptibility is written in the spectralrepresentation,1

+(q, in) =

0

d einTS(q, )S+(q, 0), (1)

1 Here = 1/kBT is the inverse temperature, is the imaginarytime, n are the Matsubara frequencies, and S(q, ) are thespin raising/lowering operators.

it is clear that the matrix elements of the spin operatorsbetween the singlet ground state and any singlet excitedstate must be zero. This means that at low tempera-tures, only triplet excitations contribute to the uniformmagnetic susceptibility, the NMR relaxation rate, Knightshift, and inelastic neutron scattering cross section. Incontrast, both singlet and triplet excitations contributeto the specific heat capacity and the thermal conductivity

at low temperatures. Hence, comparing the temperaturedependence of thermal and magnetic properties shouldallow one to distinguish options (1) and (2) above. Fur-thermore, the singlet spectrum will not shift in a mag-netic field but the triplets will split and the correspondingspectral weight be redistributed.

One important reason for wanting to understand thesedetails of the spin liquid states is that spin excitationspectrum may well be important for understanding un-conventional superconductivity. Strong coupling RVB-type theories focus on singlet excitations whereas weak-coupling antiferromagnetic spin fluctuation theories focuson triplet excitations. This important point is empha-

sized and discussed in a review on the cuprates (Norman,2006).

5. What are spinons?

A key question is what are the quantum numbers andstatistics of the lowest lying excitations. In a Neel or-dered antiferromagnet these excitations are magnonsor spin waves which have total spin one and obeyBose-Einstein statistics (Auerbach, 1994). Magnons canbe viewed as a spin flip propogating through the back-ground of Neel ordered spins. They can also be viewed as

the Goldstone modes associated with the spontaneouslybroken symmetry of the ground state.In contrast, in a one-dimensional antiferromagnetic

spin chain (which has a spin liquid ground state) thelowest lying excitations are gapless spinons which havetotal spin-1/2 and obey semion statistics which areintermediate between fermion and boson statistics (i.e.there is a phase factor of/2 associated with particle ex-change) (Haldane, 1991). The spinons are deconfinedin the sense that if a pair of them are created (for exam-ple, in an inelastic neutron scattering experiment) withdifferent momentum then they will eventually move in-finitely far apart. Definitive experimental signatures ofthis deconfinement are seen in the dynamical structurefactor S(, q) which shows a continuum of low-lying ex-citations rather than the sharp features associated withspin waves. This is clearly seen in the compound KCuF3,which is composed of linear chains of spin-1/2 copper ions(Tennant et al., 1995). The most definitive evidence forsuch excitations in a real two-dimensional material is inCs2CuCl4 (Coldea et al., 2003; Kohno et al., 2007) abovethe Neel ordering temperature. Below the Neel tempera-ture these excitations become confined into conventionalmagnons (Fjrestad et al., 2007). It is an open theoret-


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ical question as to whether there is any two-dimensionalHeisenberg model with such excitations at zero temper-ature, other than at a quantum critical point (Singh,2010).

What type of spinon statistics might be possible intwo dimensions? Wen used quantum orders and projec-tive symmetry groups, to construct hundreds of symmet-ric spin liquids, having either SU(2), U(1), or Z2 gauge

structures at low energiesr (Wen, 2002). He divided thespin liquids into four classes, based on the statistics ofthe quasi-particles and whether they were gapless:

Rigid spin liquid: spinons (and all other excitations)are fully gapped and may have either bosonic,fermionic, or fractional statistics.

Fermi spin liquid: spinons are gapless and are de-scribed by a Fermi liquid theory (the spinon-spinoninteractions vanish as the Fermi energy is ap-proached).

Algebraic spin liquid: spinons are gapless, but they

are not described by free fermionic and free bosonicquasiparticles.

Bose spin liquid: low-lying gapless excitations are de-scribed by a free-boson theory.

6. Antiferromagnetic fluctuations

It has been proposed that and instability to a d-wave superconducting state can occur in a metallic phasewhich is close to an antiferromagnetic instability. Thishas been described theoretically by an Eliashberg-typetheory in which the effective pairing interaction is pro-

portional to the dynamical spin susceptibility, (, q)(Moriya and Ueda, 2000a). If this quantity has a sig-nificant peak near some wavevector, that will signifi-cantly enhance the superconducting Tc in a specific pair-ing channel. NMR relaxation rates are also determinedby (, q) and so NMR can provide useful informationabout it. For example, a signature of large antiferromag-netic fluctuations is a dimensionless Korringa ratio thatis much larger than one.

From a local picture one would like to know thestrength of the antiferromagnetic exchange J between lo-calised spins in the Mott insulating and the bad metallicphase. In RVB theory J sets the scale for the supercon-ducting transition temperature. It is important to realisethat this is very different from picture of a glue in theEliashberg-type theories (Anderson, 2007).

7. Quantum critical points

Figure 3 shows a schematic phase diagram associatedwith a quantum critical point (Coleman and Schofield,2005; Sachdev, 1999). We will discuss the relevance ofsuch diagrams to the organic charge transfer salts below.

FIG. 3 Schematic phase diagram associated with a quan-tum critical p oint. The vertical axis is temperature and thehorizontal axis represents a coupling constant, g. Quantumfluctuations increase with increasing g and for a critical valuegc there is a quantum phase transition from an ordered phase(with broken symmetry) to a disordered phase, usually as-sociated with an energy gap, (g gc)

z where z is thedynamical critical exponent and is the critical exponent as-sociated with the correlation length |g gc|

. In thequantum critical region the only energy scale is the temper-ature and the correlation length 1/T1/z. In this regionthere are also no quasi-particles (i.e., any singularities in spec-tral functions are not isolated poles but rather branch cuts).

We will see that some of the theoretical models (such asthe Heisenberg model on an anisotropic triangular lat-tice) do undergo a quantum phase transition from a mag-netically ordered to a quantum disordered phase with anenergy gap to the lowest lying triplet excitation.

A particularly important question is whether any sig-natures of quantum critical behavior have been seen.Perhaps, the clearest evidence of quantum critical fluc-tuations come from the NMR spin relaxation rate in -(BEDT-TTF)2Cu2(CN)3, which will be discussed in Sec-tion VII.D.

B. Key consequences of frustration

We briefly list some key consequences of frustration.Many of these are discussed in more detail, later in thereview.

Frustration enhances the number of low energy ex-citations. This increases the entropy at low tem-peratures (Ramirez, 1994). The temperature de-pendence of the magnetic susceptibility is flatterand the peak occurs at a lower temperature (Sec-tion I.B.3).

Quantum fluctuations in the ground state are en-hanced due to the larger density of states at lowenergies. These fluctuations can destroy magneti-cially ordered phases (Section VI.A).


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Singlet excitations are stabilised and singlet pair-ing correlations are enhanced. Resonating valencebond states have a larger overlap with the trueground state of the system (Section VI.A).

Intersite correlations are reduced which enhancesthe accuracy of single site approximations such asCurie-Weiss theory and dynamical mean-field the-

ory. In Heisenberg models frustrating spin interactions

produce incommensurate correlations which canlead to new triplet excitations (phasons) (Chandraet al., 1990). Furthermore, these incommensuratecorrelations can lead to the emergence of new gaugefields 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).

1. Estimates of the correlation length

The temperature dependence of the correlation length(T) and the static structure factor S( Q), associated

with the classical ordering wavevector Q has been cal-culated for both the triangular lattice and square latticeHeisenberg models using high-temperature series expan-sions (Elstner et al., 1993, 1994). For the triangular lat-tice the correlation length has values of about 0.5 and 2lattice constants, at temperatures T = J and T = 0.2J,respectively. In contrast, the model on the square lat-tice has values of about 1 and 200 lattice constants, at

T = J and T = 0.2J, respectively. At T = 0.2J the staticstructure factor has values of about 1 and 3000 for the tri-angular and square lattices, respectively. These distinctdifferences in temperature dependence can be understoodin terms of frustration producing a roton like minimumin the triplet excitation spectra of the triangular latticemodel (Zheng et al., 2006).

We discuss later how the temperature dependence ofthe uniform magnetic susceptibility of several frustratedcharge transfer salts can be fit to that for the Heisenbergmodel on the triangular lattice with J = 250 K (Shimizuet al., 2003; Zheng, Singh, McKenzie and Coldea, 2005).This implies that 2a at 50 K. This is consistent withestimates of the correlation length from NMR relaxationrates.

2. Competing phases

One characteristic feature of strongly correlated elec-tron systems that, we believe, perhaps should be dis-cussed more is how sensitive they are to small pertur-bations. This is particularly true in frustrated systems.A related issue is that there are often several competing

phases which are very close in energy. This can makevariational wave functions unreliable. Getting a goodvariational energy may not be a good indication that thewave function captures the key physics. Here are twoconcrete examples to illustrate this point.

Consider the spin 1/2 Heisenberg model on theisotropic triangular on a lattice of 36 sites, and withexchange interaction J. Exact diagonalisation (Sindz-

ingre et al., 1994) gives a ground state energy per site of0.5604J and a net magnetic moment (with 120 degreeorder as in the classical model) of 0.4, compared to theclassical value of 1/2. In contrast, a variational short-range RVB wavefunction has zero magnetic moment anda ground state energy of 0.5579J, which is only 0.5%larger than the exact value. Yet, it is qualitatively incor-rect because it predicts no magnetic order (and thus nospontaneous symmetry breaking) in the thermodynamiclimit. Note, however, that the energy difference is aboutJ/500. [For details and references see Table III in (Zhenget al., 2006)].

The second example concerns the spin 1/2 Heisenberg

model on the anisotropic triangular lattice, viewed aschains with exchange J and frustrated interchain cou-pling J. For J 3J this describes the compoundCs2CuCl4. The triplet excitation spectrum of the modelhas been calculated both with a small Dzyaloshinski-Moriya interaction D, and without (D = 0). It is strikingthat even when D J/20 it induces energy changes inthe spectrum of energies as large as J/3, including newenergy gaps (Fjrestad et al., 2007).

3. Measures of frustration

Balents recently considered how to quantify theamount of frustration in an antiferromagnetic material(or model) and its tendency to have a spin liquid groundstate (Balents, 2010). He used a measure (Ramirez, 1994)f = TCW/TN, the ratio of the Curie-Weiss tempera-ture TCW to the Neel temperature, TN at which three-dimensional magnetic ordering occurs.

One limitation of this measure is it does not separateout the effects of fluctuations (both quantum and ther-mal), dimensionality, and frustration. For strictly oneor two dimensional systems, TN is zero. For quasi-two-dimensional systems the interlayer coupling determinesTN. Thus, f would be larger for a set of weakly coupledunfrustrated chains than for a layered triangular lattice

in which the layers are moderately coupled together.Section II of (Zheng, Singh, McKenzie and Coldea,

2005) contains a detailed discussion of two different mea-sures of frustration for model Hamiltonians: (1) the num-ber of degenerate ground states, and (2) the ratio ofthe ground state energy to the base energy [the sum ofall bond energies if they are independently fully satis-fied.] This measure was introduced previously for classi-cal models (Lacorre, 1987).

Figure 4 shows results that might be the basis of some


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FIG. 4 (Color online) Effect of frustration on the tempera-ture dependence of the magnetic susceptibility (T) for theHeisenberg model on an anisotropic triangular lattice (Zheng,Singh, McKenzie and Coldea, 2005). The variation of key pa-rameters is shown as a function of the ratio J/(J + J). Tpis the temperature at which the susceptibility is a maximum,with a value p (Tp). Tcw is the Curie-Weiss tempera-ture which can be extracted from the high-temperature de-pendence of the susceptibility. Bsat is the magnetic saturationfield and Ag2 is the Curie-Weiss constant. All quantities werecalculated by a high-temperature series expansion. All of thequantities plotted have extreme values for the isotropic tri-angular lattice, suggesting that in some sense it is the mostfrustrated.

alternative measures of frustration. The sensitivity ofthe temperature dependence of the susceptibility to theratio J/J has been used to estimate this ratio for specificmaterials (Zheng, Singh, McKenzie and Coldea, 2005).

In some sense then the temperature Tp at which thesusceptibility has a maximum and the magnitude of thatsusceptibility is a measure of the amount of frustration.This is consistent with some intuition (or is it just preju-dice?) that for the anistropic triangular lattice the frus-tration is largest for the isotropic case. These measuresof frustration are not dependent on dimensionality and sodo not have the same problems discussed above that the

ratio f does. On the other hand these measures reflectshort-range interactions rather than the tendency for thesystem to fail to magnetically order.

One issue that needs to be clarified is how one mightdistinguish quantum and classical frustration. In generalthe nearest neighbour spin correlation fs Si Sj willbe reduced by frustration. Entanglement measures fromquantum information theory can be used to distinguishtruly quantum from classical correlations. For a spinrotationally invariant state f is related to the entangle-

ment measure known as the concurrence C by (Cho andMcKenzie, 2006)

C = max{0, 2fs 1/2}. (2)Hence, there is maximal entanglement (C = 1) when thetwo spins are in a singlet state and are not entangled withthe rest of the spins in the system. Once the spin cor-relations decrease to f

s= 1/4 there is no entanglement

between the two spins.

4. Geometric frustration of kinetic energy

In a non-interacting electron model we are aware ofonly two proposed quantitative measures of the geomet-rical frustration of the kinetic energy. Both are basedon the observation that, for t > 0, an electron at thebottom of the band does not gain the full lattice kineticenergy, while a hole at the top of the band does. Bar-ford and Kim (Barford and Kim, 1991) suggested that fortight binding models a measure of the frustration is then

= |maxk | |mink |, where maxk and mink are the energies(relative to the energy of the system with no electrons) ofthe top and bottom of the band respectively. This frus-tration increases the density of states for positive energiesfor t > 0 (negative energies for t < 0) which representsan increased degeneracy and enhances the many-bodyeffects when the Fermi energy is in this regime.

Together with Merino we previously argued (Merinoet al., 2006) that a simpler measure of the kinetic energyfrustration is W/2z|t|, where W is the bandwidth and zis the coordination number of the lattice. The smallerthis ratio, the stronger the frustration is, while for anunfrustrated lattice W/2z|t| = 1. For example, on thetriangular lattice the kinetic energy frustration leads toa bandwidth, W = 9|t|, instead of 12|t| as one mightnavely predict from W = 2z|t| since z = 6.

We argued that geometrical frustration of the kineticenergy is a key concept for understanding the propertiesof the Hubbard model on the triangular lattice. In partic-ular it leads to particle-hole asymmetry which enhancesmany-body effects for electron (hole) doped t > 0 (t < 0)lattices.

It should be noted that geometrical frustration of thekinetic energy is a strictly quantum mechanical effectarising from quantum interference. This interferencearises from hopping around triangular plaquettes whichwill have an amplitude proportional to t3 which clearlychanges sign when t changes sign. In contrast on the, un-frustrated, square lattice the smallest possible plaquetteis the square and the associated amplitude for hoppingaround a square is independent of the sign of t as it is pro-portional to t4. Barford and Kim noted that the phasecollected by hopping around a frustrated cluster may beexactly cancelled by the Aharonov-Bohm phase associ-ated with hopping around the cluster for a particularchoice of applied magnetic field (Barford and Kim, 1991).Thus a magnetic field may be used to lift the effects of


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kinetic energy frustration. The quantum mechanical na-ture of kinetic energy frustration is in distinct contrastto geometrical frustration in antiferromagnets which canoccur for purely classical spins.

II. TOY MODELS TO ILLUSTRATE THE INTERPLAYOF FRUSTRATION AND QUANTUM FLUCTUATIONS

We now consider some model Hamiltonians on justfour lattice sites. Although such small systems are farfrom the thermodynamic limit, these models can illus-trate some of the essential physics associated with theinterplay of strong electronic correlations, frustration,and quantum fluctuations. They illustrate the quan-tum numbers of important low-lying quantum states,the dominant short-range correlations, and how frustra-tion changes the competition between these states. Fur-thermore, understanding these small clusters is a pre-requisite for cluster extensions of dynamical mean-fieldtheory (Ferrero et al., 2009) and rotationally invariantslave boson mean-field theory (Lechermann et al., 2007)which describes band selective and momentum space se-lective Mott transitions. Insight can also be gain by con-sidering two, three, and four coupled Anderson impurities(Ferrero et al., 2007). Small clusters are also the basis ofthe contractor renormalisation method (CORE) (Altmanand Auerbach, 2002; Berg et al., 2003).

A similar approach of just considering four sites hasbeen taken before when considering the ground state ofCaV4O9 which can be described by a Heisenberg modelon a depleted lattice (Ueda et al., 1996). The authorsfirst considered a single plaquette with frustration, al-beit along both diagonals (see also Section 3 in (Valkovet al., 2006)). Dai and Whangbo (Dai and Whangbo,

2004) considered the Heisenberg model on a triangle anda tetrahedra. Similar four site Heisenberg Hamiltoni-ans have also been discussed in the context of mixed va-lence metallic clusters of particular interest to chemists(Augustyniak-Jablokow et al., 2005).

A. Four site Heisenberg model

We show how this model illustrates that frustrationcan lead to energy level crossings and consequently tochanges in the quantum numbers of the ground state andlowest lying excited state.

The Hamiltonian is (see Figure 5 (a))

H = J

S1 S2 + S2 S3 + S3 S4 + S4 S1

+JS1 S3. (3)

It is helpful to introduce the total spin along each ofthe diagonals, S13 = S1 + S3 and S24 = S2 + S4, andnote that these operators commute with each other andwith the Hamiltonian. The total spin of all four sites canbe written in terms of these operators: S = S13 + S24.

Thus, total spin S, and the total spin along each of thetwo diagonals, S13 and S24 are good quantum numbers.The term in (3) associated with J can be rewritten as

J/2(S2 S213 S224). Hence, the energy eigenvalues are

E(S, S13, S24) =1

2JS(S+ 1) +

1

2(J J)S13(S13 + 1)

1

2

JS24(S24 + 1)

3

4

J. (4)

Figure 5 (c) shows a plot of these energy eigenvalues asa function of J/J. We note that the quantum numbersof the lowest lying excited state change when J = J andthe ground state changes when J = 2J.

The two singlet states can also be written as lin-ear combinations of two orthogonal valence bond states,denote |H and |V, which descibe a pair of singletsalong the horizontal and vertical directions, respectively(see Figure 5 (b)). The state with quantum numbers(S, S13, S24) = (0, 0, 0) is

|0, 0, 0

=

1

2(

|H

|V

) (5)

and the state with (S, S13, S24) = (0, 1, 1) state is

|0, 1, 1 = 12

(|H + |V) . (6)

Both of these singlet states are resonating valence bondstates (see Figure 5 (b)).

The Hamiltonian has C2v symmetry with respect tothe diagonals. The two singlet states above have A1 andA2 symmetry, respectively. However, if J = 0 there isC4v symmetry and the (0, 0, 0) and (0, 1, 1) states haveA1 and B1 symmetry, respectively. The latter connectsnaturally to the B1 symmetry of a dx2y2 superconduct-

ing order parameter on the square lattice.It is possible to relate the two singlet states to the phys-

ical states of a Z2 gauge field on a single plaquette (seeSection 3.2 of (Alet et al., 2006)). The gauge flux opera-tor on the plaquette Fp flips the bonds between horizontaland vertical. The RVB states (5) and (6) are eigenstatesof the plaquette flux operator Fp with eigenvalues 1.

1. Effect of a ring exchange iteraction

Consider adding to the Hamiltonian (3)

H = J(P

1234+ P

4321) (7)

= J(P12P34 + P14P23 P13P24 + P13 + P24 1)where J describes the ring-exchange interaction arounda single plaquette, P12 = 2S1 S2 + 1/2 permutes spins1 and 2, and P1234 is the permutation operator aroundthe plaquette (Misguich and Lhuillier, 2005; Thouless,1965).

Intuitively,

H|H = J|V H|V = J|H (8)


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0 2 4 6 8

J'/J

-6

-4

-2

0

E/J (1,1,0)

(0,1,1)

(2,1,1), (1,1,1)

(0,0,0),

(1,0,1)

(S,S13,S24)=

J

J JJ

J

1 2

34

a) b)H

+

V

= +

H

V=

c)

FIG. 5 Eigenstates of the frustrated Heisenberg model on asingle plaquette. (a) The exchange interactions in the model.(b) The two resonating valence bond states which span allthe singlet states (compare equations (5) and (6). (c) Depen-dence of the energy eigenvalues as a function of the diagonalinteraction J/J. Note that the quantum numbers of the low-est lying excited state change when J = J and the groundstate changes when J = 2J. Furthermore, the two resonatingvalence bond states become degenerate at J = 2J.

Hence, the RVB states (5) and (6) are eigenstates ofthe ring-exchange Hamiltonian with eigenvalues J andJ, respectively. Hence, ring exchange has a similar ef-fect to the diagonal interaction in that it stabilises thestate (5).

B. Four site Hubbard model

A comprehensive study of the t = 0 model (which hasC4v symmetry) has been given by Schumann (Schumann,2002). The analysis is simplified by exploiting the SU(2)symmetry associated with particle-hole symmetry (Noceand Cuoco, 1996). In particular, the Hamiltonian matrixthen decomposes into blocks of dimension 3 or less. Schu-mann has also solved the model on a tetrahedron and atriangle (Schumann, 2008). When t = 0 the SU(2) sym-metry is broken, but it may be that the SU(2) quantumnumbers may be useful to define a basis set in which todiagonalize the Hamiltonian and see the effect of t = 0.

Freericks, Falicov, and Rokshar studied an eight siteHubbard model with a next nearest neigbour hopping t

and periodic boundary conditions (Freericks et al., 1991).The model is invariant under a 128-element cluster per-mutation group. For t = t/2 the model is equivalent toan eight site triangular lattice cluster or a face-centredcubic cluster. They found that at half filling the symme-try of the ground state changed as a function of both t/tand U/t [see Figure 3 in (Freericks et al., 1991)]. Therelevant irreducible representations are denoted 1n, 2,and 4 [cf. Table V in (Freericks et al., 1991)].

Falicov and Victora (Falicov and Victora, 1984) showedthat the Hubbard model on a tetrahedron [which has Tdsymmetry] with four electrons has a singlet ground statewith E symmetry.2 Later Falicov and Proetta (Falicovand Proetto, 1993) also showed that an RVB state withcomplex pairing amplitude (and which thus breaks time-reveral symmetry) and which they state has E symmetry3

is within 0.3% of the exact ground state energy for U =

10t.More work is required to extract insights from the

above work about the role of frustration. An importantquestion is whether results on four sites can be related toa simple picture of how dx2y2 Cooper pairing emergeson the square lattice dues to antiferromagnetic interac-tions (Scalapino and Trugman, 1996). Then does thispairing symmetry change with frustration?

III. -(BEDT-TTF)2X

An important class of model systems for quantum

frustration is the organic charge transfer salts based onthe molecule bis(ethylenedithio)tetrathiafulvalene (alsoknown as BEDT-TTF or ET; shown in Fig. 6a). Thesesalts have been extensively studied and show a widerange of behaviours including, antiferromagnetism, spinliquids, (unconventional) superconductivity, Mott transi-tions, incoherent (or bad) metals, charge ordering andFermi liquid behaviour. In this section we focus on the as-pects most relevant to the quantum frustration in thesematerials, a number of such reviews focusing on differ-ent aspects of these materials are also available elsewhere(Ishiguro et al., 1998; Lang and Muller, 2003; Powell andMcKenzie, 2006; Seo et al., 2006; Singleton and Mielke,2002; Wosnitza, 2007) and in the November 2004 issue ofChemical Reviews. Further, although a number of crys-tallographic phases are observed in the BEDT-TTF salts,we will limit ourselves to the phase, which is by far themost widely studied and, in which, the most profoundeffects of frustration have been found.

The experimentally observed phase diagrams of two-BEDT-TTF salts (-(BEDT-TTF)2Cu[N(CN)2]Cl and-(BEDT-TTF)2Cu2(CN)3) are shown in Figs. 7 and 8.One should note how similar these two phase diagramsare (except for the magnetic order, or lack of order, ob-served in the Mott insulating phase). Two importantparameters are the strength of the electronic correla-tions and the degree of frustration. These parameters

are determined by the choice of anion, X, in -(BEDT-TTF)2X and the applied hydrostatic pressure. Belowwe will focus of four of the most widely studied mate-

2 We will use Mulliken notation, Falicov and Victora use Bethenotation and label this representation 3, see (Lax, 1974) fordetails.

3 Or 12 in the Bouckaert, Smoluchoski, Wigner notation (Lax,1974) that Falicov and Proetta use.


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S

S

S

S S

S

S

S

Pd

S

S S

S

S

SS

S

SS

H

H

H

H

H

H

H

H

a) BEDT-TTF

b) Pd(dmit)2

FIG. 6 The molecules BEDT-TTF and Pd(dmit)2, whichform charge transfer salts with frustrated lattices and in whichthe electrons are strongly correlated.

rials: -(BEDT-TTF)2Cu(NCS)2 and -(BEDT-TTF)2-Cu[N(CN)2]Br, which superconduct below 10 K at am-bient pressure; -(BEDT-TTF)2Cu[N(CN)2]Cl, which isan antiferromagnetic Mott insulator at ambient pressure;and -(BEDT-TTF)2Cu2(CN)3, which appears to be aspin liquid at ambient pressure. Both -(BEDT-TTF)2-Cu2(CN)3 and -(BEDT-TTF)2Cu[N(CN)2]Cl undergoMott transitions to superconducting states under mod-est pressures (a few 100 bar).

A. Crystal and electronic structure

-(BEDT-TTF)2X salts form layered crystals with aquasi-two-dimensional (q2D) band structure. Charge is

transferred from organic (BEDT-TTF) layer to the anion(X) layer; for monovalent anions, which we consider here,one electron is transferred from each dimer [(BEDT-TTF)2 unit] to each anion formula unit. Band structurecalculations predict that the anion layer is insulating, butthat the dimer layers are half-filled. Hence, these calcu-lations predict that the organic layers are metallic, incontrast to the rich phase diagram observed (Figs. 7 and8).

The phase salts of BEDT-TTF are stronglydimerised, that is the molecules stack in pairs within thecrystal, cf. Fig. 9. The frontier molecular orbitals of theBEDT-TTF molecule are orbitals, i.e., they have nodesin the plane of the molecule, cf. Fig. 11. Thus, these or-bitals overlap with the equivalent orbitals on the othermolecule in the dimer, cf. Fig. 11, more than they over-lap with the orbitals of any other BEDT-TTF molecule.This, combined with the greater physical proximity ofthe two molecules within the dimer, means that the am-plitude for an electron to hop between two moleculeswithin the same dimer has a much larger magnitude thanthe amplitude for hopping between molecules in differentdimers. This suggests that the interdimer hopping mightbe integrated out of an effective low energy Hamiltonian

0 100 200 300 400 500 600 700 800

10

20

30

40

50

60

70

80

AF

insulator

Semiconductor

Bad metal

Fermi liquid

Mott

insulator

P (bar)

T

(K)

Pc

1(T) P

c

2(T)

T*

Met ()

max

T*

Ins(d/dP)

max

FIG. 7 Pressure-temperature phase diagram of -(BEDT-TTF)2Cu[N(CN)2]Cl. At low temperatures -(BEDT-TTF)2Cu[N(CN)2]Cl undergoes a first order Mott transitionfrom an antiferromagnetic insulator (cf. section III.B) to ametal when hydrostatic pressure is applied (see section III.C).As the temperature is raised the line of first order transitionsends in a critical point with novel critical exponents (sec-tion III.C.1). In the insulating phase raising the tempera-

ture destroys the antiferromagnetic order. At the very lowesttemperatures the metallic state becomes superconducting (cf.section III.E). As the temperature is raised superconductivitygives way to a metal with coherent charge transport (sectionIII.D.2) and a pseudogap (section III.D.3). Further, raisingthe temperature results in a loss of coherence in the in-planetransport. This incoherent metallic phase is referred to asa bad-metal (discussed in section III.D). From (Limelette,Wzietek, Florens, Georges, Costi, Pasquier, Jerome, Meziere,and Batail, 2003).

(Kino and Miyazaki, 2006; McKenzie, 1998).

1. Dimer model of the band structure of -(BEDT-TTF)2X

The dimer model described above is the simplest, andmost widely studied, model of the electronic structure forthe -BEDT-TTF salts and leads to the Hubbard modelon an anisotropic lattice (Powell and McKenzie, 2006).The Hamiltonian of this model is

H = tij

cicj t[ij]

cicj + Ui

cicicici,(9)

where c()

i

creates (destroys) an electron with spin onsite (dimer) i, t and t are the hopping applitudes be-tween neighbouring dimers in the directions indicated inFig. 9, and U is the effective Coulomb repulsion betweentwo electrons on the same site (dimer). This model is,up to an overall scale factor with dimensions of energy,governed by two dimensionless ratios: t/t, which sets thestrength of the frustration in system and U/W, which de-termines the strength of electronic interactions. Here Wis the bandwidth, which is determined by the values of tand t. These two ratios can be manipulated experimen-


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Superconductor

(Fermi liquid)

Crossover

(Spin liquid) onset TC

R = R0

+ AT2

T1T = const.

(dR/dT)max

(1/T1T)

max

Mott insulator

Metal

Pressure (10-1GPa)

FIG. 8 Pressure-temperature phase diagram of -(BEDT-TTF)2Cu2(CN)3 . This is similar to that of-(BEDT-TTF)2-Cu[N(CN)2]Cl (Fig. 7), but has important differences. Mostimportantly the Mott insulating phase does not show anysigns of long range magnetic order down to 20 mK (thelowest temperature studied; see section III.B and Fig. 12).

Thus, -(BEDT-TTF)2Cu2(CN)3 is believed to be a spin liq-uid at ambient and low pressures. Further, there is no evi-dence of a pseudogap in -(BEDT-TTF)2Cu2(CN)3 (see sec-tion III.D.4). These differences are believed to result from thegreater geometrical frustration in -(BEDT-TTF)2Cu2(CN)3(cf. Table I, Eq. (9) and Fig. 9). From (Kurosaki et al.,2005).

tally by hydrostatic pressure,4 P, or by studying materi-als with different anions, X. Varying the anions is oftenreferred to as chemical pressure, as both degrees of free-dom lead to changes in the lattice constants. However,

it appears that chemical pressure causes larger variationsin t/t than hydrostatic pressure does. We will limit ourdiscussions to monovalent anions,5 in which case we have-(BEDT-TTF)+2 X

, i.e., there is, on average, one holeper dimer and the appropriate dimer Hubbard model ishalf filled.

The anisotropic triangular model extrapolates contin-uously between three widely studied models. For t = 0it is just the square lattice. For t = t we recoverthe (isotropic) triangular lattice. And for t 0 onehas quasi-one-dimensional chains with weak zig-zag in-terchain hopping. Thus, this model is a playground forexploring the effects of frustration in strongly correlatedsystems and would be of significant theoretical interesteven without the experimental realisations of the model

4 It has often been emphasised (Kanoda, 1997) that increasedhydrostatic pressures correspond to decreased correlation (de-creased U/W), but it has become increasingly clear (Caulfieldet al., 1994; Kandpal et al., 2009; Pratt, 2010) also suggests thatpressure may also induce small changes in the frustration, t/t.

5 See (Mori, 2004) for a discussion of anions with valencies otherthan one.

FIG. 9 Sketch of the band-structure of a -type BEDT-TTFsalt. Panel a shows a cross section of the crystal structure of-(BEDT-TTF)2Cu(NCS)2 in the organic layer. In panel bthe black circles mark the dimers, between which the hoppingintegral is large and which serve as a site in lattice modelsof the band structure. Lines indicate the inter-dimer hop-ping integrals in both panels b and c, which are topologicallyequivalent. [Taken from (Powell, 2006)].

in organic charge transfer salts.In order to make a direct comparison between theory

and experiment one would like to know what parametersof the anisotropic triangular lattice (i.e., what values of t,t and U) represent given specific materials. Significanteffort has therefore been to estimate these parametersfrom electronic structure calculations. The first studies ofthe electronic structure of-BEDT-TTF salts where lim-ited, by the computational power available at the time,to extended Huckel theory (Williams et al., 1992). Thisis a semi-empirical, i.e. experimentally parameterised,tight-binding model and ignores the role of the anions.However, modern computing power means that density


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functional theory (DFT) calculations are no longer pro-hibitively expensive and several DFT studies have ap-peared recently.

The large unit cells and complex anions of the phasematerials, meant that the first DFT studies of BEDT-TTF salts focused on other crystallographic phases(French and Catlow, 2004; Kasowski and Whangbo, 1990;Kino and Fukuyama, 1996; Kubler et al., 1987; Lee et al.,

2003; Miyazaki and Kino, 2003, 2006; Yamaguchi et al.,2003). However, two groups have recently reported pa-rameterisations of the tight-binding part of the Hamil-tonian from DFT6 calculations (Kandpal et al., 2009;Nakamura et al., 2009). Both groups find that the frus-tration parameter, t/t, is significantly smaller was pre-viously thought on the basis of extended Huckel calcula-tions (summarised in Table I). Note that the frustrationis least in -(BEDT-TTF)2Cu[N(CN)2]Cl, which has anantiferromagnetically ordered ground state and greatestin -(BEDT-TTF)2Cu2(CN)3, which has a spin liquidground state. However, even -(BEDT-TTF)2Cu2(CN)3is quite far from the isotropic triangular lattice (t = t),

which has been taken as the basis of a number of theo-ries of -(BEDT-TTF)2Cu2(CN)3 (discussed in SectionVI.A.4 ) on the basis of Huckel calculations (Komatsuet al., 1996).

The anisotropic triangular lattice has one site per unitcell. However, the -phase organics have two dimers perunit cell. This halves the Brillouin zone and causes theFermi surface to be split into two sheets (Merino andMcKenzie, 2000a; Powell and McKenzie, 2006). Thus,two orbits are observed in quantum oscillations experi-ments. The lower frequency orbit, known as the pocketcorresponds to a hole like orbit. A higher frequency os-cillation, known as the orbit, is only observed at higher

fields and corresponds to the magnetic breakdown orbitaround the Fermi surface of the dimer per unit cell model.The ratio of the areas of these orbits is strongly depen-dent on t/t (Pratt, 2010). Thus, estimates of t/t canbe made from quantum oscillation or angle-dependentmagnetoresistance (AMRO) experiments which allow oneto map out the Fermi surface.(Kartsovnik, 2004) In -(BEDT-TTF)2Cu(NCS)2 at ambient pressure this yieldst/t = 0.7 (Caulfield et al., 1994). In reasonable agree-ment with the calculated value. In -(BEDT-TTF)2Cu2-(CN)3 at 7.6 kbar one finds that t

/t = 1.1 (Ohmichiet al., 2009; Pratt, 2010), which is rather larger thanthe ambient pressure value calculated from DFT. At 0.75GPa DFT calculation give t/t = 0.75 (Kandpal et al.,2009), which is significantly smaller than the experimen-tal estimate. AMRO experiments give a picture of theFermi surface that is qualitatively consistent with the cal-culated Fermi surface (Ohmichi et al., 1997) (see Figure10). However, the value of t/t has not been estimatedfrom these measurements.

6 Both GGA and LDA functionals give similar results.

FIG. 10 The Fermi surface of -(BEDT-TTF)2Cu2(CN)3in the metallic phase at pressures of 2.1 kbar and 7.0kbar.(Ohmichi et al., 1997)

The area of the Fermi surface can be also determinedby quantum oscillations. For a wide range of organiccharge transfer salts the area is found to be consistentwith Luttingers theorem and the hypothesis that thesematerials are always at half filling.(Powell and McKenzie,2004a) This may put significant constraints on theoriesthat the metal-insulator transition involves self-doping.

The hopping between layers is much weaker than that

within the layers. This can be measured in two separateways: from AMROs (Singleton et al., 2002; Wosnitzaet al., 1996, 2002) or from a comparision of how disordereffects the superconducting critical temperature and theresidual resistivity (Powell and McKenzie, 2003). Bothmethods find that the interlayer hopping integral, t isa few tens of V in the -(BEDT-TTF)2X, but that tis an order of magnitude larger in the phase ET salts.DFT calculation et al. (Lee et al., 2003) find that inter-layer dispersion in -(BEDT-TTF)2I3 is 9 meV. How-ever, experimental estimates for the closely related mate-rial, -(BEDT-TTF)2IBr2, yield t 0.3 meV (Powelland McKenzie, 2003; Wosnitza et al., 1996). However,one should note that the value of t represents a very

sensitive test of theory due its small absolute value andthe small overlap of the atomic orbitals at the large dis-tances involved in interlayer hopping.

2. The Hubbard U

There is a considerable literature that discusses thecalculation of the Hubbard U in a molecular crys-tals. Notable systems for which this problem has been


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TABLE I Values of t/t of selected -BEDT-TTF salts, calculated from DFT. Huckel theory gives systematically smallervalues of 0.75, 0.84, and 1.06, for these salts, respectively. Estimates of the frustration parameter for the metallic phase canalso be made from quantum oscillation experiments for -(BEDT-TTF)2Cu(NCS)2 (Pratt, 2010) this yields t

/t = 0.6 in goodagreement with the calculated value.

Material t/t Reference

-(BEDT-TTF)2Cu[N(CN)2]Cl 0.4 (Kandpal et al., 2009)

-(BEDT-TTF)2Cu(NCS)2 0.6 (Kandpal et al., 2009; Nakamura et al., 2009)

-(BEDT-TTF)2Cu2(CN)3 0.8 (Kandpal et al., 2009; Nakamura et al., 2009)

tackled include the alkali doped fullerides (Gunnars-son, 2004), oligo-acene and thiopenes (Brocks et al.,2004), and the organic conductor tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) (Cano-Corteset al., 2007). These authors have proceeded by identify-ing two separate contributions to the Hubbard U:

U = U(v) U(p), (10)where U(v) is the contribution from the molecule (or clus-

ter) in vacuum, and U(p) is the reduction in the effectiveU when the molecule is placed in the polarisable environ-ment of the crystal.

One might think that U(v) is straightforward to calcu-late once one has a set of suitably localised orbitals asit is just the Coulomb repulsion between two holes (orelectrons) in the same orbital:

F0 =

d3r1

d3r2

(r1)(r2)

|r1 r2| , (11)

where (r) the density of spin electrons at the po-sition r in the relevant orbital. However, this is incor-

rect. When one moves from a full band structure to therelevant one (or few) band model this interaction is sig-nificantly renormalised (Freed, 1983; Gunnarsson, 2004;Iwata and Freed, 1992; Powell, 2009; Scriven and Powell,2009a). Indeed DFT calculations for a single BEDT-TTF molecule find that the renormalised U(v) is about50% smaller than F0 (Scriven and Powell, 2009a).

The first attempts to calculate U(v) from electronicstructure calculations were also based on the extendedHuckel method. It was noted (Kino and Miyazaki, 2006)that if one models the dimer as a two site Hubbard modelwhere each site represents a monomer then in the limit

U(v)m , where U(v)m is the effective Coulomb repul-

sion between two holes on the same monomer, one findsthat U(v) 2|tb1 |, where tb1 is the intra-dimer hoppingintegral. Whence, calculations of tb1 from the extendedHuckel approximation yield estimates ofU(v) ranging be-tween 0.14 eV(Rahal et al., 1997) and 2.1 eV (Simonovet al., 2005). Note that this range of Hubbard Us isnot caused just by changes in anions, but also the differ-ence between different groups, who often find differencesof more than a factor of two for the same material [foran extended discussion see (Scriven and Powell, 2009b)].More recently DFT has also been used to calculate tb1

and hence U(v) (Kandpal et al., 2009; Nakamura et al.,2009) - again there is a factor of 2 difference between thetwo different groups.

A better method of calculating U(v) is to note that

U = E0(+2) + E0(0) 2E0(+1), (12)where E0(q) is the ground state energy of the molecule orcluster with charge q. This can be understood as U is theenergy required to activate the charge disproportionationreaction 2(BEDT-TTF)+2 (BEDT-TTF)02 + (BEDT-TTF)2+2 . Equivalently, U is the difference in the chem-ical potentials for electrons and holes on the moleculeor cluster. Calculations of this type for isolated BEDT-

TTF monomers show that U(v)m is essentially the same

for all monomers in the geometries in which they arefound experimentally regardless of the anion in the salt,the crystal polymorph, or the temperature or pressure atwhich the crystal structure was measured (Scriven andPowell, 2009a). Remarkably, the same result holds forisolated dimers, consistent with the experimental findingthat the dimer is a conserved structural motif in both the and polymorphs (Scriven and Powell, 2009b).

Further, comparison of DFT calculations for monomerswith those for dimers reveals that the approximationU(v) 2|tb1 | is incorrect (Scriven and Powell, 2009b).This is because the effective Coulomb interaction be-tween two holes on different monomers within the samedimer, V

(v)m , is also large. Indeed, Scriven et al. found

that U(v)m V(v)m tb1 , in which case U(v) 12 (U

(v)m +

V(v)m ), which is in reasonable agreement with their di-

rectly calculated value of U(v).To date there are no calculations of U(p) for BEDT-

TTF salts. This problem is greatly complicated forBEDT-TTF relative to the other molecular crystals pre-viously studied (Brocks et al., 2004; Cano-Cortes et al.,2007; Gunnarsson, 2004) because of the (often) polymericanions and the fact that the intermolecular spacing issmall compared to the size of the molecule. Therefore,Nakamura et al. (Nakamura et al., 2009) have calculatedU directly from DFT band structure calculation by ex-plicitly integrating out high energy interband excitationsto leave an effective one band model. Interestingly, inorder for the value ofU to converge Nakamura et al. hadto include over 350 bands - corresponding to includingexcitations up to 16 eV above the Fermi level! Naka-


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mura et al. find that the value of U in the Mott in-sulator -(BEDT-TTF)2Cu2(CN)3 (0.83 eV) is remark-ably similar to that in the ambient pressure supercon-ductor -(BEDT-TTF)2Cu(NCS)2 (0.85 eV). However,they find that t = 55 meV for -(BEDT-TTF)2Cu2(CN)3and t = 65 70 meV for -(BEDT-TTF)2Cu(NCS)2,yielding U/t = 15.5 for -(BEDT-TTF)2Cu2(CN)3 andU/t = 12.012.8 for -(BEDT-TTF)2Cu(NCS)2, consis-tent with the experimental finding that former material isa Mott insulator that undergoes a Mott transition undermoderate pressure and the later is an ambient pressuresuperconductor.

However, these values are much larger than thatfound from comparisons of DMFT calculationsto optical conductivity measurements and on -(ET)2Cu[N(CN)2]BrxCl1x, which suggest that U = 0.3eV (Merino et al., 2008). These measurements arediscussed in more detail in section III.C.

3. The (BEDT-TTF)2 dimer

Significant insight can be gained from comparing(BEDT-TTF)2 with the hydrogen molecule. In themolecular orbital (Hartree-Fock) picture (Fulde, 1995)the ground state wavefunction of H2 is

| = 12

(|1 + |2) (|1 + |2) , (13)

where |i is a basis function for an electron with spin centred on atom i. This provides the simplest modelof the chemical bond, which results from the stabilisa-tion of the bonding combination of atomic orbitals, andimplies an increased electronic density between the two

atoms. If one includes electronic interactions the pictureis somewhat complicated as the wavefunction becomescorrelated. These correlations can be described in thetwo site Hubbard model, which is a good model for thehydrogen molecule, where each atom is treated as a site(Powell, 2009). If one compares the electronic density inthe HOMO of a single BEDT-TTF molecule (Fig. 11a)with that of the HOMO of the (BEDT-TTF)2 dimer (Fig.11b), it is clear that the (BEDT-TTF)2 dimer wave-function is close to being an antibonding combinationof molecular wavefunctions, whereas the HOMO-1 (Fig.11c) is close to being a bonding combination of molecularwavefunctions. In the charge transfer salt there are, onaverage, two electrons in the HOMO-1, but only one inthe HOMO. Therefore, the net effect is bonding.

Electronic correlations also play an important role inthe (BEDT-TTF)2 dimer. But, as in the case of H2, thetwo site Hubbard model, where each site is an BEDT-TTF molecule, provides a good description of the elec-tronic correlations in the (BEDT-TTF)2 dimer (Powelland McKenzie, 2006; Scriven and Powell, 2009b). Thisshows that the physics of the (BEDT-TTF)2 dimer isremarkably similar to that of the hydrogen molecule.Therefore, we can understand the (BEDT-TTF)2 dimer

FIG. 11 The highest occupied molecular orbital (HOMO)of (a) an BEDT-TTF molecule in the geometry found in -(BEDT-TTF)2Cu[N(CN)2]Cl, (b) an (BEDT-TTF)

2+2 dimer

in the geometry found in -(BEDT-TTF)2Cu2(CN)3 and (c)a neutral (BEDT-TTF)2 dimer in the geometry found in -(BEDT-TTF)2Cu2(CN)3. It is clear from these plots thatthe HOMO of the neutral dimer is the antibonding combina-tion of the two monomer HOMOs, whereas the HOMO of the(double) cation dimer is the bonding combination of the twomonomer HOMOs. Thus, the (BEDT-TTF)2 dimer is held to-gether by a covalent bond between the two monomers rather

than bonds between any two particular atoms. [Modified from(Scriven and Powell, 2009a) and (Scriven and Powell, 2009b)].

as being held together by a covalent bond not be-tween any two atoms, but between the two BEDT-TTFmolecules themselves. As one expects this intermolecu-lar covalent bond to be strong compared to the interac-tions between dimers, this provides a natural explanationof the conservation of the dimer motif across different


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materials.

B. Insulating phases

Both -(BEDT-TTF)2Cu[N(CN)2]Cl and -(BEDT-TTF)2Cu2(CN)3 are insulators at ambient pressure(Ishiguro et al., 1998), and undergo a metal-insulator

transition under the application of hydrostatic pressure(which we will discuss in section III.C). This can beunderstood straightforwardly, in terms of the half-filledHubbard model, introduced in section III.A, as a Mottinsulator phase (Kanoda, 1997; McKenzie, 1998). How-ever, despite these similarities in the charge sector, thespin degrees of freedom in the two materials behave verydifferently.

1. Antiferromagnetic and spin liquid phases

Shimizu et al. (Shimizu et al., 2003) measured and

compared bulk spin susceptibilities of -(BEDT-TTF)2-Cu[N(CN)2]Cl and -(BEDT-TTF)2Cu2(CN)3. Bothmaterials are described by the Hubbard model on theanisotropic triangular lattice, cf. Fig. 9. In the Mott in-sulating phase the spin degrees of freedom are describedby a Heisenberg model (Section VI.A) with exchange con-stants, J 250 K. However, -(BEDT-TTF)2Cu2(CN)3is significantly more frustrated than -(BEDT-TTF)2Cu-[N(CN)2]Cl(as expected by electronic structure calcula-tions cf. table I). -(BEDT-TTF)2Cu[N(CN)2]Cl showsa clear magnetic phase transition at 27 K. This is anantiferromagnetic transition (Kanoda, 1997; Miyagawaet al., 1995) and is only visible in the bulk spin suscep-

tibility because there is a small canting of the magneticmoment (Miyagawa et al., 2002), which gives rise to aweak ferromagnetic moment (Welp et al., 1992). In con-trast no such phase transition is visible in the suscepti-bility of -(BEDT-TTF)2Cu2(CN)3. Analyses (Shimizuet al., 2003; Zheng, Singh, McKenzie and Coldea, 2005)of the high temperature magnetic susceptibility find thatin both materials the effective Heisenberg exchange isJ 250 K. Therefore the absence of a phase transitionin -(BEDT-TTF)2Cu2(CN)3 down to 32 mK (the low-est temperature studied and four orders of magnitudesmaller than J) led Shimizu et al. to propose that -(BEDT-TTF)2Cu2(CN)3 is a spin liquid.

The form of the temperature dependence of the sus-ceptibility turns out to be quite sensitive to the amountof frustration(Zheng, Singh, McKenzie and Coldea, 2005)(cf. Figure susc). The values of both J and J can be es-timated by comparing the observed temperature depen-dence of the uniform magnetic susceptibility with hightemperature series expansions (above about J/4). For -(BEDT-TTF)2Cu2(CN)3 they agree for J 200 K andJ J. In Section VI.A.4 we discuss the possible effectsof ring exchange. Consequently, it is desirable to knowhow they may modify the temperature dependence of the

susceptibility and the values of the exchange interactionestimated from the experimental data.

Further evidence for the absence of magnetic order-ing in -(BEDT-TTF)2Cu2(CN)3 comes from its NMRspectrum. Fig. 12 compares the 1H NMR absorptionspectrum of -(BEDT-TTF)2Cu2(CN)3 with that of -(BEDT-TTF)2Cu[N(CN)2]Cl. Shimizu et al. reportedthat the difference of the spectra between the two salts

at high temperatures is explained by the difference inthe orientation of ET molecules against the applied fieldand does not matter. But, in -(BEDT-TTF)2Cu[N-(CN)2]Cl (Fig. 12b) they observe clear changes in theNMR spectrum below TC 27 K. These multiple peaksare caused by the distinct crystal environments for the1H atoms due to the antiferromagnetic ordering. In con-trast no quantitative changes are observed the spectrumof -(BEDT-TTF)2Cu2(CN)3 down to 32 mK, the low-est temperature studied (Fig. 12a), consistent with anabsence of long range magnetic ordering.

No evidence of long range magnetic order is observedin the 13C-NMR spectra of -(BEDT-TTF)2Cu2(CN)3

down to 20 mK (the lowest temperature studied) (Fig.12c). This is important because these experiments werecarried out on samples where the 13C is one of the atomsinvolved in the central C=C double bond. The electrondensity is much higher for this atom (cf. Fig. 11) the forthe H atoms, which are on the terminal ethylene groups(cf. Fig. 6). There the 13C spectra demonstrate that theabsence of long range order is genuine and not an artefactcaused by low electronic density on the H atoms.

The observed temperature dependence of the NMR re-laxation rates for -(BEDT-TTF)2Cu2(CN)3 are also in-consistent with this material having a magnetically or-dered ground state. The observed(Shimizu et al., 2006)decrease with decreasing temperature of the NMR relax-ation rate 1/T1 and the spin echo rate 1/T2 for -(BEDT-TTF)2Cu2(CN)3 is distinctly different from that ex-pected for a material with a magnetically ordered groundstate. For such materials at low temperatures, both 1/T1and 1/T2 should increase rapidly with decreasing temper-ature, rather than decrease (since 1/T1T (T)2 ). Thisis seen in -(BEDT-TTF)2Cu[N(CN)2]Cl . For materialsdescribed by the antiferromagnetic Heisenberg model ona square lattice [La2CuO4](Sandvik and Scalapino, 1995)or a chain [Sr2CuO3],(Takigawa et al., 1997) both relax-ation rates do increase monotonically as the temperaturedecreases.

It is noteworthy that for -(BEDT-TTF)2Cu2(CN)3at low temperatures, from 1 K down to 20 mK, it wasfound(Shimizu et al., 2006) that 1/T1 T3/2 and 1/T2 constant. As discussed in Section ?? this is qualita-tively similar with that expected in the quantum criticalregime, (53), with 1. However, the observed temper-ature dependence ofT1 and T2 would lead to two differentvalues for the exponent .

There is a way to check that the NMR relaxation isactually due to spin fluctuations and not another physi-cal mechanism. The magnitude of the relaxation rate at


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FIG. 12 The low temperature, ambient pressure 1H-NMR absorption spectra of (a) -(BEDT-TTF)2Cu2(CN)3 and (b) -(BEDT-TTF)2Cu[N(CN)2]Cl, and (c) the

13C-NMR absorption spectra of -(BEDT-TTF)2Cu2(CN)3. The antiferromagneticphase transition at 27 K is clear seen in -(BEDT-TTF)2Cu[N(CN)2]Cl. In contrast, no major changes occur with tempera-ture in -(BEDT-TTF)2Cu2(CN)3, consistence with the absence of long range magnetic order. However, the spectra do broadenas T is lowered in -(BEDT-TTF)2Cu2(CN)3. This broadening is seen even more dramatically in the

13C-NMR spectrum of-(BEDT-TTF)2Cu2(CN)3 (panel c). Again, no signs of long range magnetic order are seen down to 20 mK, at temperaturethat is four orders of magnitude smaller than the antiferromagnetic exchange energy, J 250 K (Shimizu et al., 2003; Zheng,Singh, McKenzie and Coldea, 2005). [Panels (a) and (b) were taken from (Shimizu et al., 2003) and panel (c) was modifiedfrom (Shimizu et al., 2006).]

high temperatures can be used to provide an independentestimate of J. Data in Ref. (Kawamoto et al., 2006) for-(BEDT-TTF)2Cu2(CN)3 at ambient pressure gives forthe outer site 1/T1 10 30/sec in the range 100-300K. From the K plot a value ofA = 0.07 T/B is de-duced for the outer site.(Shimizu et al., 2003) Using theabove values in the expression (37) gives J 200 600K, consistent with the value J = 250 deduced from thetemperature dependence of the uniform magnetic sus-ceptibility.(Shimizu et al., 2003; Zheng, Singh, McKenzieand Coldea, 2005)

We stress that 20 mK is four orders of magnitudesmaller than the exchange coupling, which suggests that-(BEDT-TTF)2Cu2(CN)3 may well be a true spin liq-uid.

However, Shimizu et al. did observe a slight broaden-ing of the 1H NMR spectrum of -(BEDT-TTF)2Cu2-(CN)3 as the temperature is lowered. They observed aneven more dramatic broadening in the 13C NMR (Fig.12c) (Shimizu et al., 2006). This is somewhat counter-intuitive and has received some theoretical interest, dis-cussed below. Spin echo 13C experiments show that thebroadening is inhomogenous (T2 ) rather than an increasein the extrinsic T2. Similar broadenings are also seen inEtMe3Sb[Pd(dmit)2]2 and EtMe3P[Pd(dmit)2]2, cf. sec-

tion IV.C and Fig. 25, which could hint that this is arather general phenomenon.

It was also observed that a magnetic field induces spa-tially non-uniform local moments.(Shimizu et al., 2006)Motrunich has given an interpretation of this observationin terms of spin liquid physics.(Motrunich, 2006)

Gregor and Motrunich(Gregor and Motrunich, 2009)performed several model calculations to attempt to ex-plain the large broadening by taking into account therole of disorder. They found they could only explainthe experimental data above about 5 K, if there is muchlarger disorder than expected and that it is stronglytemperature dependent. This is in contrast to pre-vious work by the authors where comparable calcula-tions for a kagome antiferromagnet could explain the ex-perimental data(Gregor and Motrunich, 2008). Gregorand Motrunich mention that it is hard to estimate thestrength of the disorder and the role of temperature de-pendent screening. It is desirable to connect this workto recent estimates of the strength of disorder in the -(BEDT-TTF)2X materials.(Scriven and Powell, 2009b)


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2. Is the spin liquid gapped?

Key questions about a spin liquid are: is it gappedand what are the nature of the low lying excitations?In particular, are there deconfined spinons? Two experi-ments have recently tried to address these questions in -(BEDT-TTF)2Cu2(CN)3, one by measuring the specificheat (Yamashita et al., 2008), the other by measuring the

thermal conductivity (Yamashita et al., 2009). However,as will now discuss, the two groups reached contradictoryconclusions on the basis of these different measurements.

S. Yamashita et al. (Yamashita et al., 2008) concludedthat there are gapless fermionic excitations, i.e., decon-fined spinons, in -(BEDT-TTF)2Cu2(CN)3 on the basisof their specific heat, Cp, measurements. A plot ofCp/Tagainst T2 is linear in the range 0.75-2.5 K, implyingthat Cp = T + T

3, with = 20 5 mJ K2 mol1.Moving to lower temperatures complicates heat capacitymeasurements as there is a significant Schottky anomaly.Nevertheless, the data in the temperature range 0.075- 3 K fits well to the form Cp = /T

2 + T + T3

with = 12 mJ K2

mol1

. One expects a large linearterm in the heat capacity if there are gapless fermionicexcitations. Indeed, the values of estimated by S.Yamashita et al. are the same order of magnitude asthose found in the metallic phases of -(BEDT-TTF)2Xsalts. Further, comparing this value with the previ-ous measurements of the bulk magnetic susceptibility(Shimizu et al., 2003) gives a Sommerfeld-Wilson ra-tio, RW = (

2k2B/2)(0/), of order unity (Yamashita

et al., 2008), which is what one would expect if the samefermions were responsible for both the linear term in thespecific heat and the susceptibility.(Lee et al., 2007b) Incontrast, other organic charge transfer salts which un-dergo magnetic ordering were found to have no such lin-ear term but to have a specific heat capacity that wasquadratic in temperature.

In a discussion of these results Ramirez (Ramirez,2008) pointed out that S. Yamashita et al.s data is fitequally well by Cp = /T2 + 2/3T

2/3 + T3. This isconsistent with the predictions for spinons coupled to aU(1) gauge field(Montrunich, 2005)(as discussed in Sec-tion VII. Ramirez was also concerned that the entropyassociated with the term estimated by S. Yamashita etal. is only about R ln 240 , which is only a small fraction ofthe total spin entropy. However, it is not clear to us thatthis should be a point of concern since for at temperaturesof order J/5 the entropy of a Heisenberg antiferromagnet

is already much less than the high temperature value dueto short-range spin correlations.(Elstner et al., 1994)

In contrast to the specific heat results described above,on the basis of thermal conductivity measurement, M.Yamashita et al. (Yamashita et al., 2009) concludedthat the spin liquid state in -(BEDT-TTF)2Cu2(CN)3is fully gapped. As with the heat capacity, one expectsthat for a simple metal the thermal conductivity is givenby = T+ T3 + . . . (Ziman, 1960), with the fermionsgiving rise to the linear term and phonons giving rise

A 10T

A 0T

B 0T

T2(K2)

/T

(WK

2m

1)

0.14

0.12

0.10

0.08

0.06

0.04

0.02

00.090.060.030

-

-

| | | . .

FIG. 13 The thermal conductivity, , of -(BEDT-TTF)2-Cu2(CN)3 measured in two samples (A and B) and in differ-ent magnetic fields. As with the heat capacity, simple argu-ments suggest that, at low temperatures /T = +T2+. . . .Clearly this is not what is observed. These data suggest

that 0 (a simple extrapolation gives < 0, which isunphysical). This suggests that the spin liquid state of -(BEDT-TTF)2Cu2(CN)3 is gapped. However, finite temper-atures do not lead to a quadratic increase in /T, suggestingthat the low-lying excitations may be more complex that sim-ply magnons and phonons. [Modified from (Yamashita et al.,2009).]

to the cubic term. Note particularly that, as is onlysensitive to itinerant excitations, one does not need tosubtract a Schottky term. Fig. 13 shows M. Yamashitaet al.s data in the temperature range 0.08-0.3 K plot-

ted as /T against T

2

. One immediately notices thatthe data does not lie on a straight line, which suggeststhat it is not dominated by phonons and therefore thatM. Yamashita et al. did resolve the contribution frommagnetic excitations. Further support for this assertioncomes from the field dependence of the data, which onewould not expect if the heat transport was dominated byphonons. However, more importantly, one should noticethat an extrapolation of the data to T = 0 will not give asignificant /T (indeed the simplest extrapolation, indi-cated by the arrows in the figure, gives /T < 0, which isunphysical). Therefore, M. Yamashita et al. concludedthat /T vanishes at T = 0 K. If correct this would implythat the spin liquid state of -(BEDT-TTF)2Cu2(CN)3is gapped.

M. Yamashita et al. also attempted to quantitativelyanalyse the very lowest temperature part of their data.One complication in this exercise was that one does notknow what fraction of the measured thermal conductiv-ity is due to magnetic excitations and what fraction,ph = T

3, is due to phonons. M. Yamashita et al.found that (T) cannot be well described by a power laweven if ph is large enough to represent three quarters ofthe measured at T = 100 mK, which seems a rather


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generous upper bound given their arguments (describedabove) that the phonons do not dominate the thermalconductivity. This suggests that the gap does not havenodes, which would give a small but non-zero intercept.

M. Yamashita et al. also made an Arrhenius-plot oftheir data. A reasonable fit was found for a value of that implies that about one quarter of the thermalconductivity at 100 mK is due to phonons. This fit yields

a gap of 0.46 (0.38) K in zero field (10 T). However,as M. Yamashita et al. stress, one should be cautiousabout taking this precise value too seriously as that fitwas limited to less than a decade of temperature (0.08-0.5 K) due to the low energy scales involved and currentlimitations in cryogenic technology. Nevertheless, thisanalysis does show that, if there is a gap, it is 2-3 ordersof magnitude smaller than the exchange energy J 250K.

Clearly, an important question is why these two ex-periments (specific heat and thermal conductivity) leadto such different conclusions. M. Yamashita et al. (Ya-mashita et al., 2009) argued that this disagreement re-

sults from an incorrect subtraction of the Schottky termin the heat capacity. However, this is unlikely to be thefull story because the Schottky term only dominates theheat capacity below 0.2 K. One point of interest is thatthe value of extracted from the heat capacity measuredbetween 0.75-2.5 K (in which not Schottky anomaly isevident) is almost twice that found from the fit of thedata taken between 0.075 and 3 K. The gap estimatedby M. Yamashita et al. is small compared to 0.75 K, soone would expect there to be high densities of thermallyexcited fermions in the higher temperature range. In-deed, significant densities of thermally excited fermionswould remain over most of the lower temperature range.

3. The 6 K anomaly

One thing both groups do agree on is that somethinginteresting happens at around 6 K. S. Yamashita et al.found a broad hump when they replot their data asCpT3 against T (in this plot the phonon term should

just appear as a constant offset, while the Schottky termis not relevant at these relatively high tempera

b. j. powell and ross h. mckenzie- quantum frustration in organic mott insulators: from spin liquids...

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