IAS, Tsinghua University, October 8th, 2014

QuantumQuantum spinspin liquidsliquids asas softsoft-- gapgap MottMott insulatorsinsulators

YiYi ZhouZhou (Zhejiang(Zhejiang University)University)

Reference: YZ and Tai-Kai Ng, Phys. Rev. B 88, 165130 (2013), YZ, Kazushi Kanoda, and Tai-Kai Ng, submitted, invited by RMP (2014). GeneralGeneral motivationsmotivations

 WhyWhy quantumquantum spinspin liquidliquid isis interesting?interesting?

 New states of matter in magnetic insulators

 To understand Mott physics more CondensedCondensed mattermatter physicsphysics

 TheThe centralcentral issueissue inin condensedcondensed mattermatter physicsphysics isis toto discoverdiscover andand understandunderstand newnew statesstates ofof mattersmatters..

metal insulator superconductor MagneticMagnetic insulators:insulators: AnyAny newnew statesstates ofof mattermatter ??  MagnetismMagnetism isis oneone ofof thethe oldestoldest subjectssubjects inin physicsphysics  Historically it is associated with magnetic field generated by ordered magnetic moments. MagneticMagnetic insulators:insulators: AnyAny newnew statesstates ofof mattermatter ??  MagneticMagnetic momentsmoments orderorder differentlydifferently inin variousvarious materialsmaterials

helical magnet

All these ordered states can be understood from the magnetic interaction between classical vectors (spins). MagneticMagnetic insulators:insulators: AnyAny newnew statesstates ofof mattermatter ??  WhatWhat’’ss thethe groundground statestate forfor anan antiferromagnet?antiferromagnet?  TheThe debatedebate betweenbetween NNééelel andand LandauLandau

Quantum AFM fluctuated ordered MagneticMagnetic insulators:insulators: AnyAny newnew statesstates ofof mattermatter ??  WhatWhat’’ss thethe groundground statestate forfor anan antiferromagnet?antiferromagnet?  TheThe debatedebate betweenbetween NNééelel andand LandauLandau

 Quantum fluctuations dominate over Néel’s order in one dimension → spin liquid.  Néel’s order wins at 2D square lattice → AFM order.  The general situation is still unclear.

MagneticMagnetic insulators:insulators: AnyAny newnew statesstates ofof mattermatter ??

 IItt waswas proposedproposed thatthat quantumquantum spinspin liquidliquid statesstates cancan possesspossess veryvery exoticexotic properties.properties.  Emerged particles and fields  EmergentEmergent phenomenaphenomena  New particles and fields emerge at low-energy scales but they are totally absent in the Hamiltonian that describes the initial system.  Different physics laws emerge at different scales.

Definition:Definition: QSLQSL

 Quantum spin liquid (QSL) is an insulator with an odd number of electrons per unit cell which does not order magnetically down to zero temperature due to quantum fluctuations.

 The quantum disorder is intrinsic, not induced by extrinsic impurities.

 Not a constructive definition.

FeaturesFeatures (or(or featurelessfeatureless ?)?)

 ““FeaturelessFeatureless”” MottMott insulators.insulators.

 Lattice translational symmetry is respected.

 The absence of long ranged magnetic order.

 …

 CharacterizedCharacterized byby emergentemergent phenomenaphenomena andand possiblepossible quantumquantum ordersorders EmergentEmergent particlesparticles andand fieldsfields

 Spinon: S=1/2, charge neutral, mobile objects  The spinons may obey Fermi or Bose statistics or even nonabelian statistics and there may or may not be an energy gap;  Gauge field: spin singlet fluctuations  These spinons are generally accompanied by gauge fields, U(1) or Z2 .

 How can we know which of these plausible states are “real”, especially when D>1 ?  This is a very difficult problem. I shall outline a few of different approaches to this problem. NonlinearNonlinear σσ--modelmodel  From two-spin dynamics to a rotor model

    2    2    nnSS iii 1, 1 , 212  nnJSnnnJSn 21

        ,     2 prLrprLnrr  21 ,  nnnnnL 21 p  1   r  LLr  .0,   2     2    ,0   LnJSnL  mrm 

2 2  Quantum mechanics   z   ,,,,)1( llmmLllL

Ground state: l  0

Heisenberg uncertainty: L  , L 0   NonlinearNonlinear σσ--modelmodel  Many spin problem: rotor representation ~ 1  2 ~  2 JI 1 magnetical orderedly H  i  i  nnJL j ~ 2I i , ji  2 JI 1 liquidspin state

1  2 ~  H  i  iij  nnJL j 2I i , ji  frustration  quantum disordered state  How does the magnitude of spin enters ? Topological term (F.D.M. Haldane)  I   dtS   2     SnnJn   i  ji  ' PhasesBerry  2 i , ji  

eiS Qualitative difference between ground states of integer and half-odd-integer spin chains (Haldane conjugation) ResonatingResonating ValenceValence BondBond (RVB)(RVB)  Issue: the nonlinear σ-model approach becomes too difficult to implement in D>1.

P.W. Anderson: Why not just “guess” the wave-function…

1 , ji  2

   ,,, jijijia  RVB  jiji 2211  2211  NN 2/2/ jiji 2211 

variational parameters superposition of spin-singlet pairs

The term RVB was first coined by Pauling (1949) in the context of metallic materials. RVB:RVB: GutzwillerGutzwiller projectionprojection  Problem: Is there any simple way to obtain reasonable good variational parameters in RVB wave functions?

Anderson and Zou: We may construct one from BCS wave-function…

  ccvu   ,0 P BCS   kk  kk  RVB  BCSG k

The number of electrons in a BCS wave- depend on some variational function is not fixed. There can be 0, 1, or 2 parameters, determined by electrons in a lattice site. The Gutzwiller Bogoliubov-de Gennes equation projection removes all the double occupied components. An insulator state is obtained when # of electron = # of lattice sites.

Later, physicist apply this type of wave-function to different lattice systems with different Hamiltonians and find interesting and rather good results when

the  BCS is chosen correctly. RVB:RVB: spinonsspinons andand gaugegauge fieldsfields

 How about excited states?

The spin excitations are electron-like (or superconductor quasiparticle- like, S=1/2) except that they do not carry charge, so called spinons.

  excited  P kkG ' BCS      cvcu kkkkk 

spions: S=1/2, charge neutral, mobile objects

Question: Can this simple picture survives Gutzwiller projection? RVB:RVB: spinonspinon andand gaugegauge fieldsfields

 Confinement of spinons  X. -G. Wen used the tool of lattice gauge theory to show that spinons may be confined with other spinons to form S=1 excitations after Gutzwiller projection.  Structure of projected wave-function  X. -G. Wen also invented a new mathematical tools (Projective Symmetry Group) to classify spin-liquid states within the Gutzwiller-BCS approach.  Schwinger bosons  People also use a similar approach where spins are represented by Schwinger bosons. Then spin excitations are S=1/2 bosons instead of fermions in projected BCS approach. OtherOther aspectsaspects ofof theorytheory

 Exact solvable models  Kitaev honeycomb model changed our view on spin liquids singanificantly  A spin liquid is not necessary to be spin-rotation-invariant.  The statistics of spinons can be non-abelian.  Spin-orbital effect, etc.  …  Other approaches  Analytical methods  Bethe Ansatz, Bosonization, CFT, …  Numerical methods  Exact Diagonalization, DMRG, PEPS / Tensor Network, QMC, CDMFT, …

 Spin liquids with spin S>1/2  …

for a review, see YZ, Kazushi Kanoda, Tai-Kai Ng (2014), invited by RMP ExistingExisting S=1/2S=1/2 quantumquantum spinspin liquidliquid candidatescandidates atat D>1D>1

κ -(ET)2Cu 2(CN) 3 (2003) Pd-(dmit)2(EtMe 3Sb) (2008) ZnCu3(OH) 6Cl 2 (2007)

Fake

Ba3CuSb 2O 9 (2011) Na4Ir 3O 8 (2007) quasi 1D? LiZn2Mo 3O 8 ?(2012) YZ, Kazushi Kanoda, Tai-Kai Ng (2014), invited by RMP

All existing QSL candidates are fermion-like gapless systems. ExperimentalExperimental detectiondetection ofof spinspin liquidliquid states?states?  How to identify and characterize spin liquid states in established materials?

Patrick A. Lee: All these materials may be described by some kind of projected BCS (or Fermi liquid) state at low temperature.

P.A. Lee is perhaps the strongest believer of spin liquid states. He works most closely with experimentalist to show that spin liquids exist in nature. He noticed a common feature of most of the spin liquid candidates discovered so far: they are all closed to the metal to insulator transition.  Question: What is the physical significance of this observation? FurtherFurther experimentalexperimental proposalsproposals

 Optical conductivity: gapless spinons, power law behavior Tai-Kai Ng and P. A. Lee, PRL 99, 156402 (2007). Expts: 1) κ-ET organic salt, S. Elsässer, et. al. (U. Stutggart group), PRB 86, 155150 (2012). 2) Herbertsmithite, D. V. Pilon, et. al. (MIT group), Phys. Rev. Lett. 111, 127401 (2013).  GMR-like setup: oscillatory coupling between two FMs via a QSL spacer M. R. Norman and T. Micklitz, PRL 102, 067204 (2009).  Thermal Hall effect: different responses between magnons and spinons H. Katsura, N. Nagaosa, and P. A. Lee, Phys. RRL 104, 066403 (2010). in contradiction to expts.  Sound attenuation: spinon-phonon interaction, spinon lifetime, gauge fields YZ and Patrick A. Lee, PRL. 106, 056402 (2011).  ARPES: electron spectral function for a QSL with spinon FS or Dirac cone E. Tang, M.P.A. Fisher and Patrick A. Lee, PRB, 045119 (2013).  Neutron scattering: spin chirality, DM interaction, Kagome lattice N. Nagaosa and Patrick A. Lee, PRB 87, 064423 (2013).  Spinon transport: measure spin current flow through M-QSL-M junction C.Z. Chen, Q.F.Sun, Fa Wang and X.C. Xie, PRBB 88, 041405(R) (2013). Motivation 1: A generic framework to compare theoretical predictions of QSLs to experimental data is still missing at the phenomenological level. K 0.2 K el el 1 1   ,,33.3 , ~ Tk Tk  B  B s eV106.6~ s  Hz103~cm 1 Hz103~cm eV102    5- 10

~)(   16-   ,2  ~ 1- -1         cm1 

 (2007) and P.A.Lee T.K.Ng DifficultiesDifficulties inin QSLQSL theorytheory charge ˆ   hfc ˆˆ , i ˆˆ gauge field  fef ,  i heh ˆˆ . electron U(1) gauge structure spinon  What’s the low energy effective theory for QSLs?  Confinement vs. deconfinement  Spinons are not well defined quasiparticles even though the U(1) gauge 3/2 field is deconfined. "   Lack of Renormalization-group scheme to illustrate possible effective theories as in Fermi liquid theory.

Motivation 2: Could we construct an effective theory with “electrons” or “dressed electrons” (quasiparticles) directly? SpinSpin liquidsliquids inin thethe vicinityvicinity ofof metalmetal--insulatorinsulator transitiontransition

 Pressure effect

 κ-(ET)2 Cu2 (CN)3 : Z2 QSL → superconductor

 Pd-(dmit)2 (EtMe3 Sb) : U(1) QSL → metal

 Na4 Ir3O 8 : QSL → metal  Importance of charge fluctuations

Mott transition Heisenberg model ° Fermi liquid 120 AF order 0 / tU metal insulator

Schematic phase diagram on triangular lattice QSLQSL asas aa softsoft--gapgap MottMott insulatorinsulator

Mott transition Heisenberg model Fermi liquid AFM order 0 / tU metal insulator

Schematic DOS ( from optical conductivity and other expts. )

 E)(  E)(  E)(

 E  E E  EE   ||)(  QuestionsQuestions

 What kind of charge fluctuations should be the key to QSLs ?

 Can we formulate a phenomenological theory for QSLs in the vicinity of Mott transition starting from the metallic side? FermiFermi LiquidLiquid theorytheory Quasiparticles

 When electron-electron interactions are adiabatically turned on, the low energy excited states of interacting N-electron systems evolve in a continuous way, and therefore remain one-to-one correspondence with the states of noninteracting N-electron systems.  Assumption: The same labeling scheme through fermion occupation number can be applied to fermionic QSLs. Interaction between quasiparticles

 p2  1  E   n   '  nOnnf 3  *  p  pp' p'p ' p  2m  2 pp' ' 0   0 ,   nnnFEE ppp 

Quasiparticle energy p2 ~  ' nf  p *   pp'  p' ' 2m p' ' LandauLandau parametersparameters

Spin symmetric and antisymmetric decomposition

 ' s a pp' pp'  '  fff pp''

Isotropic systems  as )( as )( 3D: pp'   l Pff l  )(cos l0

 as )( as )( 2D: pp'   l lff  )cos( l0

Dimensionless Landau parameters

as )( as )( l  )0( fNF l IdeasIdeas

 Effective theory with chargeful quasiparticles

 “Building blocks” are chargeful quasiparticles instead of spinons.  Both Fermi liquids and quantum spin liquids can be described within the same framework.  Physical quantities will be renormalized

 By the interaction between quasiparticles.

 Electrically insulating but thermally conducting state

 Can be achieved in the framework of Landau’s Fermi liquid theory with properly chosen Landau parameters. QuasiQuasi--particleparticle transporttransport (I)(I)

Particle (charge or mass) current carried by quasi-particles

0 ~ s n  n pp   n pp , vjjvJ pp   fpp' vp' p p p'  p' ~~ 0   nnn ppp ~0 ~ p nn F ( p  ) is local equilibrium occupation number

m  F s   J 1 1 J 0 *   m  d  dimension where J 0 is the current carried by corresponding non-interacting quasiparticles. QuasiQuasi--particleparticle transporttransport (I)(I)

m  F s  J 1 1 J 0 *   m  d 

Galilean invariance is broken by the periodic crystal potential

m* F s 1 1 m d

charge current is always renormalized by the interaction.

The electron system will be electrically insulating when

s * 1 dF  mm  0/or0/1 QuasiQuasi--particleparticle transporttransport (II)(II)

Thermal current carried by quasi-particles

 n0  J  n~ v  )()( v n  s nf  Q  pp p  p  pp  p'pp'  p p  p'  p 

m   JJ 0 Q m* Q

Thermal current is renormalized by the factor / mm *

A quantum spin liquid phase will be achieved through

F s m 1  0,01 strong charge (current) fluctuations d m* in the l  1 channel FrameworkFramework ofof effectiveeffective theorytheory

Effective theory = Landau Fermi-liquid-type theory with chargeful quasiparticles + singular Landau parameters

Elementary excitations (particle-hole excitations) described by Landau transport equation is chargeless at q=0 and ω=0, but charges are recovered at finite q and ω. ThermodynamicThermodynamic quantitiesquantities

Specific heat ratio C m*  V   )0( T m

Pauli susceptibility * m 1 )0(  P  a  P 1 Fm 0

Wilson ratio 22 4 k PB 1 RW  2  a g B )(3 1 F0 ElectromagneticElectromagnetic responsesresponses

Charge response function  q ),( q ),(  0d d  F s q ),(  2  q ),( 1  F s  1  0d  0 s 22   1 q ),(  F  NqvdF )0(

Transverse current response function

0t q ),( t q ),(  s F1 q ),( 0t q ),( 1 s 1 q ),(  NdF )0(

*  , 00 td : for a Fermi liquid with effective mass m in the absence of Landau interactions

A. J. Leggett, Phys. Rev. 140, A1869 (1965); Phys. Rev. 147, 119 (1966). Longitudinal current response function  2 q  q ),(),( l q2 d AC conductivity e2 q  q ),(),( tl )( i tl )(

In the limit s 1 dF  0/1

 s q  /),(1 dF Expand it as 1 ~  qq 222 N )0( lltt

UU c     l  0&0 incompressibility

s The other possibility F 0  would result in complete vanishing of charge response. Dielectric function

4e2 q 1),(  q  222  qONve 2 )()0(41~),( q2 d F

Insulator: no screening effect even though we start with the metallic side.

AC conductivity

 )( ω 0 For q=0 and small ω, )(  22  Nei 0 )(])0(/[

2    0 )(Re)(Re

Power law AC conductivity inside the Mott gap. QuasipatricleQuasipatricle scatteringscattering amplitudeamplitude

App' q ),(   f 0 'p''pp' A '' p'p qq ),(),(  fpp' 'p'

0 0 0  2/  nn qpqp 2/  vq p np where 0 'p' q ),(    2/  qpqp 2/ vq p  p

Assuming that the scattering is dominating in the l  1 channel,

p'p s fpp' ~ 2 f1 pF s  1 q  /),(1 dF 22 Using ~   q tt , after some algebra, we obtain N )0(

 d  2,1 d p'p 1  A q ),(  , g  pp' 2   )0( pN F 2  d  3, ig   t q  2 F qv ThermalThermal conductivityconductivity

Thermal resistivity for a Fermi liquid 11  n0  0 0 0 0 2  1   1 2 3 nnnnW 4 )(1)(1()4,3;2,1(  4321 )  111 uv   4 4,3,2,1  1 1  (  )  4321 ,  ,  pppp 43214321 C.J. Pethick, Phys. Rev. 177, 393 (1969)

0 ni i nn 0  i , vi is quasiparti velocity,cle u is an arbitrary unit vecto alongr T  i

Trial functions:    iii uv ,    ii  

Scattering probability:

2 W  App' q |),(|2)4,3;2,1(

 2/    qpqpqpqp 2/'2/'2/ Thermal resistivity due to inelastic scattering between quasiparticles

d  3/)1( 1  BTk     in  F 

Thermal resistivity due to elastic impurity scattering 1 At low temperature, the inelastic scattering is cut off by elastic impurity scattering rate  0  1 el  v2*  dT F 0

d 3/)4( 1    Tk  d 1    max   B  ,  kT 3   v2*   B  F  F 0 

Consistent with U(1) gauge theory, Cody P. Nave and Patrick A. Lee, Phys. Rev. B 76 235124 (2007). CollectiveCollective modesmodes

Charge sector: density fluctuations

energy shift in the direction pˆ spherical symmetry 

Equation of motion: linearized Landau transport equation

collision integral Density fluctuation modes in a system with spherical symmetry

m  0 m 1 m  2

longitudinal mode transverse mode quadrupolar mode Zero sound modes in the spin liquid phase:

ss s Three channel model with only , FF 10 and F 2 : a weakly damping zero sound s mode exists when F 2  3/10 . SchematicSchematic phasephase diagramdiagram

Different from Brinkman-Rice picture SummarySummary

 Mott transition driven by current fluctuations  An alternative picture of metal-insulator transitions to Brinkman -Rice  QSL as a soft gap Mott insulator  Phenomenological theory for both Fermi liquids and quantum spin liquids in the vicinity of Mott transition.  FL: both electrically and thermally conducting  QSL: electrically insulating but well thermally conducting  Mott physics : characterized by many intrinsic in-gap excitations in such quantum spin liquids.  Wilson ratio ~ 1, ac conductivity, dielectric function, thermal conductivity  There exist collective modes as well as “quasiparticles”

AcknowledgementAcknowledgement

 Theory  P. A. Lee (MIT)  Z. X. Liu (Tsinghua U, Beijing)  T. K. Ng (HKUST)  H. H. Tu (MPQ)  X. G. Wen (MIT / PI)  F. C. Zhang (ZJU / HKU)  Experiment  K. Kanoda (U Tokyo )

ThankThank youyou forfor attentionattention !!