Yuji MATSUDA (Kyoto University): Elementary excitations in a 2D candidate quantum spin liquid

Lecture Notes

OUTLINE:
- Introduction
- A possible qu. SL on 2D triangulat lattice:
* kappa-(BEDT-TTF)2 Cu2(CN)3 (ET)
* EtMe3Sb[Pd(dmit)2]2
- Conclusions

Introduction

Exotic spin states have been proposed in the past: liquid, ice, chiral...
Quantum spin liquid (QSL) is a state that does not break any simple symmetry (lattice or spin-roationslal).

QSL - proposed in 1973 by PW.Anderson (strong qu. fluctuations deny LRO even at T=0)

1D: QSL is firmly established (S=1.2, e=0)
2D: classical - kagome
quantum: fluctuations lift the degeneracy of the ground state, so that QSL may disappear

Exp-tal candidates:

• triangular lattice of 3-He atoms
• BEDT salts (triangular lattice)
• kagome lattice: ZnCu3(OH)6Cl2

2D triangular lattice
Possible ground states:

• three sublattice Neel state (120 degree state)
• Valence bond solid (VBS) : breaks lattice symmetry, LRO of singlets
• RVB: resonating configuration of spin singlets. See Fazekas and Anderson, Philos. Mag. 30, 423 (1974)

Neel order even at T=0 (see D. Huse at al.)

Key questions:

• How can we identify a QSL in the experiments?
  
• What is the elem excitation of QSL in 2D triang. lattice?
• Does a QSL host exotic excitations? (gapped or gapless, magnetic or non-magnetic? localized or itinerant)

A wide range of exp-tal probes: NMR, muSR, \chi, torque, Cv, thermal conductivity...

A powerful probe: Thermal conductivity.
- kappa = kappa_spin + kappa_phonon
- kappa_spin = C*v*l = specific heat * velocity of excitations * mean-free-path
E.g. - Sr2CuO3 (1D Heisenberg). kappa/T goes to 0 as T goes to 0 (gapped SL).
Signatures measurable in experiment:

• If gapless SL: kappa/T would have finite value at T=0; otherwise the value is 0.
  
• Field dependence of k_spin talls you if excitations are magnetic
  
• localized or itinerant? - from the magnitude of the mean free path

3-He atoms on triangular lattice: "4/7 phase": gapless down to T~J/300, where J~3mK (Masutomi PRL 92, 025301 (2004)).

Example 1: kappa-(BEDT-TTF)2Cu2(CN)3:
2D triangular lattice of BEDT-TTF molecules (with S=1/2 per two molecules), seperated by the layers of anions, like Cu2(CN)3.

Experimental observations:

• NMR: no internal magn. field
• \chi(T): J ~ 250K from high-T expansion of susceptibility
• mu-SR: no spin rotation, i.e. no magnetic order down to 20mK (~J/10^5)
• specific heat: Cv/T non-zero, i.e. gapless. subtrating Schottky anomaly (in other BEDT salts, it shows a gap)
• thermal conductivity: gapped by \Delta ~ 0.5K
• muSR: relaxation curve below 300mK, indication microscopic separation between gapped and (magnetic) gapless regions - see T.Goto et al.
  
• NMR 1/T1 shows stretched exponential with \alpha less than 0.5
• thermal expansion: lattice anomaly at ~6K (Manna et al., PRL'2010) - is it a structural transition involving charge degrees of freedom?
  
• frequency-dependent dielectric constant
  

Summary: genuine features of a QSL masked by inhomogeneity.

Example 2: Recently found material: EtMe3Sb[Pd(dmit)2]2
Nearly triangular lattice of EtMe3Sb units (S=1/2 per two neighbouring units). t'/t~ 0.93 (close to triangular lattice).
As a function of t'/t, different compounds in this family show a variety of phases, including AFM, charge insulator, QSL.

Experiments (Yamashita et al, Science'2010, and others):

• magnetic susceptibility measured down to 5K (and disappears below, meaning there is a spin-gap to excitation - see a Comment to this post below)
  
• magnetic torque down to T=0.3K
• no LRO down to T=0.3K (from 13C NMR)
  
• ZF mu-SR: no magnetic order down to ~J/10^5. See Ishi et al. Itou et al, Nat. Phys. (2010)
• no change in zero-field vs. field cooled (answer to the question from the audience)
  
• NMR 1/T1 shows stretched exponential with alpha that changes between 0.5 and 1 with a minimum at ~1K: sign of inhomogeneity
• specific heat after Schottky subtraction: - gapless in EtMe3Sb (dmit-131) with finite C/T at T=0 (as opposed to non-magnetic Et2Me2Sb (dmit-221) with only phonon contribution C~T^3)
• thermal conductivity: gapless, since kappa/T=0.19 W/K^2m shows finite value in dmit-131 (i.e. gapless) as opposed to zero value in the spin-singlet analogue dmit-221.
• excitations are itinerant: mean free path estimates ~1.2 micro-meter (~10^3 larger than interspin distance)
  

Question from S.-W. Cheong: how come phonons do not scatter on spin excitations?
A: irrelevant, since phonon mean-free path is comparable to the sample size at these low temperatures.

Question from P. Coleman: could you dope the sample with impurities to see how \kappa/T changes?
A: We know that the mean free path of the excitations is very long, much longer than the distance between the spins. Hence, we do not expect any scattering off impurities.

Note: QSL apparently conducts heat very well - like brass of a 5-yen coin (LOL :-)
Field-dependence of elementary excitations:
H larger 1K: linear increase in kappa_spin
H smaller 1K: spin-gap like behaviour (H_gap ~ 2 T) Interpreted as coexistence of non-magnetic gapless excitations and spin-gap like excitations that couple to magnetic field.
C.f. R.Singh and D.Huse (2007): S=1/2 on kagome lattice showing gapped excitations

Wilson ratio R~1.2 - similar to metals (!)
Is there symmetry breaking in the QSL?
NMR shows a peak in 1/(T1T) around T~1K suggesting a phase transition (to what ?)

Theories on QSL
- Hubbard model on triangular lattice for intermediate U strength.
- Heisenbeg model suggest QSL for J'/J between 0.6 and 0.8.
- ring exchange theory: suggests QSL (Misguich et al), but excitations are gapped
- O. Motrunich (2003), S. Lee and P.A. Lee (2005), Lee, Lee and Senthil (2007) suggest a spinon Fermi surface, however that would predict a large Hall angle kappa_xy/kappa_xx, which the exp-t does not see.
- algebraic spin liquid: Wen PRB (2002)
- gapless boson

Conclusions:

1. kappa-(BEDT-TTF)2 Cu2(CN)3 (ET)
   
   
   • controversial gap vs. gapless spin excitations
   • problems with homogeneity
     
2. EtMe3Sb[Pd(dmit)2]2

• a homogeneous system
• specific heat and \kappa/T shows unambiguous(?) gapless excitations
• very long mean free path - itinerant spin excitations
  
• dual nature of spin excitations (gapless at H=0, and gapped when field is applied)
  
• Wilson ratio ~ 1.2 (like in metals)
  
• Symmetry breaking in QSL?

QUESTIONS:
Q:Chubukov: Anyone did dHvA?
A: we plan it in near future

Q: S-W. Cheong. Your story is based on T below 0.3K which is not too far from the phonon peak. Are you sure of your power-law fitting?
A: Yes, we are sure.

Q: P. Coleman. kappa/T shows a peak at ~0.6K - where does it come from?
A: This is a low-lying phonon peak.

Q. Silke Paschen: Can you exclude the possibility of coupling to phonons?
A: these are very itinerant excitations

Q: Keimer: Some spin chain systems also have long m.f.p., is it therefore so unusual to see long m.f.p. in this triangular-lattice system?
A: Experimentally, you're quite right. However there are still debates on this subject.