Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf ·...

Spin liquids on the triangular lattice

HARVARDTalk online: sachdev.physics.harvard.edu

ICFCM, Sendai, Japan, Jan 11-14, 2011

Monday, January 10, 2011

Outline

κ

1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea

2. Quantum-disordering magnetic order Application to -(ET)2Cu2(CN)3

3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2 ?


Fractionalization of the electron spin and charge


�c↑c↓

�=

�z↑ −z∗↓z↓ z∗↑

� �ψ+

ψ−

�


A) Quantum “disordering” magnetic order


neutral bosonicspinons which transform to a

rotating reference from along the local antiferromagnetic order


�c↑c↓

�=

�z↑ −z∗↓z↓ z∗↑

� �ψ+

ψ−

�



spinlesscharge -efermions


�c↑c↓

�=

�z↑ −z∗↓z↓ z∗↑

� �ψ+

ψ−

�



SU(2)spin


�c↑c↓

�=

�z↑ −z∗↓z↓ z∗↑

� �ψ+

ψ−

�



U(1)charge


�c↑c↓

�=

�z↑ −z∗↓z↓ z∗↑

� �ψ+

ψ−

�



S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, 155129 (2009)


�c↑c↓

�=

�z↑ −z∗↓z↓ z∗↑

� �ψ+

ψ−

�

U × U−1

Theory has SU(2)sgauge invariance



�c↑c†↓

�=

�b∗1 b∗2−b2 b1

� �f1f†2

�


B) Projected Fermi seaX.-G. Wen, P. A. Lee,

O. Motrunich, M. P. A. Fisher


charge e spinless bosons (or “rotors”) whose fluctuations

project onto the single electron states on each site


�c↑c†↓

�=

�b∗1 b∗2−b2 b1

� �f1f†2

�

X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. FisherB) Projected Fermi sea


neutralfermionicspinons


�c↑c†↓

�=

�b∗1 b∗2−b2 b1

� �f1f†2

�





SU(2)pseudospin ⊃ U(1)charge

�c↑c†↓

�=

�b∗1 b∗2−b2 b1

� �f1f†2

�




SU(2)spin


�c↑c†↓

�=

�b∗1 b∗2−b2 b1

� �f1f†2

�





�c↑c†↓

�=

�b∗1 b∗2−b2 b1

� �f1f†2

�

U × U−1

Theory has SU(2)pgauge invariance




On the triangular lattice, this leads to aU(1) spin liquid with a Fermi surface of spinons

which has been proposed to apply to EtMe3Sb[Pd(dmit)2]2.

This spin liquid has a thermal Hall responsewhich is not observed.


�c↑c†↓

�=

�b∗1 b∗2−b2 b1

� �f1f†2

�




Complete fractionalization: separate excitations carrying spin, charge, and Fermi statistics

Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)Monday, January 10, 2011

Complete fractionalization: separate excitations carrying spin, charge, and Fermi statisticsDecompose electron operator into real fermions, χ:

c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4

Introduce a 4-component Majorana fermion ζi, i = 1 . . . 4and a SO(4) matrix R, and decompose:

χ = R ζ

Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)Monday, January 10, 2011


c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4


χ = R ζ

Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)

SO(4) ∼= SU(2)pseudospin×SU(2)spin



c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4


χ = R ζ


O ×OT

SO(4)gauge∼= SU(2)p;gauge×SU(2)s;gauge



c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4


χ = R ζ


By breaking SO(4)gauge with different Higgs fields, we can reproduce essentially all earlier theories of spin liquids.

We also find many new spin liquid phases, some with Majorana fermion excitations which carry neither spin nor charge


Outline

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H =�

�ij�

Jij�Si · �Sj + . . .

H = J

�

�ij�

�Si · �Sj ; �Si ⇒ spin operator with S = 1/2


Found in -(ET)2Cu[N(CN)2]Clκ

Anisotropic triangular lattice antiferromagnet

Classical ground state for small J’/J



Classical ground state for large J’/JFound in Cs2CuCl4


Valence bond solid


N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)Monday, January 10, 2011

M. Tamura, A. Nakao and R. Kato, J. Phys. Soc. Japan 75, 093701 (2006)Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. 99, 256403 (2007)

Observation of a valence bond solid (VBS) in ETMe3P[Pd(dmit)2]2

Spin gap ~ 40 K J ~ 250 K

X-ray scattering


Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

=

P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

Triangular lattice antiferromagnet


P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

=

Triangular lattice antiferromagnet


Excitations of the Z2 Spin liquid

=A spinon


• A characteristic property of a Z2 spin liquidis the presence of a spinon pair condensate

• A vison is an Abrikosov vortex in the paircondensate of spinons

• Visons are are the dark matter of spin liq-uids: they likely carry most of the energy,but are very hard to detect because they donot carry charge or spin.


A vison


=

-1


A vison


=

-1

-1


A vison


Mutual Chern-Simons Theory Express theory in terms of the physical excitations of the Z2 spinliquid: the spinons, zα, and the visons. After accounting for Berryphase effects, the visons can be described by complex fields va,which transforms non-trivially under the square lattice space groupoperations.

The spinons and visons have mutual semionic statistics, and thisleads to the mutual CS theory at k = 2:

L =2�

α=1

�|(∂µ − iaµ)zα|2 + sz|zα|2

�

+Nv�

a=1

�|(∂µ − ibµ)va|2 + sv|va|2

�

+ik

2π�µνλaµ∂νbλ + · · ·

Cenke Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)


Z2 spin liquidValence bond solid

(VBS)

Neel antiferromagnet

Spiral antiferromagnet

M

Phase diagram of frustrated antiferromagnetsN. Read and S. SachdevPhys. Rev. Lett. 63, 1773 (1991)




(VBS)



M


Vanishing of gap to bosonic spinon

excitations




(VBS)



M



Vanishing of gap to bosonic vison

excitations



(VBS)



M

Phase diagram of frustrated antiferromagnetsProposal:

κ-(ET)2Cu2(CN)3is here

Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett.

102, 176401 (2009)


• Originally motivated by NMR relation. The

quantum critical point has O(4) symmetry

and has 1/T1 ∼ T η with η = 1.374(12).

• Recent µSR experiments show field-induced

magnetic order at very small fields.

• Thermal conductivity is dominated by con-

tribution of visons, and activated by the vi-

son gap.

Proposal: κ-(ET)2Cu2(CN)3 is a Z2 spin liquidnear a quantum phase transition to magnetic order

Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009)Monday, January 10, 2011

Spin excitation in κ-(ET)2Cu2(CN)3

1/T1

Inhomogeneous relaxation

α in stretched exp

Low-lying spin excitation at low-T

Shimizu et al., PRB 70 (2006) 060510 13C NMR relaxation rate

Anomaly at 5-6 K

1/T1 ~ power law of T




and has 1/T1 ∼ T η with η = 1.374(12).





son gap.



Field-induced QPT from SR Line Width

!"#$%$&'#&()%&*(*%+*(,%'-.(/0!(123(45"6.'#%#+(",(7*%8%9:('&(6-;3<(=0>/?@?@=(AB@@CD

F. Pratt et al. (ISIS, UK)preprint


Measured Phase Boundary of the QPT!"#$%&'

H-H0

Tiny H0 implies that spin gap per spin1/2 is z ~ 3.5 mK !

H0 = 5.2(2) mT

F. Pratt et al. (ISIS, UK)preprint




and has 1/T1 ∼ T η with η = 1.374(12).





son gap.



H = 0 Tesla

•Arrhenius behavior for T < Δ !

•Tiny gapΔ = 0.46 K ~ J/500

M. Yamashita et al., Nature Physics 5, 44 (2009)

Thermal conductivity of κ-(ET)2Cu2(CN)3





and has 1/T1 ∼ T η with η = 1.374(12).





son gap.


Outline

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Shastry-Kitaev Majorana representation

On each site, we introduce neutral Majorana fermions

γi0, γix, γiy, γiz which obey

γiαγjβ + γjβγiα = 2δijδαβ

We write the S = 1/2 spin operators as

Sjx =i

2γjyγjz

Sjy =i

2γjzγjx

Sjz =i

2γjxγjy

along with the constraint

γj0γjxγjyγjz = 1 for all j


SU(2)-invariant spin liquids with neutral, spinful Majorana excitations

• Postulate an SU(2)-invariant effective Hamiltonian

for physical S = 1 excitations created by Majorana

fermions γix, γiy, γiz.

• Ensure there is a Projective Symmetry Group (PSG)

under which physical observables have the full trans-

lational symmetry of the triangular lattice.

• Examine stability of the effective Hamiltonian to gauge

fluctuations.

Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011

Majorana fermions on the triangular lattice

H = −i

�

α=x,y,z

�

i<j

tijγiαγjα

where tij is an anti-symmetric matrix withthe following symmetry


Fermi surfaces

Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear

1BZ

The Majorana fermionsgenerically have Fermi sur-faces with the structureshown. 6 Fermi surfacelines intersect at k = 0,and the excitation energy∼ k3 as k → 0.


Fermi surfaces


1BZ

Excitation ∼ k − kF



Fermi surfaces


1BZ

Excitation ∼ k − kF

Excitation ∼ k3



• Time-reversal symetry (T ) and rotation by 60 de-

grees (Rπ/3) are broken, but T Rπ/3 is unbroken.

• The specific heat and spin susceptibility are domi-

nated by excitations near k = 0, with CV ∼ T 2/3

and χ ∼ T−1/3.

• These power-laws are quenched by impurities: de-

tails under investigation.

• The thermal conductivity is dominated by fermions

along the Fermi line. The excitations near k = 0

move too slowly to contribute to transport.

• There is no thermal Hall transport.

Physical properties


Conclusions

Unified perspective on spin liquids obtained by projection of a Fermi sea, and by “quantum-disordering” magnetic order.

Z2 spin liquid proximate to magnetic order a viable candidate for -(ET)2Cu2(CN)3

Proposed novel spin liquid with Fermi surfaces of spinful Majorana fermions

κ


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