Spin liquids on the triangular lattice
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1 Spin liquids on the triangular lattice ICFCM, Sendai, Japan, Jan 11-14, 2011 Talk online: sachdev.physics.harvard.edu HARVARD
2 Outline 1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea 2. Quantum-disordering magnetic order Application to κ -(ET)2Cu2(CN)3 3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2?
3 Outline 1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea 2. Quantum-disordering magnetic order Application to κ -(ET)2Cu2(CN)3 3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2?
4 Fractionalization of the electron spin and charge
5 Fractionalization of the electron spin and charge A) Quantum disordering magnetic order c = z z ψ+ c z z ψ
6 Fractionalization of the electron spin and charge A) Quantum disordering magnetic order c = z z ψ+ c z z ψ neutral bosonic spinons which transform to a rotating reference from along the local antiferromagnetic order
7 Fractionalization of the electron spin and charge A) Quantum disordering magnetic order c = z z ψ+ c z z ψ spinless charge -e fermions
8 Fractionalization of the electron spin and charge A) Quantum disordering magnetic order c = z z ψ+ c z z ψ SU(2) spin
9 Fractionalization of the electron spin and charge A) Quantum disordering magnetic order c = z z ψ+ c z z ψ U(1) charge
10 Fractionalization of the electron spin and charge A) Quantum disordering magnetic order c = z z ψ+ c z z ψ U U 1 Theory has SU(2) s gauge invariance S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, (2009)
11 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2
12 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2 charge e spinless bosons (or rotors ) whose fluctuations project onto the single electron states on each site
13 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2 neutral fermionic spinons
14 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2 SU(2) pseudospin U(1) charge
15 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2 SU(2) spin
16 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2 U U 1 Theory has SU(2) p gauge invariance
17 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2 On the triangular lattice, this leads to a U(1) spin liquid with a Fermi surface of spinons which has been proposed to apply to EtMe 3 Sb[Pd(dmit) 2 ] 2. This spin liquid has a thermal Hall response which is not observed.
18 Fractionalization of the electron spin and charge B) Projected Fermi sea X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. Fisher c b 1 b 2 f1 c = b 2 b 1 f 2 On the triangular lattice, this leads to a U(1) spin liquid with a Fermi surface of spinons which has been proposed to apply to EtMe 3 Sb[Pd(dmit) 2 ] 2. This spin liquid has a thermal Hall response which is not observed.
19 Complete fractionalization: separate excitations carrying spin, charge, and Fermi statistics Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, (2010)
20 Complete fractionalization: separate excitations carrying spin, charge, and Fermi statistics Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: χ = R ζ Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, (2010)
21 Complete fractionalization: separate excitations carrying spin, charge, and Fermi statistics Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: χ = R ζ SO(4) = SU(2) pseudospin SU(2) spin Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, (2010)
22 Complete fractionalization: separate excitations carrying spin, charge, and Fermi statistics Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: χ = R ζ O O T SO(4) gauge = SU(2) p;gauge SU(2) s;gauge Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, (2010)
23 Complete fractionalization: separate excitations carrying spin, charge, and Fermi statistics Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: 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 Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, (2010)
24 Outline 1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea 2. Quantum-disordering magnetic order Application to κ -(ET)2Cu2(CN)3 3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2?
25 Outline 1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea 2. Quantum-disordering magnetic order Application to κ -(ET)2Cu2(CN)3 3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2?
26 H = ij J ij Si S j +... S i spin operator with S = 1/2
27 Anisotropic triangular lattice antiferromagnet Classical ground state for small J /J Found in κ-(et) 2 Cu[N(CN) 2 ]Cl
28 Anisotropic triangular lattice antiferromagnet Classical ground state for large J /J Found in Cs 2 CuCl 4
29 Anisotropic triangular lattice antiferromagnet Valence bond solid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
30 Anisotropic triangular lattice antiferromagnet Valence bond solid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
31 Anisotropic triangular lattice antiferromagnet Valence bond solid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
32 Anisotropic triangular lattice antiferromagnet Valence bond solid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
![Observation of a valence bond solid (VBS) in ETMe 3 P[Pd(dmit) 2 ] 2 X-ray scattering Spin gap ~ 40 K J ~ 250 K M. Tamura, A. Nakao and R.](/docs-images/90/103898845/images/33-0.jpg "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)")
33 Observation of a valence bond solid (VBS) in ETMe 3 P[Pd(dmit) 2 ] 2 X-ray scattering Spin gap ~ 40 K J ~ 250 K M. Tamura, A. Nakao and R. Kato, J. Phys. Soc. Japan 75, (2006) Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. 99, (2007)
34 Triangular lattice antiferromagnet 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).
35 Triangular lattice antiferromagnet 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).
36 Triangular lattice antiferromagnet 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).
37 Triangular lattice antiferromagnet 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).
38 Triangular lattice antiferromagnet 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).
39 Triangular lattice antiferromagnet 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).
40 Excitations of the Z2 Spin liquid A spinon =
41 Excitations of the Z2 Spin liquid A spinon =
42 Excitations of the Z2 Spin liquid A spinon =
43 Excitations of the Z2 Spin liquid A spinon =
44 Excitations of the Z2 Spin liquid A vison A characteristic property of a Z 2 spin liquid is the presence of a spinon pair condensate A vison is an Abrikosov vortex in the pair condensate of spinons Visons are are the dark matter of spin liquids: they likely carry most of the energy, but are very hard to detect because they do not carry charge or spin. N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
45 Excitations of the Z2 Spin liquid A vison = -1 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
46 Excitations of the Z2 Spin liquid A vison = -1 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
47 Excitations of the Z2 Spin liquid A vison = -1 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
48 Excitations of the Z2 Spin liquid A vison = -1 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
49 Excitations of the Z2 Spin liquid A vison = -1-1 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
50 Excitations of the Z2 Spin liquid A vison = -1-1 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
51 Mutual Chern-Simons Theory Express theory in terms of the physical excitations of the Z 2 spin liquid: the spinons, z α, and the visons. After accounting for Berry phase effects, the visons can be described by complex fields v a, which transforms non-trivially under the square lattice space group operations. The spinons and visons have mutual semionic statistics, and this leads to the mutual CS theory at k = 2: L = + 2 α=1 N v a=1 ( µ ia µ )z α 2 + s z z α 2 ( µ ib µ )v a 2 + s v v a 2 + ik 2π µνλa µ ν b λ + Cenke Xu and S. Sachdev, Phys. Rev. B 79, (2009)
52 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) Cenke Xu and S. Sachdev, Phys. Rev. B 79, (2009) Valence bond solid Z 2 spin liquid (VBS) M Neel antiferromagnet Spiral antiferromagnet
53 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) Cenke Xu and S. Sachdev, Phys. Rev. B 79, (2009) Z 2 spin liquid Valence bond solid (VBS) M Vanishing of gap to bosonic spinon excitations Neel Spiral antiferromagnet antiferromagnet
54 Phase diagram of frustrated antiferromagnets Valence bond solid Vanishing of gap to bosonic vison excitations Z 2 spin liquid N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) Cenke Xu and S. Sachdev, Phys. Rev. B 79, (2009) (VBS) M Neel antiferromagnet Spiral antiferromagnet
55 Phase diagram of frustrated antiferromagnets Valence bond solid Z 2 spin liquid Proposal: κ-(et) 2 Cu 2 (CN) 3 is here (VBS) M Neel antiferromagnet Spiral antiferromagnet Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, (2009)
56 Proposal: κ-(et) 2 Cu 2 (CN) 3 is a Z 2 spin liquid near a quantum phase transition to magnetic order Originally motivated by NMR relation. The quantum critical point has O(4) symmetry and has 1/T 1 T η with η =1.374(12). Recent µsr experiments show field-induced magnetic order at very small fields. Thermal conductivity is dominated by contribution of visons, and activated by the vison gap. Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, (2009)
57 Spin excitation in κ-(et) 2 Cu 2 (CN) 3 13 C NMR relaxation rate Shimizu et al., PRB 70 (2006) Inhomogeneous relaxation 1/T 1 α in stretched exp 1/T 1 ~ power law of T Low-lying spin excitation at low-t Anomaly at 5-6 K
58 Proposal: κ-(et) 2 Cu 2 (CN) 3 is a Z 2 spin liquid near a quantum phase transition to magnetic order Originally motivated by NMR relation. The quantum critical point has O(4) symmetry and has 1/T 1 T η with η =1.374(12). Recent µsr experiments show field-induced magnetic order at very small fields. Thermal conductivity is dominated by contribution of visons, and activated by the vison gap. Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, (2009)
59 Proposal: κ-(et) 2 Cu 2 (CN) 3 is a Z 2 spin liquid near a quantum phase transition to magnetic order Originally motivated by NMR relation. The quantum critical point has O(4) symmetry and has 1/T 1 T η with η =1.374(12). Recent µsr experiments show field-induced magnetic order at very small fields. Thermal conductivity is dominated by contribution of visons, and activated by the vison gap. Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, (2009)
60 Field-induced QPT from SR Line Width!"#$%$&'#&()%&*(*%+*(,%'-.( /0!(123(45"6.'#%#+(",( 7*%8%9:('&(6- F. Pratt et al. (ISIS, UK) preprint
61 Measured Phase Boundary of the QPT!"#$%&' H-H 0 H 0 = 5.2(2) mt F. Pratt et al. (ISIS, UK) preprint Tiny H 0 implies that spin gap per spin1/2 is z ~ 3.5 mk!
62 Proposal: κ-(et) 2 Cu 2 (CN) 3 is a Z 2 spin liquid near a quantum phase transition to magnetic order Originally motivated by NMR relation. The quantum critical point has O(4) symmetry and has 1/T 1 T η with η =1.374(12). Recent µsr experiments show field-induced magnetic order at very small fields. Thermal conductivity is dominated by contribution of visons, and activated by the vison gap. Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, (2009)
63 Proposal: κ-(et) 2 Cu 2 (CN) 3 is a Z 2 spin liquid near a quantum phase transition to magnetic order Originally motivated by NMR relation. The quantum critical point has O(4) symmetry and has 1/T 1 T η with η =1.374(12). Recent µsr experiments show field-induced magnetic order at very small fields. Thermal conductivity is dominated by contribution of visons, and activated by the vison gap. Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, (2009)
64 Thermal conductivity of κ-(et) 2 Cu 2 (CN) 3 Arrhenius behavior for T < Δ! Tiny gap Δ = 0.46 K ~ J/500 H = 0 Tesla M. Yamashita et al., Nature Physics 5, 44 (2009)
65 Proposal: κ-(et) 2 Cu 2 (CN) 3 is a Z 2 spin liquid near a quantum phase transition to magnetic order Originally motivated by NMR relation. The quantum critical point has O(4) symmetry and has 1/T 1 T η with η =1.374(12). Recent µsr experiments show field-induced magnetic order at very small fields. Thermal conductivity is dominated by contribution of visons, and activated by the vison gap. Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, (2009)
66 Outline 1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea 2. Quantum-disordering magnetic order Application to κ -(ET)2Cu2(CN)3 3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2?
67 Outline 1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea 2. Quantum-disordering magnetic order Application to κ -(ET)2Cu2(CN)3 3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2?
68 Shastry-Kitaev Majorana representation On each site, we introduce neutral Majorana fermions γ i0, γ ix, γ iy, γ iz which obey γ iα γ jβ + γ jβ γ iα =2δ ij δ αβ We write the S =1/2 spin operators as S jx = i 2 γ jyγ jz S jy = i 2 γ jzγ jx along with the constraint S jz = i 2 γ jxγ jy γ j0 γ jx γ jy γ jz = 1 for all j
69 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 γ ix, γ iy, γ iz. Ensure there is a Projective Symmetry Group (PSG) under which physical observables have the full translational symmetry of the triangular lattice. Examine stability of the effective Hamiltonian to gauge fluctuations. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
70 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 γ ix, γ iy, γ iz. Ensure there is a Projective Symmetry Group (PSG) under which physical observables have the full translational symmetry of the triangular lattice. Examine stability of the effective Hamiltonian to gauge fluctuations. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
71 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 γ ix, γ iy, γ iz. Ensure there is a Projective Symmetry Group (PSG) under which physical observables have the full translational symmetry of the triangular lattice. Examine stability of the effective Hamiltonian to gauge fluctuations. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
72 Majorana fermions on the triangular lattice H = i t ij γ iα γ jα α=x,y,z i<j where t ij is an anti-symmetric matrix with the following symmetry Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
73 Fermi surfaces The Majorana fermions generically have Fermi surfaces with the structure shown. 6 Fermi surface lines intersect at k = 0, and the excitation energy k 3 as k 0. 1BZ Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
74 Fermi surfaces The Majorana fermions generically have Fermi surfaces with the structure shown. 6 Fermi surface lines intersect at k = 0, and the excitation energy k 3 as k 0. 1BZ Excitation k k F Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
75 Fermi surfaces The Majorana fermions generically have Fermi surfaces with the structure shown. 6 Fermi surface lines intersect at k = 0, and the excitation energy k 3 as k 0. 1BZ Excitation k k F Excitation k 3 Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
76 Physical properties Time-reversal symetry (T ) and rotation by 60 degrees (R π/3 ) are broken, but T R π/3 is unbroken. The specific heat and spin susceptibility are dominated by excitations near k = 0, with C V T 2/3 and χ T 1/3. These power-laws are quenched by impurities: details 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. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
77 Physical properties Time-reversal symetry (T ) and rotation by 60 degrees (R π/3 ) are broken, but T R π/3 is unbroken. The specific heat and spin susceptibility are dominated by excitations near k = 0, with C V T 2/3 and χ T 1/3. These power-laws are quenched by impurities: details 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. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
78 Physical properties Time-reversal symetry (T ) and rotation by 60 degrees (R π/3 ) are broken, but T R π/3 is unbroken. The specific heat and spin susceptibility are dominated by excitations near k = 0, with C V T 2/3 and χ T 1/3. These power-laws are quenched by impurities: details 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. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
79 Physical properties Time-reversal symetry (T ) and rotation by 60 degrees (R π/3 ) are broken, but T R π/3 is unbroken. The specific heat and spin susceptibility are dominated by excitations near k = 0, with C V T 2/3 and χ T 1/3. These power-laws are quenched by impurities: details 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. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
80 Physical properties Time-reversal symetry (T ) and rotation by 60 degrees (R π/3 ) are broken, but T R π/3 is unbroken. The specific heat and spin susceptibility are dominated by excitations near k = 0, with C V T 2/3 and χ T 1/3. These power-laws are quenched by impurities: details 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. Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
81 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 κ -(ET)2Cu2(CN)3 Proposed novel spin liquid with Fermi surfaces of spinful Majorana fermions
82 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 κ -(ET)2Cu2(CN)3 Proposed novel spin liquid with Fermi surfaces of spinful Majorana fermions
83 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 κ -(ET)2Cu2(CN)3 Proposed novel spin liquid with Fermi surfaces of spinful Majorana fermions
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