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Detecting collective excitations of quantum spin liquids Talk online: sachdev.physics.harvard.edu Talk online: sachdev.physics.harvard.edu

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Ribhu Kaul Microsoft Max Metlitski Harvard Cenke Xu Harvard Roger Melko Waterloo arXiv:0809.0694 arXiv:0808.0495 Yang Qi Harvard

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Collective excitations of quantum matter Fermi liquid - zero sound and paramagnons Superfluid - phonons and vortices Quantum hall liquids - magnetoplasmons Antiferromagnets - spin waves

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Collective excitations of quantum matter Fermi liquid - zero sound and paramagnons Superfluid - phonons and vortices Quantum hall liquids - magnetoplasmons Antiferromagnets - spin waves Spin liquids - visons and “photons”

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Antiferromagnet

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Spin liquid =

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=

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=

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=

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=

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=

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General approach Look for spin liquids across continuous (or weakly first-order) quantum transitions from antiferromagnetically ordered states

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Outline 1. Collective excitations of spin liquids in two dimensions Photons and visons 2. Detecting the vison Thermal conductivity of -(ET) 2 Cu 2 (CN) 3 3. Detecting the photon Valence bond solid order around Zn impurities

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Outline 1. Collective excitations of spin liquids in two dimensions Photons and visons 2. Detecting the vison Thermal conductivity of -(ET) 2 Cu 2 (CN) 3 3. Detecting the photon Valence bond solid order around Zn impurities

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Ground state has long-range Néel order Square lattice antiferromagnet

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Destroy Neel order by perturbations which preserve full square lattice symmetry Square lattice antiferromagnet A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007). R.G. Melko and R.K. Kaul, Phys. Rev. Lett. 100, 017203 (2008).

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Theory for loss of Neel order

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Perturbation theory

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N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

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From the square to the triangular lattice

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Interpretation of non-collinearity N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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What is a vison ? N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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Z 2 Spin liquid = What is a vison ?

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= Z 2 Spin liquid What is a vison ? N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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= Z 2 Spin liquid What is a vison ? N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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= Z 2 Spin liquid What is a vison ? N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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= Z 2 Spin liquid What is a vison ? N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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= Z 2 Spin liquid What is a vison ? N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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= Z 2 Spin liquid What is a vison ? N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

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Global phase diagram S. Sachdev and N. Read, Int. J. Mod. Phys B 5, 219 (1991), cond-mat/0402109

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Mutual Chern-Simons Theory Cenke Xu and S. Sachdev, to appear

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Mutual Chern-Simons Theory Cenke Xu and S. Sachdev, to appear

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Global phase diagram Cenke Xu and S. Sachdev, to appear

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Mutual Chern-Simons Theory Cenke Xu and S. Sachdev, to appear

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Outline 1. Collective excitations of spin liquids in two dimensions Photons and visons 2. Detecting the vison Thermal conductivity of -(ET) 2 Cu 2 (CN) 3 3. Detecting the photon Valence bond solid order around Zn impurities

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Outline 1. Collective excitations of spin liquids in two dimensions Photons and visons 2. Detecting the vison Thermal conductivity of -(ET) 2 Cu 2 (CN) 3 3. Detecting the photon Valence bond solid order around Zn impurities

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The following slides and thermal conductivity data are from:

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Q2D organics -(ET) 2 X; spin-1/2 on triangular lattice dimer model ET layer X layer Kino & Fukuyama t’/t = 0.5 ~ 1.1 Triangular lattice Half-filled band

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face-to-face pairs of BEDT-TTF molecules form dimers by strong coupling. Dimers locate on a vertex of triangular lattice and ratio of the transfer integral is ~1. Charge +1 for each ET dimer; Half-filling Mott insulator. -(BEDT-TTF) 2 Cu 2 (CN) 3 Y. Shimizu etal, PRL 91, 107001 (2003) Cu 2 [N(CN) 2 ]Cl (t’/t = 0.75, U/t = 7.8) Néel order at T N =27 K Cu 2 (CN) 3 (t’/t = 1.06. U/t = 8.2) No sign of magnetic order down to 1.9 K. Heisenberg High-T Expansion (PRL, 71 1629 (1993) ) J ~ 250 K

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Spin excitation in -(ET) 2 Cu 2 (CN) 3 1/T 1 Inhomogeneous relaxation in stretched exp Low-lying spin excitation at low-T Shimizu et al., PRB 70 (2006) 060510 13 C NMR relaxation rate Anomaly at 5-6 K 1/T 1 ~ power law of T

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Heat capacity measurements S. Yamashita, et al., Nature Physics 4, 459 - 462 (2008) Evidence for Gapless spinon? A. P. Ramirez, Nature Physics 4, 442 (2008)

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Thermal-Transport Measurements What is the low-lying excitation of the quantum spin liquid found in -(BEDT-TTF) 2 Cu 2 (CN) 3. Gapped or Gapless spin liquid? Spinon with a Fermi surface? Only itinerant excitations carrying entropy can be measured without localized ones no impurity contamination 1/T 1, measurement ← free spins Heat capacity ← Schottky contamination Best probe to reveal the low-lying excitation at low temperatures.

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Thermal Conductivity below 10K Similar to 1/T 1 by 1 H NMR Heat capacity vs. T below 10 K Magnetic contribution to Phase transition or crossover? Chiral order transition? Instability of spinon Fermi surface? S. Yamashita, et al., Nature Physics 4, 459 - 462 (2008) Difference of heat capacity between X = Cu 2 (CN) 3 and Cu(NCS) 2 (superconductor). No structure transition has been reported.

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Thermal Conductivity below 300 mK Convex, non-T^3 dependence in Magnetic fields enhance /T vs T 2 Plot Note: phonon contribution has no effect on this conclusion.

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Arrhenius plot H = 0 Tesla Arrhenius behavior for T < Δ Tiny gap Δ = 0.46 K ~ J/500

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How do we reconcile power-law T dependence in 1/T 1 with activated thermal conductivity ?

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Yang Qi, Cenke Xu and S. Sachdev, arXiv:0809:0694

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How do we reconcile power-law T dependence in 1/T 1 with activated thermal conductivity ? Yang Qi, Cenke Xu and S. Sachdev, arXiv:0809:0694

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Thermal Conductivity below 10K Similar to 1/T 1 by 1 H NMR Heat capacity vs. T below 10 K Magnetic contribution to Phase transition or crossover? Chiral order transition? Instability of spinon Fermi surface? S. Yamashita, et al., Nature Physics 4, 459 - 462 (2008) Difference of heat capacity between X = Cu 2 (CN) 3 and Cu(NCS) 2 (superconductor). No structure transition has been reported.

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Spin excitation in -(ET) 2 Cu 2 (CN) 3 1/T 1 Inhomogeneous relaxation in stretched exp Low-lying spin excitation at low-T Shimizu et al., PRB 70 (2006) 060510 13 C NMR relaxation rate Anomaly at 5-6 K 1/T 1 ~ power law of T

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How do we reconcile power-law T dependence in 1/T 1 with activated thermal conductivity ? Yang Qi, Cenke Xu and S. Sachdev, arXiv:0809:0694

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Global phase diagram Cenke Xu and S. Sachdev, to appear

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How do we reconcile power-law T dependence in 1/T 1 with activated thermal conductivity ? Cenke Xu and S. Sachdev, to appear

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Spin excitation in -(ET) 2 Cu 2 (CN) 3 1/T 1 Inhomogeneous relaxation in stretched exp Low-lying spin excitation at low-T Shimizu et al., PRB 70 (2006) 060510 13 C NMR relaxation rate Anomaly at 5-6 K 1/T 1 ~ power law of T

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How do we reconcile power-law T dependence in 1/T 1 with activated thermal conductivity ? Yang Qi, Cenke Xu and S. Sachdev, arXiv:0809:0694

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Outline 1. Collective excitations of spin liquids in two dimensions Photons and visons 2. Detecting the vison Thermal conductivity of -(ET) 2 Cu 2 (CN) 3 3. Detecting the photon Valence bond solid order around Zn impurities

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1. Collective excitations of spin liquids in two dimensions Photons and visons 2. Detecting the vison Thermal conductivity of -(ET) 2 Cu 2 (CN) 3 3. Detecting the photon Valence bond solid order around Zn impurities Outline

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Non-perturbative effects in U( 1 ) spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1990) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

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Non-perturbative effects in U( 1 ) spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1990) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

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Non-perturbative effects in U( 1 ) spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1990) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

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Order parameter of VBS state

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Non-perturbative effects in U( 1 ) spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1990) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

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Non-perturbative effects in U( 1 ) spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1990) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

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Quantum Monte Carlo simulations display convincing evidence for a transition from a Neel state at small Q to a VBS state at large Q A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007). R.G. Melko and R.K. Kaul, Phys. Rev. Lett. 100, 017203 (2008). F.-J. Jiang, M. Nyfeler, S. Chandrasekharan, and U.-J. Wiese, arXiv:0710.3926

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Distribution of VBS order at large Q Emergent circular symmetry is evidence for U(1) photon and topological order A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007).

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Non-magnetic (Zn) impurity in the U( 1 ) spin liquid A. Kolezhuk, S. Sachdev, R. R. Biswas, and P. Chen, Phys. Rev. B 74, 165114 (2006).

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Non-magnetic (Zn) impurity in the U( 1 ) spin liquid M. A. Metlitski and S. Sachdev, Phys. Rev. B 77, 054411 (2008).

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Schematic of VBS order around impurity Bulk VBS order is columnar

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Schematic of VBS order around impurity Bulk VBS order is columnar

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Schematic of VBS order around impurity Bulk VBS order is columnar

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Schematic of VBS order around impurity Bulk VBS order is columnar

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Schematic of VBS order around impurity Bulk VBS order is columnar

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Schematic of VBS order around impurity Bulk VBS order is plaquette

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Schematic of VBS order around impurity Bulk VBS order is plaquette

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Schematic of VBS order around impurity Bulk VBS order is plaquette

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Schematic of VBS order around impurity Bulk VBS order is plaquette

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Schematic of VBS order around impurity Bulk VBS order is plaquette

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Bond order from QMC of J-Q model R. K. Kaul, R. G. Melko, M. A. Metlitski and S. Sachdev, arXiv:0808.0495

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Bond order from QMC of J-Q model R. K. Kaul, R. G. Melko, M. A. Metlitski and S. Sachdev, arXiv:0808.0495 Phase of VBS order Phase of VBS order

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Bond order from QMC of J-Q model R. K. Kaul, R. G. Melko, M. A. Metlitski and S. Sachdev, arXiv:0808.0495 Amplitude (height) and phase (color) of VBS order

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Vison and spinon excitations of a Z 2 spin liquid: possible explanation for NMR and thermal conductivity measurements on - (ET) 2 Cu 2 (CN) 3 Signature of photon in circular distribution of VBS order around Zn impurity: possibly relevant for STM studies of bond order in underdoped cuprates. Vison and spinon excitations of a Z 2 spin liquid: possible explanation for NMR and thermal conductivity measurements on - (ET) 2 Cu 2 (CN) 3 Signature of photon in circular distribution of VBS order around Zn impurity: possibly relevant for STM studies of bond order in underdoped cuprates. Conclusions

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