Quantum Spin Liquid Patrick Lee MIT

1. Quantum Spin Liquid Patrick Lee MIT Collaborators: M. Serbyn, A. Potter, T. Senthil N. Nagaosa X-G Wen Y. Ran Y. Zhou M. Hermele T. K. Ng T. Grover …. Supported by NSF.

2. Outline: • Introduction to quantum magnetism and spin liquid. • Why is spin liquid interesting? • Spin liquid is much more than the absence of ordering: Emergence of new particles and gauge fields. • 3. Spin liquid in organic compounds and kagome lattice. • 4. Low energy theory: fermion plus gauge field. • 5. Proposals for experimental detection of emergent particles and gauge fields.

3. Conventional Anti-ferromagnet (AF): Louis Néel Cliff Shull 1994 Nobel Prize 1970 Nobel Prize

4. t Undoped CuO2 plane: Mott Insulator due to e- - e- interaction Virtual hopping induces AF exchange J=4t2/U CuO2 plane with doped holes: La3+ Sr2+: La2-xSrxCuO4 strongly-correlated electron systemexample: Hi Tc cuprate. One hole per site: should be a metal according to band theory. Mott insulator.

5.   Competing visions of the antiferromagnet “….To describe antiferromagnetism, Lev landau and Cornelis Gorter suggested quantum fluctuations to mix Neel’s solution with that obtained by reversal of moments…..Using neutron diffraction, Shull confirmed (in 1950) Neel’s model. ……Neel’s difficulties with antiferromagnetism and inconclusive discussions in the Strasbourg international meeting of 1939 fostered his skepticism about the usefulness of quantum mechanics; this was one of the few limitations of this superior mind.” Jacques Friedel, Obituary of Louis Neel, Physics today, October,1991. Lev Landau | |  Quantum Classical

6. Mott against Slater debate: Mott: One electron per unit cell. Charge gap is due to correlation. Antiferromagnetism is secondary. Mott insulator violate band theory. Slater: Anti-ferromagnetic ground state. Unit cell is doubled. Then we have 2 electrons per unit cell and the system can be an insulator, consistent with band theory. Can there be a Mott insulator which does not have AF order?

7. P. W. Anderson introduced the RVB idea in 1973. Key idea: spin singlet can give a better energy than anti-ferromagnetic order. What is special about S=1/2? 1 dimensional chain: Energy per bond of singlet trial wavefunction is -(1/2)S(S+1)J = -(3/8)J vs. -(1/4)J for AF.

8. Spin liquid: destruction of Neel order due to quantum fluctuations. In 1973 Anderson proposed a spin liquid ground state (RVB) for the triangular lattice Heisenberg model.. It is a linear superposition of singlet pairs. (not restricted to nearest neighbor.) Spin liquid is more than the absence of Neel order. New emergent property of spin liquid: Excitations are spin ½ particles (called spinons), as opposed to spin 1 magnons in AF. These spinons may even form a Fermi sea. Emergent gauge field. (U(1), Z2, etc.) Topological order (X. G. Wen) in case of gapped spin liquid: ground state degeneracy, entanglement entropy. More than 30 years later, we may finally have several examples of spin liquid in higher than 1 dimension!

9. It will be very useful to have a spin liquid ground state which we can study. Two routes to spin liquid: • Geometrical frustration: spin ½ Heisenberg model on Kagome, hyper-kagome. • Proximity to Mott transition. Requirements: insulator, odd number of electron per unit cell, absence of AF order. Finally there is now a promising new candidate in the organics and also in a Kagome compound.

10. Two ways to proceed: 1. Numerical: Projected trial wavefunction. Extended Hilbert space: many to one representation. 2. Analytic: gauge theory. Introduce fermions which carry spin index Constraint of single occupation, no charge fluctuation allowed.

11. Why fermions? Can also represent spin by boson, (Schwinger boson.) Mean field theory: 1. Boson condensed: Neel order. 2. Boson not condensed: gapped state. Generally, boson representation is better for describing Neel order or gapped spin liquid, whereas fermionic representation is better for describing gapless spin liquids. The open question is which mean field theory is closer to the truth. We have no systematic way to tell ahead of time at this stage. Since the observed spin liquids appear to be gapless, we proceed with the fermionic representation.

12. Enforce constraint with Lagrange multipier l The phase of cij becomes a compact gauge field aij on link ij and il becomes the time component. Compact U(1) gauge field coupled to fermions.

13. General problem of compact gauge field coupled to fermions. Z2 gauge theory: generally gapped. Several exactly soluble examples. (Kitaev, Wen) U(1) gauge theory: gapless Dirac spinons or Fermi sea. Hermele et al (PRB) showed that deconfinement is possible if number of Dirac fermion species is large enough. (physical problem is N=4). Sung-sik Lee showed that fermi surface U(1) state is always deconfined. • Mean field (saddle point) solutions: • For cij real and constant: fermi sea. • For cij complex: flux phases and Dirac sea. • Fermion pairing: Z2 spin liquid. Enemy of spin liquid is confinement:(p flux state and SU(2) gauge field leads to chiral symmetry breaking, ie AF order) If we are in the de-confined phase, fermions and gauge fields emerge as new particles at low energy. (Fractionalization) The fictitious particles introduced formally takes on a life of its own! They are not free but interaction leads to a new critical state. This is the spin liquid.

14. Stability of gapless Mean Field State against non-perturbative effect. • U(1) instanton 1) Pure compact U(1) gauge theory : always confined. (Polyakov) 2) Compact U(1) theory + large N Dirac spinon : deconfinement phase [Hermele et al., PRB 70, 214437 (04)] 3) Compact U(1) theory + Fermi surface : more low energy fluctuations deconfined for any N. (Sung-Sik Lee, PRB 78, 085129(08).) F Ф

15. Non-compact U(1) gauge theory coupled with Fermi surface. Integrating out some high energy fermions generate a Maxwell term with coupling constant e of order unity. The spinons live in a world where coupling to E &M gauge fields are strong and speed of light given by J. Longitudinal gauge fluctuations are screened and gapped. Will focus on transverse gauge fluctuations which are not screened.

16. Physical Consequence Specific heat : C ~ T2/3 Gauge fluctuations dominate entropy at low temperatures. Non-Fermi liquid. [Reizer (89);Nagaosa and Lee (90), Motrunich (2005).]

17. Physical meaning of gauge field: gauge flux is gauge invariant b= x a Fermions hopping around a plaquette picks up a Berry’s phase due to the meandering quantization axes. The is represented by a gauge flux through the plaquette. It is related to spin chirality (Wen, Wilczek and Zee, PRB 1989)

18. Three examples: • Organic triangular lattice near the Mott transition. • Kagome lattice, more frustrated than triangle. • Hyper-Kagome, 3D. We are not talking about spin glass, spin ice etc.

19. Kagome lattice.

20. Mineral discovered in Chile in 1972 and named after H. Smith. Herbertsmithite : Spin ½ Kagome. Spin liquid in Kagome system. (Dan Nocera, Young Lee etc. MIT). Curie-Weiss T=300, fit to high T expansion gives J=170K No spin order down to mK (muSR, Keren and co-workers.)

21. Spin ½ Heisenberg on Kagome has long been suspected to be a spin liquid. (P. W. Leung and V. Elser, PRB 1993) Projected wavefunction studies. (Y. Ran, M. Hermele, PAL,X-G Wen) Effective theory: Dirac spinons with U(1) gauge fields. (ASL)

22. White, Huse and collaborators find a gapped spin liquid using DMRG. Entnglement entropy calculations (Hong-Chen Jiang and others) show that their state is a Z2 spin liquid.

23. How to understand Huse-White result? Gapped Z2 spin liquid. • Slave boson: Motrunich 2011: projected slave boson mean field. Proximity to QCP? 2. Fermion pairing: Lu, Ran and Lee: classified projected fermionic pairing state. However, recent QMC calculation by Iqbal, Becca and Poilblanc did not find energy gain by pairing. They found that the Dirac SL is remarkably stable and has energy comparable to DMRG after two Lanchoz steps.

24. Theoretically, the best estimate (Huse and White) is that there is a triplet gap of order 0.14J. Experimentally, the gap is much smaller. Specific heat, NMR (Mendels group PRL2008, 2011, T. Imai et al 2011). See also recent neutron scattering. (Y. Lee group, Nature 2012.) Caveats: Heisenberg model not sufficient. 1. Dzyaloshinskii- Moriya term: Estimated to be 5 to 10% of AF exchange. QCP between Z2 spin liquid and AF order. (Huh, Fritz and Sachdev, PRB 2010) 2. Local moments, current understanding is that 15% of the Zn sites are occupied by copper.

25. Mendels group PRL 2008 Mendels group, PRL 2012

26. Large single crystals available (Young Lee’s group at MIT). Neutron scattering possible. Science 2012. Projected Dirac S(k). Serbyn and PAL.

27. Mott insulator t t t’ Q2D organics k-(ET)2X ET dimer model X X = Cu(NCS)2, Cu[N(CN)2]Br, Cu2(CN)3….. anisotropic triangular lattice t’ / t = 0.5 ~ 1.1

28. Q2D antiferromagnet k-Cu[N(CN)2]Cl t’/t=0.75 Q2D spin liquid k-Cu2(CN)3 t’/t=1.06 No AF order down to 35mK. J=250K.

29. From Y. Nakazawa and K. Kanoda, Nature Physics 2008. g is about 15 mJ/K^2mole Wilson ratio is approx. one at T=0. Something happens around 6K. Partial gapping of spinon Fermi surface due to spinon pairing?

30. More examples have recently been reported.

32. Thermal conductivity of dmit salts. mean free path reaches 500 inter-spin spacing. M. Yamashita et al, Science 328, 1246 (2010) However, ET salt seems to develop a small gap below 0.2 K.

33. ET2Cu(NCS)2 9K sperconductor ET2Cu2(CN)3 Insulator spin liquid

34. Charge fluctuations are important near the Mott transition even in insulating phase Heisenberg model 120° AF order Numeric.[Imada and co.(2003)] Spin liquid state with ring exchange. [Motrunich, PRB72,045105(05)] + + … J ~ t2/U J’ ~ t4/U3 Importance of charge fluctuations Mott transition Fermi Liquid I n s u l a t o r Metal U/t

35. Constraint : L = -1 0 1 Slave-rotor representation of the Hubbard Model :[S. Florens and A. Georges, PRB 70, 035114 (’04), Sung-Sik Lee and PAL PRL 95,036403 (‘05)] Q. What is the low energy effective theory for mean-field state ?

36. Effective Theory : fermions and rotor coupled tocompact U(1) gauge field.Sung-sik Lee and P. A. Lee, PRL 95, 036403 (05)

37. 3 dim example? Hyper-Kagome. Okamoto ..Takagi PRL 07 Near Mott transition: becomes metallic under pressure.

38. Strong spin orbit coupling. Spin not a good quantum number but J=1/2. Approximate Heisenberg model with J if direct exchange between Ir dominates. (Chen and Balents, PRB 09, see also Micklitz and Norman PRB 2010 ) Slave fermion mean field , Zhou et al (PRL 08) Mean field and projected wavefunction. Lawler et al. (PRL 08) Conclusion: zero flux state is stable: spinon fermi surface. Low temperature pairing can give line nodes and explain T^2 specific heat.

39. Enforce constraint with Lagrange multipier l The phase of cij becomes a compact gauge field aij on link ij and il becomes the time component. Compact U(1) gauge field coupled to fermions.

40. Non-compact U(1) gauge theory coupled with Fermi surface. Integrating out some high energy fermions generate a Maxwell term with coupling constant e of order unity. The spinons live in a world where coupling to E &M gauge fields are strong and speed of light given by J. Longitudinal gauge fluctuations are screened and gapped. Will focus on transverse gauge fluctuations which are not screened.

41. RPA results: 1. Gauge field dynamics: over-damped gauge fluctuations, very soft! 2.Fermion self energy is singular. No quasi-particle pole, or z  0.

43. Only bosons with q tangent to a given patch couple. Two patch theory. This is special to 2D. In 3D bands of tangential points are coupled. Then all points are coupled.

44. Large N: Polchinski (94), Altshuler, Ioffe and Millis (94). N fermions coupled to gauge field. Minimal 2 patch model. Sung-Sik Lee, (PRB80 165102 (09) Plus opposite patch with e -> -e Note curvature of patch is kept.

45. It was believed that 1/N expansion is systematic, and D has no further singular correction, but Fermion G might. Sung-Sik Lee showed that 1/N expansion breaks down. This term is dangerous if it serves as a cut-off in a diagram. He concludes that an infinite set of diagrams contribute to a given order of 1/N.

46. Recent progress: Metlitski and Sachdev PRB82, 075127 (10) They did loop expansion anyway and found no log correction to boson up to 3 loops, but for fermion self-energy:

47. Solution: double expansion. (Mross, McGreevy,Liu and Senthil). Maxwell term. ½ filled Landau level with 1/r interaction. Expansion parameter: e=zb-2. Limit N  infinity, e 0, eN finite gives a controlled expansion. Results are similar to RPA and consistent with earlier e expansion at N=2. The double expansion is technically easer to go to higher order.

48. Conclusion: No correction to boson: z=3/2. For the gauge field problem, h is positive and sub-leading. RPA is recovered to 3 loop.

49. Sung-Sik Lee, arXiv 2013, co-dimension expansion. 2 patch theory fails for d > 2. Therefore cannot do conventional epsilon expansion. Instead, keep FS to be a line and extend the dimension perpendicular to it to d-1. He finds an expansion about d=2.5. Results are consistent with Mross et al: No correction to boson D to 3 loops. Correction to fermions: for the nematic problem For the gauge field problem, h is positive and sub-leading. RPA is recovered to 3 loop.

50. How non-Fermi liquid is it? Physical response functions for small q are Fermi liquid like, and can be described by a quantum Boltzmann equation. Y.B. Kim, P.A. Lee and X.G. Wen, PRB50, 17917 (1994) Take a hint from electron-phonon problem. 1/t=plT, but transport is Fermi liquid. If self energy is k independent, Im G is sharply peaked in k space (MDC) while broad in frequency space (EDC). Can still derive Boltzmann equation even though Landau criterion is violated.(Kadanoff and Prange). In the case of gauge field, singular mass correction is cancelled by singular landau parameters to give non-singular response functions. For example, uniform spin susceptibility is constant while specific heat gamma coefficent (mass) diverges. On the other hand, 2kf response is enhanced. (Altshuler, Ioffe and Millis, PRB 1994). May be observable as Kohn anomaly and Friedel oscilations. (Mross and Senthil)