1. Exotic Phases in Quantum Magnets MPA Fisher Outline: 2d Spin liquids: 2 Classes Topological Spin liquids Critical Spin liquids Doped Mott insulators: Conducting Non-Fermi liquids KITPC, 7/18/07 Interest: Novel Electronic phases of Mott insulators

2. 2 Quantum theory of solids: Standard Paradigm Landau Fermi Liquid Theory pypy pxpx Free Fermions Filled Fermi sea particle/hole excitations Interacting Fermions Retain a Fermi surface Luttingers Thm: Volume of Fermi sea same as for free fermions Particle/hole excitations are long lived near FS Vanishing decay rate

3. Add periodic potential from ions in crystal Plane waves become Bloch states Energy Bands and forbidden energies (gaps) Band insulators: Filled bands Metals: Partially filled highest energy band Even number of electrons/cell - (usually) a band insulator Odd number per cell - always a metal

4. Band Theory s or p shell orbitals : Broad bands Simple (eg noble) metals: Cu, Ag, Au - 4s1, 5s1, 6s1: 1 electron/unit cell Semiconductors - Si, Ge - 4sp3, 5sp3: 4 electrons/unit cell Band Insulators - Diamond: 4 electrons/unit cell Band Theory Works d or f shell electrons: Very narrow “bands” Transition Metal Oxides (Cuprates, Manganites, Chlorides, Bromides,…): Partially filled 3d and 4d bands Rare Earth and Heavy Fermion Materials: Partially filled 4f and 5f bands Electrons can ``self-localize” Breakdown

5. Mott Insulators: Insulating materials with an odd number of electrons/unit cell Correlation effects are critical! Hubbard model with one electron per site on average: electron creation/annihilation operators on sites of lattice inter-site hopping on-site repulsion t U

6. Antiferromagnetic Exchange Spin Physics For U>>t expect each electron gets self-localized on a site (this is a Mott insulator) Residual spin physics: s=1/2 operators on each site Heisenberg Hamiltonian:

7. Symmetry Breaking Mott InsulatorUnit cell doubling (“Band Insulator”) Symmetry breaking instability Magnetic Long Ranged Order (spin rotation sym breaking) Ex: 2d square Lattice AFM Spin Peierls (translation symmetry breaking) 2 electrons/cell Valence Bond (singlet) = (eg undoped cuprates La 2 CuO 4 )

8. How to suppress order (i.e., symmetry-breaking)? Low dimensionality –e.g., 1D Heisenberg chain (simplest example of critical phase) –Much harder in 2D! “almost” AFM order:  S(r)·S(0)  ~ (-1) r / r 2  Low spin (i.e., s = ½) Geometric Frustration –Triangular lattice – Kagome lattice ? Doping (eg. Hi-T c ): Conducting Non-Fermi liquids

9. Spin Liquid: Holy Grail Theorem: Mott insulators with one electron/cell have low energy excitations above the ground state with (E_1 - E_0) < ln(L)/L for system of size L by L. (Matt Hastings, 2005) Remarkable implication - Exotic Quantum Ground States are guaranteed in a Mott insulator with no broken symmetries Such quantum disordered ground states of a Mott insulator are generally referred to as “spin liquids”

10. Spin-liquids: 2 Classes Topological Spin liquids –Topological degeneracy Ground state degeneracy on torus –Short-range correlations –Gapped local excitations –Particles with fractional quantum numbers RVB state (Anderson) odd even Critical Spin liquids - Stable Critical Phase with no broken symmetries - Gapless exci tations with no free particle description -Power-law correlations -Valence bonds on many l ength scales

11. Simplest Topological Spin liquid (Z 2 ) Resonating Valence Bond “Picture” = Singlet or a Valence Bond - Gains exchange energy J 2d square lattice s=1/2 AFM Valence Bond Solid

12. Plaquette Resonance Resonating Valence Bond “Spin liquid”

13. Plaquette Resonance Resonating Valence Bond “Spin liquid”

14. Plaquette Resonance Resonating Valence Bond “Spin liquid”

15. Valence Bond Solid Gapped Spin Excitations “Break” a Valence Bond - costs energy of order J Create s=1 excitation Try to separate two s=1/2 “spinons” Energy cost is linear in separation Spinons are “Confined” in VBS

16. RVB State: Exhibits Fractionalization! Energy cost stays finite when spinons are separated Spinons are “deconfined” in the RVB state Spinon carries the electrons spin, but not its charge ! The electron is “fractionalized”.

17. J 1 =J 2 =J 3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z 2 ) J1J1 J2J2 J3J3 For J z >> J xy have 3-up and 3-down spins on each hexagon. Perturb in J xy projecting into subspace to get ring model

18. J 1 =J 2 =J 3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z 2 ) J1J1 J2J2 J3J3 For J z >> J xy have 3-up and 3-down spins on each hexagon. Perturb in J xy projecting into subspace to get ring model

19. Properties of Ring Model No sign problem! Can add a ring flip suppression term and tune to soluble Rokshar-Kivelson point Can identify “spinons” (s z =1/2) and Z 2 vortices (visons) - Z 2 Topological order Exact diagonalization shows Z 2 Phase survives in original easy-axis limit D. N. Sheng, Leon Balents Phys. Rev. Lett. 94, 146805 (2005) L. Balents, M.P.A.F., S.M. Girvin, Phys. Rev. B 65, 224412 (2002)

20. Other models with topologically ordered spin liquid phases Quantum dimer models Rotor boson models Honeycomb “Kitaev” model 3d Pyrochlore antiferromagnet Moessner, SondhiMisguich et al Motrunich, Senthil Hermele, Balents, M.P.A.F Freedman, Nayak, ShtengelKitaev (a partial list) ■ Models are not crazy but contrived. It remains a huge challenge to find these phases in the lab – and develop theoretical techniques to look for them in realistic models.

21. Critical Spin liquids T Frustration parameter: Key experimental signature: Non-vanishing magnetic susceptibility in the zero temperature limit with no magnetic (or other) symmetry breaking Typically have some magnetic ordering, say Neel, at low temperatures:

22. Organic Mott Insulator,  -(ET) 2 Cu 2 (CN) 3 : f ~ 10 4 –A weak Mott insulator - small charge gap –Nearly isotropic, large exchange energy (J ~ 250K) –No LRO detected down to 32mK : Spin-liquid ground state? Cs 2 CuCl 4 : f ~ 5-10 –Anisotropic, low exchange energy (J ~ 1-4K) – AFM order at T=0.6K T 0.62K AFMSpin liquid? 0 Triangular lattice critical spin liquids?

23. Kagome lattice critical spin liquids? Iron Jarosite, KFe 3 (OH) 6 (SO 4 ) 2 : f ~ 20 Fe 3+ s=5/2, T cw =800K Single crystals Q=0 Coplaner order at T N = 45K 2d “spinels” Kag/triang planes SrCr 8 Ga 4 O 19 f ~ 100 Cr 3+ s=3/2, T cw = 500K, Glassy ordering at T g = 3K C = T 2 for T<5K Volborthite Cu 3 V 2 O 7 (OH) 2 2H 2 O f ~ 75 Cu 2+ s=1/2 T cw = 115K Glassy at T < 2K Herbertsmithite ZnCu 3 (OH) 6 Cl 2 f > 600 Cu 2+ s=1/2, T cw = 300K, T c < 2K Ferromagnetic tendency for T low, C = T 2/3 ?? All show much reduced order - if any - and low energy spin excitations present Lattice of corner sharing triangles

24. Theoretical approaches to critical spin liquids Slave Particles: Express s=1/2 spin operator in terms of Fermionic spinons Mean field theory: Free spinons hopping on the lattice Critical spin liquids - Fermi surface or Dirac fermi points for spinons Gauge field U(1) minimally coupled to spinons For Dirac spinons: QED3 Boson/Vortex Duality plus vortex fermionization: (eg: Easy plane triangular/Kagome AFM’s)

25. Triangular/Kagome s=1/2 XY AF equivalent to bosons in “magnetic field” boson hopping on triangular lattice boson interactions pi flux thru each triangle Focus on vortices Vortex number N=1 Vortex number N=0 “Vortex” “Anti-vortex” +- Due to frustration, the dual vortices are at “half-filling”

26. Boson-Vortex Duality Exact mapping from boson to vortex variables. All non-locality is accounted for by dual U(1) gauge force Dual “magnetic” field Dual “electric” field Vortex number Vortex carries dual gauge charge

27. J J’ “Vortex” “Anti-vortex” +- Half-filled bosonic vortices w/ “electromagnetic” interactions Frustrated spins vortex hopping vortex creation/annihilation ops: Vortices see pi flux thru each hexagon Duality for triangular AFM

28. Difficult to work with half-filled bosonic vortices  fermionize! bosonic vortex fermionic vortex + 2  flux Chern-Simons flux attachment “Flux-smearing” mean-field: Half-filled fermions on honeycomb with pi-flux ~ E k Band structure: 4 Dirac points Chern-Simons Flux Attachment: Fermionic vortices

29. With log vortex interactions can eliminate Chern-Simons term Four-fermion interactions: irrelevant for N>N c “Algebraic vortex liquid” –“Critical Phase” with no free particle description –No broken symmetries - but an emergent SU(4) –Power-law correlations –Stable gapless spin-liquid (no fine tuning) N = 4 flavors Low energy Vortex field theory: QED3 with flavor SU(4) Linearize around Dirac points If N c >4 then have a stable:

30. “Decorated” Triangular Lattice XY AFM s=1/2 on Kagome, s=1 on “red” sites reduces to a Kagome s=1/2 with AFM J 1, and weak FM J 2 =J 3 J’ J J 1 >0 J 2 <0 J 3 <0 Flux-smeared mean field: Fermionic vortices hopping on “decorated” honeycomb Vortex duality Fermionized Vortices for easy-plane Kagome AFM

31. QED 3 with SU(8) Flavor Symmetry “Algebraic vortex liquid” in s=1/2 Kagome XY Model –Stable “Critical Phase” –No broken symmetries – Many gapless singlets (from Dirac nodes) – Spin correlations decay with large power law - “spin pseudogap” Vortex Band Structure: N=8 Dirac Nodes !! Provided N c <8 will have a stable:

32. Doped Mott insulators High T c Cuprates Doped Mott insulator becomes a d-wave superconductor Strange metal: Itinerant Non-Fermi liquid with “Fermi surface” Pseudo-gap: Itinerant Non-Fermi liquid with nodal fermions

33. Slave Particle approach to itinerant non-Fermi liquids Decompose the electron: spinless charge e boson and s=1/2 neutral fermionic spinon, coupled via compact U(1) gauge field Half-Filling: One boson/site - Mott insulator of bosons Spinons describes magnetism (Neel order, spin liquid,...) Dope away from half-filling: Bosons become itinerant Fermi Liquid: Bosons condense with spinons in Fermi sea Non-Fermi Liquid: Bosons form an uncondensed fluid - a “Bose metal”, with spinons in Fermi sea (say)

34. Uncondensed quantum fluid of bosons: D-wave Bose Liquid (DBL) Wavefunctions: N bosons moving in 2d: Define a ``relative single particle function” Laughlin nu=1/2 Bosons: Point nodes in ``relative particle function” Relative d+id 2-particle correlations Goal: Construct time-reversal invariant analog of Laughlin, (with relative d xy 2-particle correlations) Hint: nu=1/2 Laughlin is a determinant squared p+ip 2-body O. Motrunich/ MPAF cond-mat/0703261

35. Wavefunction for D-wave Bose Liquid (DBL) ``S-wave” Bose liquid: square the wavefunction of Fermi sea wf is non-negative and has ODLRO - a superfluid ``D-wave” Bose liquid: Product of 2 different fermi sea determinants, elongated in the x or y directions Nodal structure of DBL wavefunction: + + - - D xy relative 2-particle correlations

36. Analysis of DBL phase Equal time correlators obtained numerically from variational wavefunctions Slave fermion decomposition and mean field theory Gauge field fluctuations for slave fermions - stability of DBL, enhanced correlators “Local” variant of phase - D-wave Local Bose liquid (DLBL) Lattice Ring Hamiltonian and variational energetics

37. Properties of DBL/DLBL Stable gapless quantum fluids of uncondensed itinerant bosons Boson Greens function in DBL has oscillatory power law decay with direction dependent wavevectors and exponents, the wavevectors enclose a k-space volume determined by the total Bose density (Luttinger theorem) Boson Greens function in DLBL is spatially short-ranged Power law local Boson tunneling DOS in both DBL and DLBL DBL and DLBL are both ``metals” with resistance R(T) ~ T 4/3 Density-density correlator exhibits oscillatory power laws, also with direction dependent wavevectors and exponents in both DBL and DLBL

38. D-Wave Metal Itinerant non-Fermi liquid phase of 2d electrons Wavefunction: t-K Ring Hamiltonian (no double occupancy constraint) 1 2 34 1 2 34 Electron singlet pair “rotation” term t >> K Fermi liquid t ~ K D-metal (?)

39. Summary & Outlook Quantum spin liquids come in 2 varieties: Topological and critical, and can be accessed using slave particles, vortex duality/fermionization,... Several experimental s=1/2 triangular and Kagome AFM’s are candidates for critical spin liquids (not topological spin liquids) D-wave Bose liquid: a 2d uncondensed quantum fluid of itinerant bosons with many gapless strongly interacting excitations, metallic type transport,... Much future work: –Characterize/explore critical spin liquids –Unambiguously establish an experimental spin liquid –Explore the D-wave metal, a non-Fermi liquid of itinerant electrons