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Exotic Phases in Quantum Magnets

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

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Exotic Phases in Quantum Magnets

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

  2. Quantum theory of solids: Standard Paradigm Landau Fermi Liquid Theory py Free Fermions px 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 Breakdown • 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”

  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: on-site repulsion electron creation/annihilation operators on sites of lattice U inter-site hopping t

  6. 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: Antiferromagnetic Exchange

  7. Symmetry Breaking Mott Insulator Unit cell doubling (“Band Insulator”) Symmetry breaking instability • Magnetic Long Ranged Order (spin rotation sym breaking) Ex: 2d square Lattice AFM (eg undoped cuprates La2CuO4 ) 2 electrons/cell • Spin Peierls(translation symmetry breaking) 2 electrons/cell Valence Bond (singlet) =

  8. ? How to suppress order (i.e., symmetry-breaking)? • Low spin (i.e., s = ½) • 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/ r2 • Geometric Frustration • Triangular lattice • Kagome lattice • Doping (eg. Hi-Tc): 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 RVB state (Anderson) • Topological Spin liquids • Topological degeneracy Ground state degeneracy on torus • Short-range correlations • Gapped local excitations • Particles with fractional quantum numbers odd even odd • Critical Spin liquids - Stable Critical Phase with no broken symmetries - Gapless excitations with no free particle description • Power-law correlations • Valence bonds on many length scales

  11. Simplest Topological Spin liquid (Z2) Resonating Valence Bond “Picture” 2d square lattice s=1/2 AFM = Singlet or a Valence Bond - Gains exchange energy J 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. Gapped Spin Excitations “Break” a Valence Bond - costs energy of order J Create s=1 excitation Try to separate two s=1/2 “spinons” Valence Bond Solid 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. J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2) J2 J1 J3 For Jz >> Jxy have 3-up and 3-down spins on each hexagon. Perturb in Jxy projecting into subspace to get ring model

  18. J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2) J2 J1 J3 For Jz >> Jxy have 3-up and 3-down spins on each hexagon. Perturb in Jxy projecting into subspace to get ring model

  19. Properties of Ring Model L. Balents, M.P.A.F., S.M. Girvin, Phys. Rev. B 65, 224412 (2002) • No sign problem! • Can add a ring flip suppression term and tune to soluble Rokshar-Kivelson point • Can identify “spinons” (sz =1/2) and Z2 vortices (visons) - Z2 Topological order • Exact diagonalization shows Z2 Phase survives in original easy-axis limit D. N. Sheng, Leon Balents Phys. Rev. Lett. 94, 146805 (2005)

  20. Other models with topologically ordered spin liquid phases (a partial list) • Quantum dimer models • Rotor boson models • Honeycomb “Kitaev” model • 3d Pyrochlore antiferromagnet Moessner, Sondhi Misguich et al Motrunich, Senthil Kitaev Freedman, Nayak, Shtengel Hermele, Balents, M.P.A.F ■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 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: T Frustration parameter:

  22. Triangular lattice critical spin liquids? • Organic Mott Insulator, -(ET)2Cu2(CN)3: f ~ 104 • A weak Mott insulator - small charge gap • Nearly isotropic, large exchange energy (J ~ 250K) • No LRO detected down to 32mK : Spin-liquid ground state? • Cs2CuCl4: f ~ 5-10 • Anisotropic, low exchange energy (J ~ 1-4K) • AFM order at T=0.6K AFM Spin liquid? T 0 0.62K

  23. Kagome lattice critical spin liquids? • Iron Jarosite, KFe3 (OH)6(SO4)2: f ~ 20 Fe3+ s=5/2 , Tcw =800K Single crystals Q=0 Coplaner order at TN = 45K • 2d “spinels” Kag/triang planes SrCr8Ga4O19f ~ 100 Cr3+ s=3/2, Tcw = 500K, Glassy ordering at Tg = 3K C = T2 for T<5K • Volborthite Cu3V2O7(OH)2 2H2O f ~ 75 Cu2+ s=1/2 Tcw = 115K Glassy at T < 2K • Herbertsmithite ZnCu3(OH)6Cl2f > 600 Cu2+ s=1/2 , Tcw = 300K, Tc< 2K Ferromagnetic tendency for T low, C = T2/3 ?? Lattice of corner sharing triangles All show much reduced order - if any - and low energy spin excitations present

  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 interactions pi flux thru each triangle boson hopping on triangular lattice Focus on vortices “Vortex” Vortex number N=1 Due to frustration, the dual vortices are at “half-filling” “Anti-vortex” Vortex number N=0

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

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

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

  29. Low energy Vortex field theory: QED3 with flavor SU(4) N = 4 flavors Linearize around Dirac points With log vortex interactions can eliminate Chern-Simons term Four-fermion interactions: irrelevant for N>Nc If Nc>4 then have a stable: “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)

  30. J’ Fermionized Vortices for easy-plane Kagome AFM J • “Decorated” Triangular Lattice XY AFM • s=1/2 on Kagome, s=1 on “red” sites • reduces to a Kagome s=1/2 with AFM J1, and weak FM J2=J3 J2<0 Vortex duality J1>0 J3<0 Flux-smeared mean field: Fermionic vortices hopping on “decorated” honeycomb

  31. Vortex Band Structure:N=8 Dirac Nodes !! QED3 with SU(8) Flavor Symmetry Provided Nc <8will have a stable: • “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”

  32. Doped Mott insulators High Tc 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 toitinerant 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) O. Motrunich/ MPAF cond-mat/0703261 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 dxy 2-particle correlations) Hint: nu=1/2 Laughlin is a determinant squared p+ip 2-body

  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: - + + - Dxy 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) ~ T4/3 • 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) 4 3 4 3 2 2 1 1 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

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