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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 Quantum theory of solids: Standard Paradigm Landau Fermi Liquid Theory Quic kTime™ and a TIFF (Unc ompres sed) decompress or are needed to see this picture. py Free Fermions px particle/hole excitations Filled Fermi sea Interacting Fermions Retain a Fermi surface Particle/hole excitations are long lived near FS Luttingers Thm: Volume of Fermi sea same as for free fermions Vanishing decay rate 2 Add periodic potential from ions in crystal QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. • 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 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 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” 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 inter-site hopping t U 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 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 ) • Spin Peierls (translation symmetry breaking) Valence Bond (singlet) = 2 electrons/cell 2 electrons/cell 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! • Geometric Frustration – Triangular lattice – Kagome lattice “almost” AFM order: S(r)·S(0) ~ (-1) r / r2 ? • Doping (eg. Hi-Tc): Conducting Non-Fermi liquids 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” Spin-liquids: 2 Classes • Topological Spin liquids RVB state (Anderson) – 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 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 Plaquette Resonance Resonating Valence Bond “Spin liquid” Plaquette Resonance Resonating Valence Bond “Spin liquid” Plaquette Resonance Resonating Valence Bond “Spin liquid” 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 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”. J2 J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2) 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 J2 J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2) 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 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) Other models with topologically ordered spin liquid phases (a partial list) • Quantum dimer models • Rotor boson models Moessner, Sondhi Misguich et al Motrunich, Senthil • Honeycomb “Kitaev” model Kitaev • 3d Pyrochlore antiferromagnet Hermele, Balents, M.P.A.F Freedman, Nayak, Shtengel ■ 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. 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: 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 0 Spin liquid? 0.62K T 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 SrCr8Ga4O19 f ~ 100 Cr3+ s=3/2, Tcw = 500K, Glassy ordering at Tg = 3K C = T2 for T<5K Lattice of corner sharing triangles • Volborthite Cu3V2O7(OH)2 2H2O f ~ 75 Cu2+ s=1/2 Tcw = 115K Glassy at T < 2K • Herbertsmithite ZnCu3(OH)6Cl2 f > 600 Cu2+ s=1/2 , Tcw = 300K, Tc< 2K Ferromagnetic tendency for T low, C = T2/3 ?? All show much reduced order - if any - and low energy spin excitations present 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) Triangular/Kagome s=1/2 XY AF equivalent to bosons in “magnetic field” boson hopping on triangular lattice pi flux thru each triangle boson interactions Focus on vortices + “Vortex” Vortex number N=1 Due to frustration, the dual vortices are at “half-filling” - “Anti-vortex” Vortex number N=0 Boson-Vortex Duality • Exact mapping from boson to vortex variables. 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 Duality for triangular AFM J’ J Frustrated spins vortex creation/annihilation ops: + “Vortex” H J ij eij2 U ( a ) i2 ij i tijbi b j e - i ( aij aij0 ) Half-filled bosonic vortices w/ “electromagnetic” interactions h.c. ij vortex hopping “Anti-vortex” Vortices see pi flux thru each hexagon 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 H MF tij f i f j h.c. ~ ij • Band structure: 4 Dirac points E k 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) Fermionized Vortices for easy-plane Kagome AFM J’ J “Decorated” Triangular Lattice XY AFM J2<0 • 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 J3<0 Flux-smeared mean field: Fermionic vortices hopping on “decorated” honeycomb J1>0 Vortex duality Vortex Band Structure: N=8 Dirac Nodes !! QED3 with SU(8) Flavor Symmetry Provided Nc <8 will 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” 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 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) 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 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 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 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 D-Wave Metal Itinerant non-Fermi liquid phase of 2d electrons Wavefunction: t-K Ring Hamiltonian (no double occupancy constraint) 4 1 3 2 4 1 Electron singlet pair “rotation” term t >> K t~K Fermi liquid D-metal (?) 3 2 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