Kagome Lattice Quantum Antiferromagnets
Kagome Lattice QuantumAntiferromagnetsA Quest for Unconventional Quantum PhasesPredrag Nikolić and T. SenthilMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.1/28
Quantum PhasesSystemConventionalMetalsFermi liquidMagnetsUnconventionalnon-Fermi liquidcupratesheavy gnetspin-Peierls (VBC)plain paramagnet spin liquid (RVB)Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.2/28
Unconventional Quantum Magnets Two spins form a valence bond (singlet):Staggered VBCPlaquette VBCMassachusetts Institute of TechnologyRVB spin liquidKagome Lattice Quantum Antiferromagnets – p.3/28
RVB Spin Liquid) No broken symmetries (at Fractionalized quasiparticlesspinons Topological phase in 2 and more dimensionsvortices (visons)Motivation/Realizations:Heisenberg antiferromagnetic chains (1D)Massachusetts Institute of Technology ? Pseudo-gap region of the cuprates?Kagome Lattice Quantum Antiferromagnets – p.4/28
Geometric FrustrationCheckerboardKagomePyrochloreHuge degeneracy of classical ground statesFluctuations lift the classical degeneracyincompletely: order-by-disordercompletely: spin liquidNew physics emerges at low energy scalesMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.5/28
Kagome LatticeHeisenberg AntiferromagnetMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.6/28
kagome material Spin Kagome Experiments(SCGO) No magnetic long-range order down toSpin-glass behavior? Heat-capacity not thermally activated:gapless excitations Missing entropy Heat-capacity virtually independent of magnetic fieldsinglet excitationsAlso, QS-ferrite. . .Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.7/28
Kagome Numerics (Lhuillier, et.al) Exact diagonalization of the(samples with up to 36 sites)Heisenberg modelDisordered ground-state Small spin-gap: Short-range correlations:spin-spin, dimer-dimer, chiral-chiralin thermodynamic limitSinglet excitations form a gapless band Macroscopic number of singlet excitations below the(not Goldstone modes)spin-gap: Comparison with numerics for:large spin-gap, no gapless singletstopological effects may be importantMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.8/28
QuestionsNature of the ground state:spin liquid or order-by-disorder?Confined or deconfined spinons?Nature of the mysterious gapless singlet excitations?1. Why so large number of states below the spin-gap?2. Why gapless?Great hope for a spin liquid?Gauge bosons as low energy excitations?Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.9/28
(Marston, Zheng) Kagome Theory(Sachdev)Trimerized Kagome lattice (Mambrini, Mila)Quantum dimer model at RK point (Misguich)(exactly solvable)Quantum dimer model from overlap expansion (Elser)Chiral spin liquid (Yang, Warman, Girvin)Hidden long-range order and Goldstone modesSome other approaches. . .Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.10/28
New Theoretical ApproachDescribes the singlet states below the spin-gapLow energy effective theoryApproaches to paramagnetic phases: 1. Large-N expansion: (Sachdev, Read)A gauge theory describes fluctuationsabout the mean-field statepuregauge theory 2. Quantum dimer models:(Jalabert, Sachdev; Moessner, Sondhi)fully frustrated Ising model on the dual lattice 3. Heisenberg model as a theory offermionic spinons coupled to agauge fieldMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.11/28
Gauge Theory #" % "(" &(&'! gauge field on the Kagome bonds: Fermionic spinons on the Kagome sites: Kagome spins: Ingredients:Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.12/28
Effective Theory Interactions between the spinons are mediated by thegauge fieldHeisenberg model Massachusetts Institute of Technology Odd theory: Numerics: the spinons are gapped!Integrate out spinonspure oddgauge theoryKagome Lattice Quantum Antiferromagnets – p.13/28
Massachusetts Institute of Technology Effective Dimer Model:Kagome Lattice Quantum Antiferromagnets – p.14/28
Ground State:storder: maximum number of “perfect” hexagonsnd order: honeycomb patternmacroscopic degeneracy due to the “stars”ndorder: macroscopic degeneracy liftedValence-bondorderUnit-cell:36 sites!gapMassachusetts Institute of Technology Extremely“small” singletKagome Lattice Quantum Antiferromagnets – p.15/28
Effective Ising Model: Dual representation:fully frustrated quantum Ising model on the dice latticespin liquidare gappedGap is “large” Visons Ground state is disordered Ground State:Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.16/28
Comparison With NumericsReliable Aspects of NumericsFeature found in numericsVBSLLarge number of singlets below the spin-gapExtremely small singlet energy scaleAspects of Numerics Sensitive to Finite SizeFeature found in numericsVBSLNo long range orderDeconfined spinonsCharacter of spectrum (gaps, degeneracy. . . )Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.17/28
Proposal for Kagome Heisenberg AFExotic valence-bond ordered phase:Unit-cell as large as maximum sample size in numerics!Lowest lying excitations:gapped, heavy singletsSinglet gap is extremely small Excitations with magnetic moment:gappedmagnons(confined spinon pairs)The valence-bond order is also analytically favored. . .Similar results for the checkerboard lattice:agreement with numericsMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.18/28
Kagome LatticeIsing AntiferromagnetMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.19/28
Ising model: MotivationPhysical Motivation:More frustrated than the Heisenberg modelInteresting physics expected:valence-bond order, spin liquid? Easy-axis anisotropya route to the isotropic Heisenberg modelTheoretical Motivation:Amenable to reliable Monte Carlo numericsAnalytical methods available:U(1) gauge theory, duality, field theoryMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.20/28
Questions Nothing conserved: transverse field dynamics, , Total spin conserved: XXZ dynamicsIncluding higher order dynamical processes. . .What phases are possible?Is there a topological spin liquid phase?Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.21/28
const. A U(1) Lattice Gauge Theory local constraint Massachusetts Institute of Technology honeycomb bonds Kagome sites Minimum frustrationKagome Lattice Quantum Antiferromagnets – p.22/28
AnalysisCompact U(1) gauge theory on the honeycomb lattice Honeycomb lattice is bipartite:Kagome spinelectric field vectorplaquette spincharged bosonFixed background charge: breaks lattice symmetries Charge 1 bosonic matter fielda non-topological disordered phase exists (in 2D)Analysis uses a duality transformation, sine-Gordon theory,and field theory methods. . .Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.23/28
Phases: Transverse Field Ising ModelDisordered non-topological phaseagrees with Monte-Carlo (Moessner, Sondhi) Valence-bond ordered phasebroken translational symmetry (3-fold degeneracy)broken globalsymmetry1Ordered phase corresponds to the ofthe saturated magnetization. (plateauinmagnetizationcurve) M 7/95/91/31/9000.511.522.53h/JMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.24/28
Dynamics of Frustrated BondsApplicable to other latticesConvenient for more microscopic insightCannot handle longitudinal field 1. Frustrated bond Analysis:dimer on the (bipartite) dice lattice2. Compact U(1) gauge theory on the dice lattice3. Dual lattice height theory on the Kagome lattice(two coupled height fields)4. Large degeneracy of saddle-points lifted by fluctuations5. Study which states are entropically preferred(search for order-by-disorder )Massachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.25/28
Microscopic InsightTransverse Field Dynamics:Disordered non-topological ground stateProbability amplitude concentrated aroundmaximally flippable statesQuasi-localized excited statesWave-function is well approximated by a GutzwillerprojectionXXZ Dynamics:Valence-bond ordered ground stateMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.26/28
XXZ DynamicsAnisotropic Heisenberg modelStrong easy-axis anisotropyFrustrated BondsMassachusetts Institute of TechnologyPartial Melting?Kagome Lattice Quantum Antiferromagnets – p.27/28
ConclusionsUnconventional and fundamental physics emerges atlow energies in frustrated magnetsKagome Lattice Antiferromagnets:Isotropic Heisenberg: valence-bond orderedlarge unit cellextremely small energy scale for singlet statesEasy-axis Heisenberg: possibly valence-bond orderedIsing in external field:disordered, but not topological phasevalence-bond ordered magnetized phaseMassachusetts Institute of TechnologyKagome Lattice Quantum Antiferromagnets – p.28/28
Chiral spin liquid (Yang, Warman, Girvin) Hidden long-range order and Goldstone modes Some other approaches. Kagome Lattice Quantum Antiferromagnets – p.10/28. Massachusetts Institute of Technology New Theoretical Approach Describes the singlet states below the spin-gap
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