Spin-orbit effects on the Mott transition in frustrated pyrochlores

gabriel.physics.ucsb.edu

Spin-orbit effects on the Mott transition in frustrated pyrochlores

ong>Spinong>-ong>orbitong> ong>effectsong> on the Motttransition in frustrated pyrochloresLeon Balents, KITPCoseners, UK, September 2009Dmytro PesinUT Austin


ong>Spinong>-ong>orbitong> physicsAshcroft+Mermin: an afterthoughtIntegral in rare earthsRecently, brought to the forefront:Note:Quantum spin Hall effect in HgTe quantum wellsTopological band insulators: Bi 1-x Sb x , Bi 2 Se 347 papers on arXiv with “topological insulator” inthe title in the last yearc.f. 11 papers with “spin ice”


Topological InsulatorsL. Fu, C. Kane, E. Mele (2007); J. Moore, LB (2007)3d band insulators w/ significant SOI canhave hidden topological structure, somewhatsimilar to the IQHEExhibit “helical” surface states - 2d chiral Diracfermions (evades Fermion doubling problem!)Cannot be localized by disorderSurface Hall effect magnetoelectric responseSevral experimental examplesBi 1-x Sb x , Bi 2 Se 3 , Bi 2 Te 3


Example: Bi 2 Te 3M.Z. Hasan group - ARPES studies


Monopoles...“Image” monopole - X.-L. Qi et al


What aboutinteractions?


ong>Spinong>-ong>orbitong> and MottphysicsCoulomb correlationsreduce bandwidthong>Spinong>-ong>orbitong> enhancedrelative tobandwidthU/tMott IMott IIIn Mott insulator,compare SO to J not t.MetalTBI?λ/tschematic phase diagram


ong>Spinong>-ong>orbitong> and MottphysicsCoulomb correlationsreduce bandwidthong>Spinong>-ong>orbitong> enhancedrelative tobandwidthU/tMott IRare earth magnets...Mott IIIn Mott insulator,compare SO to J not t.MetalTBI?λ/tschematic phase diagram


Mott transition withSOIsU/tStudy this phasediagram in a concretecaseMott IMetalMott IITBI?λ/tschematic phase diagram


IntroductionLn2Ir2O7Ln 3+ : (4f) n Localized momentMagnetic frustrationPyrochlore iridatesIr 4+ : 5d 5Ir[t2g]+O[2p]pyrochlore oxidesConduction electronsconduction bandItinerant electron systemon the pyrochlore lattice! (m" cm)Resi10 610 510 410 310 7 0Formula: Ln 2 Ir 2 O 7both Ln and Iratoms occupypyrochlore latticesCubic, FCC BravaislatticeLn carry localizedmoments onlyimportant at low TIrO6O!Ln10 210 110 0(Ln=K. Matsuhira e


lator Transitions in Ln2Ir2O71 1.12 1.14)NdMetal2Ir 2 O 7PMMetalPrdepend ontism of Ln 3+ .radius ofMetal-InsulatorThese MIT are a second-order transition.The insulating phase involves a magneticTransitionordering driven by 5d electrons.Weak FMA difference of M(T) between FC and ZFCDecreasing Ir-O-Irbond angle makesmore insulatingThe resistivity depends on the ionic radius of Ln.As Ir-O-Ir bond angle becomes decreases,t2g band width becomes narrower.Lnheavy rare-earth Mott insulatorElectronic correlation effect is important.IrOIr2Continuous Metal Insulator TransitionsT MI (K)Phase Diagram16014012010080604020SemiconductorHoDyK. Matsuhira et al, 2007TbGdIr AFMInsulatorEuSemi-metalSm01 1.02 1.04 1.06 1.08 1.1 1.12 1.14Ln 3+ ionic radius (Å)Ln 2 Ir 2 O 7PMMetalNdMetalPrThese MThe insuorderingWeakA diffThe resisAs Irt2gbanLnhElecThe MIT temperature TMI does not depend onthe de Gennes factor and the magnetism of Ln 3+ .T MI tends to increase as the ionic radius ofLn 3+ becomes smaller.


ong>Spinong> ong>orbitong> couplingEstimate (?) λ ≈ 0.5 eV


Modeloctahedral Ir 4+ : (t 2g ) 5effective l=1 ong>orbitong>al degeneracyIr-O-Ir hoppingdominant V pdπ channelong>Spinong>-ong>orbitong> couplingH SOI = −λL ⃗ · ⃗SHubbard UOIr


the energy difference between the Ir d and O p states. The dimensionless hopping matrices T ii′ ′, arising frU=0 Band Structure3 x 4 = 12 doublydegenerate bandsλ2.8t: bands separateonly j=1/2 states nearFermi energythese states havecomplex wavefunctions!"#$%&'( )*+,- !/"0+! . ! " ! # $E FFIG. 2: Electronic band structure of Ir 5d electrons on the pyrochlore lattice at large spin-ong>orbitong> coupling, λ →∞. Orelevant eight (four doubly degenerate) bands are shown. Energy is counted from λ. A band gap between the filled lobands and empty upper four bands is clearly seen.


Topological BandInsulatorInversion Symmetry:Fu-Kane give simple criterion for parity eigenvaluesStrong TBI (weak invariants all zero by cubicsymmetry)Surface states4(100) surface!"#$%&'( )*+,- !."/+'0! 0 !" !"#$surface Dirac point


Phase DiagramU/t00metal2.8TBIλ/t


Very large U/tFor λ >> J ∼ t 2 /U, reduces to Heisenberg “spin”model for j=1/2 eigenstatesH spin = 4t2 ∑ [JS U⃗ i · ⃗S i ′ + D ⃗ ii ′ · ⃗S i × S ⃗ i ′ + S ⃗ i · ←→ Γ ii ′ · ⃗S]i ′ .ii ′This model has been extensively studiedElhajal et al, 2005Axis of D-vector fixed by symmetryvery large DM: |D|/J = 5460 √2 ≈ 0.6312283Ground state for |D|/J > 0.3 is definitelymagnetically orderedORDERING IN THE PYROCHLORE ANTIFERROMAGQ=0 magnetic stateFIG. 6. Ground state in the case of indirect DMI’s. Thestate for the whole pyrochlore lattice is a q=0 structure so tone tetrahedron is represented. Similar structures in the zx


Phase DiagramU/tMagnetic order?00metal2.8TBIλ/t


Intermediate USlave-rotor approximationFlorens, Georges (2004)Seems to give qualitatively reasonable results forfrustrated Hubbard models (triangular,checkerboard, hyperkagome) in agreement withseveral numerical approachesDoes not describe nesting/SDW physicsSimple to implementc † a = e iθ f † aDecouple to produce independent MF dynamics forrotors (charge) and spinonsShould be solved self-consistently


Phase DiagramU/t7.6GMIMagnetic orderTMI3.80metalTBI02.8Bandwidth suppressionλ/t


Phase DiagramU/tMagnetic order7.63.8GMITMIrotorsgappedin MIs00metal2.8TBIλ/t


Phase DiagramspinonFermisurfaceU/t7.6GMIMagnetic orderTMI3.800metal2.8TBIλ/t


Phase DiagramU/t7.6GMIMagnetic orderTMIspinonsin Z 2 TBIstate3.800metal2.8TBIλ/t


Topological MottA U(1) spin liquidInsulatorGapless photonU/tMagnetic orderA quantum analogof “Coulombphase” in spin ice7.63.8GMITMIGapless “topologicalspin metal” at surface00metal2.8TBIλ/tMagnetic monopoleexcitations carry spin?


metal-TBI transitionLong-range Coulomb: excitonsc.f. Halperin, Rice (1968)U/t7.6GMIMagnetic orderTMI3.80metalTBI02.8probably weakly magneticλ/t


Back to iridatestionpyrochlore oxidesExperiments show continuous T>0 MITsResistivity (polycrystalline samples)K. Matsuhira et al, 2007Localized momentagnetic frustration10 610 5DyHoConduction electrons10 510 4Tb[2p]conduction banderant electron systemhe pyrochlore lattice! (m" cm)10 410 310 2Eu10 7 0 50 100 150 200 250 300TbGdSmDy10 3Ho10 6 60 80 100 30010 110 0PrNdLn 2Ir 2O 7T(K)Metal Insulator Transition(Ln=Nd, Sm, Eu, Gd, Tb, Dy, Ho)


Back to iridatesGround state J-multipletAnomaly in magnetization at TMIM/H (emu/Nd mole)Sm2Ir2O7 TMI=117K Sm 3+ (4f) 5 J=5/2 Kramers doubletTMI=120K EuExperiments 3+ (4f)show 6 J=0 non-magneticcontinuous T>0 MITsEu2Ir2O70.160.140.120.10.080.060.040.02FCH=1kOeZFCM/H (emu/mol Ln)CEF GSNd2Ir2O7 TMI=36K Nd 3+ (4f) 3 J=9/2 Kramers doubletMagnetization (polycrystalline samples)0.0160.0140.0120.010.008T MI 0.006Nd 2Ir 2O 0.00470.0020}FCZFCFCZFCFrustration due to 4f momentTMIK. Matsuhira et al, 2007H=1kOeEu 2Ir 2O 7Eu 3+ ! V calcSm 2Ir 2O 701 10 100T (K)The anomalies observed at TMI have a commonality.An ordering from 5d electrons.closest to QCP3000 50 100 150 200 250 300 350 400T (K)AFM orderings with a weak FM moment(speculation)Frustrated ordered state due to 5d electrons below T MI3


metal-TBI transitionPerhaps consistent with an excitonic state?U/t7.6GMIMagnetic orderTMI3.80metalTBIλ/t02.8this transition? probably too optimistic!


Conclusionsong>Spinong>-ong>orbitong> interactions become increasinglyimportant with increased correlations due toreduction in effective bandwidthThis may lead to new phases such as the TMIWe expect this physics is critical to Motttransitions in 5d TMOsfrustrated geometries such as pyrochlore areespecially beneficial for exotic band topologyReference: arXiv:0908.2962

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