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Spin Orbital Separation Spin–orbital separation in the quasi- one-dimensional Mott insulator Sr2CuO3 Jeroen van den Brink Motzensee 11.2.2014 Spin–orbital separation in the quasi- one-dimensional Mott insulator Sr2CuO3 Jeroen van den Brink Krzysztof Wohlfeld Liviu Hozoi Satoshi Nishimoto Jean Sebastien Caux University of Amsterdam Thorsten Schmitt Swiss Light Source, PSI Justine Schlappa Nature, 485 82 (2012) Motzensee 11.2.2014 charge fractionalization in 1d systems charge fractionalization in 1d systems spin-charge separation in a spin-chain charge fractionalization in 1d systems spin-charge separation in a spin-chain spin-orbital separation detected by RIXS charge fractionalization in 1d systems 1d systems: organic molecular polymers Poly-alkane 1d systems: organic molecular polymers Poly-alkane Poly-acetalyne 1d systems: organic molecular polymers Poly-alkane Poly-acetalyne What is the charge of charge carriers? E k E Conduction band k Valence band E Conduction band k Valence band ... the charge of charge carriers? E k – remove electron E k – q, ω remove electron E k – q, ω remove electron E – q, ω hole propagating in valence band k – q, ω remove electron E – q, ω hole propagating in valence band k charge e 1 extra electron/C 1 extra electron/C 1 extra electron/C dimerization opens gap 1 extra electron/C dimerization opens gap doubled unit cell 1 extra electron/C dimerization opens gap doubled unit cell ... the charge of charge carriers? Poly-acetalyne – Poly-acetalyne – Poly-acetalyne – Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne Poly-acetalyne 2 domain walls propagating Poly-acetalyne 2 domain walls propagating each carries half the charge Poly-acetalyne 2 domain walls propagating each carries half the charge Poly-acetalyne 2 domain walls propagating – each carries half the q, ω charge Poly-acetalyne 2 domain walls propagating – each carries half the q, ω charge dimerization causes charge fractionalization Poly-acetalyne electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– + + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– + + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne –– + + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + particle with charge spin without spin electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + particle with charge spin without spin holon electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + particle particle with charge spin with spin without spin without charge holon electron fractionalization Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + particle particle with charge spin with spin without spin without charge holon electron fractionalization spinon Su, Schriefer & Heeger, PRL 42, 1698 (1979) Poly-acetalyne – + particle particle with charge spin with spin without spin without charge holon electron fractionalization spinon spin-charge separation Su, Schriefer & Heeger, PRL 42, 1698 (1979) spin fractionalization in 1d systems Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 excited state; E=J Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 ground state; E=0 excited state; E=J Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 ground state; E=0 excited state; E=J Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 ground state; E=0 excited state; E=J Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 ground state; E=0 excited state; E=J Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 ground state; E=0 excited state; E=J Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 ground state; E=0 excited state; E=J 2 propagating domain walls Ising spin system z z z z S =+ħ/2 S =-ħ/2 H = -J S i S i+1 ground state; E=0 fractionalization of spin excited state; E=J 2 propagating domain walls d dimensional system with linear extend N volume Nd d dimensional system with linear extend N domain wall: surface separating 2 volume Nd differently ordered volumes d dimensional system with linear extend N domain wall: surface separating 2 volume Nd differently ordered volumes area Nd-1 d dimensional system with linear extend N domain wall: surface separating 2 volume Nd differently ordered volumes area Nd-1 domain wall carried by N d=2 atoms/electrons/spins d dimensional system with linear extend N domain wall: surface separating 2 volume Nd differently ordered volumes area Nd-1 domain wall carried by N d=2 atoms/electrons/spins 2 domain walls carried by 1 d=1 atom/electron/spin d dimensional system with linear extend N domain wall: surface separating 2 volume Nd differently ordered volumes area Nd-1 domain wall carried by N d=2 atoms/electrons/spins 2 domain walls carried by 1 d=1 atom/electron/spin d=1: elementary excitations are topological in nature d dimensional system with linear extend N domain wall: surface separating 2 volume Nd differently ordered volumes area Nd-1 domain wall carried by N d=2 atoms/electrons/spins 2 domain walls carried by 1 d=1 atom/electron/spin d=1: elementary excitations are topological in nature elementary excitations are fractionalized Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z H = J S i S i+1 Allow for spin-exchange Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z H = J S i S i+1 Allow for spin-exchange Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z H = J S i S i+1 Allow for spin-exchange Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z H = J S i S i+1 Allow for spin-exchange Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 Allow for spin-exchange Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange ground state Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon two domain walls, each with ∆Sz= ħ/2 spinon Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon two domain walls, each with ∆Sz= ħ/2 spinon d=1: magnon fractionalizes into spinons Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon two domain walls, each with ∆Sz= ħ/2 spinon d=1: magnon fractionalizes into spinons spinon k1,ω1 magnon q, ω spinon k2,ω2 Heisenberg antiferromagnet Sz=+ħ/2 Sz=-ħ/2 z z y y x x H = J S i S i+1 +J S i S i+1 +J S i S i+1 = J Si •Si+1 Allow for spin-exchange spinground flip ∆ stateSz= ħ in d=3: magnon two domain walls, each with ∆Sz= ħ/2 spinon d=1: magnon fractionalizes into spinons spinon k1,ω1 magnon q, ω q=k1+k2, ω=ω1+ω2 spinon k2,ω2 Spinons in 1d Heisenberg antiferromagnet spinon k1,ω1 magnon q, ω q=k1+k2, ω=ω1+ω2 spinon k2,ω2 Spinons in 1d Heisenberg antiferromagnet spinon k1,ω1 magnon q, ω q=k1+k2, ω=ω1+ω2 spinon k2,ω2 relative momentum of spinons p=k2-k1 not yet determined Spinons in 1d Heisenberg antiferromagnet spinon k1,ω1 magnon q, ω q=k1+k2, ω=ω1+ω2 spinon k2,ω2 excitation relative momentum of spinons
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