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 continuum q, ω p=k2-k1 not yet determined Spinon excitations in 1d Heisenberg antiferromagnet excitation relative momentum of spinons continuum q, ω q=k2-k1 not yet determined neutron scattering magnetic excitations
Bethe-Ansatz exact solution
Klauser, Mossel, Caux, JvdB, PRL 106, 157205 (2011) spin-charge separation spin-charge separation
Remove electron from Heisenberg antiferromagnet
–
holon holon holon spinon holon spinon holon holon and spinon: moving freely & independently spinon holon holon and spinon: moving freely & independently holon = quasi-particle carrying charge (e) spinon = quasi-particle carrying spin (ħ/2) spinon holon holon and spinon: moving freely & independently holon = quasi-particle carrying charge (e) spinon = quasi-particle carrying spin (ħ/2) spin-charge separation: “e-ħ ➞ e + ħ” spinon holon
holon and spinon: moving freely & independently holon = quasi-particle carrying charge (e) spinon = quasi-particle carrying spin (ħ/2) spin-charge separation: “e-ħ ➞ e + ħ” detectable by angle resolved photoemission (ARPES) C. Kim et al, PRL 77, 4054 (1996) spin-charge separation by ARPES
B.J. Kim et al, Nature Physics (2006) spin-orbital separation spin-orbital separation
Excite electron from one orbital to another in Heisenberg chain spin-orbital separation
Excite electron from one orbital to another in Heisenberg chain
1. orbital degrees of freedom 2. resonant inelastic x-ray scattering Quasi 1D cuprate Sr2CuO3 spin chains Quasi 1D cuprate Sr2CuO3 spin chains Quasi 1D cuprate Sr2CuO3 spin chains
Cu2+ d-orbital splitting: Cu2+
Cubic Crystal field splitting • eg orbitals – Cu2+ – –– –– ––
• t orbitals –– –– 2g Cu (2+) = 3d9
eg 5x
t2g d-orbital splitting: Cu2+
Cubic Crystal field splitting • eg orbitals – Cu2+ – –– –– ––
• t orbitals –– –– 2g Cu (2+) = 3d9
eg 5x
t2g d-orbital splitting: Cu2+
Cubic Crystal field splitting • eg orbitals – Cu2+ – –– –– ––
• t orbitals –– –– 2g Cu (2+) = 3d9
eg 5x
t2g d-orbital splitting: Cu2+
Cubic Crystal field splitting • eg orbitals – Cu2+ – –– –– ––
• t orbitals –– –– 2g Cu (2+) = 3d9
eg 5x
t2g RIXS on Sr2CuO3
Resonant What is Inelastic RIXS X-ray scattering RIXS on Sr2CuO3
Resonant What is Inelastic RIXS X-ray scattering
X-ray scattering: photon in solid photon out
inelastic: ωout < ωin
resonant: tune ωin to an atomic absorption edge
Ament, Veenendaal, Devereaux, Hill, JvdB Rev. Mod. Phys. 83, 705 (2011) Cu L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p Cu L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p Cu L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p Cu L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p Cu L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p s=1/2 l=1 Cu L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p s=1/2 l=1 l•s Cu L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p s=1/2 l=1 l•s dd excitation Cu spin flip L-edge
3 d Momentum transfer ~900 eV Energy loss
2 p RIXS at Cu L3 edge of Sr2CuO3
RIXS at Cu L3 edge in Sr2CuO3
Sr2CuO3
• 1D AF x FO ground state
• Famous 1D AF with very large J ~ 0.25 eV RIXS spectrum of Sr2CuO3
Schlappa, Wohlfeld, Zhou, Mourigal, Haverkort, Strocov, Hozoi, Monney, Nishimoto, Singh, Revcolevschi, Caux, Patthey, Ronnow, JvdB, Schmitt, Nature 485, 82 (2012) RIXS spectrum of Sr2CuO3
Orbital excitations
Schlappa, Wohlfeld, Zhou, Mourigal, Haverkort, Strocov, Hozoi, Monney, Nishimoto, Singh, Revcolevschi, Caux, Patthey, Ronnow, JvdB, Schmitt, Nature 485, 82 (2012) RIXS spectrum of Sr2CuO3
Orbital excitations
Spin excitations spinons
Schlappa, Wohlfeld, Zhou, Mourigal, Haverkort, Strocov, Hozoi, Monney, Nishimoto, Singh, Revcolevschi, Caux, Patthey, Ronnow, JvdB, Schmitt, Nature 485, 82 (2012) Spinons in Sr2CuO3
Bethe-Ansatz exact solution Spinons in Sr2CuO3
spinons of Heisenberg spin chain
Bethe-Ansatz exact solution Klauser, Mossel, Caux, JvdB, PRL 106, 157205 (2011) Orbital excitations in Sr2CuO3
• So far we have considered only local orbital excitations • But: an orbital excitation can disperse
The orbiton (mobile orbital excitation) appears in RIXS
E. Saitoh et al., Nature 410, 180 (2001) How to explain large orbital dispersion? How to explain large orbital dispersion?
“Standard” theoretical approaches fail
Single ion (left) and orbital wave (right) Orbital excitations in Sr2CuO3
But we get perfect agreement with experiment... employing a new theoretical concept: spin-orbital separation
<-- Experiment Theory --> Spin - Orbital separation
Spin-orbital separation Spin-charge separation
[C. Kim et al. PRL 77, 4054 (1996)] Spin - Orbital separation
Spin-orbital separation Spin-charge separation
[C. Kim et al. PRL 77, 4054 (1996)] Spin - Orbital separation
Spin-orbital separation Spin-charge separation
[C. Kim et al. PRL 77, 4054 (1996)] Spin - Orbital separation in Sr2CuO3
Simplified theory: orbiton-spin separation Ansatz for xz orbital:
Spinon-orbiton continuum Orbiton (2 edges)
Exp. Exp. Theory Conclusions Conclusions
RIXS sensitive probe of orbital excitations and can directly probe their dispersions Conclusions
RIXS sensitive probe of orbital excitations and can directly probe their dispersions
Measures magnetic excitations and dispersions spinons orbitons splitting of spinon and orbiton
Schlappa, Wohlfeld, Zhou, Mourigal, Haverkort, Strocov, Hozoi, Monney, Nishimoto, Singh, Revcolevschi, Caux, Patthey, Ronnow, JvdB, Schmitt, Nature 485, 82 (2012) Conclusions
RIXS sensitive probe of orbital excitations and can directly probe their dispersions
Measures magnetic excitations and dispersions spinons orbitons splitting of spinon and orbiton
Schlappa, Wohlfeld, Zhou, Mourigal, Haverkort, Strocov, Hozoi, Monney, Nishimoto, Singh, Revcolevschi, Caux, Patthey, Ronnow, JvdB, Schmitt, Nature 485, 82 (2012) Promising new technique... experiments... Conclusions
RIXS sensitive probe of orbital excitations and can directly probe their dispersions
Measures magnetic excitations and dispersions spinons orbitons splitting of spinon and orbiton
Schlappa, Wohlfeld, Zhou, Mourigal, Haverkort, Strocov, Hozoi, Monney, Nishimoto, Singh, Revcolevschi, Caux, Patthey, Ronnow, JvdB, Schmitt, Nature 485, 82 (2012) Promising new technique... experiments... theory...
Detecting orbitons with RIXS
line spectra: Elementary Excitations in RIXS Elementary Excitations in RIXS
Resonant Inelastic X-ray Scattering Studies of Elementary Excitations Ament, Veenendaal, Devereaux, Hill, JvdB Rev. Mod. Phys. 83, 705 (2011) Elementary Excitations in RIXS
Resonant Inelastic X-ray Scattering Studies of Elementary Excitations Ament, Veenendaal, Devereaux, Hill, JvdB Rev. Mod. Phys. 83, 705 (2011) Atomic RIXS on Cu L-edge Atomic RIXS on Cu L-edge
Michel van Veenendaal no spin flip without orbital flip PRL 96, 117404 (2006) Atomic RIXS on Cu L-edge
ground state
Michel van Veenendaal no spin flip without orbital flip PRL 96, 117404 (2006) Atomic RIXS on Cu L-edge final states ground state
Michel van Veenendaal no spin flip without orbital flip PRL 96, 117404 (2006) Atomic RIXS on Cu L-edge final states ground state
orbital+spin flip
Michel van Veenendaal no spin flip without orbital flip PRL 96, 117404 (2006) Atomic RIXS on Cu L-edge final states ground state
orbital+spin flip spin // z
Michel van Veenendaal no spin flip without orbital flip PRL 96, 117404 (2006) Atomic RIXS on Cu L-edge final states ground state
orbital+spin flip spin // z
now: generalize this to arbitrary spin orientation Ament, Ghiringhelli, Moretti, Braicovich & JvdB, PRL 103, 117003 (2009) Magnetic L-edge RIXS on La2CuO4 Magnetic L-edge RIXS on La2CuO4
In special cases direct spin-flip scattering is allowed at Cu L-edge CuO’s are such special cases! Magnetic L-edge RIXS on La2CuO4
In special cases direct spin-flip scattering is allowed at Cu L-edge CuO’s are such special cases!
Braicovich, JvdB et al., Ament, Ghiringhelli, Moretti, PRL 104, 077002 (2010) Braicovich & JvdB, PRL 103, 117003 (2009)
magnon dispersion Magnetic L-edge RIXS on La2CuO4
In special cases direct spin-flip scattering is allowed at Cu L-edge CuO’s are such special cases!
Braicovich, JvdB et al., Ament, Ghiringhelli, Moretti, PRL 104, 077002 (2010) Braicovich & JvdB, PRL 103, 117003 (2009)
magnon dispersion Magnetic L-edge RIXS on La2CuO4
In special cases direct spin-flip scattering is allowed at Cu L-edge CuO’s are such special cases!
Braicovich, JvdB et al., Ament, Ghiringhelli, Moretti, PRL 104, 077002 (2010) Braicovich & JvdB, PRL 103, 117003 (2009)
magnon dispersion Braicovich, JvdB et al., PRB 81, 174533 (2010) RIXS magnon dispersion of Sr2CuO2Cl2
—> transferred momentum q Guarise et al., PRL 105, 157006 (2010) RIXS magnon dispersion of Sr2CuO2Cl2
deviation from simple Heisenberg
—> transferred momentum q Guarise et al., PRL 105, 157006 (2010) Magnetic L-edge RIXS on 8% doped La2-xSrxCuO4 Tc = 21K Magnetic L-edge RIXS on 8% doped La2-xSrxCuO4 Tc = 21K
d-d excitation Magnetic L-edge RIXS on 8% doped La2-xSrxCuO4 Tc = 21K
d-d excitation
Braicovich, JvdB et al. paramagnon PRL 104, 077002 (2010) magnon magnon para- magnon Magnetic RIXS vs. Inelastic Neutron Scattering
RIXS Neutrons amount of material needed magnon energy accessible
materials Magnetic RIXS vs. Inelastic Neutron Scattering
RIXS Neutrons amount of small large material needed magnon energy accessible
materials Magnetic RIXS vs. Inelastic Neutron Scattering
RIXS Neutrons amount of small large material needed magnon energy high (>25 meV) low (<25 meV) accessible
materials Magnetic RIXS vs. Inelastic Neutron Scattering
RIXS Neutrons amount of small large material needed magnon energy high (>25 meV) low (<25 meV) accessible
materials Cu, Ni, ... non-absorbers
Double spin-flip scattering in spin 1/2 chain
Bethe Ansatz (exact)
Neutron scattering
Klauser, Mossel, Caux, JvdB, PRL 106, 157205 (2011)
Forte, Cuoco, Noce, JvdB, PRB 83, 245133 (2011) Double spin-flip scattering in spin 1/2 chain
Bethe Ansatz (exact)
Neutron scattering
magnon → 2 spinons Klauser, Mossel, Caux, JvdB, PRL 106, 157205 (2011) Forte, Cuoco, Noce, JvdB, PRB 83, 245133 (2011) Double spin-flip scattering in spin 1/2 chain
Bethe Ansatz (exact)
Neutron scattering
2 spinon continuum
magnon → 2 spinons Klauser, Mossel, Caux, JvdB, PRL 106, 157205 (2011) Forte, Cuoco, Noce, JvdB, PRB 83, 245133 (2011) Double spin-flip scattering in spin 1/2 chain
Bethe Ansatz (exact)
Neutron scattering RIXS
2 spinon continuum
magnon → 2 spinons Klauser, Mossel, Caux, JvdB, PRL 106, 157205 (2011) Forte, Cuoco, Noce, JvdB, PRB 83, 245133 (2011) Double spin-flip scattering in spin 1/2 chain
Bethe Ansatz (exact)
Neutron scattering RIXS
2 spinon continuum
magnon → 2 spinons Klauser, Mossel, Caux, JvdB, PRL 106, 157205 (2011) Forte, Cuoco, Noce, JvdB, PRB 83, 245133 (2011) Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins? Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins?
Hopping amplitude t Coulomb repulsion U Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins?
Hopping amplitude t Coulomb repulsion U Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins?
Hopping amplitude t Coulomb repulsion U with core hole: Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins?
Hopping amplitude t Coulomb repulsion U with core hole: Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins?
Hopping amplitude t Coulomb repulsion U with core hole:
intermediate state at U - Uc Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins?
Hopping amplitude t Coulomb repulsion U Core-hole locally modifies superexchange J ! with core hole:
intermediate state at U - Uc Spin-photon coupling for Cu K-edge RIXS
How does intermediate state core-hole couple to spins?
Hopping amplitude t Coulomb repulsion U Core-hole locally modifies superexchange J ! with core hole:
create double spin-flipintermediate / bi-magnon state excitations at U - Uc Magnetic RIXS on Cu L-edge ⇓ hole in spin // z spin ⊥ z x2-y2 orbital
Ament, Ghiringhelli, Moretti, Braicovich & JvdB, PRL 103, 117003 (2009) Magnetic RIXS on Cu L-edge ⇓ hole in spin // z spin ⊥ z x2-y2 orbital
ground state
Ament, Ghiringhelli, Moretti, Braicovich & JvdB, PRL 103, 117003 (2009) Magnetic RIXS on Cu L-edge ⇓ hole in spin // z spin ⊥ z x2-y2 orbital orbital+spin flip
ground state
Ament, Ghiringhelli, Moretti, Braicovich & JvdB, PRL 103, 117003 (2009) Magnetic RIXS on Cu L-edge ⇓ hole in spin // z spin ⊥ z x2-y2 orbital orbital+spin flip
ground state spin flip only
Ament, Ghiringhelli, Moretti, Braicovich & JvdB, PRL 103, 117003 (2009)