Online lectures accessible at: http://neutrons.ornl.gov/neutrons-videos
Quantum magnetism in insulating solids
Collin Broholm Johns Hopkins University
Supported by U.S. DoE
Basic Energy Sciences,
Materials Sciences & Engineering
DE-FG02-08ER46544
S=1/2 S=1/2 S=1/2
Coulomb + Pauli = Heisenberg
Coulomb interactions plus Pauli principle split 4-fold spin degeneracy
The level scheme is reproduced by Heisenberg Exchange Hamiltonian
|J| 1=totS
0=totS
|J| 0=totS
1=totS
Singlet gnd. State: J > 0 Triplet gnd. State: J < 0
S=1/2 S=1/2
Neutron Scattering o Inelastic scattering o Sum-rules
Singlet formation in 1D o Alternating spin-1/2 chain (static singlets) o Uniform spin-1/2 chain (resonating singlets)
Frustrated Quantum Magnetism o Continuum scattering from kagome magnet o Quantum spin-ice o Spin-orbital systems o Magneto-structural effects
Conclusions
Overview
Neutron Scattering
n-matter Interactions: • Weak (Born limit) • Energy independent • Well characterized • similar strengths
2.072 meVÅ2k2
Understanding Inelastic Magnetic Scattering: Isolate the “interesting part” of the cross section
The dynamic correlation function (“scattering law”) :
In thermodynamic equilibrium (detailed balance):
Useful exact sum-rules: Total moment sum-rule (general result from Fourier analysis)
Correlations in space & time rearrange intensity in space leaving the total scattering
scattering unaffected
Definition of the equal time correlation function:
The wave vector dependence of the energy-integrated intensity probe a snap-shot of the
fluctuating spin configuration
Fluctuation Dissipation Theorem Define the following t-dependent “response” function:
The generalized susceptibility can be expressed as:
The “scattering law” can be expressed as:
From which, the “fluctuation-dissipation” theorem:
First Moment Sum-rule Fourier transform of expression for scattering law:
Equating these leads to the first moment sum-rule:
Time derivative of left side evaluated for t=0:
Time derivative of right side evaluated at t=0:
Neutron Scattering o Inelastic scattering o Sum-rules
Singlet formation in 1D o Alternating spin-1/2 chain (static singlets) o Uniform spin-1/2 chain (resonating singlets)
Frustrated Quantum Magnetism o Continuum scattering from kagome magnet o Quantum spin-ice o Spin-orbital systems o Magneto-structural effects
Conclusions
Overview
Cu(NO3)2.2.5D2O: Alternating spin chain
( )↓↑−↑↓2
1
( ) ↓↓↓↑+↑↓↑↑ ,2
1,
0=totS
1=totS, ,
,
↑↑ ↑↓
↓↑ ↓↓
Singlet product state on alternating spin-1/2 chain
Quantum critical spin-1/2 chain
Hammar et al. (1999)
Cu(C4H4N2)(NO3)2
Copper Pyrazine dinitrate
Probing matter through scattering fp
ipSingle Particle Process
Multi Particle Process
( )( ) ( ) fi
fi
EE −=+
−=+
21
21
QQppQQ
εε
1Q
2Q( ) fi
fi
EE −=
−=
QppQ
ε
Q
Q
ћω
Intensity
Intensity
Q
ћω
Emergent quasi-particles at QCP
Quantum Critical Spin-1/2 chain
12
12
JJJJ
+−
=δ
0
∆
Damle and Huse PRL (2002)
( )∑ +++ +⋅=n
nnnn JJ 221211222 SSSSH
1 2 4 3
1 2 4 3
Odd bond singlets Even bond singlets
Magnetic excitations near quantum criticality No order parameter from which to form harmonic modes
Excitations from quasi-particles in a strongly fluctuating medium
Angular momentum conservation precludes single particle scattering
Expect a broad continuum of scattering
Neutron Scattering o Inelastic scattering o Sum-rules
Singlet formation in 1D o Alternating spin-1/2 chain (static singlets) o Uniform spin-1/2 chain (resonating singlets)
Frustrated Quantum Magnetism o Continuum scattering from kagome magnet o Quantum spin-ice o Spin-orbital systems o Magneto-structural effects
Conclusions
Overview
Square Lattice Antiferromagnet Kagome lattice Antiferromagnet
Corner-linked simplexes
Takagi et al. WIkimedia
Balents et al.
Kagome Hyper Kagome
Pyrochlore
Herbertsmithite: (Zn0.85Cu0.15)Cu3(OD)6Cl2
Natural Herbertsmithite From Chile: Big & dirty
Synthetic HS From MIT: Small, clean, & deuterated Y. S. Lee & Tianheng Han
T. Han, et al., PRB (2010).
Correlations in quantum kagome magnet
T. Han et al., Nature 492, 406 (2012).
“Rigid” extended structures in momentum space
Consistent with structure factor for spinon
a b
a b
Hao & Tchernyshyov, PRB (2010)
Continuum in 2D quantum magnet
T. Han, J. S. Helton, S. Chu, D. Nocera, J. A. Rodriguez, C. Broholm, Y. S. Lee, Nature 492, 406 (2012).
Spinons on kagome
Z.-H. Hao and O. Tchernyshyov, PRL 103, 187203 (2009).
Y. Wan and O. Tchernysnyov, ArXiv 1301.5008v1 (2013)
Z. H. Hao and O. Tchernysnyov, ArXiv arXiv:1301.3261 (2013)
S = ½ kagome AFM has a finite
concentration of spinons in its ground
state.
Spinons are solitons with spin S = ½ and
fermionic statistics.
Exchange-mediated attraction binds spinons
into pairs with spin S=0.
The ground state appears to be a Z2 spin
liquid
g0=0.2 g0=0.6
Decoupled impurity spins
Interacting impurity spins have simple cubic connectivity. The 15% concentration is below the 31.16% percolation threshold
Han et al. to appear in PRB-RC (2016)
0.4
A gap in the spectrum of kagome planes M. Fu et al., Science (2015) Han et al., submitted to PRL (2016)
The NMR gap size of 0.9 meV is • Consistent with neutron scattering • Consistent with DMRG results
Neutron Scattering o Inelastic scattering o Sum-rules
Singlet formation in 1D o Alternating spin-1/2 chain (static singlets) o Uniform spin-1/2 chain (resonating singlets)
Frustrated Quantum Magnetism o Continuum scattering from kagome magnet o Quantum spin-ice o Spin-orbital systems o Magneto-structural effects
Conclusions
Overview
Ising FM : Spin-Ice
A. P. Ramirez et al.
Nature (1999)
Pomaranski et al.
Nature (2013)
Pinch points and monopoles in spin ice Fennell et al. Science (2009)
Experiment
Theory
Quantum Spin Ice in Pr2Zr2O7?
Pr2Zr2O7: Quantum Spin Ice
Quantum fluctuations in spin-ice Pr2Zr2O7
Powder
DCS
Only 5% of the scattering is elastic
Elastic scattering |ħω|<0.2 meV
No pinch points for inelastic scattering
Excitation in (001) magnetized Pr2Zr2O7
1
2
3
4
Elastic Scattering from (001) magnetized Pr2Zr2O7
The induced “static” ( ) magnetization is not lattice periodic!
• The level distribution function: • Singlet-singlet susceptibility:
• Sample averaged susceptibility
• Fluctuation-dissipation theorem yields
( )ρ ∆
( ) ( ) ( ) ( )( ) ( )2
/ 21
1Beg
e e Z
β
β βχ ω π µ α δ ω δ ωβ
− ∆
− ∆ − ∆
−′′ = − ∆ − + ∆′+ +
Inhomogeneous level splitting in Pr2Zr2O7?
• The E>>h the excitations result from inhomogeneous level splitting • For E~h inter-site interactions (collective physics) is important • Population of the 9.5 meV singlet may account for quasi-elastic
response for T>100 K.
Origin of inhomogeneous level splitting?
• T-independent distribution function ⇒ Static not dynamic phenomenon
• Continuous spectrum in stoichiometric sample ⇒ Large unit cell or density wave: distribution of environments ⇒ Dynamic Jahn-Teller type effect
• Distribution changes in Pr2+xZr2-xO7-x/2 ⇒ Cr stuffing narrows the distribution so exchange dominates
Neutron Scattering o Inelastic scattering o Sum-rules
Singlet formation in 1D o Alternating spin-1/2 chain (static singlets) o Uniform spin-1/2 chain (resonating singlets)
Frustrated Quantum Magnetism o Continuum scattering from kagome magnet o Quantum spin-ice o Spin-orbital systems o Magneto-structural effects
Conclusions
Overview
Fe2+, S=2
Diamond
lattice
A dynamic state of affairs in FeSc2S4 K. Plumb et al. submitted to PRX
Detection of spin and orbital order
MACS and CNCS 11BM at APS
K. Plumb et al. submitted to PRX
qm=(½½0) order in tetragonal cell K. Plumb et al. submitted to PRX
Ordered moment only 50% of the full ordered moment: Strong quantum fluctuations and proximity to spin-orbital quantum criticality
Neutron Scattering o Inelastic scattering o Sum-rules
Singlet formation in 1D o Alternating spin-1/2 chain (static singlets) o Uniform spin-1/2 chain (resonating singlets)
Frustrated Quantum Magnetism o Continuum scattering from kagome magnet o Quantum spin-ice o Spin-orbital systems o Magneto-structural effects
Conclusions
Overview
Why is Mott transition “exposed” in V2O3?
McWhan et al. PRB (1973)
Spin-Waves in AFI phase of V2O3
[-2 1 L]
E (m
eV)
0 1 2 3 4
17 meV
4 meV
80
60
40
20
0
E (m
eV)
80
60
40
20
0 -2 -1 0 1 2 3 4
[-L 1 L]
J. Leiner
A frustrated paramagnet
Frustration: Bandwidth Narrowing
Frustrated Honeycomb AFM
0 0.2 0.4 0.6 0.8 1
DFT for V2O3 (Valenti et al )
J2
J1
Albuquerque et al PRB (2011)
V
V2O3
Bao et al.
Bao et al. (1995)
2kf
Summary • Inelastic magnetic scattering probes dynamic spin
correlation function:
• Scattering theory is essential to interpret data
• Tremendous progress in instrumentation: Join us!
• Challenges in Frustrated Quantum Magnetism
– Static or dynamic valence bonds?
– Local or collective singlet in Q-spin-ice
– Static or dynamic orbital configuration?
– Rigid or compliant lattice?