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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?