Spin Liquid and Solid in Pyrochlore Antiferromagnets

Doron Bergman UCSB PhysicsGreg Fiete KITPRyuichi Shindou UCSB PhysicsSimon Trebst Q Station

“Quantum Fluids”, Nordita 2007

Outline

• Quantum spin liquids and dimer models• Realization in quantum pyrochlore

magnets • Einstein spin-lattice model• Constrained phase transitions and exotic

criticality

Spin Liquids

• Empirical definition:– A magnet in which spins are strongly correlated but do

not order• Quantitatively:

– High-T susceptibility:– Frustration factor:

• Quantum spin liquid:– f=1 (Tc=0)– Not so many models can be shown to have such

phases

Quantum Dimer Models

• “RVB” Hamiltonian– Hilbert space of dimer coverings

• D=2: lattice highly lattice dependent– E.g. square lattice phase diagram (T=0)

+

+

+

1

Ordered except exactly at v=1

O. Syljuansen 2005

D=3 Quantum Dimer Models

• Generically can support a stable spin liquid state

• vc is lattice dependent. May be positive or negative

1

Spin liquid state

ordered ordered

Chromium Spinels

ACr2O4 (A=Zn,Cd,Hg)

• spin s=3/2• no orbital degeneracy• isotropic

• Spins form pyrochlore lattice

cubic Fd3m

• Antiferromagnetic interactions

CW = -390K,-70K,-32K for A=Zn,Cd,Hg

H. TakagiS.H. Lee

Pyrochlore Antiferromagnets

• ManyMany degenerate classical configurations

• Heisenberg

• Zero field experiments (neutron scattering)-Different ordered states in ZnCr2O4, CdCr2O4,

HgCr2O4

• Evidently small differences in interactions determine ordering

Magnetization Process

• Magnetically isotropic• Low field ordered state complicated, material dependent

• Plateau at half saturation magnetization

H. Ueda et al, 2005

HgCr2O4 neutrons

M. Matsuda et al, Nature Physics 2007

• Powder data on plateau indicates quadrupled (simple cubic) unit cell with P4332 space group

• Neutron scattering can be performed on plateau because of relatively low fields in this material.

• X-ray experiments: ordering stabilized by lattice distortion- Why this order?

Collinear Spins• Half-polarization = 3 up, 1 down spin?

- Presence of plateau indicates no transverse order

Penc et al

- effective biquadratic exchange favors collinear states

• Spin-phonon coupling?- classical Einstein model

large magnetostriction

But no definite order

H. Ueda et al

3:1 States• Set of 3:1 states has thermodynamic entropy

- Less degenerate than zero field but still degenerate- Maps to dimer coverings of diamond lattice

• Effective dimer model: What splits the degeneracy?-Classical:

-further neighbor interactions?-Lattice coupling beyond Penc et al?

-Quantum fluctuations?

Dimer on diamond link =down pointing spin

Effective Hamiltonian

Ring exchange

+

• Due to 3:1 constraint and locality, must be a QDM

• How to derive it?

Spin Wave Expansion

• Henley and co.: lattices of corner-sharing simplexeskagome, checkerboardpyrochlore…

• Quantum zero point energy of magnons:- O(s) correction to energy: - favors collinear states:

- Magnetization plateaus: k down spins per simplex of q sites

• Gauge-like symmetry: O(s) energy depends only upon “Z2 flux” through plaquettes

- Pyrochlore plateau (k=2,q=4): p=+1

Ising Expansion• XXZ model: • Ising model (J =0) has collinear ground states• Apply Degenerate Perturbation Theory (DPT)

Ising expansion Spin wave theory• Can work directly at any s• Includes quantum tunneling• (Usually) completely resolves degeneracy• Only has U(1) symmetry:

- Best for larger M

• Large s • no tunneling (K=0)• gauge-like symmetry leaves degeneracy • spin-rotationally invariant

• Our group has recently developed techniques to carry out DPT for any lattice of corner sharing simplexes

Effective Hamiltonian derivation• DPT:

- Off-diagonal term is 9th order! [(6S)th order]- Diagonal term is 6th order (weakly S-dependent)!

• Checks:-Two independent techniques to sum 6th order DPT-Agrees exactly with large-s calculation (Hizi+Henley) in overlapping limit and resolves degeneracy at O(1/s)

D Bergman et al cond-mat/0607210

S

c

1/2

1.5

1

0.88

3/2

0.25

2

0.056

5/2 3

0.01 0.002Off-diagonal coefficient

+

comparabledominant negligible

Diagonal term

Comparison to large s

• Truncating Heff to O(s) reproduces exactly spin wave result of XXZ model (from Henley technique)

- O(s) ground states are degenerate “zero flux” configurations

• Can break this degeneracy by systematically including terms of higher order in 1/s:

- Unique state determined at O(1/s) (not O(1)!)

Ground state for s>5/2 has 7-fold enlargement of unit cell and trigonal symmetry

Just minority sites shown in one magnetic unit cell

Quantum Dimer Model, s · 5/2• In this range, can approximate diagonal term:

(D Bergman et al PRL 05, PRB 06)• Expected phase diagram

1

U(1) spin liquid

Maximally “resonatable” R state (columnar state)

“frozen” state(staggered state)

++

0

S · 3/2S=2S=5/2

Numerical simulations in progress: O. Sikora et al, (P. Fulde group)

R state• Unique state saturating upper bound on density of resonatable hexagons• Quadrupled (simple cubic) unit cell• Still cubic: P4332• 8-fold degenerate

• Quantum dimer model predicts this state uniquely.

Is this the physics of HgCr2O4?

• Not crazy but the effect seems a little weak:– Quantum ordering scale » |K| » 0.25J– Actual order observed at T & Tplateau/2

• We should reconsider classical degeneracy breaking by– Further neighbor couplings– Spin-lattice interactions

• C.f. “spin Jahn-Teller”: Yamashita+K.Ueda;Tchernyshyov et alConsidered identical distortions of each tetrahedral “molecule”

We would prefer a model that predicts the periodicity of the distortion

Einstein Model

• Site phonon

vector from i to j

• Optimal distortion:

• Lowest energy state maximizes u*:

Einstein model on the plateau• Only the R-state satisfies the bending rule!

• Both quantum fluctuations and spin-lattice coupling favor the same state!– Suggestion: all 3 materials show same ordered state

on the plateau– Not clear: which mechanism is more important?

Zero field • Does Einstein model work at B=0?• Yes! Reduced set of degenerate states

“bending” states preferred

• Consistent with:CdCr2O4 (up to small DM-induced deformation)

Chern, Fennie, Tchernyshyov (PRB 06)

J. H. Chung et al PRL 05

HgCr2O4 Matsuda et al, (Nat. Phys. 07)

Conclusions (I)• Both effects favor the same ordered plateau

state (though quantum fluctuations could stabilize a spin liquid)– Suggestion: the plateau state in CdCr2O4 may be the

same as in HgCr2O4, though the zero field state is different

• ZnCr2O4 appears to have weakest spin-lattice coupling– B=0 order is highly non-collinear (S.H. Lee, private

communication)– Largest frustration (relieved by spin-lattice coupling)– Spin liquid state possible here?

Constrained Phase Transitions• Schematic phase diagram:

0 1U(1) spin liquid

R state

T

“frozen” stateClassical spin liquid

Classical (thermal) phase transition

Magnetization plateau developsT CW

• Local constraint changes the nature of the “paramagnetic”=“classical spin liquid” state

- Youngblood+Axe (81): dipolar correlations in “ice-like” models

• Landau-theory assumes paramagnetic state is disordered- Local constraint in many models implies non-Landau classical criticality Bergman et al, PRB 2006

Dimer model = gauge theory

A

B • Can consistently assign direction to dimers pointing from A ! B on any bipartite lattice

• Dimer constraint Gauss’ Law

• Spin fluctuations, like polarization fluctuations in a dielectric, have power-law dipolar form reflecting charge conservation

A simple constrained classical critical point

• Classical cubic dimer model

• Hamiltonian

• Model has unique ground state – no symmetry breaking.• Nevertheless there is a continuous phase transition!

- Analogous to SC-N transition at which magnetic fluctuations are quenched (Meissner effect)- Without constraint there is only a crossover.

Numerics (courtesy S. Trebst)

Specific heat

C

T/V

“Crossings”

ConclusionsConclusions• We derived a general theory of quantum

fluctuations around Ising states in corner-sharing simplex lattices

• Spin-lattice coupling probably is dominant in HgCr2O4, and a simple Einstein model predicts a unique and definite state (R state), consistent with experiment– Probably spin-lattice coupling plays a key role in

numerous other chromium spinels of current interest (possible multiferroics).

• Local constraints can lead to exotic critical behavior even at classical thermal phase transitions. – Experimental realization needed! Ordering in spin ice?