Degeneracy Breaking in Some Frustrated Magnets

Doron Bergman UCSB Physics

Greg Fiete KITP

Ryuichi Shindou UCSB Physics

Simon Trebst Q Station

HFM Osaka, August 2006

cond-mat:0510202 (prl)0511176 (prb)0605467 0607210 0608131

Outline

• Chromium spinels and magnetization plateau

• Ising expansion for effective models of quantum fluctuations

• Einstein spin-lattice model

• Constrained phase transitions and exotic criticality

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

TakagiS.H. Lee

Pyrochlore Antiferromagnets

• ManyMany degenerate classical configurations

• Heisenberg

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

-HgCr2O4?

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

• 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M. Matsuda et al, unpublished

• 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.

• S.H. Lee talk: 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?

Spin Wave Expansion

• Henley and co.: lattices of corner-sharing simplexes

kagome, 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• 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

Form of effective Hamiltonian

• The leading diagonal term assigns energy Ea(s) to plaquette “type” a: the same for any such lattice at any applicable M

kagome,pyrochlore checkerboard

• The leading off-diagonal term also depends only on plaquette size and s. It becomes very high order for large s.

• Energies are a little complicated

e.g. hexagonal plaquettes:

Some results

• Checkerboard lattice at M=1/2:- “columnar” state for all s.

• Kagome lattice at M=1/3:- state for s>1

Pyrochlore plateau case

State

Diagonal term:

+ +

Extrapolated V -2.3K

Dominant?

• 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)

for s=3/2

Resolution of spin wave degeneracy

• 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>3/2 has 7-fold enlargement of unit cell and trigonal symmetry

Just minority sites shown in one magnetic unit cell

Quantum Dimer Model

+ +

• Expected T=0 phase diagram (various arguments)

0 1Rokhsar-Kivelson Point

U(1) spin liquid

Maximally “resonatable” R state

“frozen” state

-2.3

on diamond lattice

• Interesting phase transition between R state and spin liquid! Will return to this.

Quantum dimer model is expected to yield the R state structure

S=1

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?

• Probably not:– Quantum ordering scale » |V| » 0.02J

– 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 al

Considered 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*:

• “bending rule”

Bending Rule States

• At 1/2 magnetization, only the R state satisfies the bending rule globally

- Einstein model predicts R state!

• Zero field classical spin-lattice ground states?

• collinear states with bending rule satisfied for both polarizations

• ground state remains degenerate

Consistent with different zero field ground states for A=Zn,Cd,Hg Simplest “bending rule” state (weakly perturbed by DM) appears to be consistent with CdCr2O4 Chern et al, cond-mat/0606039

SH Lee talk

Constrained Phase Transitions• Schematic phase diagram:

0 1U(1) spin liquid

R state

T

“frozen” stateClassical spin liquid

Classical (thermal) phase transition

Magnetization plateau develops

T 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?