Doron Bergman et al- Degeneracy Breaking in Some Frustrated Magnets

8/3/2019 Doron Bergman et al- Degeneracy Breaking in Some Frustrated Magnets

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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


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Outline

Chromium spinels and magnetization

plateau

Ising expansion for effective models ofquantum fluctuations

Einstein spin-lattice model

Constrained phase transitions and exoticcriticality


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Chromium Spinels

ACr2O4

(A=Zn,Cd,Hg)

spin s=3/2

no orbital degeneracy

isotropic

Spins form pyrochlore lattice

cubic Fd3m

Antiferromagnetic interactions

5CW = -390K,-70K,-32K

for A=Zn,Cd,Hg

Takagi

S.H. Lee


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Pyrochlore Antiferromagnets

ManyMany degenerate

classical configurations

Heisenberg

Zero field experiments (neutron scattering)

-Different ordered states in ZnCr2O4, CdCr2O4-HgCr2O4?

c.f. 5CW = -390K,-70K,-32Kfor A=Zn,Cd,Hg

Evidently small differences in interactions

determine ordering


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Magnetization Process

Magnetically isotropic

Low field ordered state complicated, material dependent

Plateau at half saturation magnetization

H. Ueda et al, 2005


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HgCr2O4neutrons

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unpublished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?


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Collinear Spins

Half-polarization = 3 up, 1 down spin?

- Presence ofplateau 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


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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?


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Spin Wave Expansion

Henley and co.: lattices of corner-sharing simplexes

kagome, checkerboard

pyrochlore

Quantum zero point energy of magnons:

- O(s) correction to energy:

- favors collinear states:

- Magnetization plateaus: kdown 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): Xp=+1


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Ising Expansion

XXZ model:

Ising model (JB =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


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Form of effective Hamiltonian

The leading diagonal term assigns energy Ea(s) to plaquettetype 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:


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Some results

Checkerboard lattice at M=1/2:

- columnar state for all s.

Kagome lattice at M=1/3:

- state for s>1


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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


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Resolution of spin wave

degeneracy Truncating Heffto O(s) reproduces exactlyspin

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


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Quantum Dimer Model

+ +

Expected T=0 phase diagram (various arguments)

0 1Rokhsar-Kivelson Point

U(1) spin

liquid

Maximally

resonatable R statefrozen 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


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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.


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Is this the physics of HgCr2O4?

Probably not:

Quantum ordering scale |V| 0.02J

Actual order observed at T & Tplateau/2 We should reconsiderclassicaldegeneracy

breaking by

Further neighbor couplings

Spin-lattice interactions

C.f. spin Jahn-Teller: Yamashita+K.Ueda;Tchernyshyov et al

Considered identicaldistortions of each tetrahedral molecule

We would prefer a model thatpredicts the periodicity of the distortion


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Einstein Model

Site

phonon

vector from i to j

Optimal distortion:

Lowest energy state maximizes u*:

bending

rule


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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 differentzero field ground states for A=Zn,Cd,Hg

Simplest bending rule state (weakly perturbed by DM) appears to be

consistent with CdCr2

O4Chern et al, cond-mat/0606039

SH Lee talk


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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} 5CW

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


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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


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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.


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Numerics (courtesy S. Trebst)

Specific heat

C

T/V

Crossings


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ConclusionsConclusions

We derived a general theory of quantumfluctuations around Ising states in corner-sharingsimplex lattices

Spin-lattice coupling probably is dominant in

HgCr2O4, and a simple Einstein model predicts aunique and definite state (R state), consistent withexperiment Probably spin-lattice coupling plays a key role in

numerous other chromium spinels of current interest

(possible multiferroics). Local constraints can lead to exoticcritical behavior

even at classical thermal phase transitions. Experimental realization needed! Ordering in spin ice?

Doron Bergman et al- Degeneracy Breaking in Some Frustrated Magnets

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