Order and disorder in dilute dipolar magnets

Moshe Schechter (BGU)

Helmut Katzgraber (Texas A&M)

Juan Carlos Andresen (KTH, Sweden)

Creighton Thomas (Google)

Vadim Oganesyan (CUNY)

Nicolas Laflorencie (Toulouse)

Philip Stamp (UBC)

Introduction

zj

ziij

ijJ H

1 .Moderate dilutions: Competing interactions with FM mean – disordering by random fields different

from simple mechanism, different from Imry-Ma

2 .Extreme dilution in dipolar Ising – is there SG phase?

Relation to experiments in LiHoxY1-xF4

Dilute dipolar Ising model

J. C. Andresen, C. Thomas, H. Katzgraber, M. S., PRL 111, 177202 (2013)

J. C. Andresen, H. Katzgraber, V. Oganesyan, M. S., PRX 4, 041016 (2014)

Outline

• Standard Imry-Ma

• LiHo - competing interactions and random fields

• Experimental results

• Disordering of FM with competing interactions

Competing interactions:

Dipolar glass in dilute regime:

• Overview

• Results

RFIM and Imry Ma

zj

ziij

J H

Flip a droplet

Energy gain:

Energy cost:

Large droplets flip – FM phase disordered!

Lower critical dimension - two

Imry and Ma, PRL 35, 1399 (1975)

12/ dd

1dJL2/dhL

d≤2 : infinitesimal random field, large FM domains

zii

ih

d>2 : disordering at h≈J, single spins reorient

Competing interactionszj

ziij

ijJ H

σ – standard deviation. Mean=1

Competing interactionszj

ziij

ijJ H

σ – standard deviation. Mean=1

zii

ih

Competing interactionszj

ziij

ijJ H

σ – standard deviation. Mean=1

FM d>2

FM d≤2C.I. d=3

hcJ≈0?

mechanismSingle spins

FM domains

?

zii

ih

Competing interactionszj

ziij

ijJ H

σ – standard deviation. Mean=1

FM d>2

FM d≤2C.I. d=3

hcJ≈0?

mechanismSingle spins

FM domains

?

zii

ih

LiHoF4 - Dipolar Ising model

zj

zi

ijjiJHIs

SSV jiij

ijHH cfD

iSD zi2

cfH

S0

-S

LiHoF4 with hyperfine interactions

iSD zi2

cfH

S0

-S

LiHo

27,a

27,a

Hyperfine spacing: 200 mK

K100

LiHoxY1-xF4 - Continuous dilution

LiHoxY1-xF4

zj

ziij

ijJ H

σ – standard deviation. Mean=1

LiHoF4 - Transverse field Ising model

i

xi

zj

zi

ijjiJ HIs

i

xiji

ijij SHSSV xcfD HH

iSD zi2

cfH

S0

-S

QPT in dipolar magnets

Bitko, Rosenbaum, Aeppli PRL 77, 940 (1996)

Thermal and quantum transitions

MF of TFIM

MF with hyperfine

zj

ziij

ijJ H i

xi

Ferromagnetic RFIM

S -S

M. S. and N. Laflorencie, PRL 97, 137204 (2006)

M. S., PRB 77, 020401(R) (2008)

SSVSD jiij

iji

zi

2

DH

i

xix SH

i

ziSth )(||

i

xiz

jziij

ijJ H zii

ih i

zitH )(

Ferromagnetic RFIM

S -S

SSVSD jiij

iji

zi

2

DH

i

xix SH

i

ziSth )(||

i

xiz

jziij

ijJ H zii

ih i

zitH )(

j

zxij

xBzi V

HSh

0

2

Ferromagnetic RFIM

S0

-S

H xS2

Hh x

i

xiz

jziij

ijJ H zii

ih i

zitH )(

1x

- Independently tunable random and transverse fields!- Classical RFIM despite applied transverse field

j

zxij

xBzi V

HSh

0

2

Imry Ma for SG – correlation length

2/2/)1( dd

Flip a droplet –

gain vs. cost:

Fisher and Huse PRL 56, 1601 (1986); PRB 38, 386 (1988)

Lower critical dimension – infinity!

Droplet size –

Correlation length

Imry and Ma, PRL 35, 1399 (1975)

)2/3/(1)/( hJ

JLhL 2/3

Dilution: quantum spin-glass

-Thermal vs. Quantum disorder-Cusp diminishes as T lowered

Wu, Bitko, Rosenbaum, Aeppli, PRL 71, 1919 (1993)

VTc

Vc

VTc

SG unstable to transverse field !

Finite, transverse field dependent correlation length

SG

quasi

M. S. and N. Laflorencie, PRL 97, 137204 (2006)

Young, Katzgraber, PRL 93, 207203 (2004)

i

zxij

xBzj V

HSh

0

2

)2/3/(1)/(0

xBH

Disordering of FM at x=0.44

Silevitch et al., Nature 448, 567 (2007)

Sharp transition at high T, Rounding at low T (high transverse fields)

Decrease of critical Temperature with random Field (roughly) linear

FM and SG phases in random field

)3/()2/3()3/()2/3()]([ Jxxhcc

FMSGcEExxhJhh )()/(/ )2/3/(2/332/3

FM and SG phases in random field

)()/( )2/3/(2/3cxxhJh

)3/()2/3()3/()2/3()]([ Jxxhcc

Disordering of the FM phase in 3D by finite, yet SMALL random field Jh

c

Disordered phase: SG domains of size )2/3/(1)/( hJ

Finite temperature

Sharp transition at high T, Rounding at low T (high transverse fields)

Decrease of critical Temperature with random Field (roughly) linear

)2/3/(1)/( hJSilevitch et al., Nature 448, 567 (2007)Andresen et. al., PRL 111, 177202 (2013)

Finite temperature

Finite temperature

T>0.3K : FM to PM transition

Andresen et. al., PRL 111, 177202 (2013)

T<0.3K : Intermediate frozen QSG phase

Silevitch et al., Nature 448, 567 (2007)

Finite temperature“Standard”PM Glassy

domains

Conclusions 1• Generalized Imry-Ma: 3D FM with competing

interactions – disordering at small random field• Disordered phase: spin glass domains with typical

size depending on random field• At low temperatures disordered phase is frozen,

explains rounding off of the susceptibility cusp• Critical temperature linear with field down to small

fields

Dilute Dipolar Glass

Reich et al, PRB 42, 4631 (1990)

Dilute Dipolar Glass

Ghosh, Parthasarathy, Rosenbaum, Aeppli Science 296, 2195 (2002)

Quilliam, Meng, Kycia,

PRB 85, 184415 (2012)

Dilute Dipolar Glass

Experiment: Anti-glass? Spin liquid?

Analytics

• Mean field theory (Stephen and Aharony): Tc linear in x• Fluctuations are large, could dominate• RG ineffective

Numerical

• Fluctuations increase with dilution• Snider and Yu, Biltmo and Henelius – no glass phase • Tam and Gingras – Glass phase down to 6.25%

Finite size scaling

J. C. Andresen, H. Katzgraber, V. Oganesyan, M. S., PRX 4, 041016 (2014)

• Parallel tempering Monte Carlo

• Combine single spin flip with cluster renormalization algorithm

Strong fluctuations

Algorithm: 1. cluster all spins with

Algorithm efficiency concentration independent!! Size limited by Ewald sums

.

for all n until left only with pairs.2. Repeat for

3. Sweep spins, flip: single spin - 75%, random cluster – 25%

Strong fluctuations coming from nearby spins do not effect thermodynamic transition, nor efficiency of algorithm

Cluster strongly interacting spins, standard MC for resulting entities

Can not use one method for strongly interacting and typical spins.

Tc linear in dipole concentration

Tc=ax; a=0.59(1)

LiHoxY1-xF4 - phase diagram

1 .FM-SG boundary at x=0.3. No significant reentrant SG regime

2 .SG phase down to x=0, with linear Tc

Dilute dipolar glass - broad distribution of random fields

Assume: 1n;00 Jh -Random field dictated by rare events of nearest neighbor impurities.

-Interactions are dictated by typical strength.

1/ 00 nJh

Typical random field:)2/3/(1)/( hJ

)2/3/(12/1 )/( hJn

)2/3(2/1)/( hJ00 / Jhn

M.S. and P. Stamp, EPL 88, 66002 (2009)

Conclusions 2• Dilute dipolar Ising spins order at any small

concentration with Tc linear in concentration• Induced random field – non trivial dependence of

domain size on concentration

Dipolar glasses – scaling with dilution

Interaction allows diliution – expect scaling ),( 00 nhnJf