Quantum phase transitions Quantum phase transitions in anisotropic dipolar in anisotropic dipolar

magnetsmagnets

In collaboration with: Philip Stamp, Nicolas laflorencie

Moshe Schechter

University of British Columbia

LiHoY FLiHoY Fx 1-x 4

1. Transverse field Ising model:zj

ziij

ijJ H i

xi

LiHoY FLiHoY Fx 1-x 4

Reich et al, PRB 42, 4631 (1990)

1. Transverse field Ising model:zj

ziij

ijJ H i

xi

2. Dilution!

QPT in dipolar magnetsQPT 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

Ronnow et. Al. Science 308, 389 (2005)

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

Brooke, Bitko, Rosenbaum, Aeppli Science 284, 779 (1999)

Giraud et. Al. PRL 87, 057203 (2001)

Various dilutionsVarious dilutions

LiHoF - a model quantum LiHoF - a model quantum magnetmagnet4

S. Sachdev, Physics World 12, 33 (1999)

Dilution: quantum spin-glassDilution: quantum spin-glass

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

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

VTc

Vc

VTc

Fall and rise of QPT in Fall and rise of QPT in dilute dipolar magnetsdilute dipolar magnets

Hyperfine interactions and off-Hyperfine interactions and off-diagonal dipolar termsdiagonal dipolar terms

No QPT in spin-glass regimeNo QPT in spin-glass regime In FM regime can study classical and In FM regime can study classical and

quantum phase transitions with quantum phase transitions with controlled disorder and with coupling controlled disorder and with coupling to spin bathto spin bath

Anisotropic dipolar magnetsAnisotropic dipolar magnets

zj

zi

ijjiJHIs

SSV jiij

ijHH cfD

iSD zi2

cfH

Large spin, strong lattice anisotropy

S0

-S

Anisotropic dipolar magnetsAnisotropic dipolar magnets

zj

zi

ijjiJHIs

SSV jiij

ijHH cfD

iSD zi2

cfH

Large spin, strong lattice anisotropy

S0

-S

Magnetic insulators

Single molecular magnets

Anisotropic dipolar magnets - Anisotropic dipolar magnets - TFIMTFIM

i

xi

zj

zi

ijjiJ HIs

i

xiji

ijij SSSV HH cfD

iSD zi2

cfH

Large spin, strong lattice anisotropy

S0

-S

Hyperfine interaction: electro-Hyperfine interaction: electro-nuclear Ising statesnuclear Ising states

K100 i

xiji

ijij SSSV HH cfLH

i

xi

zj

zi

ijjiJ HIs

2

Hyperfine interaction: electro-Hyperfine interaction: electro-nuclear Ising statesnuclear Ising states

ccS z2221

~

27,a

27,a

27,b

27,b

K100

K4.12 A

Hyperfine spacing: 200 mK

SJJ zeff

~2

i

xiji

ijij SSSV HH cfLH

)( SISIASIA iiii

iJzi

i

ziJ

- M.S. and P. Stamp, PRL 95, 267208 (2005)

2/7I

Phase diagram – transverse Phase diagram – transverse hyperfine and dipolar hyperfine and dipolar

interactionsinteractions

i

xi

zj

zi

ijeffeffJ HIs

J eff

eff

SG

PM

No off. dip.

With off. dip.

Experiment

Splitting

- M.S. and P. Stamp, PRL 95, 267208 (2005)

eff

J eff

VTc

AJ eff ~

0Hc

Anisotropic dipolar systems – Anisotropic dipolar systems – offdiagonal termsoffdiagonal terms

S0

-S

SSVSD zj

zi

ij

zzij

i

zi 2

DH

i

xiS SSV x

izj

ij

zxij

SS zz SS symmetry symmetry

Anisotropic dipolar systems – Anisotropic dipolar systems – offdiagonal termsoffdiagonal terms

S0

-S

SSVSD zj

zi

ij

zzij

i

zi 2

DH

i

xiS SSV x

izj

ij

zxij

SS zz SS symmetry symmetry

i

SVE

zjj

zxij

0

2)(

i

zxij

zj VSh

0

2

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

Imry-Ma argumentImry-Ma argument

LJ d 1 Lh d 2/

Ground state:

2/1 dd hLJL

Domain: '

Energy cost Energy gain

(spins down)

(all spins up)

Spontaneous formation of domains

Critical dimension: 2 (for Heisenberg interaction: 4)

zj

ziij

ijJ H zii

ih

Y. Imry and S. K. Ma, PRL 35, 1399 (1975)

Spin glass – correlation lengthSpin glass – correlation length

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

hL2/3

Energy gain: 0

2/32

LVS

i

zxij

zj VSh

0

2 X0

2 VSj

Y. Imry and S. K. Ma, PRL 35, 1399 (1975)

Spin glass – correlation lengthSpin glass – correlation length

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

JL

hL2/3

Energy cost:

Energy gain:

LVS 20

2/32

LVS

i

zxij

zj VSh

0

2 X0

2 VSj

Y. Imry and S. K. Ma, PRL 35, 1399 (1975)

Spin glass – correlation lengthSpin glass – correlation length

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

Fisher, Huse PRL 56, 1601 (86); PRB 38, 386 (88)

JL

hL2/3

Energy cost:

Energy gain:

LVS 20

2/32

LVS

2/2/)1( dd Only extra sqrt of surface bonds are satisfied, can optimize boundary.

i

zxij

zj VSh

0

2 X0

2 VSj

Spin glass – correlation lengthSpin glass – correlation length

LVSLVS 2

0

2/32

Flip a droplet – gain vs. cost:

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

Fisher, Huse PRL 56, 1601 (86); PRB 38, 386 (88)

JL

hL2/3

Energy cost:

Energy gain:

LVS 20

2/32

LVS

2/2/)1( dd Only extra sqrt of surface bonds are satisfied, can optimize boundary.

LJhL2/3

i

zxij

zj VSh

0

2 X0

2 VSj

Spin glass – correlation lengthSpin glass – correlation length

LVSLVS 2

0

2/32

Flip a droplet – gain vs. cost:

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

Fisher, Huse PRL 56, 1601 (86); PRB 38, 386 (88)

JL

hL2/3

Energy cost:

Energy gain:

LVS 20

2/32

LVS

2/2/)1( dd Only extra sqrt of surface bonds are satisfied, can optimize boundary.

LJhL2/3

i

zxij

zj VSh

0

2 X0

2 VSj

Droplet size –

Correlation length)2/3/(1)/( hJ )2/3/(1)/(

0

SG unstable to transverse SG unstable to transverse field!field!

Finite, transverse field dependent correlation length

SG

quasi

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

Enhanced transverse field – Enhanced transverse field – phase diagramphase diagram

eff

SG

PM

No off. dip.

With off. dip.

Experiment

V||

V||

M.S. and P. Stamp, PRL 95, 267208 (2005)

Quantum disordering harder than thermal disordering

Main reason – hyperfine interactions

Off-diagonal dipolar terms in transverse field – also enhanced effective transverse field

i

SVE

zjj

zxij

0

2)(

i

zxij

zj VSh

0

2

Random fields not particular to Random fields not particular to SG!SG!

Reich et al, PRB 42, 4631 (1990)

Interest in FM RFIMInterest in FM RFIMzj

ziij

ijJ H zii

ih

Diluted anti-ferromagnets:

- Equivalence only near transition

- No constant field in the staggered magnetization

- Not FM - applications

Interest in FM RFIMInterest in FM RFIM

Verifying interesting results on DAFMVerifying interesting results on DAFM Experimental techniquesExperimental techniques Novel fundamental research (away from Novel fundamental research (away from

transition, conjugate field, quantum transition, conjugate field, quantum term)term)

Applications in ferromagnets, e.g. Applications in ferromagnets, e.g. domain wall dynamics in random fieldsdomain wall dynamics in random fields

i

xiz

jziij

ijJ H zii

ih i

zitH )(

Are the fields random?Are the fields random?

Square of energy gain vs. N, different dilutions

Inset: Slope as Function of dilution

M. S., cond-mat/0611063

i

zxij

zj VSh

0

2 x0

2 VSj

Random field and quantum Random field and quantum term are independently term are independently

tunable!tunable!

S0

-S

M. S. and P. Stamp, PRL 95, 267208 (2005)

M. S., cond-mat/0611063

h

1x

i

zxij

zj VSh

0

2

S2

Ferromagnetic RFIMFerromagnetic RFIM

S0

-S

M. S. and P. Stamp, PRL 95, 267208 (2005)

M. S., cond-mat/0611063

SSVSD jiij

iji

zi

2

DH

i

xiS

i

ziSth )(||

Ferromagnetic RFIMFerromagnetic RFIM

S0

-S

M. S. and P. Stamp, PRL 95, 267208 (2005)

M. S., cond-mat/0611063

S2h

SSVSD jiij

iji

zi

2

DH

i

xiS

i

ziSth )(||

i

xiz

jziij

ijJ H zii

ih i

zitH )(

1x

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

Realization of FM RFIMRealization of FM RFIM

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

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

ConclusionsConclusions

Strong hyperfine interactions in LiHo result in Strong hyperfine interactions in LiHo result in electro-nuclear Ising states. Dictates quantum electro-nuclear Ising states. Dictates quantum dynamics and phase diagram in various dynamics and phase diagram in various dilutionsdilutions

Ising model with tunable quantum and random Ising model with tunable quantum and random effective fields can be realized in anisotropic effective fields can be realized in anisotropic dipolar systemsdipolar systems

SG unstable to transverse field, no SG-PM QPTSG unstable to transverse field, no SG-PM QPT First FM RFIM – implications to fundamental First FM RFIM – implications to fundamental

research and applicationsresearch and applications