coupled cluster theories of quantum...

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CoupledClusterTheoriesofQuantumMagnetism

DamianJJFarnell

Dfarnell@glam.ac.uk

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OutlineofTalk• Averybriefoverviewofthehigh‐orderCCM

• Comparisontoresultsofothermethodsforvarious“Archimedean”lattices

• “Spiral”modelstates:exampleforananisotropicsquare(“interpolating”square/triangular)model

• Specificexamplesofmagneticmodels:– Thetriangularlatticeantiferromagnetinanexternalmagneticfield

– Thespin‐halfJ1‐J2Modelonthesquarelattice

• Conclusionsandfutureresearch

CCMOverview–GroundState

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SSeSe!"#=$#%"=%$

~~;

!! "++==

IIIICSSCSS~

1~

;

0|],[~|0||~ =!!"

#

#=!!"

#

# +$$$ S

I

s

I

Ss

I

I

eCHeSS

HHeeC

S

H

!!= ||~HH!""#=$ % SS

g HeeEES ..

A similar approach exists for the excited state – See Parkinson and Farnell,“An Introduction to Quantum Spin Systems” LNP 816 Chapter 10.

ApproximationSchemes• SUBm:Allm‐bodyorlower‐orderclusters.

• LSUBm:Allclustersinalocaledefinedbym.

• SUBn‐m:Alln‐bodyorlower‐orderclustersinalocaledefinedbym.

Weextrapolatetheseresultsinthelimit,m→∞.4/25/12

CoupledClusterTheoriesofQuantumMagnetism

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E.g., SUB4-4 for thes=1/2 and s=1square-latticeferrimagnet

CCMFlowchart

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ArchimedeanLattices

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(a) Square(b) Honeycomb(c) CAVO(d) SrCuBO(e) Triangular(f) Maple-Leaf(g) Kagome

CCMresultsforunfrustratedlatticescomparedtosomeresultsofotherapproximatemethods

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0.212 (linear)

-0.3584 (linear)

0

0.2418 (2nd)

-0.365929 (2nd)

0.8043

-1.1641

2.316 (1st order)

0.06426(1)

0.175

0

0.3037

-0.335233

SWT

-1.1646-1.1640(1)EG/bond

2.335 (SUB2)2.28(1)SW Velocity

Square HAFs=1/2

0.020.0504Gap

Model Parameter QMC Series ED CCM

Square HAFs=1

EG/bond -0.334719(3) -0.33465(5) -0.335 -0.3348

Order Param <sz> 0.3070(3) 0.307(1) 0.3173 0.307

Gap 0 0 0.0247 -0.001

Spin Stiffness 0.175(2) 0.182 0.183 0.181

Mag. Suscept. χ 0.0669(7) 0.0659(10) 0.0700(6).

Order Param <sz> 0.8039(4) 0.8049

HoneycombHAF s=1/2

EG/bond -0.363 -0.3629 -0.3632 -0.363155

Order Param <sz> 0.235 0.266 0.2788 0.274066

Spin Stiffness 0.1379

CAVO HAFs=1/2

EG/bond -0.363 -0.3629 -0.3682 -0.36888

Order Param <sz> 0.178 0.2303 0.204

CCMresultsforfrustratedlatticescomparedtosomeresultsofotherapproximatemethods

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-0.2353*

0.203†

-0.231†

0.0794*

0.08*

0.2387*

-0.154*

SWT* (1st order)/Swinger Boson†

< 0~ 0Order Param <sz>

Model Parameter Series ED CCM

Triangle

EG/bond -0.1842 -0.1842 -0.18147

Order Param <sz> 0.20 0.193 0.189331

Spin Stiffness 0.05 0.0564Mag. Suscept. χ 0.065(23)

SrCuBOEG/bond -0.231 -0.2310 -0.2311

Order Param <sz> 0.200 0.2280 0.211

KagomeEG/bond -0.2172 -0.2126

Spin Stiffness -0.023

See “Quantum Magnetism” LNP 645 (chapter 2) and “An Introduction toQuantum Spin Systems” LNP 816 (chapters 10 & 11) for more details about allof these approximate calculations. See also, “Series Expansions Methods for

… Lattice Models” by Oitmaa et al. (Cambridge university Press, 2006).

“Spiral”modelstates:theinterpolatingsquare/triangularlatticemodel(AKA“anisotropicsquare”)

(Bishop,Li,etal.,PRB79,174405(2009))

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J1J2

(a) Néel(b) Spiral(c) Striped

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Triangular‐latticeantiferromagnetsinanexternalmagneticfield

Co2+ = spin-1/2 atoms Ni2+ = spin-1 atoms

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Background

• Co2+atomsinBa3CoSb2O9andNi2+atomsinBa3NiSb2O9forms=1/2ands=1atriangularlattice.

• QuantumspinsinteractviaanantiferromagneticHeisenbergexchanceinteraction.

• Aninterestingeffectof“spinplateau”isseenasoneimposesanexternalmagneticfieldofstrengthλ.

• TherelevantHamiltonianisgivenby:

– H=J∑s∙s+λ∑s z

Triangular lattice forBa3CoSb2O9 and Ba3NiSb2O9

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SpinPlateau(see:Farnell,Richter,&Zinke,J.Phys.:Condens.Matt.21,406002(2009))

Experiment

CCM

Co2+ = spin-1/2 atomsShirata, Tanaka et al. PRL 108,

057205 (2012). arXiv:1110.3889v1

Ni2+ = spin-1 atomsShirata, Tanaka et al. J. Phys.Soc. Japan 80 (2011) 093702

↑ ↑ ↓

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TheJ1‐J2ModelontheSquareLattice• NearestandNext‐NearestNeighbourBondsonthesquarelattice;<…>indicateseachbondcountedonce.

J2

J1!

H = J1

s i " s j + J2

s i " s ki,k

N

#i, j

N

#

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BackgroundtotheJ1‐J2model• Variousquasi‐2DmaterialsaredescribedbytheJ1‐J2model,e.g.,

Li2VOSiO4andLi2VOGeO4–J2/J1inrange5to10.Seelater…

• Itisa“canonicalmodel”usedtoinvestigatequantummagneticsystems,especiallyforcaseofstrongfrustration

• Non‐magneticquantumphase(quantumparamagnet)

– DoesthisphasesurviveforT>0?

• Thenatureofquantumphaseisstillnotfullyresolved

• Possibledeconfinedquantumcriticalityatquantumphasetransitions

• Noexactsolutionsforthis2Dmodel–itisagoodtestofapproximatetechniques.

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Ground‐StateEnergy

CCM. R. Darradiet al. PRB 78,214415 (2008)

ED: N=40. Richter &Schulenburg. Eur. Phys.J. B 73, 117–124 (2010)

Series Expansions, Sirker et al., PRB B 73, 184420 2006

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OrderParameters

CCM. R. Darradi et al.PRB 78, 214415 (2008)

ED: N=40. Richter &Schulenburg. Eur. Phys.J. B 73, 117–124 (2010)

Series Expansions,Oitmaa et al. PRB 54 3022 (1996)

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The“SpinStiffness”‐theresponsetoa“twist”inlocalaxesofspins,whichso

indicatesstabilityofgroundstate

CCM. R. Darradi et al.PRB 78, 214415 (2008)

ED + LSWT, Einarsson,PRB 51, 6151 (1995)

ED: N=40. Richter &Schulenburg. Eur. Phys.J. B 73, 117–124 (2010)

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ExcitationEnergyGap

CCM: DJJF Series: Kotov et al.PRB 60 14 613 (1999)

ED: N=40. Richter &Schulenburg. Eur. Phys.J. B 73, 117–124 (2010)

Gap opens at J2/1≈0.5. But is thisresult too large...

Higher LSUBm please!!

InitialResultsforthe“QuantumFidelity”

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)//()/()/()/(

)//()/(

12121221212

121212

JJJJJJJJJJJ

JJJJJJF

!!

!

+"+"""

+""= δ=0.001 above

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SignRuleforSmallJ2/J1

( ) I

1

1 c 0

Sites on A go over one sublattice

A

A

A

A

NM z

I A i

I i

c I M s I

=

! = " = # $

%%%%%%%

& &B A B

A B A

A B A

“Hard” sign rule: rule above can be proven exactly at J2=0 by noting that theabove formulation results in a negative definite matrix formulation of theSchrodinger equation and so for which cI>0.

“Soft” sign rule: rule can’t be proven exactly (often due to frustration),although it is seen up to some value of J2 numerically via ED and CCM results.

Weight of states: Sum over states I for which the rule holds such that:weight=ΣI |cI|2

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Marshall‐PeierlsSignRule

2

1

MN

I

i

weight c!

=

="2 2

1 1

/

fMNN

I I

i i

weight c c= =

=! !

CCM: DJJF SWT & ED: Ivanov& richter, J. Phys.:Condens. Matter 6(1994) 3785-3792

ED: N=40. Richter &Schulenburg. Eur. Phys.J. B 73, 117–124 (2010)

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RangeofParamagneticRegime

Method J2M J2c1 J2c2

ED1 0.2‐0.3 0.35 0.66

Series2 ‐‐ 0.38 0.60

DimerBosonModel3 ‐‐ 0.38 0.62

CCM4 0.2‐0.34 0.44(±0.01)5 0.59(±0.01)5

1Richter and J. Schulenburg, Eur. Phys. J. B 73, 117–124 (2010)2Sushkov et al. PRB 63, 104420 (2001)

3Kotov et al. PRB 60 14 613 (1999)4Bishop, Farnell, Parkinson PRB 58 6394 (1998).

5Darradi et al. PRB 78, 214415 (2008)

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Natureoftheparamagneticregime

Sushkov et al. PRB 63, 104420 (2001).

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“Inverse”MagneticSusceptibilitiesSeries: Sirker PRB 73,

184420 (2006)CCM + ED: Darradi, PRB 78,

214415 (2008)

Taken from Sushkov, PRB 63,104420 (2001)

?

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ResultsforJ1<0

CCM

ED

Richter et al. PRB 81, 174429 (2010)

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Spin‐SpinCorrelationFunctions

J1<0:Results(thickerlines)indicatesinglePTpointatJ2≈‐0.4J1(probablyfirst‐order)J1>0:Results(thinnerlines)indicatePTpointsatJ2≈0.6J1(probablyfirst‐order)

CCM ED

See: Richter et al. PRB 81, 174429 (2010)

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QuantumPhaseDiagramatT=0

J2

J2c1 ≈ 0.4J1

J2 ≈ -0.4J1

GaplessGaplessJ1

Gapless

J2c2

≈ 0.6J1

Gapped

J2M ≈ 0.2J1 to 0.3J1

Phase diagram from R. Nath, A.Tsirlin, H. Rosner, and C. Geibel,Phys. Rev. B 78, 064422 (2008).

Phase diagram from a synthesisall of the results presented here.

Conclusions• TheCCMprovidesausefultoolintackling(especially2D)problemsinquantummagnetism.

• Itisaccurate,reliable,andflexible.• Canbeusedtodetectphasetransitionpointpoints,andpossiblyeventheirorder.

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AfterwordandThanks• Final words in The Theory Of Magnetism Made Simple by DC

Mattis:

– “Icannotresistpointingouttheobvious: independentlyofthetechnicalimportance of magnetic substances – which cannot be denied – thetheory of magnetism continues to lie at the core of practically all ofcontemporary theoretical physics. Its study continues tobe aperpetualdelightandinspiration.”

• Thanksto:WE‐HereausStiftung.AllatthePhysikszentrum,BadHonnef.

• Also to: Prof. Raymond F. Bishop (Manchester), Prof. Johannes Richter(Magdeburg, Germany), Dr. Peggy Li (Manchester), Dr. John B. Parkinson(Manchester), Dr. Joerg Schulenburg (Magdeburg), Dr. Sven Krueger(Magdeburg),Dr.KlausGernoth(Manchester),Dr.RonaldZinke(Magdeburg),Dr. Rachid Darradi (Technische Universität Braunschweig, Germany), andProf.CharlesCampbell(Minnesota,USA),et(manymore)al.

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