Coupled Cluster Theories of Quantum Magnetismobelix.physik.uni-bielefeld.de/~schnack/conferences/WEH-504/Materi… · Coupled Cluster Theories of Quantum Magnetism 2 Outline of Talk

4/25/12CoupledClusterTheoriesofQuantum

Magnetism1

CoupledClusterTheoriesofQuantumMagnetism

DamianJJFarnell

[email protected]


Magnetism2

OutlineofTalk• Averybriefoverviewofthehigh‐orderCCM

• Comparisontoresultsofothermethodsforvarious“Archimedean”lattices

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

• Specificexamplesofmagneticmodels:– Thetriangularlatticeantiferromagnetinanexternalmagneticfield

– Thespin‐halfJ1‐J2Modelonthesquarelattice

• Conclusionsandfutureresearch

CCMOverview–GroundState


Magnetism3

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

4

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

CCMFlowchart


Magnetism5

ArchimedeanLattices


Magnetism6

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

CCMresultsforunfrustratedlatticescomparedtosomeresultsofotherapproximatemethods


Magnetism7

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


Magnetism8

-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))


Magnetism9

J1J2

(a) Néel(b) Spiral(c) Striped


Magnetism10

Triangular‐latticeantiferromagnetsinanexternalmagneticfield

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


Magnetism11

Background

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

• QuantumspinsinteractviaanantiferromagneticHeisenbergexchanceinteraction.

• Aninterestingeffectof“spinplateau”isseenasoneimposesanexternalmagneticfieldofstrengthλ.

• TherelevantHamiltonianisgivenby:

– H=J∑s∙s+λ∑s z

Triangular lattice forBa3CoSb2O9 and Ba3NiSb2O9


Magnetism12

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

↑ ↑ ↓


Magnetism13

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

J2

J1!

H = J1

s i " s j + J2

s i " s ki,k

N

#i, j

N

#


Magnetism14

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

Li2VOSiO4andLi2VOGeO4–J2/J1inrange5to10.Seelater…

• Itisa“canonicalmodel”usedtoinvestigatequantummagneticsystems,especiallyforcaseofstrongfrustration

• Non‐magneticquantumphase(quantumparamagnet)

– DoesthisphasesurviveforT>0?

• Thenatureofquantumphaseisstillnotfullyresolved

• Possibledeconfinedquantumcriticalityatquantumphasetransitions

• Noexactsolutionsforthis2Dmodel–itisagoodtestofapproximatetechniques.


Magnetism15

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


Magnetism16

OrderParameters

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


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


Magnetism17

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

indicatesstabilityofgroundstate

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

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



Magnetism18

ExcitationEnergyGap

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


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

Higher LSUBm please!!

InitialResultsforthe“QuantumFidelity”


Magnetism19

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

)//()/(

12121221212

121212

JJJJJJJJJJJ

JJJJJJF

!!

!

+"+"""

+""= δ=0.001 above


Magnetism20

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


Magnetism21

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



Magnetism22

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)


Magnetism23

Natureoftheparamagneticregime

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


Magnetism24

“Inverse”MagneticSusceptibilitiesSeries: Sirker PRB 73,

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

214415 (2008)

Taken from Sushkov, PRB 63,104420 (2001)

?


Magnetism25

ResultsforJ1<0

CCM

ED

Richter et al. PRB 81, 174429 (2010)


Magnetism26

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)


Magnetism27

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.


Magnetism28

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.


Magnetism29

Coupled Cluster Theories of Quantum Magnetismobelix.physik.uni-bielefeld.de/~schnack/conferences/WEH-504/Materi… · Coupled Cluster Theories of Quantum Magnetism 2 Outline of Talk

Documents