The density matrix renormalization group · 2018-02-08 · The density-matrix renormaliza1on group, Rev. Mod. Phys. 77, 259 (2005). -Karen Hallberg . Density Matrix Renormaliza1on:

Thedensitymatrixrenormaliza2ongroup

Adrian Feiguin

Sourcecode(PythonandC++)

h3p://www.github.com/afeiguin/h3p://www.github.com/afeiguin/comp-phys

ALPSlibrariesandcode:h3p://alps.comp-phys.org

Theselectures:

Someliterature-S.R.White:.Densitymatrixformula1onforquantumrenormaliza1ongroups,Phys.Rev.Le3.69,2863(1992)..Density-matrixalgorithmsforquantumrenormaliza1ongroups,Phys.Rev.B48,10345(1993).-U.Schollwöck.Thedensity-matrixrenormaliza1ongroup,Rev.Mod.Phys.77,259(2005).-KarenHallberg.DensityMatrixRenormaliza1on:AReviewoftheMethodanditsApplica1onsinTheoreYcalMethodsforStronglyCorrelatedElectrons,CRMSeriesinMathemaYcalPhysics,DavidSenechal,Andre-MarieTremblayandClaudeBourbonnais(eds.),Springer,NewYork,2003.NewTrendsinDensityMatrixRenormaliza1on,Adv.Phys.55,477(2006).-The“DMRGBOOK”:Density-MatrixRenormaliza1on-ANewNumericalMethodinPhysics:LecturesofaSeminarandWorkshopheldattheMax-Planck-Ins1tutfürPhysik...18th,1998(LectureNotesinPhysics)byIngoPeschel,XiaoqunWang,Ma3hiasKaulkeandKarenHallberg-R.NoackandS.Manmana:Diagonaliza1on-andNumericalRenormaliza1on-Group-BasedMethodsforInterac1ngQuantumSystemsProceedingsofthe"IX.TrainingCourseinthePhysicsofCorrelatedElectronSystemsandHigh-TcSuperconductors",VietrisulMare(Salerno,Italy,October2004)AIPConf.Proc.789,93-163(2005)-A.E.Feiguin:TheDensityMatrixRenormaliza1onGroupandits1me-dependentvariantsVietriLectureNotes,AIPConferenceProceedings1419,5(2011);h3ps://doi.org/10.1063/1.3667323-F.Verstraete,V.Murg&J.I.Cirac:Matrixproductstates,projectedentangledpairstates,andvariaYonalrenormalizaYongroupmethodsforquantumspinsystems,AdvancesinPhysics,57:2,143-224(2008)-U.Schollwock,Thedensity-matricrenormalizaYongroupintheageofmatricproductstates,AnnalsofPhysics326,96(2011)-FrankPollman’slecturenotes:EfficientNumericalSimulaYonsUsingMatrix-ProductStates

-RomanOrus:ApracYcalintroducYontotensornetworks:Matrixproductstatesandprojectedentangledpairstates,AnnalsofPhysics349(2014)117-158(2014)

Briefhistoryandmilestones•  (1992)SteveWhiteintroducestheDMRG.•  (1995-…)DynamicalDMRG.(Hallberg,Ramaseshaetal,KuhnerandWhite,

Jeckelmann)•  (1995)Nishinointroducesthetransfer-matrixDMRG(TMRG)forclassicalsystems.•  (1996-97)Bursill,WangandXiang,Shibata,generalizetheTMRGtoquantum

problems.•  (1996)XiangadaptsDMRGtomomentumspace.•  (2001)ShibataandYoshiokastudyFQHsystems.•  (2004)VidalintroducestheTEBD.(Yme-evolvingblockdecimaYon)•  (2005)VerstraeteandCiracintroduceanalternaYvealgorithmforMPS’sand

explainproblemwithDMRGandPBC.•  (2006)WhiteandAEF,andDaley,Kollath,Schollwoeck,Vidalgeneralizetheideas

withinaDMRGframework:adapYvetDMRG.

…theDMRGhasbeenusedinavarietyoffieldsandcontexts,fromclassicalsystemstoquantumchemistry,tonuclearphysics…

Singlepar2clevs.manybodypicture

4 configurations 4x4 matrix

EF

The state of the system is a “product state” of single particle states. We only need to solve the one-particle

problem.Think “Hydrogen atom”.

And similarly for the “down” electrons

Singlepar2clevs.manybodypicture

16 configurations 16x16 matrix

The state of the system is “highly entangled”. It cannot be written as a “product state”, and the behavior of each electron is dictated by the behavior of the rest.(Notice that in some case Bethe Ansatz tells us that some many-body states can still be reduced to product states)

Exact diagonalization “brute force” diagonalization of the Hamiltonian matrix.

…anythingyouwanttoknow…but…onlysmallsystems

H |x〉 = E |x〉 H : Hamiltonian operator |x〉 : eigenstate E : eigenvalue (ENERGY)

Schrödinger's Equation:

AllweneedtodoistopickabasisandwritetheHamiltonianmatrixinthatbasis

Symmetries SH=HS

Reflections

Translations

D' = D / N

D' = D / 2

Particle number conservation => Ntotal Spin conservations => Sz

total

Spin reversal => |↑↓〉 ± |↓↑〉

|ψk〉 = (1/M) ∑iaki Ti |φ〉; aki =exp(ikxi)

Block diagonalization

0

0 0

0

EDExample:Heisenbergchain

Geometry:1D chain

Basis:

HHeis = J ∑<i,j> SizSj

z+ 1/2 (Si-Sj

++Si+Sj

-)

ModelHamiltonian:

|↑↓↑↓〉; |↓↑↓↑〉; |↑↑↓↓〉; |↓↑↑↓〉; |↓↓↑↑〉; |↑↓↓↑〉

ApplyingtranslaYons:

|1〉=1/(2√2){(1+ ei2k )|↑↓↑↓〉+ eik(1+ei2k)|↓↑↓↑〉} |2〉=1/2{|↑↑↓↓〉+eik|↓↑↑↓〉+ei2k|↓↓↑↑〉+ei3k|↑↓↓↑〉}

TranslaYons

Withk=0,-π/2,π/2,π

k=0) |1〉=1/√2{|↑↓↑↓〉+|↓↑↓↑〉} |2〉=1/2{|↑↑↓↓〉+|↓↑↑↓〉+|↓↓↑↑〉+|↑↓↓↑〉}

k=-π/2) |2〉=1/2{|↑↑↓↓〉+e-iπ/2|↓↑↑↓〉-|↓↓↑↑〉+eiπ/2 |↑↓↓↑〉}

k=π/2) |2〉=1/2{|↑↑↓↓〉+eiπ/2|↓↑↑↓〉-|↓↓↑↑〉+e-iπ/2 |↑↓↓↑〉}

k=π) |1〉=1/√2{|↑↓↑↓〉-|↓↑↓↑〉} |2〉=1/2{|↑↑↓↓〉-|↓↑↑↓〉+|↓↓↑↑〉-|↑↓↓↑〉}

Limita2ons:smalllaAces•  Hubbardmodel:20sitesathalffilling,10↑and10↓,D=20!(10!

10!)x20!(10!10!)=2.4e+10.AuersymmetriesD'=1.1e+8•  t-Jmodel(only|o〉, |↑〉 and|↓〉states):32siteswith4holes,

14↑and14↓,D=32!/(14!18!)x18!/(14!4!)=1.4e+12;D'=5.6e9•  Heisenbergmodel(only|↑〉 and|↓〉states):36sites,18↑and

18↓,D=36!/(18!18!)=9075135300;D'=D/(36x2x2x2x2)=1.5e6states

Exact diag. is limited by system size… How can we overcome

this problem?

Po’ man’s solution: What about truncating the basis?

“Classical”analogyImagecompressionalgorithms(e.g.Jpeg)

Wewanttoachieve“losslesscompression”…oratleastminimizethelossofinforma1on

Idea 1: Truncated diagonalization

|gs〉 =∑ ai|xi〉

Usually, only a few important states possess most of the weight

, ∑ |ai|2 = 1

Error = 1-∑' |ai|2

Cuthere

Idea 2: Change of basis

Can we rotate our basis to one where the weights are more concentrated, to minimize the error?

|gs〉 =∑ ai|xi〉 , ∑ |ai|2 = 1 Error = 1-∑' |ai|

2

Cuthere Cuthere

Whatdoesitmean“totruncatethebasis”

Ifwetruncate

ThistransformaYonisnolongerunitary,doesnotpreservenorm->lossofinformaYon

Thecaseofspins The two-site basis is given by the states

|σσ’〉 ={|↑↑〉;|↑↓〉;|↓↑〉; |↓↓〉}

We can easily diagonalize the Hamiltonian by rotating with the matrix:

That yields the eigenstates:

Thecaseofspins…

NumericalRenormaliza2onGroup

Let’sconsiderthe1dHeisenbergmodel

( )∑∑ ++

−−+

+++ ++=⋅=

iiiii

zi

zi

iii SSSSSSSSH 1111 2

1!!

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

−= −+

1000

;0100

;2/10

02/1000 SSS z

Forasinglesite,theoperatormatricesare:

WealsoneedtodefinetheidenYtyonablockoflsites

22 dimensions with ;10

01ll

lI ×⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

= !

BuildingtheHamiltonianalaNRG

+−−+ ⊗+⊗+⊗= 0000002 21

21 SSSSSSH zz

1 2

ll-1

32 ⎥⎦

⎤⎢⎣

⎡ ⊗+⊗+⊗⊗+⊗= +−−+0000001123 2

121 SSSSSSIIHH zz

221 HIIHH llll ⊗+⊗=→ −−

Thisrecursionwillgeneratea2lx2lHamiltonianmatrixthatwecaneasilydiagonalize

Anotherwaytoputit…

ll-1

( ) ( ) ( )+−

−−+

−−−

+−−

−+−−−

+−−+−−

⊗+⊗+⊗+⊗=

⊗⊗+⊗⊗+⊗⊗+⊗=

⎥⎦

⎤⎢⎣

⎡ ⊗+⊗+⊗⊗+⊗=

01010111

00210200211

000000211

21

21

21

21

21

21

SSSSSSIH

SSISSISSIIH

SSSSSSIIHH

llzz

ll

llzz

ll

zzlll

01 OIO ll ⊗= −with

Addingasinglesitetotheblock

1+lαlα

1+ls

BeforetruncaYngwebuildthenewbasisas:

11 ++ ⊗= lll sαα

AndtheHamiltonianforthenewblockas

...'0,01,1, +⊗+⊗+⊗=+ OOHIIHH lLllLlL

01, OIO llL ⊗= −with

|ψ〉 = ∑ijψij|i〉| j〉

Dim=2L

Dimensionoftheblockgrowsexponen2ally

Idea 3: Density Matrix Renormalization Group S.R. White, Phys. Rev. Lett. 69, 2863(1992), Phys. Rev. B 48, 10345 (1993)

BlockdecimaYon

Dim=2NDim=m

constant

|ψ〉 = ∑ijψij|i〉| j〉

Thedensitymatrixprojec2on

Universe

system

|i〉environment

| j〉

We need to find the transformation

that minimizes the distance

S=||ψ'〉 -|ψ〉|2

|ψ〉 = ∑ijψij|i〉| j〉 |ψ'〉 = ∑mαjaαj|α〉| j〉

Solution: The optimal states are the eigenvectors of the reduced density matrix

ρii' = ∑jψ*ijψi'j Tr ρ = 1

with the m largest eigenvalues ωα

Understandingthedensity-matrixprojecYon

∑=ij

BAijABjiψψ

∑==→

=

jjiijAAAiiA

ABABBA

ii *'' ')(

tr

ψψρρ

ψψρThereduceddensitymatrixisdefinedas:

Universe

system

|i〉environment

| j〉RegionA RegionB

ProperYesofthedensitymatrixψψρ

ABABBA tr=

•  HermiYan->eigenvaluesarereal•  Eigenvaluesarenon-negaYve•  Thetraceequalstounity->TrρA=1•  Eigenvectorsformanorthonormalbasis.

1 and 0 with ; =≥= ∑∑α

ααα

α ωωααωρAAA

ThesingularvaluedecomposiYon(SVD)Consideramatrix

ψij= dimA

dimB

WecandecomposeitintotheproductofthreematricesU,D,V:ψ =UDV†

•  Uisa(dimAxdimB)matrixwithorthonormalcolumns->UU†=1;U=U†•  Disa(dimBxdimB)diagonalmatrixwithnon-negaYveelementsλα•  Visa(dimBxdimB)unitarymatrix->VV†=1

= xxψ U D V

(wearechoosingdimB<dimAforconvenience)

( )

( ) ( )

∑

∑∑ ∑∑

∑∑

∑∑

=→

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=

=

==

B

BAAB

B

BA

B

jBj

iAi

ij

B

BAjiAB

B

ji

B

jt

iij

jViU

jiVU

VUVU

dim

dimdim*

dim*

dim*

dim

αα

αα

αααα

αααα

αααα

αααα

ααλψ

αλαλ

λψ

λλψ

{ }{ } l!orthonorma are , bases theHereBA

αα

Thisisalsocalledthe“SchmidtdecomposiYon”ofthestate

TheSVDandthedensitymatrix

)dim,min(dim with BA

r

BAABr ==∑

αα ααλψ

Ingeneral:

IntheSchmidtbasis,thereduceddensitymatrixis

∑

∑

=

==

r

BBB

r

AAABABBA

αα

αα

ααλρ

ααλψψρ

2

2

and

tr

•  Thesingularvaluesaretheeigenvaluesofthereducedd.m.squaredωi=λi 2

•  Thetworeduceddensitymatricessharethespectrum•  thesingularvectorsaretheeigenvectorsofthedensitymatrix.

OpYmizingthewave-funcYon

S=||ψ'〉 -|ψ〉|2

Wewanttominimizethedistancebetweenthetwostates

where|ψ〉istheactualgroundstate,and|ψ’〉isthevariaYonalapproximaYonauerrotaYngtoanewbasisandtruncaYng

|ψ'〉 = ∑mαjaαj|α〉| j〉

WereformulatethequesYonas:Givenamatrixψ ,whatistheopYmalmatrixψ’withfixedrankrthatminimizestheFrobeniusdistancebetweenthetwomatrices.Itturnout,thisisawellknownproblem,calledthe“lowrankmatrixapproximaYon”or“pricipalcomponentapproximaYon”(PCA)inmachinelearning.Ifweordertheeigenvaluesofthedensitymatrixindescendingorderω1, ω2,…,ωm,…, ωrweobtain

S=||ψ'〉 -|ψ〉|2 = ∑+

r

mi

1ω TruncaYonerror!

DMRG:TheAlgorithmHowdowebuildthereducedbasisofstates?

Wegrowourbasissystema2cally,addingsitestooursystemateachstep,andusingthedensitymatrixprojec2ontotruncate

2) We diagonalize the system and obtain the ground state |gs〉=∑ψ1234|α1〉|s2〉|s3〉|β4〉

3) We calculate the reduced density matrix ρ for blocks 1-2 and 3-4.

4) We diagonalize ρ obtaining the eigenvectors and eigenvalues ωi

1) We start from a small superblock with 4 sites/blocks, each with a dimension mi , small enough to be easily diagonalized

1 2 3 4 m1

H1

The Algorithm

∑=43

*34'2'1123421121 ''

β

ψψαραs

ss

5) We add a new site to blocks 1 and 4, expanding the basis for each block to m'1 = m m2 and m'4 = m3 m

m2 m m'1=m m2

7) We repeat starting from 2) replacing H1 by H'1 and H4 by H'4

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

6) We rotate the Hamiltonian and operators to the new basis keeping the m states with larger eigenvalues (notice that we no longer are in the occupation number representation)

We add one site at a time, until we reach the desired system size

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Thefinitesizealgorithm

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

We sweep from left to right We sweep from right to left

Thefinitesizealgorithm

…Until we converge

Finite-sizeDMRGFlowchart

The discarded weight 1- ∑mα=1ωα measures

the accuracy of the truncation to m states

Observa2ons•  SweepingisessenYaltoachieveconvergence•  Runthefinite-sizeDMRGandextrapolatetothethermodynamiclimit.

•  Foreachsystemsize,extrapolatetheresultswiththenumberofstatesm,orfixthetruncaYonerrorbelowcertaintolerance.

DensityMatrixRenormaliza2onGroup

Avaria2onalmethodwithouta-prioriassump2onsaboutthephysics.

• Similarcapabili1esasexactdiagonaliza1on.

• Cancalculateproper1esofverylargesystems(1Dandquasi-2D)withunprecedentedaccuracy.

• Resultsarevaria1onal,but“quasi-exact”:Accuracyisfinite,butundercontrol.

AdvantagesoftheDMRG

•  DMRGisveryversaYle,andeasytoadapttocomplexgeometriesandHamiltonians.

•  Canbeusedtostudymodelsofspins,bosons,orfermions.

•  Generalandreusablecode:Asingleprogramcanbeusedtorunarbitrarymodelswithoutchangingasingleline(e.g.ALPSDMRG)

•  Symmetriesareeasytoimplement.

Limita2onsoftheDMRG

•  DMRGisthemethodofchoicein1dandquasi-1dsystems,butitislessefficientinhigherdimensions.

•  Problemswith(i)criYcalsystems,(ii)longrangeinteracYons,and(iii)periodicboundarycondiYons.

•  TheselimitaYonsaredueto:–  ThestructureofthevariaYonalwavefuncYonusedbytheDMRG(theMPSansatz).

–  Entanglemententropyfollowsarealaw.

TechnicaliYes…Addingasinglesitetotheblock

1+lαlα

1+ls

BeforetruncaYngwebuildthenewbasisas:

11 ++ ⊗= lll sαα

AndtheHamiltonianforthenewblockas

...'0,01,1, +⊗+⊗+⊗=+ OOHIIHH lLllLlL

01, OIO llL ⊗= −with

..andfortherightblock

BeforetruncaYngwebuildthenewbasisas:

433 +++ ⊗= lll s ββ

AndtheHamiltonianforthenewblockas

...' 4,0)4(04,13, +⊗+⊗+⊗= ++−++ lRlLlRlR OOIHHIH

3+lβ4+lβ

3+ls

)1(0, +−⊗= lLlR IOOwith

PuzngeverythingtogethertobuildtheHamiltonian…

...' 3,1,

3,1,3,1,

+⊗+

⊗+⊗=

++

++++

lRlL

lRlLlRlL

OOHIIHH

1+lα 3+lβ

Trunca2onWhen we add a site to the left block we represent the new basis states as:

( )∑∑+

++

+

++

++++ ⊗=⊗=ll

lllll s

llslL

sllllll sUss

ααα

α

ααααα,

1,1

,1111

111

1

1+lαlα

1+ls

Similarly for the right block:

( )∑∑++

+++

++

+++

++++++ ⊗=⊗=43

34343 ,

43,3

,433433

lllll

ll slls

lR

sllllll sUss

βββ

β

βββββ

3+lβ4+lβ

3+ls

MeasuringobservablesSupposewehaveachainandwewanttomeasureacorrelaYonbetweensitesiandj

iO jO'

i j

WehavetwoopYons:1.  MeasurethecorrelaYonbystoringthecompositeoperatorinablock2.  Measurewhenthetwooperatorsareonseparateblocks

Weshallgoforop2on(2)forthemoment:simplerandmoreefficient

Operatorsonseparateblocks

iO jO'ˆ

i j

WeonlymeasurewhenwehavethefollowingsituaYon:

Then,itiseasytoseethat:

ββααψψ

αββαψψ

ψψ

βααβαββα

βααβαββα

ji

ji

jiji

OO

OO

OOOO

'''

'''

''

'',

*''

'',

*''

∑

∑

=

=

==

Wecannotdothisifthetwooperatorsareinthesameblock!!!

OperatorsonthesameblockiO jO'ˆ

i j

Weneedtopropagatetheproductoperatorintotheblock,thesamewayaswedofortheHamiltonian

Doneverdothis:

ααααψψ

ψψ

ααβαββα ji

jiji

OO

OOOO

'''

''

',

*'∑=

≠=

Targe2ngstatesinDMRG

If we target the ground state only, we cannot expect to have a good representation of excited states (dynamics).

If the error is strictly controlled by the DMRG truncation error, we say that the algorithm is “quasiexact”. Non quasiexact algorithms seem to be the source of almost all DMRG “mistakes”. For instance, the infinite system algorithm applied to finite systems is not quasiexact.

Our DMRG basis is only guaranteed to represent targeted states, and those only after enough sweeps!

Excitedstates

gsgsHHH Λ+=→ '

b)AteachstepoftheDMRGsweep,targetthegroundstate,andthegroundstateofthemodifiedHamiltonian:

FortargeYngthetwostates,weusethedensitymatrix:

1121

21

+= gsgsρ

a)Ifweusequantumnumbers,wecancalculatethegroundstatesindifferentsectors,forinstanceS=0,andS=1,toobtainthespingap

2DGeneraliza2on

WhydoestheDMRGwork???ωα

α

good!

bad!

Inotherwords:whatmakesthedensitymatrixeigenvaluesbehavesonicely?

EntanglementWesaythatatwoquantumsystemsAandBare“entangled”whenwecannotdescribethewavefuncYonasaproductstateofawavefuncYonforsystemA,andawavefuncYonforasystemBForinstance,letusassumewehavetwospins,andwriteastatesuchas:

|ψ〉 =|↑↓〉 + |↓↑〉 + |↑↑〉 + |↓↓〉

Wecanreadilyseethatthisisequivalentto:

|ψ〉 =(|↑〉+|↓〉)⊗(|↑〉+|↓〉)=|↑〉x ⊗ |↓〉x ->Thetwospinsarenotentangled!Thetwosubsystemscarryinforma2on

independentlyInstead,thisstate: |ψ〉 =|↑↓〉 + |↓↑〉

is“maximallyentangled”.ThestateofsubsystemAhasALLthe

informa2onaboutthestateofsubsystemB

TheSchmidtdecomposiYon

Universe

system

|i〉environment

| j〉

∑=ij

BAijABjiψψ

WeassumethebasisfortheleusubsystemhasdimensiondimA,andtheright,dimB.ThatmeansthatwehavedimAxdimBcoefficients.WegobacktotheoriginalDMRGpremise:Canwesimplifythisstatebychangingtoanewbasis?(whatdowemeanwith“simplifying”,anyway?)

TheSchmidtdecomposiYonWehaveseenthatthroughaSVDdecomposiYon,wecanrewirethestateas:

∑=r

BAABα

α ααλψ

Where

lorthonorma are ; and 0 );dim,min(dimBABAr ααλα ≥=

NoYcethatiftheSchmidtrankr=1,thenthewave-funcYonreducestoaproductstate,andwehave“disentangled”thetwosubsystems.

AuertheSchmidtdecomposiYon,thereduceddensitymatricesforthetwosubsystemsread:

∑=r

BABABAα

α ααλρ//

2/

TheSchmidtdecomposiYon,entanglementandDMRG

ItisclearthattheefficiencyofDMRGwillbedeterminedbythespectrumofthedensitymatrices(the“entanglementspectrum”),whicharerelatedtotheSchmidtcoefficients:•  Ifthecoefficientsdecayveryfast(exponenYally,forinstance),thenweintroduceveryli3leerrorbydiscardingthesmallerones.

•  Fewcoefficientsmeanlessentanglement.Intheextremecaseofasinglenon-zerocoefficient,thewavefuncYonisaproductstateanditcompletelydisentangled.

•  NRGminimizestheenergy…DMRGconcentratesentanglementinafewstates.Thetrickistodisentanglethequantummanybodystate!

QuanYfyingentanglementIngeneral,wewritethestateofabiparYtesystemas:

∑=ij

ij jiψψ

Wesawpreviouslythatwecanpickandorthonormalbasisfor“leu”and“right”systemssuchthat

∑=α

α ααλψ RL

Wedefinethe“vonNeumannentanglemententropy”as:

22 log αα

α λλ∑−=SOr,intermsofthereduceddensitymatrix:

( )LLLLL S ρρααλρα

α logTr2 −=→=∑

EntanglemententropyLetusgobacktothestate:

|ψ〉 =|↑↓〉 + |↓↑〉

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2/1002/1

Lρ

Weobtainthereduceddensitymatrixforthefirstspin,bytracingoverthesecondspin(andauernormalizing):

Wesaythatthestateis“maximallyentangled”whenthereduceddensitymatrixispropor2onaltotheiden2ty.

2log21log

21

21log

21

=−−=S

Entanglemententropy•  Ifthestateisaproductstate:

{ } 0,...0,0,1 =→=→= SwRL αααψ

•  Ifthestatemaximallyentangled,allthewαareequal

{ } DSDDDw log,...1,1,1 =→=→ α

whereDis

{ }RL HHD dim,dimmin=

Arealaw:IntuiYvepictureConsideravalencebondsolidin2D

singlet

2logcut) bonds of(#2log LS ≈×=

TheentanglemententropyisproporYonaltotheareaoftheboundaryseparaYngbothregions.Thisistheprototypicalbehavioringappedsystems.NoYcethatthisimpliesthattheentropyin1DisindependentofthesizeoftheparYYon

CriYcalsystemsin1Dcisthe“centralcharge”ofthesystem,ameasureofthenumberofgaplessmodes

EntropyandDMRGThenumberofstatesthatweneedtokeepisrelatedtotheentanglemententropy:

Sm exp≈

•  Gappedsystemin1D:m=const.•  CriYcalsystemin1D:m=Lα•  Gappedsystemin2D:m=exp(L)•  In2Dingeneral,mostsystemsobeythearealaw(notfreefermions,or

fermionicsystemswitha1DFermisurface,forinstance)…•  PeriodicboundarycondiYonsin1D:twicethearea->m2

Thewave-func2ontransforma2onBefore the transformation, the superblock state is written as:

∑+++

++++++ ⊗⊗⊗=321 ,,,

321321llll ss

llllllll ssssβα

βαψβαψ

lα

1+ls

After the transformation, we add a site to the left block, and we “spit out” one from the right block

∑ ≈l

llα

αα 1

3+lβ

2+ls

∑++++

++++++++ ⊗⊗⊗=4321 ,,,

43214321llll ss

llllllll ssssβα

βαψβαψ

After some algebra, and assuming , one readily obtains:

∑++

++++++++++++ ≈31 ,,

343321114321lll s

llllllllllllll ssssssβα

ββψβαααψβα

Whathaveweleuout?

…ExploiYngquantumnumbers