TheDMRGandMatrixProductStates

Adrian Feiguin

WhydoestheDMRGwork???ωα

α

good!

bad!

Inotherwords:whatmakesthedensitymatrixeigenvaluesbehavesonicely?

EntanglementWesaythatatwoquantumsystemsAandBare“entangled”whenwecannotdescribethewavefuncConasaproductstateofawavefuncConforsystemA,andawavefuncConforasystemBForinstance,letusassumewehavetwospins,andwriteastatesuchas:

|ψ〉 =|↑↓〉 + |↓↑〉 + |↑↑〉 + |↓↓〉

Wecanreadilyseethatthisisequivalentto:

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

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

is“maximallyentangled”.ThestateofsubsystemAhasALLthe

informaDonaboutthestateofsubsystemB

TheSchmidtdecomposiCon

Universe

system

|i〉environment

| j〉

∑=ij

BAijABjiψψ

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

TheSchmidtdecomposiConWehaveseenthatthroughaSVDdecomposiCon,wecanrewirethestateas:

∑=r

BAABα

α ααλψ

Where

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

NoCcethatiftheSchmidtrankr=1,thenthewave-funcConreducestoaproductstate,andwehave“disentangled”thetwosubsystems.

AIertheSchmidtdecomposiCon,thereduceddensitymatricesforthetwosubsystemsread:

∑=r

BABABAα

α ααλρ//

2/

TheSchmidtdecomposiCon,entanglementandDMRG

ItisclearthattheefficiencyofDMRGwillbedeterminedbythespectrumofthedensitymatrices(the“entanglementspectrum”),whicharerelatedtotheSchmidtcoefficients:•  Ifthecoefficientsdecayveryfast(exponenCally,forinstance),thenweintroduceveryliXleerrorbydiscardingthesmallerones.

•  Fewcoefficientsmeanlessentanglement.Intheextremecaseofasinglenon-zerocoefficient,thewavefuncConisaproductstateanditcompletelydisentangled.

•  NRGminimizestheenergy…DMRGconcentratesentanglementinafewstates.Thetrickistodisentanglethequantummanybodystate!

QuanCfyingentanglementIngeneral,wewritethestateofabiparCtesystemas:

∑=ij

ij jiψψ

Wesawpreviouslythatwecanpickandorthonormalbasisfor“leI”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(andaIernormalizing):

Wesaythatthestateis“maximallyentangled”whenthereduceddensitymatrixisproporDonaltotheidenDty.

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:IntuiCvepictureConsideravalencebondsolidin2D

singlet

2logcut) bonds of(#2log LS ≈×=

TheentanglemententropyisproporConaltotheareaoftheboundaryseparaCngbothregions.Thisistheprototypicalbehavioringappedsystems.NoCcethatthisimpliesthattheentropyin1DisindependentofthesizeoftheparCCon

CriCcalsystemsin1Dcisthe“centralcharge”ofthesystem,ameasureofthenumberofgaplessmodes

EntropyandDMRGThenumberofstatesthatweneedtokeepisrelatedtotheentanglemententropy:

Sm exp≈

•  Gappedsystemin1D:m=const.•  CriCcalsystemin1D:m=Lα•  Gappedsystemin2D:m=exp(L)•  In2Dingeneral,mostsystemsobeythearealaw(notfreefermions,or

fermionicsystemswitha1DFermisurface,forinstance)…•  PeriodicboundarycondiConsin1D:twicethearea->m2

Thewave-funcDontransformaDonBefore 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

TheDMRGtransformaConWhenweaddasitetotheblockweobtainthewavefuncConforthelargerblockas:

[ ]∑−

−⊗=→ −

11

,1,

llll

sllll ssA

ααα αα

Let’schangethenotaCon…

A sl[ ]αl ,αl−1 ≡ αl ULl αl−1sl

∑−

⊗= −−

1,11

llsllll

lLll ssU

α

αααα

WecanrepeatthistransformaConforeachl,andrecursivelywefind

∑ 〉=〉−

}{1,21 ...|][...][][|

1211s

lll sssAsAsAll αααααα

NoCcethesingleindex.Thematrixcorrespondingtotheopenendisactuallyavector!

SomeproperCestheAmatricesRecallthatthematricesAinourcasecomefromtherotaConmatricesU

A= 2m

m

AtA= X =1

ThisisnotnecessarilythecaseforarbitraryMPS’s,andnormalizaConisusuallyabigissue!

LeIcanonicalrepresentaCon

TheDMRGwave-funcConinmoredetail…

∑

∑

〉

=〉〉=

++++− +++

++

}{1,2,11,21

,11

...|][...][][][...][][

||

32211211s

LLlllll

llll

sssBsBsBsAsAsALllllll

ll

βββββααααα

βα

ψβα

βαψβαψ

∑ 〉=〉+++ +

}{,2, ...|][...][][|

321s

LlLlll sssBsBsBLllll ββββββ

WecanrepeatthepreviousrecursionfromleItoright…

Atagivenpointwemayhave

Withoutlossofgenerality,wecanrewriteit:

∑ 〉=−

}{1,2,21 ...|][][...][][

1211s

LL sssMsMsMsMLLL ααααααψ

MPSwave-funcConforopenboundarycondiCons

DiagrammaDcrepresentaDonofMPS

ThematricescanberepresenteddiagrammaCcallyas

≡αβ][sA α βs

αs

≡α][sA

ThedimensionDoftheleIandrightindicesiscalledthe“bonddimension”

AndthecontracCons,as:

1α 2α 3α

s1 s2

MPSforopenboundarycondiCons

∑∏

∑

∑

〉=

〉=

〉=

=

−

}{1

1

}{121

}{1,2,21

...|][

...|][]...[][

...|][][...][][1211

sL

L

ll

sLL

sLL

sssM

sssMsMsM

sssMsMsMsMLLL ααααααψ

1α 2α Lα

s1 s2 s3 s4 … sL

MPSforperiodicboundarycondiCons

( )

∑ ∏

∑

∑

〉⎟⎟⎠

⎞⎜⎜⎝

⎛=

〉=

〉=

=

−

}{1

1

}{121

}{1,,2,21

...|][Tr

...|][]...[][Tr

...|][][...][][11211

sL

L

ll

sLL

sLL

sssM

sssMsMsM

sssMsMsMsMLLLL ααααααααψ

1α 2α 3α Lα 1α

s1 s2 s3 s4 … sL

ProperCesofMatrixProductStates

Innerproduct:

1α 2α Lαs1 s2 s3 s4 … sL

1'α 2'α L'α

AddiCon:

⎟⎟⎠

⎞⎜⎜⎝

⎛=→

⎟⎟⎠

⎞⎜⎜⎝

⎛=

〉=+→

〉=〉=

∑

∑∑

L

LL

sLL

sLL

sLL

MMMMMM

NNN

MM

N

ssNNN

ssMMMssMMM

~...~~...

...

~00

with

...|...

...|~...~~;...|...

21

2121

}{121

}{121

}{121

ϕψ

ϕψ

ψ

ϕ

ψϕ

GaugeTransformaCon

α β γ α γ= X X-1

TherearemorethanonewaytowritethesameMPS.ThisgivesyouatooltoothonormalizetheMPSbasis

Operators

α

O

β

Theoperatoractsonthespinindexonly

' elementsh matrix wit a is sOsO

s1 s2 s3 sN

PairwiseunitarytranformaDonsThe two-site time-evolution operator will act as:

1α 2α 3α Nα 1α

U

s4 s5

Which translates as:

65

54

54

5454]'[]'[ 55

','

,','44 αααα sAUsA

ss

ssss∑

s1 s2 s3

1α 2α 3α

U

Nα 1α4α 4α 5α 6α 6α

s4 s5

s6 sN

Matrixproductbasis∑ 〉=〉

−}{

1,21 ...|][...][][|1211

slll sssAsAsA

ll αααααα

1α 2α lαs1 s2 s3 s4 sl

∑ 〉=〉+++ +

}{,2, ...|][...][][|

321s

LlLlll sssBsBsBLllll ββββββ

1+lβ 2+lβlβsl+1 sl+2 sl+3 sl+4 sL

Lβ

=〉ll '|ααAswesawbefore,inthedmrgbasisweget:

1α 2α Lα

1'α 2'αL'α

ll ',ααδ=

“leIcanonical”

“rightcanonical”

TheDMRGw.f.indiagrams

∑

∑

〉

=〉

=

+++++−

++++−

++

+++

}{1,2,1,21

}{1,2,11,21

...|][...][][][...][][

...|][...][][][...][][

322111211

32211211

sLLlll

sLLlllll

sssBsBsBsAsAsA

sssBsBsBsAsAsA

Lllllllll

Lllllll

ββββββαααααα

βββββααααα

ψ

ψβα

ψ

1α 2α lαs1 s2 s3 s4 sl

2+lβ 3+lβ1+lβsl+1 sl+2 sl+3 sL

Lβlα βl+1ψ

(It’sajustliXlemorecomplicatedifweaddthetwositesinthecenter)

TheAKLTState( ) 1 with

31 2

11 =⋅+⋅=∑ ++ SSSSSHi

iiiiAKLT

WereplacethespinsS=1byapairofspinsS=1/2thatarecompletelysymmetrized

( )

ibiai

ibiaibiai

ibiai

↓↓=−

↑↓+↓↑=

↑↑=+

210

…andthespinsondifferentsitesareformingasinglet

( )biaibiai ,1,,1,2

1++

↑↓−↓↑

a b

TheAKLTasaMPS

ThelocalprojecConoperatorsontothephysicalS=1statesare

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛= −+

1000

;0

21

210

;0001 0

ababab MMM

ThemappingonthespinS=1chainthenreads

∑∑}{ ,

,,, },{}{...2

22

1

11s ba

sba

sba

sba basMMM L

LL

111

2

32

1

21

132

2

2221

1

11

,,,}{

,,,

}{,,,,,,

with }{...

}{...

++Σ==

ΣΣΣ==

∑

∑Σ

ll

l

ll

l

ll

L

L

L

L

LL

absba

saa

s

saa

saa

saaAKLT

sab

sbaab

sbaab

sbaAKLT

MAsAAA

sMMMP

ψ

ψψProjecCngthesingletwave-funcConweobtain

ThesingletwavefuncConwithsingletonallbondsis

with },{...}{ ,

,,, 13221∑∑ ΣΣΣ=Σs ba

ababab baL

ψ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−=Σ

021

210

ab

VariaConalMPSWecanpostulateavariaConalprinciple,starCngfromtheassumpConthattheMPSisagoodwaytorepresentastate.EachmatrixAhasDxDelementsandwecanconsidereachofthemasavariaConalparameter.Thus,wehavetominimizetheenergywithrespecttothesecoefficients,leadingtothefollowingopCmizaConproblem:

[ ]ψψλψψα

−HAmin

DMRGdoessomethingveryclosetothis…