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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…