The DMRG and Matrix Product States. Adrian Feiguin

Transcription

1 The DRG nd trix Product Stte Adrin Feiguin

2 Why doe the DRG work??? ω In other word: wht mke the denity mtrix eigenvue ehve o nicey? good! d!

3 Entngement We y tht two quntum ytem A nd re entnged when we cnnot decrie the wve funccon product tte of wve funccon for ytem A nd wve funccon for ytem For intnce et u ume we hve two pin nd write tte uch : We cn rediy ee tht thi i equivent to: ( ) ( ) x x -> The two pin re not entnged! The two uytem crry informdon independenty Inted thi tte: i mximy entnged. The tte of uytem A h A the informdon out the tte of uytem

4 The Schmidt decompoicon Univere ytem i environment j A ij ij We ume the i for the ei uytem h dimenion dim A nd the right dim. Tht men tht we hve dim A x dim coefficient. We go ck to the origin DRG premie: Cn we impify thi tte y chnging to new i? (wht do we men with impifying nywy?) i A j

5 The Schmidt decompoicon We hve een tht through SVD decompoicon we cn rewire the tte : A r λ A Where r min(dim A dim ); λ 0 nd A ; re orthonorm NoCce tht if the Schmidt rnk r then the wve-funccon reduce to product tte nd we hve dientnged the two uytem. AIer the Schmidt decompoicon the reduced denity mtrice for the two uytem red: r ρ A/ λ A/ A/

6 The Schmidt decompoicon entngement nd DRG It i cer tht the efficiency of DRG wi e determined y the pectrum of the denity mtrice (the entngement pectrum ) which re reted to the Schmidt coefficient: If the coefficient decy very ft (exponency for intnce) then we introduce very ixe error y dicrding the mer one. Few coefficient men e entngement. In the extreme ce of inge non-zero coefficient the wve funccon i product tte nd it competey dientnged. NRG minimize the energy DRG concentrte entngement in few tte. The trick i to dientnge the quntum mny ody tte!

7 QunCfying entngement In gener we write the tte of iprcte ytem : ij ij We w previouy tht we cn pick nd orthonorm i for ei nd right ytem uch tht λ We define the von Neumnn entngement entropy : S i j λ λ R og Or in term of the reduced denity mtrix: ρ ( ρ og ) λ S Tr ρ

8 Entngement entropy et u go ck to the tte: We otin the reduced denity mtrix for the firt pin y trcing over the econd pin (nd Ier normizing): ρ / 0 0 / We y tht the tte i mximy entnged when the reduced denity mtrix i propordon to the idendty. S og og og

9 Entngement entropy If the tte i product tte: 00 0 w S R If the tte mximy entnged the w re equ w S og D D D D where D i D min dim dim H H R

10 Are w: IntuiCve picture Conider vence ond oid in D inget S og (#of ond cut) og The entngement entropy i proporcon to the re of the oundry eprcng oth region. Thi i the prototypic ehvior in gpped ytem. NoCce tht thi impie tht the entropy in D i independent of the ize of the prccon

11 CriCc ytem in D c i the centr chrge of the ytem meure of the numer of gpe mode

12 Entropy nd DRG The numer of tte tht we need to keep i reted to the entngement entropy: m exp S Gpped ytem in D: mcont. CriCc ytem in D: m Gpped ytem in D: mexp() In D in gener mot ytem oey the re w (not free fermion or fermionic ytem with D Fermi urfce for intnce) Periodic oundry condicon in D: twice the re -> m

13 The wve-funcdon trnformdon efore the trnformtion the uperock tte i written : After the trnformtion we dd ite to the eft ock nd we pit out one from the right ock After ome ger nd uming one rediy otin: 4 4

14 The DRG trnformcon When we dd ite to the ock we otin the wve funccon for the rger ock : et chnge the notcon A U U A We cn repet thi trnformcon for ech nd recurivey we find A A A NoCce the inge index. The mtrix correponding to the open end i ctuy vector!

15 Some properce the A mtrice Rec tht the mtrice A in our ce come from the rotcon mtrice U m A m A t A X Thi i not neceriy the ce for ritrry PS nd normizcon i uuy ig iue!

16 ei cnonic repreentcon

17 The DRG wve-funccon in more deti A A A We cn repet the previou recurion from ei to right At given point we my hve Without o of generity we cn rewrite it: PS wve-funccon for open oundry condicon

18 DigrmmDc repreentdon of PS The mtrice cn e repreented digrmmccy A A And the contrccon : The dimenion D of the ei nd right indice i ced the ond dimenion

19 PS for open oundry condicon 4

20 PS for periodic oundry condicon ( ) Tr Tr 4

21 ProperCe of trix Product Stte Inner product: 4 ' ' ' AddiCon: N N N N N N N ~ ~ ~ ~ 0 0 with ~ ~ ~ ; ϕ ϕ ϕ ϕ

22 Guge TrnformCon γ γ X X - There re more thn one wy to write the me PS. Thi give you too to othonormize the PS i

23 Opertor O i mtrix with eement O ' O The opertor ct on the pin index ony

24 Pirwie unitry trnformdon The two-ite time-evoution opertor wi ct : U N Which trnte : 4 5 N N U N ' ' 4 5 A ' ' 4 U 5 ' 4 ' A

25 trix product i A A A 4 ei cnonic 4 right cnonic ' ' ' A we w efore in the dmrg i we get: ' δ '

26 The DRG w.f. in digrm A A A A A A 4 (It jut ixe more compicted if we dd the two ite in the center)

27 H AKT The AKT Stte S Si Si i i i ( S S ) with We repce the pin S y pir of pin S/ tht re competey ymmetrized 0 i i i i ( ) nd the pin on different ite re forming inget i i i i ( ) i i i i i i i

28 The AKT PS The oc projeccon opertor onto the phyic S tte re ; 0 0 ; The mpping on the pin S chin then red with Σ Σ Σ Σ Σ AKT AKT A A A A P ProjecCng the inget wve-funccon we otin The inget wve funccon with inget on ond i with Σ Σ Σ Σ Σ 0 0

29 VriCon PS We cn potute vricon principe trcng from the umpcon tht the PS i good wy to repreent tte. Ech mtrix A h DxD eement nd we cn conider ech of them vricon prmeter. Thu we hve to minimize the energy with repect to thee coefficient eding to the foowing opcmizcon proem: min A H λ DRG doe omething very coe to thi