Hiroaki Matsueda (Sendai National College of Tech.)

Transcription

1 Jun 2 26@YITP 4-das confrnc: oograph & Quantum Informaton Snapshot ntrop: An atrnatv hoographc ntangmnt ntrop roa atsuda Snda Natona Cog of Tch.

2 Purpos of ths wor Purpos: Stratg: Entangmnt hoograph RG crtcat tpcat SVD snguar vau dcomposton Ta a spn confguraton ont Caro snapshot for 2D cassca Isng & q3 Potts mod SVD for th snapshot matr Cacuat th snapshot ntrop S Drv th scang formua of S as a functon of nar sstm s L What dos ths scang man?

3 ont Caro Smuaton of th 2D Isng od J Cassca Isng Spn od: ± Snapshots at varous tmpraturs j j a T. 52J b T T c 2. 27J c T 3. 2J L 256 2J T c J og 2

4 Crtcat Fracta and amount of nformaton 2D Isng mod J j j fracta- spn structur Tpcat?: A st of parta sstms rough rprsnt th nformaton of a possb thrma fuctuaton. A tpca snapshot of th Isng mod T2.26J Sng Informaton of partton functon

5 Dnst matr of a snapshot A snapshot dtrmnd b ont Caro smuaton ρ Y ρ X atr product trac ovr parta dgr of frdom X Y

6 Λ V U Snguar Vau Dcomposton SVD of matr Ψ Snapshot Data Snapshot Entrop boundar aw not tnsv S S Y X λ λ og Λ Λ λ / Snguar Vau Dcomposton SVD : snguar vau non-ngatv unqu dtrmnd Λ Λ Λ Y X V V U U ρ ρ V U : untar matrcs varous chocs

7 V U Λ L 4 prboc structur hddn n our SVD mthod Phs. Rv. E ± ncodng

8 Scang raton for th snapshot ntrop <S> - /3 n L ncodng <S >/nh TT c..2 /h and D.Oa Phs. Rv. E <S> - /3 n L b - b T/J 2D Isng modc/ T/J 2D 3-stats Potts modc4/5 Scang formua: S og L 3 2 Smar to CFT rsut Orgn of th scang ar numbr of sca dcomposton Consstnt wth RT formua S EE~c S

9 { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } βλ β βλ β βλ β βλ β β J J J J Z J λ { } { } βλ βλ A tanh n 2 p A 2βλ snh 2 I R K K A Z p Suu-Trottr dcomposton Trottr formua for non-commutatv oprators A and B B A B A m D transvrs-fd Isng mod 2D cassca Isng mod

10 ± ncodng ε ncodng

11

12 - n λ n a <S> - n L n n - b T/J c d ± ncodng Chng-ua L and Y. ashum 24

13 SVD spctrum agbrac dca nar Tc * ponnt anomaous dmnson f λ n S λ δ λ λ n Aλ χ χ a n n χ η λ n n λ n N n α 2 η α η { γ } n χ n χ N gh-t mt RT S L N S L n L γ γ n n L π 4 Ounsh s wor

14 Tnsor-product constructon of Srpns carpt Factord form h h3 3 unt c h L N Fracta mag L L matr N dffrnt scas

15 SVD spctrum of Srpns carpt γ nγ ± n h c 3 Two non-ro gnvaus of 2 : Γ± 4 ± 2 3 Normaaton of Γ: γ ± ± λ j N j j γ γ Egnvaus of 2 : j γ γ γ γ Dgnrac : C Snapshot ntrop ntangmnt ntrop of D fr frmons N j N j S N C j j n j n N N λ λ γ γ γ nγ j ± ± C..L Y.Yamada K.Kumamoto JPSJ I. Psch J. Phs. A: ath. Gn. 36 L25 23 n n L h

16 Snapshot ntrop as a functon of ar numbr N Numrca cacuaton C..L Y.Yamada K.Kumamoto JPSJ

17 Coars-grand snapshot ntrop C..L Y.Yamada K.Kumamoto JPSJ

18 Fnt-χ scang Fracta dgnrat gnvaus W focus on th frst N-th gnvaus S λ3 λ λ 2 N χ λ 2 λ λog λ og χ 2 m χ λog λ λ 2 og λ 2 S χ S S χ cf. fnt-ntangmnt scang nar D quantum crtcat S χ cκ og χ og χ 6 2 / c Whn w oo at th ovra structur th scang raton sms to b ogarthmc. Thr dffrnc ma com from voaton on fu conforma smmtr on th fracta mag that has just sca nvaranc. χ

19 Summar Th SVD spctra for th snapshots of th 2D Isng & q3 Potts mods rprsnt th nformaton of th two-pont corrator of spns. Thus th SVD data ar good bnchmars for th phas transton. Nar Tc th snapshot ntrop obs th ogarthmc scang whch s consstnt wth th CFT formua. Th SVD data contan th nformaton smar to th hoographc ntrop formua.