Tensor Renormalization Group and its Application

Transcription

1 Tensor Renormalzaton Group and ts Applcaton Tao Xang ( 向濤 ) Insttute of Physcs / Insttute of Theoretcal Physcs Chnese Academy of Scences txang@aphy.phy.ac.cn

2 Motvaton: Challenges n the study of strongly correlated systems Non-perturbatve: quantum feld theory not always useful Exponental Wall: total degree of freedom grows exponentally wth system sze Wheat chessboard problem 1st square: 2nd square: 3rd square: 4th square: 64th square: 1 gran 2 grans 4 grans 8 grans 2 63 grans = = grans

3 Weak Couplng Approach Treat many-body nteractons by certan mean-feld approxmaton Densty Functonal Theory(1960s) Most successful numercal method for treatng weak couplng systems Walter Kohn Based on LDA or smlar approxmaton

4 Strong Couplng Approach 2 64 a partcularly selected set of bass states Quantum Monte Carlo Numercal Renormalzaton Group K. Wlson Wlson NRG 1960s 1980s Kondo problem Densty-matrx renormalzaton group (Whte 1992) Most accurate numercal method for studyng 1D quatnum systems Poor performance n 2D

5 Characterstc Energy Scales Numercal Renormalzaton Group Densty Functonal Theory Quantum Monte Carlo Dynamcal Mean Feld Energy T(K) Cold atoms 3 He-superflud 4 He-superflud CMR smple metal heavy fermons QHE hgh-tc semconductor superconductor Strong couplng Weak couplng low Dmensonalty hgh strong Quantum fluctuatons weak weak Coulomb screenng strong

6 Concept of renormalzaton group H 0 H 1 H 2 Energy H eff RG teraton 1943 Ernst Stueckelberg ntalzed a renormalzaton program to attack the problems of nfntes n QED but hs paper was rejected by Physcal Revew Ernst Stueckelberg and Andre Petermann opened the feld of renormalzaton group Ernst Stueckelberg

7 Block Spn Renormalzaton Group Leo P. Kadanoff 1966 Kadanoff ntroduced the block spn RG scheme n combne wth the scalng nvarance to solve the problem of phase transton and crtcal phenomena

8 Numercal Renormalzaton Group Method 1974 Wlson opened the feld of numercal RG by combnng the dea of renormalzaton group wth numercal calculatons. Kenneth Wlson He solved the famous sngle-mpurty Kondo model (Kondo effect).

9 Basc Idea of Numercal Renormalzaton Group To represent a targeted state ψ 0 = l= 1 f n l l by an approxmate wavefuncton usng a lmted number of bass states ψ% 0 M f % n l= 1 l l such that ther overlap s maxmzed ψ% ψ 0 0 M = % l= 1 ff l l Refne a bass set by performng a seres of bass transformatons

10 Wlson block-spn NRG H 2 keep the lowest D states of H 2 H = H + H + H L R LR keep the lowest D states of H 4 H = H + H + H L R LR 2n n n 2n Dagonalze H,keep the states accordng to ther energy

11 Reasons for the Falure of Wlson block-spn NRG 1. Interface effect s too bg 2. Bg truncaton error: total D 2 states, but only D of them retaned 3. Crteron of truncaton: energy s not a good ndcator of bass states ( + ) + 1. H = a a + hc Example: Partcle on a lattce

12 Improved NRG method 1. Boundary error s reduced 2. Truncaton error s reduced Retan D states from 2D states 3. Crteron of truncaton: Energy T. Xang and G. A. Gehrng, J. Mag. Mag. Mat , 861 (1992).

13 Densty Matrx Renormalzaton Group (DMRG) System Envronment m 1 m 2 n 2 m 3 S. Whte, PRL (1992) Use Envronment as a pump to probe the bass states n System n 3 m 4 sys j env ψ = f j, j j sys env

14 NRG versus DMRG Wlson NRG System Sys DMRG System Env Energy s the only quantty that can be used to measure the weght of a bass state ρ = e β H Use a sub-system as a pump to probe the other part of the system ρ sys = Trenv e β H The weght s measured by the entanglement between Sys and Env

15 What s a Densty Matrx measurement? System Envronment ψ = f j, j j sys env sys j env Quantum Informaton: Schmdt decomposton ψ = Λ n n sys env Λ n2 s the egenvalue of reduced densty matrx n n Mathematcan: Sngular value decomposton j n, n j, n n= 1 D n= 1 N f = U Λ V U Λ V n, n j, n

16 DMRG Wavefuncton: Matrx Product State S Ostlund, S Rommer, PRL 1995 The wavefuncton generated by the DMRG teraton s a matrx-product state It can be also regarded as a varatonal ansatz of many-body wavefuncton ( ) Ψ = Tr Am [ ]... Am [ ] m... m m Lm 1 L 1 L 1 L A xx [ m ] D D matrx x x +1 bond varables m local bass

17 Valence bond sold state S=1 1 H = S 1 ( ) 2 2 S+ 1 S S Ψ = m Lm 1 L ( ) Tr Am [ ]... Am [ ] m... m 1 L 1 L A[ 1] = A[0] = A[1] = S=1/2 S=1/2 Affleck, Kennedy, Leb, Tasak, PRL 59, 799 (1988)

18 Haldane Conjecture Integer Hesenberg spn chan has a fnte exctaton gap = H S S + 1 Haldane N 2+ :S = 1 Energy gap Y 2 BaNO 5

19 Hdden Order Hdden antferromagnetc order Is there any order n the Haldane spn chan? S = -1 ( ), 0, 1 ( ) Nd 2 BaNO 5 Antferromagnet Neel order Neel Nobel 1970 YBa 2 Cu 3 O 6

20 Accuracy of the DMRG n 1D DMRG: Can calculate all statc, thermodynamc, dynamc quanttes 1D S=1 Hesenberg model Ground state energy Quantum Monte Carlo (5) DMRG (4) (keep D = 100 states) Hesenberg N H = J S r S r + 1 = 1

21 Extend DMRG to 2D: Map 2D lattce to 1D DMRG s essentally a 1D method Mult-chan mappng: The wdth of the lattce s fxed Dagonal-path: Lattce grows n both drectons T Xang, J Z Lou, Z B Su, PRB 64, (2001)

22 Ground state energy of the 2D Hesenberg model Ground State Energy Square Lattce Ground State Energy Trangle Lattce /L Square Trangle DMRG MC SW /L Free boundary condtons ΔE(L) ~1/L Perodc boundary condtons: ΔE(L) ~ (1/L) 3

23 Why s the performance of DMRG poor n 2D? S 1 S 2 S 3 S 4 S 5 S 6 1D D D 1D mappng Introduce long-range couplng Break the square lattce symmetry Demand grows exponentally wth lattce sze n order to satsfy the Area Law

24 Area Law of Entanglement Entropy For a gapped system d 1 Sentanglement ~ L ~lndmn envronment system ( ) Ψ = Tr Am [ ]... Am [ ] m... m m Lm 1 L 1 L 1 1D (d=1) D mn ~ L 0 L A xx [ m ] D D matrx 2D (d=2) D mn ~ e L D mn grows exponentally wth the system sze

25 Tensor-Network Wavefuncton Ψ = Tr λ j black whte x λ y λ z A x y y [ m ] B x j y j y j [ m j ] m m j keep the localty of local nteractons satsfy the area law: The number of danglng bonds s proportonal to the cross secton Tensor network state = tensor product state : vertex state model Nggemann, Zttartz, 96 : projected entangled-par state (PEPS) Crac, Verstraete, 04

26 Two Problems Need To Be Solved 1. How to determne the local tensor? 2. How to evaluate the expectaton values, gven a tensor-product wavefuncton? = Ψ whte j black j j y y x y y x z y x m m m B m A Tr j j j ] [ ] [ λ λ λ

27 How to determne the local tensor? [ ] Ψ = Tr T m m x yz 1. Varatonal approach to mnmze Convergng slowly The bond dmenson that can be treated s small Ψ H Ψ Ψ Ψ D 5 Gu, Levn, Wen, PRB (2008)

28 How to determne the local tensor? Ψ = Tr λ j black whte x λ y λ z A x y y [ m ] B x j y j y j [ m j ] m m j 2. Projecton approach lm e βh Ψ = ground state β Very accurate Fast convergng Large bond dmenson can be treated (more f symmetry s consdered) Projecton Operator D ~ 70 (honeycomb lattce) D ~ 20 (square or Kagome lattce) Jang, Weng, Xang, PRL 101, (2008) 1D: Vdal, PRL 98, (2007)

29 Projecton Approach lm β e βh Ψ = ground state lm M ( τh ) M e Ψ = ground state Hesenberg model H = H = H + H + H j H = JS S j j j x y z

30 1. One teraton 2. Repeat the above teraton untl converged ~ Ψ = Ψ Ψ = Ψ Ψ = Ψ z y x H H H e e e τ τ τ Trotter-Suzuk decomposton Projecton Iteraton ),, ( ) (, 2 z y x H H o e e e e black H H H H x y z = = + + α τ α α τ τ τ τ

31 Projecton: x-bond τ H x, j e Ψ = Tr mm e mm λλλa [ m ] B [ m ] mm j j j black j=+ xˆ τ H j j x y z x yy x y y j j Step I Step I Step II Step II Step III Step III Truncate bass space SVD: sngular value decomposton

32 How accurate s ths projecton approach λ MF Wthout performng the canoncal transformaton for the matrx product state Usng the canoncal representaton of the matrx product state: λ s the egenvalue of the densty matrx 1D Hesenberg model

33 Expectaton Value Ψ = Tr T [ m ] m ΨΨ = Tr xyy xx, yy, zz A = T [ m] T [ m] xx, yy, zz xyz x y z m Bond dmenson D 2 A To evaluate Ψ Ψ and Ψ O Ψ s equvalent to evaluatng the partton functons of classcal statstcal models, such as the S=1/2 Isng model

34 Coarse Gran Tensor Renormalzaton Group Z= Tr T xyz Rewrng Decmaton Levn & Nave, PRL (2007)

35 Step I: Rewrng M = kj, l mj mlk m N n= 1 T T = U Λ V kj, n n l, n Bond feld: measures the entanglement between U and V

36 Step II: Decmaton

37 Accuracy of TRG D = 24 Isng model on trangular lattce

38 Second renormalzaton of tensor-network state (SRG) TRG: truncaton error of M s mnmzed ( env Z=Tr MM ) What needs to be mnmzed s the error of Z! SRG: The renormalzaton effect of M env to M s consdered system envronment Xe et al, PRL 103, (2009)

39 I. Poor Man s SRG: entanglement mean-feld approach ( env ) j, kl, Z=Tr MM = M M jkl env kl j M = U Λ V kj, l kj, n n l, n 4 n= 1... D M Λ Λ Λ Λ env 1/ 2 1/ 2 1/ 2 1/ 2 kl, j k l j Mean feld (or cavty) approxmaton Bond feld mean feld varable Λ Λ = Λ 1/2 Λ 1/2 From envronment From system

40 Accuracy of Poor Man s SRG D = 24 Isng model on trangular lattce

41 II. More accurate treatment of SRG Evaluate the envronment contrbuton M env usng TRG TRG M env

42 ( n 1) M jkl 1. Forward teraton M M (0) (1) L M ( N ) ( n) M ' j ' kl ' ' 2. Backward teraton M M ( N) ( N 1) M (0) L = M env M M S S S S = ( n 1) ( n) jkl ' j ' k' l' k' jp j ' p ' lq l ' qk ' j' k' l' pq

43 Accuracy of SRG D = 24 Isng model on a trangular lattce

44 Error versus D

45 Specfc Heat of the Isng model on Trangular Lattces D = 24

46 Quantum Hesenberg Model on Honeycomb Lattce H = J S r S r Ground State Energy j j SRG D = 14 E = SRG D = 30 E = Monte Carlo: E = ( ± 20 ) Lattce sze 30 N = 2 3 U. Low, Condensed Matter Physcs 2009 Vol 12, 497

47 Hesenberg Model on Honeycomb Lattce Quantum Monte Carlo Result E = (± 13) per bond = ( ± 20 ) per ste U. Low, Condensed Matter Physcs 2009 Vol 12, 497

48 Staggered Magnetzaton SRG D = 15 M = Monte Carlo: M = U. Low, Condensed Matter Physcs 2009 Vol 12, 497 M = 0.22 Reger, Rera, Young, JPC 1989 Spn Wave: M = 0.24 Seres expanson M = 0.27

49 Staggered Magnetzaton The tensor-network state cuts the longrange correlaton The bond dmenson s roughly of the order of the correlaton length of the tensor-network state The logarthmc correcton to the Area Law s mportant here

50 Staggered Magnetzaton 4 th order polynomal ft M = Monte Carlo: M =

51 Square Lattce

52 Kagome Lattce

53 Summary DMRG s an accurate numercal method for studyng 1D quantum systems In 2D, the tensor-network representaton of quantum manybody states s a good startng pont The quantum tensor-network wavefuncton can be accurately and effcently evaluated by the projecton method The partton functon or expectaton values of tensornetwork model/state can be accurately determned by the SRG method Acknowledgement: Zhyuan Xe, Qaon Chen, Huha Zhao IOP / ITP, CAS Zhengyu Weng, Hongchen Jang Tsnghua Unversty

54 Tensor-Network Representaton of Classcal Statstcal Model H= -J S S j Isng model j Z= Tr T xyz Tensor-network model n dual lattce Trangular lattce Dual lattce: honeycomb lattce

55 S 1 S 2 S 3 σ 1 Tensor-network representaton H= -J S S j σ 3 σ j 2 Z = Trexp βh = Tr exp βh Δ ( ) ( ) σ σ σ = SS = SS = SS H ( Δ = J σ1+ σ2 + σ3)/ 2 σσσ = SSSSSS = Δ

56 Tensor-network representaton Z= Tr T xyz ( ) Jβσ+ σ + σ / Tσσσ = e δ σ σ2σ3 1 ( ) S 1 σ 3 σ 2 S 2 S 3 σ 1