WORM ALGORITHM. Nikolay Prokofiev, Umass, Amherst. Boris Svistunov, Umass, Amherst Igor Tupitsyn, PITP, Vancouver

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1 WOR ALGORTH Nkolay Prokofev, Umass, Amherst asha ra Bors Svstunov, Umass, Amherst gor Tuptsyn, PTP, Vancouver assmo Bonnsegn, UAlerta, Edmonton Los Angeles, January 23, 2006

2 Why other wth algorthms? Effcency PhD whle stll young Better accuracy Large system sze ore complex systems Fnte-sze scalng Crtcal phenomena Phase dagrams New quanttes, more theoretcal tools to address physcs Grand canoncal ensemle Off-dagonal correlatons Sngle-partcle and/or condensate wave functons Wndng numers and ρ S N ( µ ) G( r, τ ) ϕ ( r) Relaly! Examples from: superflud-nsulator transton, spn chans, helum, deconfned crtcalty, polarons, resonant fermons, holes n the t-j model,

3 Worm algorthm dea Consder: - confguraton space = artrary closed loops - each cnf. has a weght factor W c n f - quantty of nterest A c n f A = c n f A c n f c n f W W c n f c n f

4 conventonal samplng scheme: local shape change Add/delete small loops No samplng of topologcal classes (non-ergodc) can not evolve to Crtcal slowng down (large loops are related to crtcal modes) N updates L d L z dynamcal crtcal exponent z 2 n many cases

5 Worm algorthm dea draw and erase: ra asha asha or ra asha + keep drawng asha Topologcal classes are sampled (whatever you can draw!) No crtcal slowng down n most cases Dsconnected loops are related to correlaton functons and are not merely an algorthm trck!

6 H Hgh-T expanson for the sng model = K σ σ ( σ = ± 1) T < j> j H / T K σσ j Z = e e K K { σ } { σ } =< j > { σ } =< j > cosh (1 + tanh σ σ ) N N ~ ( tanh K σ σ j ) = tanh K σ { σ } =< j > N = 0,1 { N } = < j > σ = ± 1 j ~ tanh { N } = loops =< j > N K = N = even =< j > < j > Graphcally: N = numer of lnes; contnuty (enter/ext) = even

7 G Spn-spn correlaton functon: g =, Z G = { σ } H / T e σ σ G N tanh +δ +δ K σ { N } =< j> σ =± 1 ~ tanh { N } = loops =< j> + worm N K Worm algorthm cnf. space = Z G

8 Complete algorthm: =, - f, select a new ste for at random - select drecton to move, let t e ond N = 0 1 N = new - f accept wth pro. 0 1 mn(1, tanh( K)) R = 1 mn(1, tanh ( K)) Easer to mplement then sngle spn-flp!

9 = C estmators G( ) = G( ) + 1 Z = Z + δ, Nonds = Nonds + N Correlaton functon: g( ) = G( ) / Z 2 agnetzaton fluctuatons: ( ) 2 = σ = N g( ) Energy: ether E = JNd σ1σ 2 = JNdg(1) or 2 E = J tanh( K) dn + Nonds snh ( K)

10 sng XY-model = K cos( θ θ j ) H T < j> ( d ) ( d ) Z = d e e F j e 2π j = H / T K cos( θ θ ) ( θ θ ) j θ k θ k θ k ( ) k 0 =< kn > =< kn > j = k n k n { j = } 2π θ k k Z = F ( j ) dθ k e { j = } 0 m j mk k = jk, k + ν = 0 ν = j km { loops} = F j { j } ( ) Flux n = Flux out closed orented loops of nteger j-currents; zero-dvergence constrant k j kn n e e ( θ θ ) j ( θ θ ) j m k mk k m km

11 Worm algorthm cnf. space = Z G θ θ g( ) = e e = G( ) Z (one open loop) Same algorthm: = - f, select a new ste for at random - select drecton to move, let t e ond new j = j 1 - for postve/negatve drectons; j + 1 accept wth pro. = R new F( j ) = mn 1, F( j )

12 H = H = U n n n H1 j j µ t( n, n j ) j j < j> Lattce path-ntegrals for osons and spns are dagrams of closed loops! βh βh Z = Tr e Tr e e β 0 0 H 1 ( τ ) dτ β β β = 1 τ τ + 1 τ 1 τ τ τ τ βh0 Tr e 1 H ( ) d H ( ) H ( ) d d '... β magnary tme + t(1, 2) τ' τ + 0 j j n = 0,1,2,0

13 Dagrams for - Z= Tr e β H G Dagrams for = Tr T τ ( τ ) ( τ )e -β H β β magnary tme magnary tme 0 lattce ste 0 lattce ste The rest s conventonal worm algorthm n contnuous tme (there s no prolem to work wth artrary numer of contnuous varales as long as an expanson s well defned)

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15 Path-ntegrals n contnuous space are consst of closed loops too! p H = + V ( r r ) 2 2m < j j P=β / τ 2 m( R + 1 R ) Z = dr1... drp exp + U ( R) τ = 1 2τ Feynman path-ntegral P τ = β P r r,1, 2 R = ( r, r,..., r ), 1, 2, N 2 1 P

16 ZG ψ + ( r ', t ') ψ ( r, t) Perform smulatons n the extended confguraton space Z G

17 ZG ψ + ( r ', t ') ψ ( r, t) (open/close update)

18 ZG ra asha (nsert/remove update)

19 G ra ra asha (advance/recede update)

20 G asha ra ra (swap update)

21 Not necessarly for closed loops! H = ε ( k ) + V ( r r ) σ j, σ= < j Feynman (space-tme) dagrams for fermons wth contact nteracton (attractve) = U Par correlaton functon a ( r, τ ) a ( r, τ ) a ( r, τ ) a ( r, τ ) General deas: 1. Enlarge the confguraton space (usng correlaton functons, source operators, even unphyscal structures!) 2. Perform local updates through the specal ponts

22 ore: wndng numers and superflud densty W β = 0 [ partcle numer flux] µ µ τ (cross-secton ndependent n Z-sector) d z y L z x L y W = 0 W = fractonal W = + 1 β 0 ρs = ( m / βdl ) W d 2 2

23 Grand canoncal ensemle (a must for dsorder prolems!) R1 1 RR2 2 V(x) β β R1 R2 0 0 exponentally rare event 2 pdx e

24 deal equlrum crystals of He-4 Vacancy & ntersttal gaps from exponental decay of E, v G( p, τ ) Ze τ E V = E = T = K n = A N = , , 800 (meltng densty)

25 Current standard for smulatons of osons n optcal lattces and n traps: all expermental parameters as s, ncludng partcle numer N ~ 10 6 Contnuous varales (any numer of them) s not a prolem.

26 Let ur r r r r r r ur A y d x d x d x D x x x y ( ) ( ) 1 2 K n n ξ; 1, 2, K n, = n= 0 ξ Dagram order Same-order dagrams Contruton to the answer or the dagram weght (postve defnte, f possle) ntegraton varales ENTER

27 Balance Equaton: f the desred proalty densty dstruton of dagrams n the stochastc sum s P ν then the C process of updatng dagrams should e statonary wth respect to P ν (equlrum condton) (n most cases t s the same as the dagram weght ) D ν D W R D W R ν ν' ν ν ( ν ') accept = ν' ν '( ν ) updates ν ν ' updates ν' ν ν' ν accept Flux out ofν Flux toν W ν ( ' ) ν s the proalty densty of makng new varales, f any Detaled Balance: solve t for each par of updates separately. D W R D W R ν ν' ν ν ( ν ') accept = ν' ν '( ν ) ν' ν accept

28 Equaton: D r r r r r r r r x1, K, xn ( d x) W xn+ 1, K, xn+ m ( d x) p R = D x1, K, xn+ m r ( d x) p R ( ) ( ) accept ( ) n m n n+ m n+ m n+ m n n n+ m n+ m n accept Soluton: R R D x x p = = R D x x W x x p n n+ m accept n+ m( 1, K, n+ m) 1 n n+ m n accept n( 1, K, n) ( n+ 1, K, n+ m) n+ m Example: Create/Delete & W = D D ν ν ' 2 2 τ = C e p D ν τ = 1/ K p = 1 ntervals K [ U ( n 1) + U ( n) µ ][ τ τ ] G/ Z ntervals Ths s t! τ D ν ' τ