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

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1 WOR ALGORITH asha kolay Prokofev, Umass, Amherst Ira Bors Svstunov, Umass, Amherst Igor Tuptsyn, PITP, Vancouver Vladmr Kashurnkov, EPI, oscow assmo Bonnsegn, UAlberta, Edmonton Evgen Burovsk, Umass, Amherst atthas Troyer, ETH ASA ISSP, August 006

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

3 Worm algorthm dea Consder: - confguraton space = arbtrary 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 o samplng of topologcal classes (non-ergodc) can not evolve to Crtcal slowng down (large loops are related to crtcal modes) updates L d L z dynamcal crtcal exponent z n many cases

5 Worm algorthm dea draw and erase: Ira asha asha or Ira asha + keep drawng asha Topologcal are sampled (whatever you can draw!) o 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 Isng model = K σ σ ( σ = ± 1) T < j> j H / T K σσ j Z = e e K K { σ } { σ } b =< j > { σ } b =< j > cosh (1 + tanh σ σ ) b b ~ ( tanh K σ σ j ) = tanh K σ { σ } b =< j > b = 0,1 { b = 0,1} b =< j > σ = ± 1 j ~ tanh { b } = loops b =< j > b K = = even b =< j > < j > 4 Graphcally: = b number of lnes; contnuty (enter/ext) = even

7 G I Spn-spn correlaton functon: g I =, Z G I = { σ } H / T e σ σ I G tanh b +δ I +δ K σ { } b=< j> σ =± 1 b ~ tanh { b } = loops b=< j> + I worm b K Worm algorthm cnf. space = Z Same as for generalzed partton ZW = Z + κg G I 1 3 4

8 Complete algorthm: I = I, - If, select a new ste for at random - select drecton to move, let t be bond b b = 0 1 b If accept wth prob. R = mn(1, tanh( K)) 1 mn(1, tanh ( K)) Easer to mplement then sngle-flp!

9 C estmators I= I G( I ) = G( I ) + 1 Z = Z + δ I, bonds = bonds + b b Correlaton functon: g( ) = G( ) / Z agnetzaton fluctuatons: ( ) = σ = g( ) Energy: ether E = Jd σ1σ = Jdg(1) or E = J tanh( K) d + snh ( K) bonds

10 Isng lattce feld theory H = t ψ ψ + µ ψ U ψ T 4 + ν, ν=± ( x, y, z) Z = dψ e H / T expand e +ν t = = 0 ( ψ ψ )! tψ ψ +ν ν t Z = dψ ψ ψ e { },! ν ν ν ( ( ) ) 4 1 µ ψ U ψ 1 e ϕ( ) = = 1 Q( ) 0 f1 where Q( ) = µ x Ux π dx x e 0 = closed orented loops tabulated numbers

11 ψ ν ν ( ψ ) ν +ν, ν + ν ' +ν ', ν ' Flux n = Flux out closed orented loops of nteger -currents ν + ν G( I ) g( I ) = = ψiψ Z (one open loop) I Worm algorthm cnf. space = Z G

12 Same algorthm: I= Z G sectors, prob. to accept R z G Q( I + 1) mn 1, Q( ) = I + 1 ' ν ν draw R t Q( + 1) = mn 1, ( ν + 1) Q ( ' ) +ν, ν +ν, ν 1 erase R ( ) ( 1) +ν, ν Q = mn 1, t Q( ) Keep drawng/erasng

13 ult-component gauge feld-theory (deconfned crtcalty, XY-VBS and eel-vbs quantum phase transtons H ( ) t Aν = ψ a, +νψ a, e + µ ψa, U ab ψa, ψb, κ[ Aν ( )] T a; ν a; ab; A 3 A 4 +A +A 1 U11 = U U1 XY-VBS transton no DCP, always frst-order U11 = U = U1 eel-vbs transton, unknown!

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

15 Dagrams for - Z= Tr e β H G Dagrams for = Tr T τ b ( τ ) b ( τ )e -β H I I I β β magnary tme magnary tme I 0 0 lattce ste lattce ste The rest s conventonal worm algorthm n contnuous tme (there s no problem to work wth arbtrary number of contnuous varables as long as an expanson s well defned)

16 Dagrammatc onte Carlo (not n ths lecture) ur r r r r r r ur A y d x d x d x D x x x y ( ) ( ) 1 K n n ξ; 1,, K n, = n= 0 ξ Dagram order Same-order dagrams Contrbuton to the answer or the dagram weght (postve defnte, please) Integraton varables ETER

17 I I I I

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

19 ZG ψ + ( r ', t ') ψ ( r, t) dagrammatc expanson for V ( r ) < 0

20 ot 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 ) connect vortexes wth G and G K K n n Dn = ( U ) G G G G ( drdτ ) ( 1) r perm ( n!) sum over all possble connectons ξ Dn n ( ξ ) = ( U ) G ( x, x j ) n det ( drdτ ) 0 r r r

21 G space s OT necessarly physcal Doman walls n D Isng model are loops! Dsconnected loop s unphyscal!? Ira? asha Worm cnf. space = Z ( G = dsconnected? loop )?

22 Quantum spn chans magnetzaton curves, gaps, spn wave spectra H = [ J x ( S jxsx + S jysy ) + J z S jzsz ] H Sz < j> S=1/ Hesenberg chan C data Bethe ansatz (. Takahash)

23 Lne s for the effectve fermon theory wth spectrum ε ( p = π n / L) = + cp = (1) c =.48(1) devatons are due to magnon-magnon nteractons Lou, Qn, g, Su, Affleck 99

24 Energy gaps: One dmensonal S=1 chan wth J / J = 0.43 z x G( p, τ) e dx S S px = T τ (x, τ ) I (0) α' EG Eα ' ( p ) = ( ) τ τ S e Z e Gα' τ J >> 1 Spn gap = (5) Z -factor Z = 0.980(5) G( p, τ >> 1) Z e + e 1+ e τ ( β τ) β

25 Spn waves spectrum: One dmensonal S=1 Hesenberg chan

26 Conclusons: Worm algorthm = cnf. space of + updates based on Z G asha Ira Can be formulated for: - classcal and quantum models (Bose/Ferm/Spn) - dfferent representatons (path-ntegrals, Feynam dagrams,sse) - non-local solutons of balance Eqn. (clusters, drected loops, geom. worm)

27 Wndng numbers A Homogeneous gauge n x-drecton: t te, A x( r) = ϕ / L, A x( r) dx = ϕ = ϕwx Z e Z W x W d F = T ln Z = F(0) + L Λ x S ( ϕ/ L) W = ( ) / L = 1 A e x, x + x, x A e e A T W d F x Λ S = L = d ϕ L Ceperley Pollock 86