Advanced School on Quantum Monte Carlo Methods in Physics and Chemistry. 21 January - 1 February, Worm algorithm

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1 Advanced School on Quantum onte Carlo ethods n Physcs and Chemstry 21 January - 1 February, 2008 Worm algorthm. Prokofev Unversty of assachusetts, Amherst

2 WOR ALGORITH FOR CLASSICAL AD QUATU STATISTICAL ODELS kolay Prokofev, Umass, Amherst Collaborators on major algorthm developments Bors Svstunov Uass, Amherst Igor Tuptsyn PITP, Vancouver assmo Bonnsegn UAlberta Treste, January 2008 ASA

3 Why bother wth algorthms? Effcency PhD whle stll young PhD whle stll young Better accuracy Large system sze ore complex systems Fnte-sze scalng Crtcal phenomena Phase dagrams Relably! ew physcs 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 ) Grτ (,) ϕ ( r) Applcatons: classcal and quantum crtcal phenomena, latttce spn systems, cold atoms (bosons & fermons), lqud&sold Helum-4...

4 Standard onte Carlo setup: Worm algorthm dea - confguraton space = (depends on the model and t s representaton) arbtrary closed loops ( more or less anythng you can draw wthout loose ends ) - each cnf. has a weght factor W cnf - quantty of nterest A cnf cnf / e E T A = cnf A cnf cnf W W cnf cnf

5 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 n many cases z 2

6 Worm algorthm dea draw and erase: asha asha Ira or Ira asha + keep drawng asha Topologcal classes are sampled effcently (whatever you can draw!) o crtcal slowng down n most cases Dsconnected loops relate to mportant physcs (correlaton functons) and are not merely an algorthm trck!

7 H Hgh-T expanson for the Isng model = K σ σ ( σ = ± 1) T < j> j K σσ j b j K K j b Z e ( ) < > σ σ = = e σσ j { σ } { } { } 0! σ b=< j> σ b=< j> b = b K where { }! b b=< j> b=< j> b σ =± 1 < j> b σ = = even b K 2 { }! b = loops b= < j > b = b number of lnes; enter/ext rule = even

8 I Spn-spn correlaton functon: =, g I G Z / G e H T = σ σ { σ } I G b K + δ I +δ σ! { } b=< j> b σ=± 1 b b K 2 b=< j> b! { b } = loops + Ira asha worm same as before Worm algorthm cnf. space = Z G I Same as for generalzed partton ZW = Z +κg

9 K Gettng more practcal: snce 1 2 e σσ = cosh ( K) [ 1+ tanh( K) σ σ ] loops d b Z = cosh ( K) tanh ( K) { = 0, 1} b b 1 2 I tanh( ) K no-overlaps Complete algorthm : I = I, - If, select a new ste for them 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))

10 Solvng the crtcal slowng down problem: Queston: What are the sgnatures of the phase transton (crtcal modes)? spn representaton large domans of smlarly orented spns of lnear sze ~ L sngle-spn flps are not effcent n updatng them! loop representaton I large loops of lnear sze ~ L (long-range correlatons between spns = large dstance between I and ) draw large loops!

11 GI ( ) = GI ( ) + 1 I= I Z = Z +δ I, lnks = lnks + b b Correlaton functon: g() = G()/ Z 2 agnetzaton fluctuatons: = ( σ ) 2 = σσ j = g() Energy: ether E = Jd σσ 1 2 = Jdg(1) or 2 E = J tanh( K) d + lnks snh ( K) j

12 Isng ψ 4 lattce-feld theory H = t ψ ψ +µ ψ U ψ T * ν ν=± ( xyz),, (XY-model n the µ = 2U lmt) Start as before Z = dψ Integrate over phases ψ = xe ϕ e H / T expand on each bond e ν * t ψ ψ t ( ψ+νψ) * +ν = = 0 ν t Z = dψ ψ ψ e! ν ν ν where Q ( ) = ν! ( ( ) ) µψ U ψ 0 0 ν ϕ( 1 2) e = = Q( ) µ x Ux π dx x e f = ψ ψ * +ν * 1 2 closed orented loops tabulated numbers ψ ψ +ν + ν

13 ν ν ψ ψ ν ( ) +ν, ν + ν ' +ν ', ν ' Flux n = Flux out closed orented loops of nteger -currents ν + ν GI ( ) gi ( ) = = ψiψ Z I (one open loop) Z-confguratons have I =

14 Same algorthm: I= Z G sectors, prob. to accept R z G Q ( I + 1) = mn 1, Q ( I ) + 1 ' ν ν draw tq( + 1) R = mn 1, ( ν + 1) Q ( ' ) +ν, ν +ν, ν 1 erase R ( +ν, ν ) Q( 1) = mn 1, tq( ) Keep drawng/erasng

15 ult-component gauge feld-theory: H = t ψ ψ e +µ ψ U ψ ψ κ [ Aν ( )] T A () ν a, +ν a, a, ab a, b, a ; ν a ; ab; plaquette sum 2 A 3 A 4 +A 2 +A 1 sold-lqud transtons, deconfned crtcalty, XY-VBS and eel-vbs quantum phase transtons, etc. and fnte-t quantum models

16 Interactng partcles on a lattce: H = H U n βh βh Z = Tr e Tr e e 0 + H1 = j nj µ n tn (, nj) b + jb j < j> β 0 0 H 1 ( τ ) dτ dagonal off-dagonal H () τ = e H e βh β β β = 1 τ τ + 1 τ 1 τ τ τ + 0 τ' 0 βh0 Tr e 1 H ( ) d H ( ) H ( ') d d '... βh { n} = { n, n, n,...} In the dagonal bass set (occupaton number representaton): β Z = n e e H e dτ+ e H e H e dτ dτ + n βh0 ( β τ ) H0 τ H0 ( β τ ) H0 ( τ τ ') H0 τ ' H0 { } '... { } { n } 0 τ' 0 β β Each term descrbes a partcular evoluton of { n } as magnary tme ncreases

17 0-order term one of the 2-order terms β magnary tme 0 j + t(1, 2) j τ' τ + n ths example { n } = 0,1, 2,0 U({ n ( τ)}) dτ K = τk + 1 τk τk { n ( τ )} k = 1 0 Z e n H d n all possble trajectores for partcles wth K hoppng transtons β potental energy contrbuton off-dagonal matrx elements for the trajectory wth K knks at tmes (ordered sequence on the -cylnder) β > τ >... > τ > τ > 0 β K { ( 0)} ( ) { ( 0)} 2 1 n ths example, for K=2, t equals for bosons t 2 t 2

18 β 0 hgh-order term for - Z=Tre β H Smlar expanson n hoppng terms for G =Tr (, ) (, )e H I b τ b τ -β I I I + two specal ponts for Ira and asha β τ τ I 0 I I Z G The rest s worm algorthm n ths I confguraton space: draw and erase lnes usng exclusvely Ira and asha

19 ergodc set of local updates tme shft: τ Ira or asha τ ' j j space shft ( partcle type): j space shft ( hole type): j Insert/delete Ira and asha: Z G Z G connects and confguraton spaces

20 Fg.1 Fg.2 Fg.3 Fg.4 Fg.5

21 Path-ntegrals n contnuous space P=β/ τ 2 mr ( + 1 R) Z = dr1... drp exp + U( R) τ = 1 2τ P τ r r,1, 2 R = ( r, r,..., r ), 1, 2, 2 1 P

22

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24 G Ira Ira asha (advance/recede update)

25 G asha Ira Ira (swap update)

26 ot necessarly for closed loops! Feynman (space tme) dagrams for fermons wth contact nteracton (attractve) = U (n=1 postve Hubbard model too) Par correlaton functon a ( r, τ ) a ( r, τ ) a ( r, τ ) a ( r, τ ) Z G The rest s worm algorthm n ths I confguraton space: draw and erase nteracton vertexes usng exclusvely Ira and asha

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

28 Grand canoncal ensemble (a must for dsorder problems!) V(x) R1 R2 Fg.1 Fg.2

29 Some examples: Weakly nteractng Bose gas: T ( n a)/ T 3 3 C 1/3 (0) C na = 510 ho, Landau 04 T / T = 1.078(1)? C 0 Worm algorthm: Plat, Gorgn, P 100,000 correct value Imperfect crossng due to correctons to scalng

30 ott nsulator superflud T=0 phase dagram: ( µ / U, t/ U) plane, 3D case gaps control the exponental decay of the Green s functon G( p= 0, τ ) n tme I SF ( µ / U ) ± determne gaps for addng/removng partcles from the I state wth n = 1 Otherwse, good luck n calculatng energy dfferences 3 E ( ± 1) E ( ) for = Lwth L= 40

31 Current standard for smulatons of bosons n optcal lattces and n traps: all expermental parameters as s, ncludng partcle number ~10 6

32 Quantum spn chans gaps, spn wave spectra, magnetzaton curves H = [ J x ( SjxSx + S jysy ) + Jz S jzsz ] H Sz < j> Energy gap: One dmensonal S=1 chan wth J / J = 0.43 z x Spn gap = (5) Z -factor Z = 0.980(5) G( p, τ>> 1) Z e τ + e 1+ e ( β τ) β

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

34 magnetzaton curves S=1/2 Hesenberg chan C data Bethe ansatz

35 magnetzaton curves

36 ore tools: 1.Densty matrx nr ( ', r) = ψ ( r', τ ) ψ ( r, τ ) (and the condensate fracton) s as cheap as energy 2. µ s an nput parameter, and µ s a smple dagonal property 3. But also compressblty ( ) 2 κ VT = µ P ( ) = P ( ) e µ µ ' µ ( ' µ ) / T 4. Added partcle wavefuncton: G r r G r G G r G r r ( β /2,, ') = ψ ( ) 1 1ψ( ') = ϕ( ) ϕ( ') moblty thresholds, partcpaton rato, etc.

37 Why bother wth algorthms? Effcency PhD whle stll young PhD whle stll young Better accuracy Large system sze ore complex systems Fnte-sze scalng Crtcal phenomena Phase dagrams Relably! ew physcs 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 ) Grτ (,) ϕ ( r)

38 Wave functon of the added partcle Complete phase dagram L = 160 /U Gap n the Ideal system t/ U > It s a theorem that for the compressblty s fnte E GAP