A Quick introduction to Quantum Monte Carlo methods

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Florence Charles
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1 A Quck ntroducton to Quantum Mont Carlo mthods Fabn Alt LPT, Unv. Paul abatr Toulous Contact : alt@rsamc.us-tls.fr ALP Tutoral PI 08/09/006

2 Quantum Mont Carlo What s Quantum Mont Carlo (QMC)? Most gnral dfnton: A stochastc mthod to solv th chrödngr quaton QMC coms n many dffrnt flavours Zro Tmratur or T 0 tatstcs of artcls : Frmons, Bosons or Boltmannons Contnuum or dscrt sac Fnt numbr of artcls or Thrmodynamc lmt 3 Today n modls on lattc : fnt T, Bosons (sns), Dscrt sac, Thrmodynamc lmt Prformancs Larg systms ( thrmodynamc lmt, T0) : u to sns! Quanttatv modlng of quantum magnts At last 4 dffrnt QMC communts! Accurat dscrton of has transtons, Asymtotc rgm (crtcal xonnts) 0 6 F. Alt (Toulous) Introducton to QMC ALP Tutoral PI

3 QMC a larg famly tr «Varatonal Mont Carlo» famly «Dffuson Mont Carlo» famly Proctor Mont Carlo, Grn Functon Mont Carlo, Dffuson Mont Carlo ( Fxd Nod, tochastc Rconfguraton) (Partal) Rf. for lattc modls :. orlla and L. Carott, Phys. Rv. B 6, 599 (000) «Dtrmnntal Mont Carlo» famly Auxlary Fld Mont Carlo, Dtrmnntal Mont Carlo, Hrsch-Fy (Partal) Rf. for lattc modls : F. F. Assaad, Lctur Nots, NIC srs 0, 99 (003) «Path ntgral Mont Carlo» famly Lattc systms (clustr algorthms) Basc Ida : Mang Quantum systm n dmnson d Classcal systm n dmnson d Thn do classcal Mont Carlo on th quvalnt roblm F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 3

4 A Quck ntroducton to Quantum Mont Carlo mthods for lattc quantum sn modls Drvaton of confguraton sac : Path ntgral aroach Mont Carlo movs : quantum clustr algorthms Th sgn roblm Practcal consdratons ALP Tutoral PI 08/09/006 4

5 Quantum Mont Carlo smulatons Not as «asy» as classcal Mont Carlo Z Calculatng th nrgy gnvalu E c solvng th roblm Nd to fnd a mang of th quantum artton functon to a classcal roblm Dffrnt aroachs βh Tr Path ntgrals (tm-dndnt rturbaton thory n magnary tm) tochastc rs Exanson (hgh tmratur xanson) gn roblm f som c < 0 (thus try to avod ths) Thn nd ffcnt udats for th quvalnt classcal roblm c Z βe c Tr βh c c F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 5

6 6 F. Alt (Toulous) Introducton to QMC ALP Tutoral PI Hamltonan of sn ½ modls Hamltonan of sn ½ modls Examl : XXZ modl n a fld Ansotroc xchang ntractons : XY, Z Magntc fld h Hsnbrg modl : XY Z Hamltonan matrx n -st bass XY y y x x XY XXZ h h H,, ) ( ) ( h H, r r { },,, h h H xy xy

7 Trottr dcomoston Bass of most QMC algorthms Hr : Gnrc mang of a quantum sn systm onto a classcal Isng modl Not lmtd to scal cass lt Hamltonan nto two asly dagonalabl cs H H H εh εh εh ε O( ) Obtan a dcomoston of th artton functon H H H H () H () H (3) H (4) Z Tr Tr[( βh ΔτH Tr Tr[( β ( H H ) Δτ ( H H ) ΔτH M ) ] O( Δ τ ) M ) ] ( Δτ β / M ) Insrt M sts of comlt bass stats,..., M ΔτH M M ΔτH M 3 ΔτH ΔτH F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 7

8 Examl : n ½ Hsnbrg modl Quantum roblm n d dmnsons mas onto a classcal roblm n d Exand th stats α n th gnbass Effctv Isng-modl n d dmnsons wth - and 4-sts ntracton trms Z,..., M ΔτH M M ΔτH M 3 ΔτH ΔτH Each of th matrx lmnts ΔτH, corrsonds to a row of shadd laqutts and quals th roduct ovr thos laqutts F. Alt (Toulous) Introducton to QMC ALP Tutoral PI Consrvaton of magntaton : Contnuous worldlns 8

9 Wghts for th sn ½ Hsnbrg modl Th artton functon bcoms a sum of roducts of laqutt wghts Z W ( C) C w( C Th only allowd laqutt-confguratons ar (hr h0) C laqutts ) Δτ / 4 Δτ /4 ch( Δτ / ) Δτ /4 sh( Δτ /) Frromagnt (<0) : All wghts ar ostv Antfrromagnt on a bartt lattc : rform a gaug transformaton on on sublattc ± ± ( ) ( ) ( ) Frustratd antfrromagnt : w hav a sgn roblm F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 9

10 Worldln aroach : ummary Each vald confguraton contnuous worldlns on chckrboard W ( C) laqutts w( C ) / 4 Δτ Δτ /4 ch( Δτ / ) Δτ /4 sh( Δτ /) Worldln QMC amlng ovr all (mortant) worldln confguratons Accordng to th abov wght Try to gnrat a nw confguraton from a gvn on Alrady roblms ars F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 0

11 F. Alt (Toulous) Introducton to QMC ALP Tutoral PI Intrmo : tochastc rs Exanson (E) Intrmo : tochastc rs Exanson (E) Altrnatv aroach : Exanson n nvrs tmratur Usng th bond Hamltonans mlar to ath ntgral aroach (Mnor) dffrnc n th tratmnt of dagonal trms ( ) α α β β α β n b b b n n n n n H n H n H n Z,..., 0 0 ) (! ) Tr(! ) Tr( ), ( b H b H H d (,) confguraton orator H o (3,4) H d (3,4) H o (3,4) 3 4 H d (,) ( ) d xy o h H H ), ( ), ( ) ( (andvk, 99)

st solvd roblm : th contnuous tm lmt ystmatc rror du to fnt valu of Δτ («Trottr rror») Nd to rform an xtraolaton to Δτ 0 from smulatons wth dffrnt valus of Δτ (or Trottr numbr M) Th lmt Δτ 0 can b

12 st solvd roblm : th contnuous tm lmt ystmatc rror du to fnt valu of Δτ («Trottr rror») Nd to rform an xtraolaton to Δτ 0 from smulatons wth dffrnt valus of Δτ (or Trottr numbr M) Th lmt Δτ 0 can b takn drctly n th constructon of th algorthm! (Prokof v t al., 996) Numbr of changs Δτ β N c M stays fnt as Δτ 0 Dffrnt comutatonal aroach: Dscrt tm : stor confguraton at all tm sts Contnuous tm : stor tms at whch confguraton changs ( ntal stat) F. Alt (Toulous) Introducton to QMC ALP Tutoral PI

13 nd solvd roblm solvd hft a knk Insrt or rmov two knks (knk-antknk ar craton rocss) Problms wth local udats Rstrctd to canoncal nsmbl No chang of magntaton, artcl numbr, wndng numbr Crtcal slowng down oluton for classcal Mont Carlo was clustr algorthms Gnralaton to quantum cas s ossbl! Loo, Drctd loos, Worm algorthms

14 Worm algorthm : ntutv vw Q : How to udat non-locally ths confguraton? F. Alt (Toulous) Introducton to QMC ALP Tutoral PI (E Rrsntaton) 4

15 Worm algorthm : ntutv vw ) Insrt a worm (had and tal) F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 5

16 Worm algorthm : ntutv vw ) Insrt a worm (had and tal) ) Mov th worm had Modfy locally th confguraton on th fly F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 6

17 Worm algorthm : ntutv vw ) Insrt a worm (had and tal) ) Mov th worm had Modfy locally th confguraton on th fly 3) At vry vrtx : Choos an xt straght um turn bounc F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 7

18 Worm algorthm : ntutv vw ) Insrt a worm (had and tal) ) Mov th worm had Modfy locally th confguraton on th fly 3) At vry vrtx : Choos an xt straght, um, turn, bounc F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 8

19 Worm algorthm : ntutv vw ) Insrt a worm (had and tal) ) Mov th worm had Modfy locally th confguraton on th fly 3) At vry vrtx : Choos an xt straght, um, turn, bounc F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 9

20 Worm algorthm : ntutv vw ) Insrt a worm (had and tal) ) Mov th worm had Modfy locally th confguraton on th fly 3) At vry vrtx : Choos an xt straght, um, turn, bounc 4) Annhlat worms F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 0

21 Worm algorthm : ntutv vw ) Insrt a worm (had and tal) ) Mov th worm had Modfy locally th confguraton on th fly 3) At vry vrtx : Choos an xt straght, um, turn, bounc 4) Annhlat worms NEW CONFIGURATION F. Alt (Toulous) Introducton to QMC ALP Tutoral PI

22 Intrmo : How to choos th worm s movs? Gvn a vrtx (laqutt) confguraton and an ntranc lg, whr to xt? Consdr xt lg, gvn ntranc lg at a vrtx n confguraton c wght w(c) Assgn ths ath a robablty P(, c) um ovr all aths must qual unty P(, c) w( c) Z h Choos xt lg wth robablty P (, c) P (, c) P ( 3, c) P( 4, c) P(, c) Consdr th rvrtd ath P(, c), ladng back to c ladng to confguraton c wght w(c) Imos Local dtald balanc P (, c) w( c) P(, c) w( c) F. Alt (Toulous) Introducton to QMC ALP Tutoral PI

23 Quantum clustr algorthms Tny dffrncs n how th worm xactly movs lad to dffrnt algorthms Worm algorthm : random Mtrools mov Drctd loo algorthm : locally mrovd «clvr» mov Loo algorthm : dtrmnstc mov Dffrnc n Rrsntaton E : usually slghlty fastr Path Intgral (PI) : good whn larg dagonal trms Cods n ALP : Whch qlgorthm to choos? loor Loo algorthm n PI/E : For modls wth sn nvrson symmtry.g. : Hsnbrg, XY, XXZ modl whout fld drloo-ss Drctd loos n E : Modls wthout nvrson symmtry (most gnra.g. : modls wth fld, boson modls tc worm Worm algorthm n PI : Modls wth larg dagonal lmnts.g. : larg Isng ansotroy All ths algorthms gv accurat rsults for larg systms

24 Intrmo 3 : Th sgn roblm In mang of quantum to classcal systm «gn roblm» f som of th <0 Cannot ntrrt as robablts Aars n smulaton of frmons and frustratd magnts Way out : Prform smulatons usng and masur th sgn : amlng A A Tr Tr [ Ax( βh )] [ x( βh )] A Z A sgn sgn A A gn gn F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 4

25 Intrmo 3 : Th sgn roblm Th avrag sgn bcoms vry small gn sgn Z Z Z βvδf Both n systm s and nvrs tmratur Ths s th orgn of th sgn roblm! Th rror of th sgn: Δgn gn Nd of th ordr gn N gn gn N x( βvδf ) N gn masurmnts for suffcnt accuracy mlar roblm occurs for th obsrvabls Exonntal growth! Imossbl to trat larg systms or low tmraturs βvδf N F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 5

26 In ractc, what can I do? mulat larg samls of lattc quantum sn modls For non-frustratd modls (Not : Dagonal (Isng) frustraton s OK) Ths ncluds most Hsnbrg-lk modls XY, Isng ansotroy, magntc fld, all valus of, sngl-on ansotroy, Frustratd modls ar ossbl : but quckly sgn roblm arss In all dmnsons on all lattcs (on can dfn ts own scfc lattc) For all valus of T (ncludng T 0) Whch quantts ar accssbl? A lot of obsrvabls : Enrgy, cfc hat, usctblts, tructur factor, n stffnss, Grn functons In som cass : on can dfn ts own obsrvabl How much tm? Tycally smulaton tm scals as Numbr of sns x Invrs Tmratur For mor dtals : s hands-on ssson F. Alt (Toulous) Introducton to QMC ALP Tutoral PI 6

27 Thanks! ALP Tutoral PI 08/09/006 7

Grand Canonical Ensemble

Grand Canonical Ensemble Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls