BOLD DIAGRAMMATIC MONTE CARLO: From polarons to path- integrals to skeleton Feynman diagrams

Transcription

1 BOLD DIAGRAMMATIC MONTE CARLO: From polarons to path- integrals to skeleton Feynman diagrams Nikolay Prokofiev, Umass, Amherst Tallahassee, NHMFL (January 2012)

2 High-energy Standard model physics High-Tc Hubbard superconductors model Quantum chemistry Coulomb & gas band structure Quantum Heisenberg magnetism model Periodic Anderson Heavy fermion & Kondo materials lattice models - Introduced in mid 1960s or earlier - Still not solved (just a reminder, today is 01/13/2012) - Admit description in terms of Feynman diagrams

3 Feynman Diagrams & Physics of strongly correlated many- body systems In the absence of small parameters, are they useful in higher orders? N.Abel, Yes, 1828: with sign-blessing Divergent for regularized series are skeleton the devil's graphs! invention... And if they are, how to handle millions and billions of skeleton graphs? Steven Sample Weinberg, them Physics with Today, Diagrammatic Aug Monte : Also, Carlo it was techniques easy to imagine (teach computers rules of quantum any number field theory) of quantum field theories of strong interactions but what could anyone do with From current strong-coupling Unbiased them? solutions beased theories based on one lowest on millions of graphs with order skeleton graph (MF, extrapolation to the infinite RPA, GW, SCBA, GG 0, GG, diagram order

4 Answering Weinberg s question: Equation of State for ultracold fermions & neutron matter at unitarity MIT group: MrRn Zwierlein, Mark Ku, Ariel Sommer, Lawrence Cheuk, Andre Schirotzek BDMC results Kris Van Houcke, Felix Werner, Evgeny Kozik, Boris Svistunov, NP Uncertainty due to locaron of the a resonance B 834 ± 1.5 G s virial expansion (3d order) Ideal Fermi gas QMC for connected Feynman diagrams NOT particles! Sign blessing Sign problem

5 Conventional Sign-problem vs Sign-blessing Sign-problem: (diagrams for Z ) Computational complexity is exponential in system volume tcpu exp{# L d β} exptrapolation to L and error bars explode before a reliable can be made Feynamn diagrams: No (for ln Z ) L limit to take, selfconsistent formulation, admit analytic results and partial resummations. n 3/2 n!2 n n Sign-blessing: Number of diagram of order is factorial thus the (diagrams for ln Z ) only hope for good series convergence properties is sign alter- nation of diagrams leading to their cancellation. Still, t n!2 n n CPU 3/2 i.e. Smaller and smaller error bars are likely to come at exponential price (unless convergence is exponential).

6 Standard Monte Carlo setup: - configuraron space (depends on the model and it s representaron) - each cnf. has a weight factor cnf / We E T cnf - quanrty of interest Acnf A cnf cnf A W cnf W cnf cnf

7 Statistics: { states} e E state / T O state Monte Carlo MC O state { states} states generated from probability distribution state / e E T Anything: { ν any set of variables} F( ν) O( ν) Monte Carlo MC i arg[ F ( ν )] { ν } e O ( ν ) states generated from probability distribution F( ν ) Connected Feynman Anything diagrams, e.g. for the proper self-energy ν diagram order topology internal variables Monte Carlo Answer to S. Weinberg s question MC Σ sign( F( v)) ν

8 Classical MC ur r r r r r r ur Z y dx dx dx W x x x y ( ) ( ) 1 2K N 1, 2, K N, the number of variables N is constant Quantum MC (o`en) ur r r r r r r ur Z y dx dx dx D x x x y ( ) ( ) 1 2K n n ξ; 1, 2, K n, n 0 ξ term order different terms of of the same order IntegraRon variables ContribuRon to the answer or weight (with differenral measures!)

9 ur r r r r r r ur A y dx dx dx D x x x y D ( ) ( ) 1 2K n n ξ; 1, 2, K n, n 0 ξ ν ν Monte Carlo (Metropolis) cycle: Diagram W ν suggest a change Accept with probability R acc W W ν ' ν Same order diagrams: r Dν ' ( dx) r D ( dx) ν n n O(1) Business as usual UpdaRng the diagram order: r n m D ' ( dx) r ν m r ( dx) n D ( dx) ν Ooops

10 Balance EquaRon: If the desired probability density distriburon of different terms in the stochasrc sum is P ν then the updarng process has to be staronary with respect to (equilibrium condiron). O`en P ν P ν W ν D R D R ν ν' ν Ων ( ν ') accept ν' Ων '( ν ) updates ν ν ' updates ν' ν ν' ν accept ν ν Flux out of Flux to Ω ( ') ν ν Is the probability of proposing an update transforming ν to ν ' Detailed Balance: solve equaron for each pair of updates separately D R D R ν ν' ν Ων ( ν ') accept ν' Ων '( ν ) ν' ν accept

11 Let us be more specific. EquaRon to solve: r r r r r r r r r n D x1, K, xn ( dx) Ω x1, K, xn m ( dx) R D x1, K, xn m ( dx) Ω R ( ) nn m( ) accept ( ) m n n m n m n m n n, n m n m, n accept D ν Ω ( ') ν ν r x r 1, K, x new variables n n m are proposed from the normalized probability distribution D ν Ω ( ) ν' ν SoluRon: R R D x x Ω R D ( x,, x ) ( x,, x ) n n m accept n m( 1, K, n m) n m, n n m n accept n 1 K n Ωn, n m 1 K n m All differential measures are gone! Efficiency rules: - try to keep R ~1 - simple analytic function Ω ( x, K, x ) nn, m n 1 n m ENTER

12 ConfiguraRon space (diagram order, topology and types of lines, internal variables) Diagram order MC update v { q, τ, p r } i i i Diagram topology This is NOT: write/enumerate diagram after diagram, compute its value, and then sum

13 Polaron problem: H H H H quasiparticle particle environment coupling E( p 0), m, G( p, t),... * Electrons in semiconducting crystals (electron-phonon polarons) e e H ε ( p) a a p q p ω( p)( b b 1/ 2) q ( Va q p qab p q hc..) pq p q electron phonons el.- ph. interacron

14 ( ) ( )( 1/2) ( ε..) p p ω p bb q q Vq p q pbq H p a a a a hc p q electron phonons el.- ph. interacron p q p pq q p q Green funcron: Gp (, τ ) a(0) a ( τ ) a e a e - τh p p p p Sum of all Feynman diagrams PosiRve definite series in the ur ( p, τ ) τh representaron 0 p τ

15 q 2 q 1 q 3 q 2 Gpτ (, ) Feynman digrams p 0 τ1 p q 1 τ ' 1 p q 2 τ 4 ' p τ Graph-to-math correspondence: G ur p r r r ξ; r, r, r x, ur, τ where u r x( q r, τ, τ ') (, τ ) dxdx ( ) 1 2K dxn D x1 x2 K n p n 0 PosiRve definite series in the ξ q i V qi ur ( p, τ ) n i i i i is a product of τ1 pi representaron τ 1 ' τ 1 ( p i )( i ' i ) e ε τ τ ω( q)( τi' τ i) e q i τ 1 '

16 0 Diagrams for: bq (0) bq (0) ap q q ( 0) ap q q( τ) bq ( τ) bq ( τ) there are also diagrams for oprcal conducrvity, etc. Doing MC in the Feynman diagram configuration space is an endless fun!

17 The simplest algorithm has three updates: Insert/Delete pair (increasing/decreasing the diagram order) q 2 q 1 q 1 p1 p 2 p p3 τ τ ' 1 1 p p p ' p ' p4' p2 5 τ 2 τ 1 τ ' 2 τ ' 1 p p D ν D ν ' D / D e ν ' ν 2 ω( q2)( τ2 ' τ2) ( ( p2') ( p2)) ( 1 2 ) p3 p2 2 1 Vq e ε ε τ τ e ε ε τ τ 2 ( ( ') ( ))( ' ) R D Ω Dν' 1 D Ω ( x, K, x ) D ( n 1) Ω ( x, K, x ) ν' n 1, n ν nn, 1 1 n 1 ν nn, 1 1 n 1 In Delete select the phonon line to be deleted at random

18 The optimal choice of Ω ( x, K, x ) nn, 1 1 n 1 depends on the model Frohlich polaron: ε 2 p /2 m, ω q ω0, Vq α/ q D / [( p2') p2 ]( τ1 τ2) [( p3') p2 ]( τ2' τ1) q ω0( τ2' τ2) 2m D e e dqd d d ν' ν 2 q 2 ϕ θ τ τ (0, τ ) 1. Select anywhere on the interval from uniform prob. density 2 2. Select τ ' 0( 2' 2) 2 anywhere to the left of τ 2 from prob. density e ω τ τ (if reject the update) q 2 τ 2 ' > τ ( q2 /2 m)( τ2' τ2) 3. Select from Gaussian prob. density, i.e. e 2 Ω q2 m 2 2 ω ( τ ' τ ) ( /2 )( τ ' τ ) nn, 1( τ2, τ2', q2) e e

19 New τ : p τ last τ τ ' Standard heat bath probability density ~ Always accepted, R 1 ( p)( ' last ) e ε τ τ This is it! Collect statistics for G( p, τ). Analyze it using Gp (, τ ) Ze p E( p) τ, etc.

20 Diagrammatic Monte Carlo in the generic many-body setup Feynman diagrams for free energy density G ( p, τ ) U( q) x G ( p, τ ) x x x G (0) (0) G G Σ G U U - U Π U

21 Bold (self-consistent) Diagrammatic Monte Carlo DiagrammaRc technique for ln(z ) diagrams: admits parral summaron and self- consistent formularon No need to compute all diagrams for G and U : G( p, τ ) Σ( p, τ τ ) Σ... Dyson Equation: Gpτ (,) Screening: U U Π Calculate irreducible diagrams for Σ, Π, to get G, U,. from Dyson equations

22 Σ Π In terms of exact propagators Dyson Equation: Σ Screening: Π

23 More tools: Incorporating DMFT solutions for Σ [ G ] loc loc Σ ' Σ loc all electron propagator lines in all graphs are local, G Gloc Grrδ rr ' at least one electron propagator in the graph is non-local, i.e. the rest of graphs Impurity solver MC simulation

24 More tools: Build diagrams using ladders: (contact potential) Γ (0) U (0) G (0) G Σ Π[ - ] In terms of exact propagators Dyson Equations: Σ Π

25 Fully dressed skeleton graphs (Heidin): Σ Σ Γ 3 Π Π.. Irreducible 3-point vertex: Γ 3 1 all accounted for already!...

26 H H U nn n 0 H1 ij i j µ i i t( ni, nj) bjbi ij i < ij> Laece path- integrals for bosons and spins are diagrams 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 0 0 τ βh0 Tr e 1 H ( ) d H ( ) H ( ) d d '... β imaginary Rme t(1, 2) τ' τ 0 i j i j n i 0,1, 2,0

27 Diagrams for - ZTr e β H G Diagrams for Tr T τ b ( τ ) b ( τ )e -β H IM M M I I β β imaginary Rme imaginary Rme M I 0 laece site 0 laece site The rest is convenronal worm algorithm in conrnuous Rme

28 Pβ/ τ 2 mr ( i 1 Ri) Z dr1... drp exp U( R) τ i 1 2τ Path- integrals in conrnuous space are diagrams of closed loops too! P τ r r i,1 i, 2 R ( r, r,..., r ) i i, 1 i, 2 i, N 2 1 P

29 Diagrams for the attractive tail in Ur () : N τ Ur ( r) θ( r r r) << 1 If and N>>1 all the effort is for something small! j i j i j c Masha V( r ) τ V( r ) τ ij ij e 1 ( e 1) 1 p ij statistical interpretation p ij Ira V( r ij ) ignore : stat. weight 1 i j Account for ( ) V r ij : stat. weight p Faster than conventional scheme for N>30, scalable (size independent) updates with exact account of interactions between all particles

30 Other applications: Continuous-time QMC solves (impurity solvers) are standard DMC schemes Fermions with contact interaction ν { n; r, τ ; r, τ ;...; r, τ } n n det G ij G G... G p G G... G p G G... G p1 p2 pp!!!!! p p Z... ( drd ) ( U ) det G ( xi, xj)det G ( xi, xj) p 0 τ Rubtsov (2003) more in A. Millis s talk

31 Skeleton diagrams up to high- order: do they make sense for g 1? NO g Diverge for large even if are convergent for small. g Dyson: Expansion in powers of g is asymptotic if for some (e.g. complex) g one finds pathological behavior. Electron gas: Bosons: e ie U U [collapse to infinite density] Math. Statement: # of skeleton graphs 3/2 2 n n n! asymptotic series with zero conv. radius (n! beats any power) A N Asymptotic series for g 1 with zero convergence radius 1/ N

32 Skeleton diagrams up to high- order: do they make sense for g 1? YES Divergent series outside of finite convergence radius can be re-summed. Dyson: - Does not apply to the resonant Fermi gas and the Fermi-Hubbard model at finite T. - not known if it applies to skeleton graphs which are NOT series in bare g : cf. the BCS answer (one lowest-order diagram) - Regularization techniques Δ e 1/ g # of graphs is 3/2 2 n n n! but due to sign-blessing they may compensate each other to accuracy better then 1/ n! leading to finite conv. radius

33 Re-summation of divergent series with finite convergence radius. Example: A c 3 9 / / 4... n 0 n бред какой то Define a function such that: f nn, fnn, 1 for n<< N fnn, 0 for n> N 1 N N a b n f f nn, nn, e n 2 / N nln( n) e ε (Gauss) (Lindeloef) A c f lim A N Construct sums N n n, N and extrapolate to get N n 0 A A N ln 4 1/ N