Second Lecture: Quantum Monte Carlo Techniques

Size: px

Start display at page:

Download "Second Lecture: Quantum Monte Carlo Techniques"

Crystal McLaughlin
5 years ago
Views:

Second Lecture: Quantum Monte Carlo Techniques Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden http://www.pks.

1 Second Lecture: Quantum Monte Carlo Techniques Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden aml@pks.mpg.de Lecture Notes at Modern theories of correlated electron systems - Les Houches - 19/5/2009

2 Outline of today s lecture Quantum Monte Carlo (World line) Loop Algorithm Stochastic Series Expansion Worm Algorithm Green s function Monte Carlo Sign Problem What is not treated here: Auxiliary field Monte Carlo for fermions Diagrammatic Monte Carlo CT-QMC for Quantum impurity problems Real Time Quantum Monte Carlo

3 1. Quantum Monte Carlo (World lines)

4 Quantum Monte Carlo In quantum statistical mechanics the canonical partition function now reads: Z(T ) = Tr e H/T Expectation values: O = 1 Z(T ) Tr[Oe H/T ] Tr[Oρ(T )] We need to find a mapping onto a classical problem in order to perform MC World-line methods (Suzuki-Trotter, Loop algorithm, Worm algorithm,...) Stochastic Series Expansions Potential Problem: The mapping can give probabilities which are negative infamous sign problem. Common cause is frustration or fermionic statistics.

5 Quantum Monte Carlo Feynman (1953) lays foundation for quantum Monte Carlo Map d dimensional quantum system to classical world lines in d+1 dimensions

6 Quantum Monte Carlo Feynman (1953) lays foundation for quantum Monte Carlo Map d dimensional quantum system to classical world lines in d+1 dimensions classical particles space

7 Quantum Monte Carlo Feynman (1953) lays foundation for quantum Monte Carlo Map d dimensional quantum system to classical world lines in d+1 dimensions classical imaginary time quantum mechanical particles space world lines

8 Quantum Monte Carlo Feynman (1953) lays foundation for quantum Monte Carlo Map d dimensional quantum system to classical world lines in d+1 dimensions classical imaginary time quantum mechanical particles space world lines Use Metropolis algorithm to update world lines

9 The Suzuki-Trotter Decomposition Generic mapping of a quantum spin system to Ising model (vertex model) basis of most discrete time QMC algorithms not limited to special models Split Hamiltonian into two easily diagonalized pieces H = H 1 + H 2 e εh = e ε ( H 1 + H 2 ) = e εh 1 e εh 2 + O(ε2 ) Obtain the checkerboard decomposition H H 1 H 2 = + Z = Tr[ exp( βh) ] = Tr[ e β ( H 1 + H 2 ) ] = Tr[ e (β / M )H 1 e (β / M )H ] M 2 + O(β 3 / M 2 ) imaginary time space direction

10 Path integral QMC Use Trotter-Suzuki or a simple low-order formula gives a mapping to a (d+1)-dimensional classical model imaginary time Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) i 1 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i 1 {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 space direction partition function of quantum system is sum over classical world lines

11 Path integral QMC Use Trotter-Suzuki or a simple low-order formula gives a mapping to a (d+1)-dimensional classical model imaginary time Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) i 1 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i 1 {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 space direction place particles (spins) partition function of quantum system is sum over classical world lines

12 Path integral QMC Use Trotter-Suzuki or a simple low-order formula gives a mapping to a (d+1)-dimensional classical model imaginary time Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) i 1 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i 1 {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 space direction place particles (spins) for Hamiltonians conserving particle number (magnetization) we get world lines partition function of quantum system is sum over classical world lines

13 Calculating configuration weights Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 Examples: particles with nearest neighbor repulsion H = t i,j (a i a j + a j a i)+v i,j n i n j

14 Calculating configuration weights Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 Examples: particles with nearest neighbor repulsion H = t i,j (a i a j + a j a i)+v i,j n i n j 1

15 Calculating configuration weights Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 Examples: particles with nearest neighbor repulsion H = t i,j (a i a j + a j a i)+v i,j n i n j 1 ( τt) 2

16 Calculating configuration weights Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 Examples: particles with nearest neighbor repulsion H = t i,j (a i a j + a j a i)+v i,j n i n j 1 ( τt) 2 (1 τv ) 3

17 Calculating configuration weights Z = Tre βh = Tr e MΔτH = Tr( e ΔτH ) M = Tr( 1 ΔτH) M + O(βΔτ) {(i 1...i M )} = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 Examples: particles with nearest neighbor repulsion H = t i,j (a i a j + a j a i)+v i,j n i n j 1 ( τt) 2 (1 τv ) 3 ( τt) 2

18 Monte Carlo updates just move the world lines locally probabilities given by matrix element of Hamiltonian example: tight binding model H = t( c i c i+1 + c i+1 c ) i i, j

19 Monte Carlo updates just move the world lines locally probabilities given by matrix element of Hamiltonian example: tight binding model H = t( c i c i+1 + c i+1 c ) i i, j introduce or remove two kinks:

20 Monte Carlo updates just move the world lines locally probabilities given by matrix element of Hamiltonian example: tight binding model introduce or remove two kinks: H = t( c i c i+1 + c i+1 c ) i i, j shift a kink:

21 Monte Carlo updates just move the world lines locally probabilities given by matrix element of Hamiltonian example: tight binding model introduce or remove two kinks: H = t( c i c i+1 + c i+1 c ) i i, j shift a kink: P =1 P = ( Δτt) 2 P = Δτt

Monte Carlo updates just move the world lines locally probabilities given by matrix element of Hamiltonian example: tight binding model introduce

22 Monte Carlo updates just move the world lines locally probabilities given by matrix element of Hamiltonian example: tight binding model introduce or remove two kinks: H = t( c i c i+1 + c i+1 c ) i i, j shift a kink: P =1 P = ( Δτt) 2 P = Δτt P = min[ 1,(Δτt) 2 ] P = min[ 1,1/(Δτt) 2 ] P = P =1

64, 853 (1996)] i 1 imaginary time i 8 i 7 i 6 i 5 i 4 i 3 imaginary time τ 5 τ 6 τ 4 τ 2 τ

23 The continuous time limit the limit Δτ 0 can be taken in the algorithm [Prokof'ev et al., Pis'ma v Zh.Eks. Teor. Fiz. 64, 853 (1996)] i 1 imaginary time i 8 i 7 i 6 i 5 i 4 i 3 imaginary time τ 5 τ 6 τ 4 τ 2 τ 3 i 2 τ 1 i 1 space direction space direction discrete time: store configuration at all time steps continuous time: store times at which configuration changes

24 Advantages of continuous time No need to extrapolate in time step a single simulation is sufficient no additional errors from extrapolation Less memory and CPU time required Instead of a time step Δτ << t we only have to store changes in the configuration happening at mean distances t Speedup of 1 / Δτ 10 Conceptual advantage we directly sample a diagrammatic perturbation expansion

25 2. Cluster updates: The loop algorithm

26 Problems with local updates

27 Problems with local updates Local updates cannot change global topological properties number of world lines (particles, magnetization) conserved winding conserved braiding conserved cannot sample grand-canonical ensemble

28 Problems with local updates Local updates cannot change global topological properties number of world lines (particles, magnetization) conserved winding conserved braiding conserved cannot sample grand-canonical ensemble Critical slowing down at second order phase transitions solved by cluster updates

29 Problems with local updates Local updates cannot change global topological properties number of world lines (particles, magnetization) conserved winding conserved braiding conserved cannot sample grand-canonical ensemble Critical slowing down at second order phase transitions solved by cluster updates Tunneling problem at first order phase transitions solved by extended sampling techniques

30 Cluster algorithms: the formal explanation Z = W (C) = W (C,G) with W (C) = W (C,G) C C G G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = 1 graph G allowed for C 0 otherwise C i (C i,g) G (C i+1,g) C i+1

31 Cluster algorithms: the formal explanation Extend the phase space to configurations + graphs (C,G) Choose graph weights independent of configuration (C) Perform updates Z = W (C) = W (C,G) with W (C) = W (C,G) C C G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = 1 graph G allowed for C 0 otherwise G C i (C i,g) G (C i+1,g) C i+1

32 Cluster algorithms: the formal explanation Extend the phase space to configurations + graphs (C,G) Choose graph weights independent of configuration (C) Perform updates Z = W (C) = W (C,G) with W (C) = W (C,G) C C G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = 1 graph G allowed for C 0 otherwise G C i (C i,g) G (C i+1,g) C i+1 1. Pick a graph G P[G] = V (G) W (C)

33 Cluster algorithms: the formal explanation Extend the phase space to configurations + graphs (C,G) Z = W (C) = W (C,G) with W (C) = W (C,G) C C Choose graph weights independent of configuration (C) G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = 1 graph G allowed for C 0 otherwise G Perform updates 2. Discard configuration C i (C i,g) G (C i+1,g) C i+1 1. Pick a graph G P[G] = V (G) W (C)

34 Cluster algorithms: the formal explanation Extend the phase space to configurations + graphs (C,G) Z = W (C) = W (C,G) with W (C) = W (C,G) C C Choose graph weights independent of configuration (C) G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = 1 graph G allowed for C 0 otherwise G Perform updates 2. Discard configuration C i (C i,g) G (C i+1,g) C i+1 1. Pick a graph G P[G] = V (G) W (C) 3. Pick any allowed new configuration

35 Cluster algorithms: the formal explanation Extend the phase space to configurations + graphs (C,G) Z = W (C) = W (C,G) with W (C) = W (C,G) C C Choose graph weights independent of configuration (C) G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = 1 graph G allowed for C 0 otherwise G Perform updates 2. Discard configuration 4. Discard graph C i (C i,g) G (C i+1,g) C i+1 1. Pick a graph G P[G] = V (G) W (C) 3. Pick any allowed new configuration

36 Cluster algorithms: the formal explanation Extend the phase space to configurations + graphs (C,G) Z = W (C) = W (C,G) with W (C) = W (C,G) C C Choose graph weights independent of configuration (C) G W (C,G) = Δ(C,G)V (G) where Δ(C,G) = 1 graph G allowed for C 0 otherwise G Perform updates 2. Discard configuration 4. Discard graph Detailed balance is satisfied C i (C i,g) G (C i+1,g) C i+1 1. Pick a graph G P[G] = V (G) W (C) 3. Pick any allowed new configuration P[(C i,g) (C i+1,g)] P[(C i+1,g) (C i,g)] = 1/N C =1= Δ(C i+1,g)v (G) 1/N C Δ(C i,g)v (G) = P[(C i+1,g)] P[(C i,g)]

37 Cluster algorithms: Ising model We need to find Δ(C,G) and V(G) to fulfill W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) o-o o o W(C), 1 1 e +βj, 0 1 e βj W(G) e +βj -e βj e βj This means for: C i (C i,g) G Parallel spins: pick connected graph o-o with Antiparallel spins: always pick open graph o o P( o-o ) = e+βj + e βj e +βj =1 e 2βJ And for: G (C i+1,g) C i+1 Configuration must be allowed connected spins must be parallel connected spins flipped as one cluster

39 The loop algorithm (Evertz et al) Swendsen-Wang cluster algorithm for the Ising model two choices on each bond: connected (flip both spins) or disconnected all connected spins are flipped together Loop algorithm is a generalization to quantum systems world lines may not be broken always 2 or 4 spins must be flipped together four different connection types

40 The loop algorithm (Evertz et al) Swendsen-Wang cluster algorithm for the Ising model two choices on each bond: connected (flip both spins) or disconnected all connected spins are flipped together Loop algorithm is a generalization to quantum systems world lines may not be broken always 2 or 4 spins must be flipped together four different connection types

41 The loop algorithm (Evertz et al) Swendsen-Wang cluster algorithm for the Ising model two choices on each bond: connected (flip both spins) or disconnected all connected spins are flipped together Loop algorithm is a generalization to quantum systems world lines may not be broken always 2 or 4 spins must be flipped together four different connection types

42 The loop algorithm (Evertz et al) Swendsen-Wang cluster algorithm for the Ising model two choices on each bond: connected (flip both spins) or disconnected all connected spins are flipped together Loop algorithm is a generalization to quantum systems world lines may not be broken always 2 or 4 spins must be flipped together four different connection types

43 The loop algorithm (Evertz et al) Swendsen-Wang cluster algorithm for the Ising model two choices on each bond: connected (flip both spins) or disconnected all connected spins are flipped together Loop algorithm is a generalization to quantum systems world lines may not be broken always 2 or 4 spins must be flipped together four different connection types

44 The loop algorithm (Evertz et al) Swendsen-Wang cluster algorithm for the Ising model two choices on each bond: connected (flip both spins) or disconnected all connected spins are flipped together Loop algorithm is a generalization to quantum systems world lines may not be broken always 2 or 4 spins must be flipped together four different connection types

45 Hamiltonian of spin-1/2 models Consider a 2-site quantum spin-1/2 model H XXZ = J xz (S x 1 S x 2 + S y 1 S y 2 ) + J z S z 1 S z 2 h( S z z 1 + S ) 2 = J xz 2 (S + 1 S 2 + S 1 S + 2 ) + J z S z 1 S z 2 h( S z z 1 + S ) 2 Heisenberg model if J xy = J z = J H = J S 1 S 2 h S 1 z + S 2 z ( ) Hamiltonian matrix in 2-site basis,,, { } H XXZ = J z 4 + h J xy J z 4 J 0 xy 2 J z h J z

46 Cluster building rules: XY-like antiferromagnet H XXZ = J xz 2 (S + i S j + S i S + j ) + J z S z z i S j i, j with 0 J z J xy i, j W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) (J z /4) dτ (J z /4) dτ 1 1 (J xy /2) dτ V(G) 1 (J z /4) dτ (J z /2) dτ (J xy - J z )/2 dτ Connected spins form a cluster and have to be flipped together

47 Loop-cluster updates 1. Connect spins according to loop-custer building rules

48 Loop-cluster updates 1. Connect spins according to loop-custer building rules

49 Loop-cluster updates 1. Connect spins according to loop-custer building rules 2. Build and flip loop-cluster

50 Loop-cluster updates 1. Connect spins according to loop-custer building rules 2. Build and flip loop-cluster

51 Heisenberg antiferromagnet H Heisenberg = J i, j S i S j W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) (J/4) dτ (J/4) dτ 0 1 (J/2) dτ V(G) 1 (J/4) dτ (J/2) dτ Connected spins form a cluster and have to be flipped together Very simple and deterministic for Heisenberg model

52 Ising-like ferromagnet H XXZ = J xz 2 (S + i S j + S i S + j ) J z S z z i S j i, j with 0 J xy J z i, j W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) (J z /4) dτ (J z /4) dτ (J xy /2) dτ V(G) 1 (J z /4) dτ (J xy /2) dτ (J z -J xy )/2 dτ Now 4-spin freezing graph is needed: connects (freezes) loops

53 Ising ferromagnet W (C) = W (C,G) = Δ(C,G)V (G) G G H Ising = J S z z i S j = J 4 i, j i, j σ i σ j Δ(C,G) W(C) (J/4) dτ (J/4) dτ V(G) 1 (J/4) dτ (J/2) dτ Two spins are frozen if there is any freezing graph along the world line P no freezing = lim M (1 (β / M)J /2) M = exp( βj /2) = exp( 2βJ classical ) We recover the Swendsen Wang algorithm: probability for no freezing

54 Extensions of the loop algorithm The loop algorithm can also extended to other models Hubbard model (Kawashima Gubernatis, Evertz, PRB 1994) higher spin S > 1/2 (Kawashima, J. Stat. Phys. 1996) t-j model (Ammon, Evertz, Kawashima, Troyer, PRB 1998) SU(N) models (Harada and Kawashima, JPSJ, 2001) dissipative quantum models (Werner and Troyer, PRL 2005) Josepshon junction arrays (Werner and Troyer, PRL 2005)...

55 3. Stochastic Series Expansion (SSE)

Path integral representation Split the Hamiltonian into diagonal term H0

0 + V, H 0 = J z ij S z i S z j hs z i, V = J xy ij (S x x i S j + S y i

= Tr(e βh 0 β (1 dτv (τ ) + dτ 1 dτ 2 V (τ 1 ) V (τ 2 ) +.

56 Path integral representation Split the Hamiltonian into diagonal term H0 and perturbation V Then perform time-dependent perturbation theory H = H 0 + V, H 0 = J z ij S z i S z j hs z i, V = J xy ij (S x x i S j + S y i S y j ) <i, j > i < i, j > Z = Tr(e βh ) = Tr(e βh 0 β dτv (τ ) 0 Te ) Z = Tr(e βh 0 β (1 dτv (τ ) + dτ 1 dτ 2 V (τ 1 ) V (τ 2 ) +...)) 0 β 0 Each term is represented by a diagram (world line configuration) β τ 1 τ τ 2 τ 1

57 Stochastic Series Expansion based on high temperature expansion, developed by Sandvik Z = Tr(e βh ) = = with H = β n n! n= 0 i H i ( β) n Tr(H n ) n= 0 α ( H bi ) α α ( ) b 1,...,b n n i=1 H b1 H b2 H b1 Similar to world line representation but without times assigned

58 Splitting the Hamiltonian for SSE Break up the Hamiltonian into offdiagonal and diagonal bond terms i, j o H = H (i, j ) d + H (i, j ) Example: Heisenberg antiferromagnet H XXZ = J xz 2 (S + i S j + S i S + j ) + J z S z z z i S j h S 1 i, j i, j i convert site terms into bond terms with = J xz 2 (S + i S j + S i S + j ) + J z S z z i S j i, j = o d H (i, j ) + H (i, j ) i, j i, j i, j h z i, j ( S z z i + S ) j split into diagonal and offdiagonal bond terms o H (i, j ) = J xz 2 (S + i S j + S i S + j ) d H (i, j ) = J z S z i S z j h ( z S z z i + S ) j

59 Ensuring positivity of diagonal bond weights Recall the SSE expansion: Z = β n n! n= 0 α ( H bi ) α α ( ) b 1,...,b n n i=1 Negative matrix elements of H are the weights Need to make all matrix elements non-positive Diagonal matrix elements: subtract an energy shift d H (i, j ) = d H (i, j ) = J z S z i S z j h ( z S z z i + S ) j C J z 4 + h C J z 4 C J z 4 C h C J z C J z 4 + h

60 Positivity of off-diagonal bond weights Energy shift will not help with off-diagonal matrix elements o H (i, j ) = J xz 2 (S + i S j + S i S + j ) Ferromagnet (J xy < 0) is no problem Antiferromagnet on bipartite lattice o H (i, j ) J 0 0 xy 0 = 2 J 0 xy perform a gauge transformation on one sublattice J xy 2 S + i S j + S + i S j ( ) S ± i ( 1) i ± S i J xy 2 S + i S j + S + i S j ( ) Frustrated antiferromagnet: we have a sign problem

61 Fixed length operator strings SSE sampling requires variable length n operator strings Z = β n n! n= 0 α ( H bi ) α α ( ) b 1,...,b n n i=1 i, j d H bi H (i, j ) { } o,h (i, j ) Extend operator string to fixed length by adding extra identity operators: n number of non-unit operators Z = Λ n= 0 (Λ n)!β n Λ α ( H bi ) α Λ! α ( ) b 1,...,b Λ i=1 H id = 1 H bi d { H id } H (i, j ) i, j { } o,h (i, j ) pick Λ large enough during thermalization And now just perform updates

62 Step 1: Local diagonal updates Recall the weight of a configuration: Z = Λ n= 0 Walk through operator string (Λ n)!β n α ( H bi ) α Λ! α ( ) b 1,...,b Λ Λ i=1 Propose to insert diagonal operators instead of unit operators d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) α Λ n Propose to remove diagonal operators d P[H (i, j ) 1] = min 1, Λ n +1 d βn bonds α H (i, j ) α configuration index operator H d (1,2) H o (3,4) 1 H o (3,4)

63 Step 1: Local diagonal updates Recall the weight of a configuration: Z = Λ n= 0 Walk through operator string (Λ n)!β n α ( H bi ) α Λ! α ( ) b 1,...,b Λ Λ i=1 Propose to insert diagonal operators instead of unit operators d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) α Λ n Propose to remove diagonal operators d P[H (i, j ) 1] = min 1, Λ n +1 d βn bonds α H (i, j ) α configuration index operator H d (1,2) H o (3,4) 1 H o (3,4)

64 Step 1: Local diagonal updates Recall the weight of a configuration: Z = Λ n= 0 Walk through operator string (Λ n)!β n α ( H bi ) α Λ! α ( ) b 1,...,b Λ Λ i=1 Propose to insert diagonal operators instead of unit operators d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) α Λ n Propose to remove diagonal operators d P[H (i, j ) 1] = min 1, Λ n +1 d βn bonds α H (i, j ) α configuration index operator H d (1,2) H o (3,4) 1 H o (3,4) H d (1,2)

65 Step 1: Local diagonal updates Recall the weight of a configuration: Z = Λ n= 0 Walk through operator string (Λ n)!β n α ( H bi ) α Λ! α ( ) b 1,...,b Λ Λ i=1 Propose to insert diagonal operators instead of unit operators d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) α Λ n Propose to remove diagonal operators d P[H (i, j ) 1] = min 1, Λ n +1 d βn bonds α H (i, j ) α configuration index operator H o (3,4) 1 H o (3,4) H d (1,2)

66 Step 1: Local diagonal updates Recall the weight of a configuration: Z = Λ n= 0 Walk through operator string (Λ n)!β n α ( H bi ) α Λ! α ( ) b 1,...,b Λ Λ i=1 Propose to insert diagonal operators instead of unit operators d P[1 H (i, j ) ] = min 1, βn d bonds α H (i, j ) α Λ n Propose to remove diagonal operators d P[H (i, j ) 1] = min 1, Λ n +1 d βn bonds α H (i, j ) α configuration index operator 1 H o (3,4) 1 H o (3,4) H d (1,2)

67 Step 2: Offdiagonal updates (local) Are very easy, can be done in any representation Problem 1: only local changes Nonergodic No change of magnetization, particle number, winding number Problem 2: Critical slowing down Solution: loop algorithm

68 Loop building rules in SSE Example: XY-like AFM: H XXZ = J xz 2 (S + i S j + S i S + j ) + J z S z z i S j i, j with 0 J z J xy i, j W (C) = W (C,G) = Δ(C,G)V (G) G G Δ(C,G) W(C) 1 0 J z / J xy /2 W(G) J z /2 (J xy- J z )/2

69 Loop updates in path integrals

70 Loop updates in path integrals 1. Define breakups (graphs) for exchange processes

71 Loop updates in path integrals 1. Define breakups (graphs) for exchange processes 2. Insert decay graphs

72 Loop updates in path integrals 1. Define breakups (graphs) for exchange processes 2. Insert decay graphs 3. Build and flip one or more loops

73 Loop updates in path integrals 1. Define breakups (graphs) for exchange processes 2. Insert decay graphs 3. Build and flip one or more loops

74 Loop updates in SSE

75 Loop updates in SSE 1. Insert/remove diagonal operators

76 Loop updates in SSE 1. Insert/remove diagonal operators 2. Decide breakups (graphs)

77 Loop updates in SSE 1. Insert/remove diagonal operators 2. Decide breakups (graphs) 3. Build and flip one or more loops

78 Loop updates in SSE 1. Insert/remove diagonal operators 2. Decide breakups (graphs) 3. Build and flip one or more loops

79 Loop updates in SSE 1. Insert/remove diagonal operators 2. Decide breakups (graphs) Similarity is no chance: 3. Build and flip one or more loops

80 Loop updates in SSE 1. Insert/remove diagonal operators 2. Decide breakups (graphs) Similarity is no chance: 3. Build and flip one or more loops an exact mapping exists

81 Measurements All diagonal operators can be measured easily diagonal spin correlation functions magnetization diagonal components of the energy Green s function: The S + - S - off-diagonal Green s function can be measured during loop construction. Imaginary time displaced correlation functions can also be calculated and give access to dynamical correlation functions after an analytical continuation.

82 4. The worm algorithm

83 Loop algorithm in a magnetic field Loop cluster algorithm requires spin inversion symmetry Magnetic field implemented by a-posteriori acceptance rate Example: spin dimer at J = h =1 ( ) H = JS 1 S 2 h S z z 1 + S 2

84 Loop algorithm in a magnetic field Loop cluster algorithm requires spin inversion symmetry Magnetic field implemented by a-posteriori acceptance rate Example: spin dimer at J = h =1 ( ) H = JS 1 S 2 h S z z 1 + S 2 Triplet E = J/4 - h = -3/4

85 Loop algorithm in a magnetic field Loop cluster algorithm requires spin inversion symmetry Magnetic field implemented by a-posteriori acceptance rate Example: spin dimer at J = h =1 ( ) H = JS 1 S 2 h S z z 1 + S 2 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

86 Loop algorithm in a magnetic field Loop cluster algorithm requires spin inversion symmetry Magnetic field implemented by a-posteriori acceptance rate Example: spin dimer at J = h =1 ( ) H = JS 1 S 2 h S z z 1 + S 2 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

87 Loop algorithm in a magnetic field Loop cluster algorithm requires spin inversion symmetry Magnetic field implemented by a-posteriori acceptance rate Example: spin dimer at J = h =1 ( ) H = JS 1 S 2 h S z z 1 + S 2 p = exp( βj /2) Triplet E = J/4 - h = -3/4 E = -J/4 Singlet E = -3J/4 = -3/4

88 Loop algorithm in a magnetic field Loop cluster algorithm requires spin inversion symmetry Magnetic field implemented by a-posteriori acceptance rate Example: spin dimer at J = h =1 ( ) H = JS 1 S 2 h S z z 1 + S 2 p = exp( βj /2) Triplet E = J/4 - h = -3/4 E = -J/4 Singlet E = -3J/4 = -3/4 Loop algorithm must go through high energy intermediate state Exponential slowdown

89 Worm updates (Prokof ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators H H + η c i + c i H H + η S + i + S i move these operators ( Ira and Masha ) using local moves until Ira and Masha meet i ( ) i ( )

90 Worm updates (Prokof ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators H H + η c i + c i H H + η S + i + S i move these operators ( Ira and Masha ) using local moves until Ira and Masha meet i ( ) i ( )

91 Worm updates (Prokof ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators H H + η c i + c i H H + η S + i + S i move these operators ( Ira and Masha ) using local moves until Ira and Masha meet insert worm i ( ) i ( )

92 Worm updates (Prokof ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators H H + η c i + c i H H + η S + i + S i move these operators ( Ira and Masha ) using local moves until Ira and Masha meet insert worm move worm i ( ) i ( ) shift

93 Worm updates (Prokof ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators H H + η c i + c i H H + η S + i + S i move these operators ( Ira and Masha ) using local moves until Ira and Masha meet insert worm i move worm ( ) i ( ) shift insert jump

94 Worm updates (Prokof ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators H H + η c i + c i H H + η S + i + S i move these operators ( Ira and Masha ) using local moves until Ira and Masha meet insert worm i move worm ( ) i ( ) shift remove jump insert jump

95 Worm updates (Prokof ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators H H + η c i + c i H H + η S + i + S i move these operators ( Ira and Masha ) using local moves until Ira and Masha meet insert worm i move worm ( ) i ( ) continue until head and tail meet shift remove jump insert jump

96 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

97 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

98 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

99 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

100 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

101 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

102 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

103 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

104 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

105 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

106 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

107 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

108 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

109 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

110 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

111 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 Singlet E = -3J/4 = -3/4

112 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 No high energy intermediate state Efficient update in presence of a magnetic field Singlet E = -3J/4 = -3/4

113 Worm algorithm in a magnetic field Worm algorithm performs a random walk Change of configuration done in small steps Example: spin dimer at J = h =1 Triplet E = J/4 - h = -3/4 No high energy intermediate state Efficient update in presence of a magnetic field Singlet E = -3J/4 = -3/4

114 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

115 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

116 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

117 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

118 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

119 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

120 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

121 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method

122 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method weight J z 4 + C J xy 2 0 J z 4 + h + C

123 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method weight J z 4 + C straight J xy 2 jump 0 turn J z 4 + h + C bounce

124 Directed loops in SSE Instead of following a pre-chosen path given by graphs we pick randomly E.g. using heat bath method weight J z 4 + C straight probability J z 4 + C 2C + J xy 2 + h J xy 2 0 jump turn J xy 2 2C + J xy 2 + h 0 J z 4 + h + C bounce J z 4 + h + C 2C + J xy 2 + h

125 Better directed loop schemes

126 Better directed loop schemes Bounces are bad since they undo the last change If bounce can be eliminated loop algorithm possible Directed loops give loop algorithm as a limit for some models Bounce path can be minimized In models where there is no loop algorithm Better choices for path selection O.F. Syljuåsen and A.W. Sandvik, PRE (2002) O.F. Syljuåsen, PRE (2003) F. Alet, S. Wessel and M. Troyer, PRE (2004)

127 When to use which algorithm? Stochastic Series Expansion (SSE) is simpler to implement Continuous-time path integrals needs lower orders Use SSE for local actions with not too large diagonal terms Loop algorithm Worm algorithm SSE Spin models Spin models in magnetic field Path Integrals Spin models with dissipation Bose-Hubbard models

128 Summary: Non local QMC algorithms The loop algorithm: A generalization of Swendsen-Wang algorithm to quantum case Solves problem of critical slowing down Implementation choices : Discrete / Continuous time, Single / Multi loops, SSE or path integrals Exponential slowing down for non particle-hole symmetric systems (magnetic field) Worm and directed loop algorithms: Relax loop-building rules : A partial loop ( worm ) performs a random walk on the lattice Detailed balance fulfilled at each step : Many solutions and room for optimization (directed loops) Once head and tail meet, a loop is finished Very efficient even for non-particle hole symmetric systems Relation to loop algorithm : loop algorithm smoothly recovered as field vanishes Might not work well when the relevant Green s function is short ranged (solids, paired states).

129 Applications of Quantum Monte Carlo Supersolids on the triangular lattice Antiferromagnetic Ising Model on the triangular lattice augmented by a ferromagnetic transverse exchange SSE QMC is possible without a sign problem! AF Ising model has a macroscopic degeneracy at T=0 (Wannier, Houtappel). In a whole region of magnetizations the system becomes supersolid upon switching on the ferromagnetic transverse exchange! PS supersolid PS PS superfluid PS solid!=2/3 solid!=1/ t/v 1/3 < m/m sat < 1/3 In a bosonic language a supersolid is a state which shows charge order and superfluidity at the same time G. Murthy et al, Phys. Rev. B 55, 3104 (1997). S. Wessel & M. Troyer, Phys. Rev. Lett (2005). D. Heidarian & K. Damle, Rev. Lett (2005). R. Melko et al, Phys. Rev. Lett (2005).

130 5. Green s function Monte Carlo

131 Green s function Monte Carlo Rather different approach compared to the finite temperature QMC methods we have seen. Stochastic application of the Hamiltonian matrix on a wave function in order to filter out the ground state. aka Diffusion Monte Carlo, Projector Monte Carlo Currently the only QMC method able to treat Quantum Dimer models on reasonably large lattices, even in the absence of a sign problem. Relies on the quality of a guiding function. When guiding function is exact, the Quantum Monte Carlo process becomes Classical. When formulated in continuous time it becomes possible to study the imaginary time evolution of a quantum system.

132 Green s function MC at the RK point At the RK point the guiding function is exact. Let us study then the dimer excitation spectrum at the RK point

133 Green s function MC at the RK point At the RK point the guiding function is exact. Let us study then the dimer excitation spectrum at the RK point AML, S. Capponi, F. Assaad, JSTAT 08

134 Green s function MC at the RK point At the RK point the guiding function is exact. Let us study then the dimer excitation spectrum at the RK point AML, S. Capponi, F. Assaad, JSTAT 08

135 Green s function MC at the RK point At the RK point the guiding function is exact. Let us study then the dimer excitation spectrum at the RK point AML, S. Capponi, F. Assaad, JSTAT 08

136 6. The negative sign problem in QMC

137 Quantum Monte Carlo Not as easy as classical Monte Carlo Z = c e E c / k B T Calculating the eigenvalues E c is equivalent to solving the problem Need to find a mapping of the quantum partition function to a classical problem Z = Tre βh Negative sign problem if some p c < 0 c p c

138 The negative sign problem In mapping of quantum to classical system Z = Tre βh = i p i there is a sign problem if some of the pi < 0 Appears e.g. in simulation of electrons when two electrons exchange places (Pauli principle) i 1 > i 4 > i 3 > i 2 > i 1 >

139 The negative sign problem Sample with respect to absolute values of the weights A = i A i p i p i = i i i A i sgn p i p i sgn p i p i p i i p i i A sign p sign p Exponentially growing cancellation in the sign sign = i p i i p i = Z/Z p = e βv (f f p ) Exponential growth of errors sign sign = sign2 sign 2 M sign eβv (f f p ) M NP-hard problem (no general solution) [Troyer and Wiese, PRL 2005]

140 Is the sign problem exponentially hard? The sign problem is basis-dependent Diagonalize the Hamiltonian matrix A = Tr[ Aexp(!"H) ] Tr[ exp(!"h) ] = i A i i exp(!"# i ) All weights are positive But this is an exponentially hard problem since dim(h)=2 N! Good news: the sign problem is basis-dependent! But: the sign problem is still not solved Despite decades of attempts Reminiscent of the NP-hard problems No proof that they are exponentially hard No polynomial solution either $ $ exp(!"# i ) i i

141 Solving an NP-hard problem by QMC Take 3D Ising spin glass H = J ij σ j σ j with J ij = 0,±1 i, j View it as a quantum problem in basis where H it is not diagonal H (SG ) = J ij σ x jσ x j with J ij = 0,±1 i, j The randomness ends up in the sign of offdiagonal matrix elements Ignoring the sign gives the ferromagnet and loop algorithm is in P H (FM ) = i, j σ x jσ x j The sign problem causes NP-hardness solving the sign problem solves all the NP-complete problems and prove NP=P

142 Summary: negative sign problem

143 Summary: negative sign problem A solution to the sign problem solves all problems in NP

144 Summary: negative sign problem A solution to the sign problem solves all problems in NP Hence a general solution to the sign problem does not exist unless P=NP If you still find one and thus prove that NP=P you will get 1 million US $ from the Clay foundation! A Nobel prize? A Fields medal?

145 Summary: negative sign problem A solution to the sign problem solves all problems in NP Hence a general solution to the sign problem does not exist unless P=NP If you still find one and thus prove that NP=P you will get 1 million US $ from the Clay foundation! A Nobel prize? A Fields medal? What does this imply? A general method cannot exist Look for specific solutions to the sign problem or model-specific methods e.g. Meron algorithm

146 The origin of the sign problem

147 The origin of the sign problem We sample with the wrong distribution by ignoring the sign!

148 The origin of the sign problem We sample with the wrong distribution by ignoring the sign! We simulate bosons and expect to learn about fermions? will only work in insulators and superfluids

149 The origin of the sign problem We sample with the wrong distribution by ignoring the sign! We simulate bosons and expect to learn about fermions? will only work in insulators and superfluids We simulate a ferromagnet and expect to learn something useful about a frustrated antiferromagnet?

150 The origin of the sign problem We sample with the wrong distribution by ignoring the sign! We simulate bosons and expect to learn about fermions? will only work in insulators and superfluids We simulate a ferromagnet and expect to learn something useful about a frustrated antiferromagnet? We simulate a ferromagnet and expect to learn something about a spin glass? This is the idea behind the proof of NP-hardness

151 Working around the sign problem

152 Working around the sign problem 1. Simulate bosonic systems Bosonic atoms in optical lattices, Fermions with n.n. hopping in 1D Helium-4 supersolids Nonfrustrated magnets

153 Working around the sign problem 1. Simulate bosonic systems Bosonic atoms in optical lattices, Fermions with n.n. hopping in 1D Helium-4 supersolids Nonfrustrated magnets 2. Simulate sign-problem free fermionic systems Attractive on-site interactions Half-filled Mott insulators

154 Working around the sign problem 1. Simulate bosonic systems Bosonic atoms in optical lattices, Fermions with n.n. hopping in 1D Helium-4 supersolids Nonfrustrated magnets 2. Simulate sign-problem free fermionic systems Attractive on-site interactions Half-filled Mott insulators 3. Restriction to quasi-1d systems Use the density matrix renormalization group method (DMRG)

155 Working around the sign problem 1. Simulate bosonic systems Bosonic atoms in optical lattices, Fermions with n.n. hopping in 1D Helium-4 supersolids Nonfrustrated magnets 2. Simulate sign-problem free fermionic systems Attractive on-site interactions Half-filled Mott insulators 3. Restriction to quasi-1d systems Use the density matrix renormalization group method (DMRG) 4. Use approximate methods Dynamical mean field theory (DMFT)

156 Summary of today s lecture Quantum Monte Carlo (World line) Loop Algorithm Stochastic Series Expansion Worm Algorithm Green s function Monte Carlo Sign Problem

157 Thank you!

Classical Monte Carlo Simulations

Classical Monte Carlo Simulations Hyejin Ju April 17, 2012 1 Introduction Why do we need numerics? One of the main goals of condensed matter is to compute expectation values O = 1 Z Tr{O e βĥ} (1) and