! Second Lecture: Quantum Monte Carlo Techniques

Andreas Läuchli, “New states of quantum matter” MPI für Physik komplexer Systeme - Dresden http://www.pks.mpg.de/~aml [email protected]

Lecture Notes at http:www.pks.mpg.de/~aml/LesHouches

“Modern theories of correlated electron systems” - Les Houches - 19/5/2009 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 1. Quantum Monte Carlo (World lines) Quantum Monte Carlo

In quantum statistical mechanics the canonical partition function now reads: − Z(T ) = Tr e H/T

Expectation values:

1 − !O" = Tr[Oe H/T ] ≡ Tr[Oρ(T )] Z(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. Quantum Monte Carlo

Feynman (1953) lays foundation for quantum Monte Carlo

Map d dimensional quantum system to classical world lines in d+1 dimensions 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 Quantum Monte Carlo

Feynman (1953) lays foundation for quantum Monte Carlo

Map d dimensional quantum system to classical world lines in d+1 dimensions “imaginary time” “imaginary classical quantum mechanical

particles world lines space Quantum Monte Carlo

Feynman (1953) lays foundation for quantum Monte Carlo

Map d dimensional quantum system to classical world lines in d+1 dimensions “imaginary time” “imaginary classical quantum mechanical

particles world lines space

Use Metropolis algorithm to update world lines 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 = H1 + H2 = H1 + −εH −ε ( H1 + H 2 ) −εH1 −εH 2 2 e = e = e e + O(ε ) H2

Obtain €the checkerboard decomposition

−β ( H + H ) € Z = Tr[exp(−βH)] = Tr[e 1 2 ] M = Tr[e−(β / M )H1 e−(β / M )H 2 ] + O(β 3 / M 2) imaginary time

space direction

€ Path integral QMC Use Trotter-Suzuki or a simple low-order formula M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )}

gives a mapping to a (d+1)-dimensional classical model |i 〉 € 1 |i8〉 |i7〉 |i6〉

|i5〉 |i4〉

|i3〉 imaginary time |i2〉 |i1〉 space direction partition function of quantum system is sum over classical world lines Path integral QMC Use Trotter-Suzuki or a simple low-order formula M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )}

gives a mapping to a (d+1)-dimensional classical model |i 〉 € 1 |i8〉 place particles (spins) |i7〉 |i6〉

|i5〉 |i4〉

|i3〉 imaginary time |i2〉 |i1〉 space direction partition function of quantum system is sum over classical world lines Path integral QMC Use Trotter-Suzuki or a simple low-order formula M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )}

gives a mapping to a (d+1)-dimensional classical model |i 〉 € 1 |i8〉 place particles (spins) |i7〉 |i6〉 |i5〉 for Hamiltonians conserving |i4〉 particle number (magnetization) |i3〉 imaginary time |i2〉 we get world lines |i1〉 space direction partition function of quantum system is sum over classical world lines Calculating configuration weights

M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )} Examples: particles with nearest neighbor repulsion

H −t a†a a†a V n n = !( i j + j i)+ ! i j € !i,j" !i,j" Calculating configuration weights

M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )} Examples: particles with nearest neighbor repulsion

H −t a†a a†a V n n = !( i j + j i)+ ! i j € !i,j" !i,j"

1 Calculating configuration weights

M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )} Examples: particles with nearest neighbor repulsion

H −t a†a a†a V n n = !( i j + j i)+ ! i j € !i,j" !i,j"

2 1 (∆τt) Calculating configuration weights

M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )} Examples: particles with nearest neighbor repulsion

H −t a†a a†a V n n = !( i j + j i)+ ! i j € !i,j" !i,j"

2 3 1 (∆τt) (1 − ∆τV ) Calculating configuration weights

M M Z = Tre−βH = Tre−MΔτH = Tr(e−ΔτH ) = Tr(1− ΔτH) + O(βΔτ)

= ∑ i1 1− ΔτH i2 i2 1− ΔτH i3  iM 1− ΔτH i1 {(i1 ...iM )} Examples: particles with nearest neighbor repulsion

H −t a†a a†a V n n = !( i j + j i)+ ! i j € !i,j" !i,j"

2 3 2 1 (∆τt) (1 − ∆τV ) (∆τt) Monte Carlo updates just move the world lines locally

probabilities given by matrix element of Hamiltonian

H = −t c†c + c† c example: tight binding model ∑( i i+1 i+1 i) i, j

€ Monte Carlo updates just move the world lines locally

probabilities given by matrix element of Hamiltonian

H = −t c†c + c† c example: tight binding model ∑( i i+1 i+1 i) i, j introduce or remove two kinks:

€ Monte Carlo updates just move the world lines locally

probabilities given by matrix element of Hamiltonian

H = −t c†c + c† c example: tight binding model ∑( i i+1 i+1 i) i, j introduce or remove two kinks: shift a kink:

€ Monte Carlo updates just move the world lines locally

probabilities given by matrix element of Hamiltonian

H = −t c†c + c† c example: tight binding model ∑( i i+1 i+1 i) i, j introduce or remove two kinks: shift a kink:

€

2 P =1 P = (Δτt) P = Δτt Monte Carlo updates just move the world lines locally

probabilities given by matrix element of Hamiltonian

H = −t c†c + c† c example: tight binding model ∑( i i+1 i+1 i) i, j introduce or remove two kinks: shift a kink:

€

2 P =1 P = (Δτt) P = Δτt

2 P→ = min[1,(Δτt) ] P→ = P← =1 2 P← = min[1,1/(Δτt) ] 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)]

|i1〉 τ6 |i8〉 i 〉 τ5 | 7 τ4 |i6〉 τ |i5〉 3 τ |i4〉 2

|i3〉 imaginary time imaginary time τ |i2〉 1

|i1〉 space direction space direction discrete time: store configuration at all time steps

continuous time: store times at which configuration changes 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 2. Cluster updates: The loop algorithm Problems with local updates 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 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 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 Cluster algorithms: the formal explanation

Z = ∑W (C) = ∑∑W (C,G) with W (C) = ∑W (C,G) C C G G

1 graph G allowed for C €W (C,G) = Δ(C,G)V (G) where Δ(C,G) =   0 otherwise

€ Ci → (Ci,G) → G → (Ci+1,G) → Ci+1

€ 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 G G Choose graph weights independent of configuration (C) 1 graph G allowed for C €W (C,G) = Δ(C,G)V (G) where Δ(C,G) =   0 otherwise

Perform updates € Ci → (Ci,G) → G → (Ci+1,G) → Ci+1

€ 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 G G Choose graph weights independent of configuration (C) 1 graph G allowed for C €W (C,G) = Δ(C,G)V (G) where Δ(C,G) =   0 otherwise

Perform updates € Ci → (Ci,G) → G → (Ci+1,G) → Ci+1 V (G) 1. Pick a graph G P[G] = W (C)

€ € 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 G G Choose graph weights independent of configuration (C) 1 graph G allowed for C €W (C,G) = Δ(C,G)V (G) where Δ(C,G) =   0 otherwise

Perform updates 2. Discard configuration € Ci → (Ci,G) → G → (Ci+1,G) → Ci+1 V (G) 1. Pick a graph G P[G] = W (C)

€ € 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 G G Choose graph weights independent of configuration (C) 1 graph G allowed for C €W (C,G) = Δ(C,G)V (G) where Δ(C,G) =   0 otherwise

Perform updates 2. Discard configuration € Ci → (Ci,G) → G → (Ci+1,G) → Ci+1 V (G) 1. Pick a graph G P[G] = 3. Pick any allowed new configuration W (C)

€ € 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 G G Choose graph weights independent of configuration (C) 1 graph G allowed for C €W (C,G) = Δ(C,G)V (G) where Δ(C,G) =   0 otherwise

Perform updates 2. Discard configuration 4. Discard graph € Ci → (Ci,G) → G → (Ci+1,G) → Ci+1 V (G) 1. Pick a graph G P[G] = 3. Pick any allowed new configuration W (C)

€ € 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 G G Choose graph weights independent of configuration (C) 1 graph G allowed for C €W (C,G) = Δ(C,G)V (G) where Δ(C,G) =   0 otherwise

Perform updates 2. Discard configuration 4. Discard graph € Ci → (Ci,G) → G → (Ci+1,G) → Ci+1 V (G) 1. Pick a graph G P[G] = 3. Pick any allowed new configuration W (C)

Detailed€ balance is satisfied P€[(C ,G) → (C ,G)] 1/N Δ(C ,G)V (G) P[(C ,G)] i i+1 = C =1= i+1 = i+1 P[(Ci+1,G) → (Ci,G)] 1/NC Δ(Ci,G)V (G) P[(Ci,G)]

€ 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: Ci → (Ci,G) → G e+βJ + e−βJ Parallel spins: pick connected graph o-o with P ( o-o ) = =1− e−2βJ e+βJ Antiparallel spins: always pick open graph o o € And for: G → (C ,G) → C i+1 € i+1 Configuration must be allowed ⇒ connected spins must be parallel ⇒ connected spins flipped as one cluster €

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 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 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 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 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 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 Hamiltonian of spin-1/2 models

Consider a 2-site quantum spin-1/2 model

x x y y z z z z HXXZ = Jxz (S1 S2 + S1 S2 ) + JzS1 S2 − h(S1 + S2 ) J = xz (S +S− + S−S + ) + J S zS z − h S z + S z 2 1 2 1 2 z 1 2 ( 1 2 )

Heisenberg model if Jxy = Jz = J €   H = JS S − h S z + S z 1 2 ( 1 2 )  J  z + h 0 0 0  4   J J  Hamiltonian matrix in 2-site basis  0 − z xy 0   4 2  HXXZ =  Jxy Jz  € 0 − 0 { ↑↑ , ↑↓ , ↓↑ , ↓↓ }  2 4   J   0 0 0 z − h  4 

€ € Cluster building rules: XY-like antiferromagnet J H = xz (S +S− + S−S + ) + J S zS z XXZ 2 ∑ i j i j z ∑ i j i, j i, j W (C) = ∑W (C,G) = ∑Δ(C,G)V(G) G G with 0 ≤ Jz ≤ Jxy

Δ(C,G) W(C) € € 1 1 – 1+ (Jz/4) dτ

1 – 0 1-(Jz/4) dτ

– 1 1 (Jxy/2) dτ

V(G) 1−(Jz/4) dτ (Jz/2) dτ (Jxy- Jz)/2 dτ

Connected spins form a cluster and have to be flipped together Loop-cluster updates 1. Connect spins according to loop-custer building rules Loop-cluster updates 1. Connect spins according to loop-custer building rules Loop-cluster updates 1. Connect spins according to loop-custer building rules 2. Build and flip loop-cluster Loop-cluster updates 1. Connect spins according to loop-custer building rules 2. Build and flip loop-cluster   HHeisenberg = J∑S iS j Heisenberg antiferromagnet i, j W (C) = ∑W (C,G) = ∑Δ(C,G)V(G) G G € Δ(C,G) W(C) € 1 1 1+ (J/4) dτ

1 0 1-(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 Jxz + − − + z z HXXZ = − ∑(Si S j + Si S j ) − Jz ∑Si S j 2 i, j i, j

Ising-like ferromagnet with 0 ≤ Jxy ≤ Jz W (C) = ∑W (C,G) = ∑Δ(C,G)V(G) G G € Δ(C,G) W(C) €

1 0 0 1- (Jz/4) dτ

1 1 1 1+ (Jz/4) dτ

0 1 0 (Jxy/2) dτ

V(G) 1−(Jz/4) dτ (Jxy/2) dτ (Jz-Jxy)/2 dτ

Now 4-spin freezing graph is needed: connects (freezes) loops z z J HIsing = −J∑Si S j = − ∑σ iσ j Ising ferromagnet i, j 4 i, j W (C) = ∑W (C,G) = ∑Δ(C,G)V(G) G G Δ(C,G) € W(C) € 1 0 1- (J/4) dτ

1 1 1+ (J/4) dτ

0 0 0 V(G) 1−(J/4) dτ (J/2) dτ

Two spins are frozen if there is any freezing graph along the world line

M Pno freezing = lim (1− (β / M)J /2) = exp(−βJ /2) = exp(−2βJclassical ) M →∞ We recover the Swendsen Wang algorithm: probability for no freezing

€ 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) ... 3. Stochastic Series Expansion (SSE) Path integral representation Split the Hamiltonian into diagonal term H0 and perturbation V Then perform time-dependent perturbation theory

z z z z xy x x y y H = H0 + V, H0 = ∑ JijSi S j − ∑hSi , V = ∑ Jij (Si Sj + Si Sj ) i < i, j >

β − dτV (τ ) Z = Tr(e− βH ) = Tr(e− βH0 Te ∫0 ) β β β Z = Tr(e− βH0 (1 − dτV (τ ) + dτ dτ V (τ )V (τ ) + ...)) ∫ ∫ 1 ∫ 2 1 2 0 0 τ 1 Each term is represented by a diagram (world line configuration)

τ τ2

τ1 Stochastic Series Expansion based on high temperature expansion, developed by Sandvik ∞ Z = Tr(e−βH ) = ∑(−β)n Tr(H n ) n= 0 ∞ β n n = α (−H ) α ∑ n! ∑ ∑ ∏ bi n= 0 α (b1 ,...,bn ) i=1

with H = ∑Hi i

H b2 € Hb 1 H b1

Similar to world line representation but without times assigned Splitting the Hamiltonian for SSE Break up the Hamiltonian into offdiagonal and diagonal bond terms o d H = ∑H(i, j ) + H(i, j ) i, j Example: Heisenberg antiferromagnet J H = xz (S +S− + S−S + ) + J S zS z − h S z € XXZ 2 ∑ i j i j z ∑ i j ∑ 1 convert site terms i, j i, j i into bond terms J h = xz (S +S− + S−S + ) + J S zS z − S z + S z 2 ∑ i j i j z ∑ i j z ∑( i j ) i, j i, j i, j split into diagonal and o d offdiagonal bond terms = ∑H(i, j ) + ∑H(i, j ) i, j i, j with o Jxz + − − + H(i, j ) = (Si S j + Si S j ) € 2 h H d = J S zS z − S z + S z (i, j ) z i j z ( i j )

€ Ensuring positivity of diagonal bond weights Recall the SSE expansion:

∞ β n n Z = α (−H ) α ∑ n! ∑ ∑ ∏ bi n= 0 α (b1 ,...,bn ) 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

h J H d = J S zS z − S z + S z − C C ≥ z + h (i, j ) z i j z ( i j ) 4  J  z + h − C 0 0 0  4    Jz €  0 − − C 0 0  € H d =  4  (i, j ) J  0 0 − z − C 0   4   J   0 0 0 z − h − C  4 

€ Positivity of off-diagonal bond weights

Energy shift will not help with off-diagonal matrix elements

J  0 0 0 0 H o xz (S +S− S−S + ) (i, j ) = i j + i j  Jxy  2  0 0 0 o 2 H(i, j ) =   Jxy Ferromagnet (Jxy < 0) is no problem  0 0 0  2  0 0 0 0 €   Antiferromagnet on bipartite lattice

€ perform a gauge transformation on one sublattice

± i ± Jxy S → (−1) S Jxy S +S− + S−S +  i    i → − S +S− + S−S + 2 ( i j i j ) 2 ( i j i j )

Frustrated antiferromagnet: we have a sign problem € Fixed length operator strings SSE sampling requires variable length n operator strings

∞ n n β H ∈ H d ,H o Z = α (−H ) α bi { (i, j ) (i, j )} ∑ ∑ ∑ ∏ bi  n! i, j n= 0 α (b1 ,...,bn ) i=1 Extend operator string to fixed length by adding extra identity operators: n … number of non-unit operators € € Λ n Λ (Λ − n)!β Hid = −1 Z = α (−H ) α ∑∑ ∑ ∏ bi Λ! d o n= 0 α (b1 ,...,bΛ ) i=1 H ∈ {H }∪ H ,H bi id { (i, j ) (i, j )} i, j pick Λ large enough during thermalization€ € € And now just perform updates Step 1: Local diagonal updates Recall the weight of a configuration: Λ (Λ − n)!β n Λ Z = α (−H ) α ∑∑ ∑ Λ! ∏ bi n= 0 α (b1 ,...,bΛ ) i=1 Walk through operator string

Propose to insert diagonal operators instead of unit operators €  d  d βNbonds α H(i, j ) α P[1→ H(i, j )] = min1 ,   Λ − n  operator configuration index Propose to remove diagonal operators d 5 H (1,2) 4 € Ho  Λ − n +1  3 (3,4) P[H d →1] = min 1, (i, j )  d  2 1  βNbonds α H(i, j ) α  o 1 H (3,4) 1

1 2 3 4 € Step 1: Local diagonal updates Recall the weight of a configuration: Λ (Λ − n)!β n Λ Z = α (−H ) α ∑∑ ∑ Λ! ∏ bi n= 0 α (b1 ,...,bΛ ) i=1 Walk through operator string

Propose to insert diagonal operators instead of unit operators €  d  d βNbonds α H(i, j ) α P[1→ H(i, j )] = min1 ,   Λ − n  operator configuration index Propose to remove diagonal operators d 5 H (1,2) 4 € Ho  Λ − n +1  3 (3,4) P[H d →1] = min 1, (i, j )  d  2 1  βNbonds α H(i, j ) α  o 1 H (3,4)

1 2 3 4 € Step 1: Local diagonal updates Recall the weight of a configuration: Λ (Λ − n)!β n Λ Z = α (−H ) α ∑∑ ∑ Λ! ∏ bi n= 0 α (b1 ,...,bΛ ) i=1 Walk through operator string

Propose to insert diagonal operators instead of unit operators €  d  d βNbonds α H(i, j ) α P[1→ H(i, j )] = min1 ,   Λ − n  operator configuration index Propose to remove diagonal operators d 5 H (1,2) 4 € Ho  Λ − n +1  3 (3,4) P[H d →1] = min 1, (i, j )  d  2 1  βNbonds α H(i, j ) α  o 1 H (3,4) d H (1,2)

1 2 3 4 € Step 1: Local diagonal updates Recall the weight of a configuration: Λ (Λ − n)!β n Λ Z = α (−H ) α ∑∑ ∑ Λ! ∏ bi n= 0 α (b1 ,...,bΛ ) i=1 Walk through operator string

Propose to insert diagonal operators instead of unit operators €  d  d βNbonds α H(i, j ) α P[1→ H(i, j )] = min1 ,   Λ − n  operator configuration index Propose to remove diagonal operators 5 4 € Ho  Λ − n +1  3 (3,4) P[H d →1] = min 1, (i, j )  d  2 1  βNbonds α H(i, j ) α  o 1 H (3,4) d H (1,2)

1 2 3 4 € Step 1: Local diagonal updates Recall the weight of a configuration: Λ (Λ − n)!β n Λ Z = α (−H ) α ∑∑ ∑ Λ! ∏ bi n= 0 α (b1 ,...,bΛ ) i=1 Walk through operator string

Propose to insert diagonal operators instead of unit operators €  d  d βNbonds α H(i, j ) α P[1→ H(i, j )] = min1 ,   Λ − n  operator configuration index Propose to remove diagonal operators 5 1 4 € Ho  Λ − n +1  3 (3,4) P[H d →1] = min 1, (i, j )  d  2 1  βNbonds α H(i, j ) α  o 1 H (3,4) d H (1,2)

1 2 3 4 € 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 Loop building rules in SSE Example: XY-like AFM: W (C) = ∑W (C,G) = ∑Δ(C,G)V(G) G G Jxz + − − + z z HXXZ = ∑(Si S j + Si S j ) + Jz ∑Si S j 2 i, j i, j with 0 ≤ J ≤ J z xy € Δ(C,G) W(C) € 1 0 Jz/2 0 0 0

1 1 Jxy/2

W(G) Jz/2 (Jxy- Jz)/2 Loop updates in path integrals Loop updates in path integrals 1. Define “breakups” (graphs) for exchange processes Loop updates in path integrals 1. Define “breakups” (graphs) for exchange processes 2. Insert “decay” graphs Loop updates in path integrals 1. Define “breakups” (graphs) for exchange processes 2. Insert “decay” graphs 3. Build and flip one or more loops Loop updates in path integrals 1. Define “breakups” (graphs) for exchange processes 2. Insert “decay” graphs 3. Build and flip one or more loops Loop updates in SSE Loop updates in SSE

1. Insert/remove diagonal operators Loop updates in SSE

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

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

1. Insert/remove diagonal operators 2. Decide “breakups” (graphs) 3. Build and flip one or more loops 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 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 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. 4. The worm algorithm 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 S − h S z + S z 1 2 ( 1 2 ) 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 S − h S z + S z 1 2 ( 1 2 )

Triplet E = J/4 - h = -3/4 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 S − h S z + S z 1 2 ( 1 2 )

Triplet Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 S − h S z + S z 1 2 ( 1 2 )

Triplet Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 S − h S z + S z 1 2 ( 1 2 )

p = exp(−βJ /2)

Triplet Singlet E = J/4 - h = -3/4 E = -J/4 E = -3J/4 = -3/4 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 S − h S z + S z 1 2 ( 1 2 )

p = exp(−βJ /2)

Triplet Singlet E = J/4 - h = -3/4 E = -J/4 E = -3J/4 = -3/4

Loop algorithm must go through high energy intermediate state Exponential slowdown Worm updates (Prokof’ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators † H H S + S− H ← H + η∑(ci + ci) ← + η∑( i + i ) i i move these operators (“Ira” and “Masha”) using local moves until Ira and Masha meet Worm updates (Prokof’ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators † H H S + S− H ← H + η∑(ci + ci) ← + η∑( i + i ) i i move these operators (“Ira” and “Masha”) using local moves until Ira and Masha meet Worm updates (Prokof’ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators † H H S + S− H ← H + η∑(ci + ci) ← + η∑( i + i ) i i move these operators (“Ira” and “Masha”) using local moves until Ira and Masha meet insert worm Worm updates (Prokof’ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators † H H S + S− H ← H + η∑(ci + ci) ← + η∑( i + i ) i i move these operators (“Ira” and “Masha”) using local moves until Ira and Masha meet insert worm move worm

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

shift

insert jump Worm updates (Prokof’ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators † H H S + S− H ← H + η∑(ci + ci) ← + η∑( i + i ) i i move these operators (“Ira” and “Masha”) using local moves until Ira and Masha meet insert worm move worm

shift

remove jump

insert jump Worm updates (Prokof’ev et al, 1998) Break a world line by inserting a pair of creation/annihilation operators † H H S + S− H ← H + η∑(ci + ci) ← + η∑( i + i ) i i move these operators (“Ira” and “Masha”) using local moves until Ira and Masha meet insert worm move worm continue until head and tail meet

shift

remove jump

insert jump 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 No high energy intermediate state Efficient update in presence of a magnetic field 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 Singlet E = J/4 - h = -3/4 E = -3J/4 = -3/4 No high energy intermediate state Efficient update in presence of a magnetic field Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method 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 + C 4

€ Jxy 2

0 €

J − z + h + C 4

€ 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 + C “straight” 4

€ Jxy 2 “jump”

0 “turn” €

Jz − + h + C “bounce” 4

€ Directed loops in SSE

Instead of following a pre-chosen path given by graphs we pick randomly

E.g. using heat bath method weight probability

Jz + C “straight” Jz 4 + C 4 2C + Jxy 2 + h

Jxy € Jxy 2 2 “jump”€ 2C + Jxy 2 + h

0 “turn” 0 € €

Jz J 4 h C − + h + C “bounce” − z + + 4 2C + Jxy 2 + h

€ € Better directed loop schemes 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) 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

SSE Path Integrals Loop Spin models Spin models algorithm with dissipation

Worm Spin models Bose-Hubbard models algorithm in magnetic field 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). Applications of Quantum Monte Carlo Supersolids on the triangular lattice

Antiferromagnetic Ising Model on the triangular lattice augmented by a ferromagnetic transverse exchange

Supersolid hardcore bosons SSEon th eQMCtriang uisla rpossiblelattice without a sign problem !

Stefan Wessel(1) and Matthias Troyer(2) (1)Institut fu¨r Theoretische Physik III, Universit¨at Stuttgart, 70550 Stuttgart, Germany and (2)Theoretische Physik, ETH Zu¨ric hAF, CH -8Ising093 Zu¨r imodelch, Switzerl ahasnd a macroscopic degeneracy at T=0 (Wannier, Houtappel). (Dated: May 27, 2006) We determine the phase diagram of hardcore bosons on a triangular lattice with nearest neighbor repulsion, paying special attention to the stability of the supersolid phase. Similar to the same model on a square lattice we find that for densities ρ < 1/3 or ρ > 2/3 a supersolid phase is unstable and −1 3 1 3 the transition between a commensurate solid and thIne s uape rwholefluid is of firegionrst order. A tofin temagnetizationsrmediate / < m/msat < / fillings 1/3 < ρ < 2/3 we find an extended supersol idthephas esystemeven at half fibecomeslling ρ = 1/2. supersolid upon switching on the ferromagnetic PACS numbers: transverse exchange Next to the widely observed superfluid and Bose- 1 In a bosonic language a supersolid condensed phases with broken U(1) symmetry and “crys- talline” density wave ordered phases with broken trans- 0.8 is a state which shows charge order lational symmetry, the supersolid phase, breaking both PS PS solid !=2/3 the U(1) symmetry and translational symmetry has been 0.6 and superfluidity at the same time a widely discussed phase that is hard to find both in

! supersolid superfluid experiments and in theoretical models. Experimentally, evidence for a possible supersolid phase in bulk 4He has 0.4 G. Murthy et al, Phys. Rev. B 55, 3104 (1997). PS solid !=1/3 recently been presented [1], but the question of whether PS S. Wessel & M. Troyer, Phys. Rev. Lett. 95 127205 (2005). a true supersolid has been observed is far from being set- 0.2 D. Heidarian & K. Damle, Rev. Lett. 95 127206 (2005). tled [2, 3], leaving the old question of supersolid behavior R. Melko et al, Phys. Rev. Lett. 95 127207 (2005). in translation invariant systems [4, 5] unsettled for now. 0 More precise statements for a supersolid phase can be 0 0.1 0.2 0.3 0.4 0.5 t/V made for bosons on regular lattices. It has been pro- posed that such bosonic lattice models can be realized FIG. 1: Zero-temperature phase diagram of hardcore bosons by loading ultracold bosonic atoms into an optical lattice, on the triangular lattice in the canonical ensemble obtained where the required longer range interaction between the from quantum Monte Carlo simulations. The regions of phase bosons could be induced by using the dipolar interaction separation are denoted by PS. The insets exhibit the density in chromium condensates [6], or an interaction mediated distribution inside the solid phases for ρ = 1/3 (lower panel), by fermionic atoms in a mixture of bosonic and fermionic and ρ = 2/3 (upper panel). atoms [7]. With the recent realization of a Bose-Einstein condensate (BEC) in Chromium atoms [8], these exper- supersolid phase is unstable and phase separates into su- iments have now become feasible, raising the interest in perfluid and solid domains at a first order (quantum) phase diagrams of lattice boson model, and particularly phase transition. Recently, this occurrence of a first or- in the stability of supersolids on lattices. der phase transition was explained by showing that a The question if a supersolid phase is a stable ther- uniform supersolid phase in a hardcore boson model is arXiv:cond-mat/0505298 v1 11 May 2005 modynamic phase for lattice boson models has been unstable towards the introduction of domain walls, low- controversial for many years. Analytical calculations ering the kinetic energy of the system by enhancing the using mean-field and renormalization group methods mobility of the bosons on the domain wall [17]. In a re- [9, 10, 11, 12] have predicted supersolid phases for many lated work it has been proposed that superfluid domain models, including for the simplest model of hardcore walls might be an explanation for the experimental ob- bosons with nearest neighbor repulsion on a square lat- servation of possible supersolidity in Helium [3, 18]. tice with Hamiltonian To stabilize a supersolid on the square lattice, the ki- † † H = t a aj + a ai + V ninj µ ni, (1) netic energy of the bosons in the supersolid has to be − ! " i j # ! − ! !i,j" !i,j" i enhanced either by sufficiently reducing the on-site in- teraction to be less than 4V [17], by adding additional † where ai (ai) creates (destroys) a particle on site i, next-nearest-neighbor hopping terms [16], or by forming t denotes the nearest-neighbor hopping, V a nearest- striped solid phases with additional longer-ranged repul- neighbor repulsion, and µ the chemical potential. Sub- sions [13, 19]. sequent numerical investigations using exact diagonal- In this Letter we will consider the interplay of super- ization and quantum Monte Carlo (QMC) algorithms solidity and frustration by studying the hardcore boson [13, 14, 15, 16, 17] have shown that for this model, the model (1) on a triangular lattice. In the classical limit 5. Green’s function Monte Carlo 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. 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 Green’s function MC at the RK point

At the RK point the guiding function is exact.

Let us study then theDynamical dimer dimer excitation correlations at bipartite spectrum and non-bipartite at Rokhsar–Kivelsonthe RK point points .Sa.Mech. Stat. J. 20)P01010 (2008) AML, S. Capponi, F. Assaad, JSTAT 08 Figure 1. Square lattice. Upper panel: dynamical dimer spectral functions x(Q, ω) plotted along the Brillouin zone path (0, 0)–(π, 0)–(π, π)–(0, 0) on a D Dx system of linear extension L = 64. The first moment ωfm (Q) of the distribution— equivalent to the SMA prediction—is shown by the white filled circles. Lower panel: equal time dimer structure factor Dx(Q) along the same path. The peak at (π, 0) is logarithmically divergent with system size.

can lead to a non-divergent equal time response, as is the case for example in a generalized classical model [32]. As expected the SMA prediction matches the full spectral function very well in the close vicinity of the (π, π) point. However in the rest of the Brillouin zone the intensity is by no means concentrated near the SMA result. We can however identify traces of intensity above the SMA curve, loosely following it, for example in the neighborhood of Γ = (0, 0). On the other hand there is also some intensity at low frequencies and approaching zero energy as Q Γ. Further investigation is needed to clarify whether this low energy signal is a ‘shadow→ band’ feature of the (π, 0) or the (π, π) structure. We now focus on the low energy features around the (π, 0) and the (π, π) region, which are presented in a zoomed view in the two panels of figure 2. First we briefly discuss the behavior in the (π, π) region shown in the right panel. As stated before the SMA predicts a quadratically dispersing mode in the vicinity of this point. Indeed our numerical results follow the refined analytical prediction (12) presented by the dashed line very closely and the whole spectral function is actually dominated by the weight at this

doi:10.1088/1742-5468/2008/01/P01010 7 Green’s function MC at the RK point

At the RK point the guiding function is exact.

Let us study then the dimerDynamical excitation dimer correlations spectrum at bipartite at the and non-bipartiteRK point Rokhsar–Kivelson points .Sa.Mech. Stat. J.

AML, S. Capponi, F. Assaad, JSTAT 08 Figure 2. Square lattice. Left panel: low energy zoom of x(Q, ω) into the (π, 0) region. Right panel: similar zoom into the (π, π) region.D The dashed white lines Dx with the filled circles denotes the measured ωfm (Q) for both panels. The dash– dotted line in the left panel is the analytical prediction (15) for the maximum intensity at low energy, while in the right panel the dash–dotted line shows the

analytical prediction (12) for the resonon dispersion. P01010 (2008)

energy. It is difficult based on our numerical data for the spectral functions to address the linewidth of the mode, but based on the following argument we expect a finite lifetime for this quadratic mode. As a general remark for a quadratic dispersion, it can easily be understood that for any Q, the gap should vanish in the thermodynamic limit. Indeed, by combining many low lying excitations, such that the total momentum is Q, an arbitrary small energy state can be obtained. An equivalent statement is that a given Q excitation could decay into many lower energy excitations (when there is no symmetry preventing this process). Therefore, if we observe a quadratic behavior over a large region, this is because the spectral weight on the lower energies states is tiny, due to density of states and matrix element effects. The long wavelength, long time behavior of the (π, 0) feature has been discussed in [6]. On the basis of the analysis of the height model representation the asymptotic form of the imaginary time correlation is known in the vicinity of (π, 0) f(k, τ) exp( k√τ), (13) ∼ − where now Q = k+(π, 0), and k is the modulus of k. This expression can be transformed by an inverse Laplace transform giving: k exp( k2/4ω) F (k, ω)= − . (14) 2√πω3/2 The expression (13) is supposed to be valid only at long times, as can be seen from the fact that the first moment of F (k, ω) does not exist due to the infinite τ derivative of

doi:10.1088/1742-5468/2008/01/P01010 8 Green’s function MC at the RK point

At the RK point the guiding function is exact.

Let us study then the dimerDynamical excitation dimer correlations spectrum at bipartite at the and non-bipartiteRK point Rokhsar–Kivelson points .Sa.Mech. Stat. J.

AML, S. Capponi, F.Figure Assaad, 4. JSTATCubic 08 lattice. Left panel: low energy zoom ω [0, 0.3] of (Q, ω) along the same path in the Brillouin zone as in figure 3.∈ Right panel: Dx supplementary low energy data for x(Q, ω) along the path (π, π, 0) (π, π, π). D →Dx The dashed white lines with the filled circles denotes the measured ωfm (Q) for both panels, and is in good agreement with the expected k2 dispersion close to

(π, π, π). Note that the maximum intensity of x(Q, ω) rapidly deviates from P01010 (2008) the expected k2-law and becomes linear, revealingD the remnant of the linearly dispersing ‘photon’ mode of the U(1)-liquid Coulomb phase.

point respectively. The equal time structure factor is smooth on the triangular lattice, due to the short ranged nature of the dimer correlations in real space. Looking more closely at the low energy part of the spectral functions in figure 6 one recognizes that the intensity carrying modes at the onset of the dimer spectrum are much less dispersive than the SMA prediction suggests. Furthermore the broadening can vary through the Brillouin zone. For instance, the spectrum is very broad at the X point, while it is much sharper close to the M point, see figure 7. These observations strongly suggest that the dimer waves excited by the operator (2) are not the true ‘elementary’ excitations of this dimer model. Indeed the elementary excitations are known to consist of so-called visons which are non-local in terms of dimers and for which the dispersion has been computed [4]. Real dimer excitations can be understood as excitations of two or more visons. Here, the lowest energy dimer excitation (at the M point) is formed by a bound state of two visons [4]. On the basis of the single-vison dispersion relation [4], we have plotted in figure 6 the bottom of the two-vison continuum. By comparing to the spectral weight of the dimer correlations, we observe that: (a) close to K and Γ points, the spectral weight nicely follows the two-vison continuum; (b) the lowest energy state at M is slightly below the two-vison gap, which is consistent with the interpretation as a two- vison bound state; (c) at the X point, the spectral weight is very broad, we cannot reveal any well formed excitation so that a dimer excitation probably disintegrates

doi:10.1088/1742-5468/2008/01/P01010 11 6. The negative sign problem in QMC Quantum Monte Carlo

Not as easy as classical Monte Carlo

Z = ∑e−Ec / kBT c

Calculating the eigenvalues Ec is equivalent to solving the problem

€ Need to find a mapping of the quantum partition function to a classical problem

−βH Z = Tre ≡ ∑ pc c

“Negative sign” problem if some pc < 0

€ The negative sign problem In mapping of quantum to classical system

= Tr −βH = Z e ! pi 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)

|i1>

|i4>

|i3>

|i2>

|i1> The negative sign problem Sample with respect to absolute values of the weights

A sgn p p p ∑ i i i ∑ i A ⋅ sign A = A p p = i i ≡ p ∑ i i ∑ i sign i i ∑sgn pi pi ∑ pi p i i

Exponentially growing cancellation in the sign € ! pi = i = = −βV (f−f|p|) !sign" Z/Z|p| e !i |pi|

Exponential growth of errors − ∆sign ! sign2 sign 2 eβV (f f|p|) = ! "−! " sign √M sign ≈ √M ! " ! "

NP-hard problem (no general solution) [Troyer and Wiese, PRL 2005] Is the sign problem exponentially hard? The sign problem is basis-dependent Diagonalize the Hamiltonian matrix

A = Tr[Aexp(!"H)] Tr[exp(!"H)] = $ i Ai i exp(!"#i ) $exp(!"#i ) i i All weights are positive

But this is an exponentially hard problem since dim(H)=2N !

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 Solving an NP-hard problem by QMC

Take 3D Ising spin glass H = ∑Jijσ jσ j with Jij = 0,±1 i, j

View it as a quantum problem in basis where H it is not diagonal

€ (SG ) x x H = ∑Jijσ jσ j with Jij = 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

(FM ) x x H = −∑σ jσ j i, j

The sign problem causes NP-hardness

solving the sign problem€ solves all the NP-complete problems and prove NP=P Summary: negative sign problem Summary: negative sign problem A “solution to the sign problem” solves all problems in NP 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? 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 The origin of the sign problem The origin of the sign problem

We sample with the wrong distribution by ignoring the sign! 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 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? 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 Working around the sign problem 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 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 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) 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) Summary of today’s lecture

Quantum Monte Carlo (World line) Loop Algorithm Stochastic Series Expansion Worm Algorithm Green’s function Monte Carlo Sign Problem Thank you !