Download Introduction to Quantum Monte Carlo

Introduction to
Quantum Monte Carlo
Stefan Wessel
Institut für Theoretische Physik III
Universität Stuttgart
Outline
♦ Examples of QMC applications
♦ Representations and local updates
♦ World-line representations
♦ Discrete time
♦ Continuous time
♦ Path-integral representation
♦ Stochastic series expansion (SSE) representation
♦ Comparison
♦ The sign problem
What can modern QMC algorithms do?
♦ Local updates (before 1994)
♦ 200 spins
♦ T=0.1
♦ Cluster algorithms (after 1995)
♦ 2D quantum phase transition: 20’000 spins at T=0.005
♦ 2D square lattice: 1’000’000 spins at T=0.2
♦ Extended ensembles (Quantum Wang-Landau, Parallel Tempering)
♦ Allows efficient simulation of 1st order quantum phase transitions
♦ Determination of the free energy of a quantum system
♦ These algorithms allow
♦ Accurate simulation of phase transitions in quantum systems
♦ Quantitative modeling of quantum magnets and bosonic systems
Example 1: Quantum phase transitions
♦ Bilayer antiferromagnet
v
v
v v
H = J ∑ ∑ Si , p ⋅ S j , p + J ⊥ ∑ Si ,1 ⋅ Si , 2
2
p =1 <i , j >
i
J << J⊥: spin gap, no long range order
J >> J⊥ : long range order
Quantum phase transition at J⊥ / J ≈ 2.524(2)
Spin gap vanishes
Magnetic order vanishes
Universal properties
Example 1: Critical exponents
♦ 2D quantum phase transition in a quantum Heisenberg antiferromagnet
♦ Simulations of 20 000 spins at low temperatures
Troyer, Imada and Ueda, J. Phys. Soc. Jpn. 1997
Magnetization
Spin stiffnes
Model
QMC results
no assumption
β
0.345 ±
0.025
ν
0.685 ±
0.035
3D classical
Heisenberg
0.3639 ± 0.7048 ±
0.0035 0.0030
0.034 ±
0.005
Mean field
1/2
0
1
η
0.015 ±
0.020
z
1.018 ±
0.02
0.1
β = 0.345 ± 0.021
zν = 0.695 ± 0.032
0.01
0.01
δ =(J 0/J 1)-((J 0/J 1) c
♦ Consistent with classical 3D Heisenberg model exponents
♦ Can do quantum simulations with the same accuracy as classical
0.1
Example 2: Trapped atoms on optical lattices
♦ Bose-Hubbard model
♦ Both soft- and hardcore
U
♦ Harmonic trapping potential
t
♦ Realistic modeling of 1D, 2D, 3D traps
♦ Density profiles, Momentum distribution
U / t = 6 .7
U / t = 25.0
Lattice depth
S. Wessel, F. Alet, M. Troyer, G.G. Batrouni, PRA (2004).
Review: Classical Monte Carlo simulations
♦ We want to calculate a thermal average
A = ∑ Ac e − βEc / Z with Z = ∑ e − βEc
c
c
♦ Exponentially large number of configurations
⇒ draw a representative statistical sample by importance sampling
♦ Pick M configurations ci with probability
♦ Calculate statistical average
♦ Within a statistical error
pc i = e
− βE ci
/Z
M
1
A ≈ A = ∑ Ac i
M i=1
∆A = (1 + 2τ A )
♦ Problem: we cannot calculate pc = e
i
− βE ci
Var A
M
/Z since we do not know Z
Review: Markov chains and Metropolis
♦ Metropolis algorithm builds a Markov chain
c1 → c 2 → ... → c i → c i+1 → ...
♦ Transition probabilities Wx,y for transition x → y need to fulfill
♦ Ergodicity: any configuration reachable from any other
∀x, y ∃n : (W n ) ≠ 0
x,y
♦ Detailed balance:
W x,y py
=
W y,x px
♦ Simplest algorithm due to Metropolis (1953):
W x,y = min[1, py px ]
♦ Needs only relative probabilities (energy differences)
py px = e
− β (E y − E x )
Quantum Monte Carlo simulations
♦ Not as “easy” as classical Monte Carlo
Z = Tr e − βH = ∑ e − βEc
c
♦ Calculating the energy eigenvalue Ec is equivalent to solving the problem
♦ Need to find a mapping of the quantum partition function
to a classical problem
Z = Tr e− βH ≡ ∑ pc
♦ Different approaches
c
♦ Path integrals (time-dependent perturbation theory in imaginary time)
♦ Stochastic Series Expansion (high temperature expansion)
♦ Sign problem if some pc < 0 (thus try to avoid this)
♦ Then need efficient updates for the effective classical problem
Hamiltonian of spin-1/2 models
♦ Example: two sites
♦ Anisotropic exchange interactions Jxy, Jz
♦ Magnetic field h
H XXZ = J xz (S1x S2x + S1y S2y ) + J z S1z S2z − h(S1z + S2z )
J xz + −
(S1 S2 + S1− S2+ ) + J z S1z S2z − h (S1z + S2z )
2
♦ Heisenberg model: Jxy = Jz = J
=
rr
H = JS1S2 − h (S1z + S2z )
♦ Hamiltonian matrix in 2-site basis
{ ↑↑
, ↑↓ , ↓↑ , ↓↓
}
H XXZ
 Jz
0
 +h
4

J
 0
− z

4
=
J xy
 0
2

 0
0


0
J xy
2
J
− z
4
0



0 


0 

Jz
− h 
4

0
The world line approach
♦ Representation based on mapping of a quantum spin-1/2 system onto a
classical Ising model
♦ Traditionally employing local updates using Metropolis sampling
♦ Continuous time version can be constructed (Prokof'ev et al. 1996)
♦ Cluster updates using the loop algorithm (Evertz et al. 1993)
The Trotter-Suzuki decomposition
♦ Generic mapping of a quantum spin system onto a classical Ising model
♦ basis of most QMC algorithms
♦ not limited to special cases
♦ Split Hamiltonian into two easily diagonalizable pieces
H
H = H1 + H 2
e
−εH
2
= e−ε ( H1 + H 2 ) = e−εH1 e−εH 2 + O(ε )
H(1)
H1
H2
H(2)
H(3)
H(4)
=
+
♦ Obtain a decomposition of the partition function
Z = Tr e − βH = Tr e − β ( H1 + H 2 ) = lim Tr[(e − ∆τ ( H1 + H 2 ) ) M ]
M →∞
(∆τ = β / M )
= Tr[(e − ∆τH1 e − ∆τH 2 ) M ] + O(∆τ 2 )
♦ Insert 2M sets of complete basis states
=
∑
i1 ,...,i2 M
i1 e − ∆τH1 i2 M i2 M e − ∆τH 2 i2 M −1 ⋅ ⋅ ⋅ i3 e − ∆τH1 i2 i2 e − ∆τH 2 i1
Example: Spin-1/2 Heisenberg model
♦ Quantum problem in d dimensions maps onto a classical problem in d+1
♦ Expand the states iα
in the Sz eigenbasis
♦ Effective Ising-model in d+1 dimensions with 2- and 4-sites interaction terms
Z=
∑i
1
e − ∆τH1 i2 M i2 M e − ∆τH 2 i2 M −1 ⋅ ⋅ ⋅ i3 e − ∆τH1 i2 i2 e − ∆τH 2 i1
i1 ,...,i2 M
♦ Each of the
matrix elements
i j +1 e
− ∆τH1, 2
ij
corresponds to a
row of shaded
plaquettes and equals
the product over
those plaquettes
The weights for the spin-1/2 Heisenberg model
♦ The partition function becomes a sum of products of plaquette weights
Z = ∑ W (C ) =∑
C
∏
w(C p )
C plaquettes p
♦ Conservation of magnetization on each bond
♦ The only allowed plaquette-configurations are
∆τJ / 4
e∆τJ / 4ch(∆τJ / 2) e sh(−∆τJ / 2)
♦ Ferromagnet (J<0)
♦ All weights are positive
♦ Antiferromagnet on a bipartite lattice
♦ perform a gauge transformation on one sublattice
J
2
(S
+
i
−
j
−
i
S +S S
+
j
)
J
Si± → (−1) Si±
    → − Si+ S −j + Si− S +j
2
♦ Frustrated antiferromagnet:
♦ we have a sign problem
i
(
)
Worldlines
♦ Each valid configuration is represented by continuous worldlines
W (C ) =
∏
plaquettes p
♦ Sampling over all (important) worldline configurations
♦ According to the above weight
♦ Try to generate a new configurations from a given one
w(C p )
Local updates
♦ Move the world lines locally using Metropolis
♦ probabilities given by the resulting plaquette weights
♦ Consider the limit of small ∆τ
♦ Insert or remove two kinks
P→ = min[1, (∆τ J / 2) 2 ]
P← = min[1,1 /( ∆τ J / 2) 2 ]
P =1
♦ Shift a kink
P = (∆τ J / 2) 2
P→ = P← = 1
P = ∆τ J / 2
Local updates
♦ Problems:
♦ Restricted to canonical ensemble
♦ No change of magnetization, particle number, winding number
♦ Critical slowing down
♦ Solution for classical Monte Carlo: cluster algorithms
♦ Try the same for the quantum case
♦ Loop algorithm
♦ Worm update
♦ Directed loops
(talk by H.G. Evertz)
(talk by S. Trebst)
(talk by F. Alet)
The continuous time limit
♦ Systematic error due to finite value of ∆τ (“Trotter error”)
♦ Need to perform an extrapolation to ∆τ → 0 from simulations with different
values of ∆τ (or Trotter number M)
(∆τ = β / M )
♦ The limit ∆τ → 0 can be taken in the construction of the algorithm!
(Prokof'ev et al. 1996)
♦ Number of changes
βJ
∆τJ
Nc = M
→
2
2
stays finite as
∆τ → 0
♦ Different computational approach:
♦ Discrete time:
store configuration at all time steps
♦ Continuous time: store times at which configuration changes
Path integral representation
♦ Based on the perturbation expansion of the path integral
♦
♦
♦
♦
Continuous time representation
Discrete local objects (kinks, changes in wordline-configuration)
Local updates of kinks using Metropolis
Improved update scheme using Worm update (Prokofev et al., 1997)
Talk by S. Trebst
Path integral representation
♦ Perturbation expansion:
H = H0 +V , H0 =
Z = Tr(e
− βH
) = Tr(e
z z z
z
xy
x x
y y
J
S
S
hS
,
V
J
(
S
S
S
−
=
+
∑ ij i j ∑ i
∑ ij i j i S j )
<i , j >
− βH 0
i
<i , j >
β
dτV (τ )
∫
0
)
Te
−
β
β
τ2
0
0
0
Z = Tr(e − βH 0 (1 − ∫ dτV (τ ) + ∫ dτ 2 ∫ dτ 1V (τ 2 )V (τ 1 ) + ...))
♦ Each term represented by a world line configuration
τ
τ2
τ1
Local updates in continuous time
♦ Shift a kink
♦ Insert or remove two kinks (kink-antikink pair creation process)
P =1
♦ Vanishing acceptance rate:
P = (∆τ J / 2) 2 → 0
P→ = min[1, (∆τ J / 2) 2 ] → 0
♦ Solution: Integrate over all possible insertion in a finite time window
ΛΛ
Λ2 J 2
P→ = ∫ ∫ ( J / 2) dτ 2 dτ 1 →
≠0
8
0 τ1
2
Stochastic series expansion (SSE) approach
♦ Based on a high temperature series expansion of the partition function
♦ Original formulation based on local updates using Metropolis
♦ Cluster updates using the operator loop update (Sandvik 1999)
♦ Improved updates using directed loops (Syljuåsen and Sandvik 2002)
Talk by F. Alet
Decomposing the Hamiltonian for SSE
♦ Break up the Hamiltonian into offdiagonal and diagonal bond terms
H = ∑ H (oi , j ) + H (di , j )
i, j
♦ Example: Heisenberg antiferromagnet
H XXZ =
=
J xy
2
J xy
2
∑ (S
+
i
i, j
S −j + Si− S +j ) + J z ∑ Siz S jz − h∑ S1z convert site terms
i, j
∑ (Si+ S −j + Si− S +j ) + J z ∑ Siz S jz −
i, j
i, j
= ∑ H (oi , j ) + ∑ H (di , j )
i, j
with
H
o
(i , j )
=
2
( Si+ S −j + Si− S +j )
H (di , j ) = J z Siz S jz −
(
h z
Si + S jz
z
into bond terms
(
h
Siz + S jz
∑
z i, j
)
split into diagonal and
offdiagonal bond terms
i, j
J xy
i
)
High temperature series expansion
♦ Expansion in inverse temperature
∞
Z = Tr(e − βH ) = ∑
n =0
∞
=∑
n =0
βn
n!
Tr( − H n )
βn
n
α ∏ (− H
∑
∑
n! α (
)
b1 ,...,bn
i =1
♦ Using the bond Hamiltonians
{
H bi ∈ U H (di , j ) , H (oi , j )
i, j
}
bi
)α
Ensuring positive diagonal bond weights
♦ SSE expansion:
∞
Z = ∑∑
n =0 α
∞
= ∑∑
n =0 α
βn
∑
(
) n!
b1 ,...,bn
∑
(
)
α
n
∏ (− H
i =1
bi
)α
W ( α , (b1 ,..., bn ))
b1 ,...,bn
♦ Negative matrix elements are the bond weights
♦ Need to make all matrix elements non-positive
♦ Diagonal matrix elements: subtract an energy shift C ≥
♦ Does not change the physics
H(i,d j ) = J z Siz S zj −
h z
Si + S zj )− C
(
z
Jz
+h
4
Positivity of off-diagonal bond weights
♦ Energy shift will not help with off-diagonal matrix elements
H
o
(i , j )
=
J xy
2
( Si+ S −j + Si− S +j )
♦ Ferromagnet (Jxy < 0)
♦ no problem
♦ Antiferromagnet on a bipartite lattice
♦ perform a gauge transformation on one sublattice
J xy
2
(S
+
i
S −j + Si− S +j
)
J xy + −
Si± → (−1) S i±
    → −
Si S j + Si− S +j
2
i
(
)
♦ Frustrated antiferromagnet:
♦ we have a sign problem, similar to the world-line approach!
Fixed length operator strings
♦ SSE sampling requires variable length n operator strings
∞
Z=∑
n= 0
βn
n
α ∏ (−H
∑
∑
n!
α (b1 ,...,b n )
bi
)α
i=1
H bi ∈
d
o
H
,H
U { (i, j ) (i, j )}
i, j
♦ Extend operator string to fixed length Λ by adding extra unit operators:
H id = −1
n: number of non-unit operators
Λ
Λ
(Λ − n)!β n
Z = ∑∑ ∑
α ∏ (−H bi ) α
Λ!
n= 0 α (b1 ,...,b Λ )
i=1
H bi ∈ {H id }∪ U {H(i,d j ),H(i,o j )}
i, j
♦ Ensure Λ large enough during thermalization
♦ Such that e.g.
nmax <
3
Λ
4
nmax > n ∝ βV
The SSE configuration space
♦ Each SSE configuration is given by a initial state and a fixed length
operator string
(Λ − n)! β n
Z = ∑∑
α
Λ!
α SΛ
Λ
∏ (− H
i =1
♦ Example for a four site system:
Λ=5
bi
)α
configuration index
5
4
3
2
1
n=4
α =
S Λ = ( H (d1, 2) , H (o3, 4) , 1 , H (o3, 4) , H (d1, 2) )
1
2
3
4
♦ Sample over all possible configurations
♦ According to the above weights
♦ Need a way to generate new configurations from a given one
operator
Hd(1,2)
Ho (3,4)
1
Ho (3,4)
Hd(1,2)
Measurements in SSE
♦ Some observables are very simple:
♦ Energy:
1
E= H =−
β
n
configuration
α (0 ) = α (5)
♦ Specific Heat:
2
CV = n 2 − n − n
α (4 )
α (3)
♦ Uniform Susceptibility:
χ =β α
z
S
∑ iα
α (2 )
α (1)
2
α (0)
i
1
♦ Some look a bit more involved:
2
3
index
operator
5
4
3
2
1
Hd(1,2)
Ho (3,4)
1
Ho (3,4)
Hd(1,2)
4
♦ Equal time diagonal correlations:
1 n
D1 D2 =
∑ d 2 [ p]d1[ p] , where di [ p] = α ( p) Di α ( p)
n + 1 p =0
♦ Imaginary time depended diagonal correlations:
 n  τ
D1 (τ ) D2 (0) = ∑  
∆p = 0 ∆p  β
n



∆p
 τ 
1 − 
 β
n − ∆p
1 n
C12 (∆p ) , C12 (∆p) =
d1[ p + ∆p ]d 2 [ p]
∑
n + 1 p =0
Comparing path integrals and SSE
H = H0 +V
♦ Perturbation Expansion:
β
β
τ2
0
0
0
Z = Tr(e − βH 0 (1 − ∫ dτV (τ ) + ∫ dτ 2 ∫ dτ 1V (τ 2 )V (τ 1 ) + ...))
♦ Decomposition into bond terms:
V = ∑ H bo
b
∞
Z = ∑∑
n =0 α
(
∑
b o1 ,...,b o n
β
τn
τ2
)0
0
0
o
d
τ
d
τ
...
d
τ
W
(
α
,
{
b
}, {τ })
∫ n ∫ n−1 ∫ 1
n

−τ p ( E p − E p −1 ) 
− β E0
o
n
α
W (α , {b }, {τ }) = (−1)  e ∏ e

=
1
p


♦ Compare to SSE
∞
Z = ∑∑
n =0 α
∑
(
)
b1 ,...,bn
W ( α , (b1 ,..., bn ))
n
o
H
∏ bp α
p =1
Comparing path integrals and SSE
♦ Relation between the weights:
♦ Given:
{b o } = (b o1 ,..., b o n )
∑ ∑
β
τ2
W (α , {b}) = ∫ dτ n ... ∫ dτ 1 W (α , {b }, {τ })
m {b}= ( b1 ,...,bm )
{b}|o ={b o }
o
0
0
♦ Operator string longer in SSE, including diagonal terms
♦ In perturbation expansion need to also sample over the continuum times
(Sandvik et al. 1997)
Comparing path integrals and SSE
♦ World lines in path integrals
♦ World lines in SSE
Λ
integer index
imaginary time
β
0
space direction
♦ Advantage
♦ Diagonal terms treated exactly
♦ Disadvantage
♦ Continuous imaginary time
1
space direction
♦ Disadvantage
♦ Perturbation also in diagonal terms
♦ Advantage
♦ Integer index instead of time
The negative sign problem
♦ In mapping of quantum to classical system
Ap
∑
Tr[A exp(− β H )]
=
=
Tr[exp(− β H )]
∑p
i
A
i
i
i
i
♦ “Sign problem” if some of the pi < 0
♦ Cannot interpret pi as probabilities
♦ Appears in simulation of fermions and frustrated magnets
♦ “Way out”: Perform simulations using |pi | and measure the sign:
∑ A p ∑ A sgn p p ∑ p
=
=
p
∑
∑ sgn p p ∑ p
i
A
i
i
i
i
i
i
i
i
i
♦ Sampling
i
i
i
Z | p| = ∑ pi
i
i
i
i
≡
A Sign
Sign
p
p
The negative sign problem
♦ The average sign becomes very small:
Sign
p
1
=
Zp
Z
− βV∆f
sgn
p
p
=
=
e
∑i
i
i
Zp
♦ Both in system size and inverse temperature
♦ This is the origin of the sign problem!
♦ The error of the sign:
∆Sign
=
Sign p
Sign 2
p
− Sign
N Sign
p
2
p
≈
1
p
N Sign
p
e βV∆f
=
N
♦ Need of the order N = exp(2 β V∆f ) measurements for sufficient accuracy
♦ Similar problem occurs for the observables
♦ Exponential growth! Impossible to treat large systems or low temperatures
Is the sign problem exponentially hard?
♦ The sign problem is basis-dependent
♦ Diagonalize the Hamiltonian matrix
H i = εi i
A = Tr [Aexp(−βH)] Tr [exp(−βH)] = ∑ i Ai i exp(−βεi )
i
∑ exp(−βε )
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
Complexity of decision problems
♦ Some problems are harder than others:
♦ Complexity class P
♦ Can be solved in polynomial time on a Turing machine
♦ Eulerian circuit problem
♦ Minimum spanning Tree (decision version)
♦ Complexity class NP
♦ Polynomial complexity using non-deterministic algorithms
♦ Hamiltonian cirlce problem
♦ Traveling salesman problem (decision version)
The complexity class P
♦ The Eulerian circuit problem
♦ Seven bridges in Königsberg (now Kaliningrad) crossed the river Pregel
♦ Can we do a roundtrip by crossing each bridge exactly once?
♦ Is there a closed walk on the graph going through each edge exactly once?
♦
♦
♦
♦
Looks like an expensive task by testing all possible paths.
Euler: Desired path exits only if the coordination of each edge is even.
This is of order O(N2)
Concering Königsberg: NO!
The complexity class NP
♦ The Hamiltonian cycle problem
♦ Sir Hamilton's Icosian game:
♦ Is there a closed walk on the graph going through each vertex exactly once?
♦
♦
♦
♦
Looks like an expensive task by testing all possible paths.
No polynomial algorithm is known
Nor a proof that it cannot be constructed
Concerning the Icosian game: YES!
NP-hardness and NP-completeness
♦ Polynomial reduction
♦ Two decision problems Q and P:
♦ Q ≤ P : there is an polynomial algorithm for Q, provided there is one for P
♦ Typical proof: Use the algorithm for P as a subroutine in an algorithm for P
♦ Many problems have been reduced to other problems
♦ NP-hardness
♦ A problem P is NP-hard,
if ∀Q ∈ NP : Q ≤ P
♦ NP-completeness
♦ A problem P is NP-complete,
if P is NP-hard and P ∈ NP
♦ Most Problems in NP were
shown to be NP-complete
The P versus NP problem
♦ Hundreds of important NP-complete problems in computer science
♦ Despite decades of research no polynomial time algorithm has been found
♦ Exponential complexity has not been proven either
♦ The P versus NP problem
♦ Is P=NP or is P≠NP ?
♦ One of the millenium challenges
of the Clay Math Foundation
http://www.claymath.org
♦ 1 million US$ for proving
either P=NP or P≠NP
♦ The situation is similar to the sign problem
♦ Despite decades of research no general
polynomial time algorithm has been found
♦ Exponential complexity has not been proven either
?
The sign problem is NP-hard
♦ See: M. Troyer and U. Wiese, Report cond-mat/0408370
♦ Use the following theorem:
♦ A problem is NP-hard, if ∃Q : Q NP − complete
♦ Q is the 3D classical spin glass problem
♦ This is bad news!
♦ Or not - if you solve it, you will get both
♦ The Nobel price (??)
♦ And the 1 Million US$...
∧ Q≤P
How can we deal with the sign problem ?
♦ The sign problem is NP-hard (worst-case complexity)
♦ A general solution is almost certainly impossible
♦ What can we do?
♦ Simulate models without a sign problem
♦ Non-frustrated quantum magnets
♦ Bosonic models (atomic BEC condensates)
♦ Hubbard model in 1D
♦ Brute force-approach
♦ Live with the exponential scaling of the sign problem and stay on small lattices
♦ Other exact algorithms
♦ DMRG, exact diagonalization, or series expansion might be better
♦ Special solutions for certain models
♦ Meron-Cluster approach
♦ Special choise of basis