Lectures on
Quantum Monte Carlo Methods
B.B. Beard
Christian Brothers University
Firenze, Italy
25.06.2001-06.07.2001
5
Progression of Lectures
1 Stochastic Integration
6 The continuum limit
2 Random Numbers
7 Observables and
Estimators
3 Classical Statistical
Mechanical Simulations
8 Finite-size scaling
4 Cluster algorithms for
classical models
9 More about the
correlation length
5 Quantum Monte Carlo
10 Survey of other
applications
5. Quantum Monte Carlo
Or, how quantum maps to classical
with a little trickery.
“Quantum Monte Carlo”
is a vast subject
l Variational
Monte Carlo
ðoptimize trial wave functions ψ (x,α)
l Diffusion
Monte Carlo
ðindependent “walkers” search Hilbert space
l Green
Function Monte Carlo
ðUse (part of) the explicit propagator
l Path
Integral Monte Carlo
→ cluster algorithms
Classical SM ↔ Quantum SM
states µ
states n
Hamiltonian functional
Hamiltonian operator
H ( p, q )
H ( p$ , q$ )
Z = ∑ exp( − βH )
µ
(
= ∑ exp − βE µ
µ
)
(
)
Z = Tr exp( − βH )
= ∑ n exp( − βH ) n
n
Quantum Mech ↔ Quantum SM
imaginary time τ=it
inverse temperature β= 1 T
time evolution operator
Boltzmann factor
exp( − iHt ) = exp ( − H τ)
exp( − βH )
ground state
low temperature
Quantum Heisenberg Models
generalize classical spin models
r r
r r
H = ∑ J S ⋅S → H = J ∑ S ⋅S isotropic
xy
x
y
xy
x
xy
[
y
]
r
S x is a spin operator for site x → S xi , S yj = i δxy εijk S xk
r r
S x ⋅S x = S ( S + 1), if S =
1
2
then S i =
l Anisotropic generalizations
H = ∑ J x Si1 S 1j + J y Si2 S 2j + J z Si3 S 3j
ij
1
2
σi
 J x = J y ≠ J z ( = 0 ?)

J x ≠ J y ≠ J z

etc.
Heisenberg model connects
several important concepts
l FHM⇒
ferromagnetism
l AFHM⇒ precursors of high-Tc superconductors
l AFHM is asymptotically free
ðtoy model for QCD
σ model
l Quantum XY model ⇔ U(1) gauge theory
l Non-linear
Heisenberg interactions can be
studied in any dimension d
l 1d
chain
ðexact solution S=1/2 from Bethe ansatz
l In-between
1d & 2d: Spin ladders
l 2d
ðAFHM on square lattice: high-Tc SC, NLσM
ðtriangular, hexagonal lattices: frustration
l Nd
ð “N-vector model”, large-N expansions, etc.
Example: 2-spin interaction
(S=1/2)
r r
l Two-spin interaction: H = JS1 ⋅S 2
1

0
− 41 βJ 
n exp( − βH ) n ′= e
0

0
 ↑↑


 ↑↓
where n = 
 ↓↑


 ↓↓
= 1,1
=
1
2
=
1
2
( 1,0
( 1,0
= 1,− 1
1
2
(1 −
0
eβJ )
0
0

βJ
1
2 (1 − e ) 0
βJ
1
e
+
1
(
) 0
2

0
1
)
" up - down basis"
0,0 )
+ 0,0
−
0
βJ
1
e
+
1
(
)
2
Fundamental difficulty:
H = Σ many non-commuting terms
r r
l Consider, e.g. HM on 1d chain: H = J ∑ S ⋅S
x
ðsites •ó‚ó ƒ ó „ó
l Direct
etc.
xy
evaluation of trace Z=Tr(exp(-βH))
scales exponentially with L
l Explicit diagonalization only feasible for
small systems
y
Trotter-Suzuki expansion
underlies a large class of solutions
l Consider
1d spin chain:
ðbreak H into two sets of commuting operators
H1 = J ∑
x odd
(
r r
S x ⋅S x + 1$
)
( (
H2 = J
= lim
N→ ∞
∑ ∏
{ nk }
k
x even
))
Z = Tr exp( − βH ) = ∑ n exp − εβ( H1 + H 2 )
n
∑
r r
S x ⋅S x + 1$
N
n
where N ε = 1
nk exp( − εβH1 ) nk + 1 nk + 1 exp( − εβH 2 ) nk + 2
1/ε = N = “Trotter number”; error is O(ε2)
Quantum trace corresponds to path
integral over classical variables
l d-dimensional
quantum⇔ (d+1) classical
l Extra ‘euclidean time’dimension corresponds
to inverse temperature β
ðdifferent from ‘Monte Carlo time’
ðalternating layers of Trotter-Suzuki “sandwich”
give “checkerboard” of plaquettes
l Transfer
matrix propagates configuration from
one euclidean time slice to next
Integrating over path space
(
)
exp( − βH ) ↔ exp − S []
φ
n
[]
S φ1
discretized euclidean time ti
Integrating over path space
(
)
exp( − βH ) ↔ exp − S []
φ
n
[]
S φ2
discretized euclidean time ti
Spin-time space comprises N
TS “sandwiches”
N =4
N t = 2dN = 8
time
spin
TS sandwich
{
{
{
{
Problem now focuses on
transfer matrix
l Like
S=1/2
2-spin matrix, but with “ε”
nk exp( − εβH ) nk + 1
l ONLY
1

0
1
= exp( − 4 εβJ )
0

0
(1 +
1
2 (1 −
1
2
0
e εβJ )
(1 −
e εβJ ) 21 (1 +
0
6 NON-ZERO ELEMENTS
ðnon-zero=“finite action”
l Total
1
2
magnetization conserved
0
e εβJ )
e εβJ )
0
0

0
0

1k ,k + 1
Finite-action plaquettes for FHM
w =1
time
spin
w=
1
2
(1 +
e εβJ )
w=
1
2
(1 −
e εβJ )
Typical discrete-time
1d FHM configuration
note
every
row has
same
number
of “up”
spins
time
spin
Boltzmann weight of this configuration =
( (1 −
1
2
e
εβJ
)) ( (1 +
6
1
2
e
εβJ
))
18
Typical discrete-time
1d FHM configuration
note
every
row has
same
number
of “up”
spins
time
spin
Boltzmann weight of this configuration =
( (1 −
1
2
e
εβJ
)) ( (1 +
6
1
2
e
εβJ
))
18
Typical discrete-time
1d FHM configuration
note
every
row has
same
number
of “up”
spins
time
spin
Boltzmann weight of this configuration =
( (1 −
1
2
e
εβJ
)) ( (1 +
6
1
2
e
εβJ
))
18
How should we sample the
Boltzmann distribution?
l Local
Metropolis sampling?
ðsingle flips not allowed! L
ðergodicity requires moves that change total
magnetization L
ðdynamical critical exponent z ≈2 L
l Cluster
algorithm
ðeliminates critical slowing down J
ðimproved estimators J
Choose FHM transition probabilities
to satisfy detailed balance
w =1
1
2
εβJ
1
+
e
(
)
1
2
(1 −
e εβJ )
Choose FHM transition probabilities
to satisfy detailed balance
w =1
1
2
εβJ
1
+
e
(
)
1
2
(1 −
pcontinue = 1
e εβJ )
p cross = 1
Choose FHM transition probabilities
to satisfy detailed balance
w =1
p continue
1 + e εβJ
=
2
1
2
p cross
εβJ
1
+
e
(
)
1 − e εβ J
=
2
1
2
(1 −
pcontinue = 1
e εβJ )
p cross = 1
Choose FHM transition probabilities
to satisfy detailed balance
w =1
p continue
1 + e εβJ
=
2
1
2
p cross
εβJ
1
+
e
(
)
1 − e εβ J
=
2
with J<0, both these are < 1
1
2
(1 −
pcontinue = 1
e εβJ )
p cross = 1
Fixing the sign problem for the
AFHM (on some lattices)
l Note
if J > 0 (antiferromagnetic), then offεβJ
1
w
=
1
−
e
)< 0
diagonal weight
2 (
ð but Boltzmann factor must be > 0
l Solution
for bipartite lattices:
ðbasis change: apply U=diag(i,-i,i,-i)
ðequivalent to rotating every other spin 180o
ðnon-bipartite (e.g. triangular) ⇒ frustration
εβJ
1
w
=
e
− 1) > 0
l Gives off-diagonal
2 (
Finite-action plaquettes for AFHM
w =1
time
spin
w=
1
2
(e εβJ + 1)
w=
1
2
(e εβJ − 1)
Typical discrete-time
1d AFHM configuration
note
every
row has
same
number
of “up”
spins
time
spin
Boltzmann weight of this configuration =
( (e
1
2
εβJ
) ( (e
− 1)
6
1
2
εβJ
)
+ 1)
18
Typical discrete-time
1d AFHM configuration
note
every
row has
same
number
of “up”
spins
time
spin
Boltzmann weight of this configuration =
( (e
1
2
εβJ
) ( (e
− 1)
6
1
2
εβJ
)
+ 1)
18
Typical discrete-time
1d AFHM configuration
note
every
row has
same
number
of “up”
spins
time
spin
Boltzmann weight of this configuration =
( (e
1
2
εβJ
) ( (e
− 1)
6
1
2
εβJ
)
+ 1)
18
Choose AFHM transition probabilities
to satisfy detailed balance
w =1
1
2
εβJ
e
( + 1)
1
2
(e εβJ − 1)
Choose AFHM transition probabilities
to satisfy detailed balance
w =1
pcontinue = 1
1
2
εβJ
e
( + 1)
1
2
(e εβJ − 1)
p jump = 1
Choose AFHM transition probabilities
to satisfy detailed balance
w =1
pcontinue = 1
p continue =
1
2
εβJ
e
( + 1)
2
e εβJ + 1
p jump
1
2
(e εβJ − 1)
e εβJ − 1
= εβJ
e + 1
p jump = 1
Example program: 1dAFHM.exe
l All
the ingredients for DTCA are assembled
ðpick random starting site in spin-time lattice
ðfollow cluster-building rules until loop closes
ðflip spins
specify β,L
l Demo defaults to Nt = L (not required)
l Can
CtrlCtrl-ShiftShift-A
1dAFHM: Noteworthy
l Cluster
loop is self-avoiding
ð“1d object” unlike Wolff clusters in FIM
l Loop
can wrap in periodic time
ðchanges total magnetization
ðergodic
l Loop
can wrap in space
ðrelated to helicity modulus
More dimensions:
split H into more parts
H4
H2
H1 H3
Trotter-Suzuki Sandwich for 2d Square-Lattice AFHM
Higher spin S>1/2: Add layers
and projection operators
The bad news:
Trotter error requires treatment
is O(ε2)
l Typical routine involves repeating
simulation for various N values
l Extrapolate to continuum limit N→ ∞
l Error
Quantum systems are amenable
to cluster algorithms
quantum ⇔ (d+1) classical
l Ising-like variables
l “Worldline” configurations reflect
conserved quantities
l Cluster updates are efficient and ergodic
l Trotter error is controllable
l d-dimensional
ðbut we can do better...
Progression of Lectures
1 Stochastic Integration
6 The continuum limit
2 Random Numbers
7 Observables and
Estimators
3 Classical Statistical
Mechanical Simulations
8 Finite-size scaling
4 Cluster algorithms for
classical models
9 More about the
correlation length
5 Quantum Monte Carlo
10 Survey of other
applications