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 1k ,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 (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