Lectures on Quantum Monte Carlo Methods

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 S...
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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 (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