Quantum Monte Carlo (QMC)users.jyu.fi/~veapaja/QMC/slides.pdf{ Simulate nite temperature quantum systems { Manifest Feynman’s path integral representation of quantum ... I Quantum

Quantum Monte Carlo (QMC)

December 14, 2018

Quantum Monte Carlo (QMC) December 14, 2018 1 / 241

Motivation

We simulate many-body quantum systems by solving the Schrodingerequation stochastically, because there are too many degrees of freedom inanything but trivial systemsBefore you begin, you need:• Hamiltonian (fix particle masses and their interactions)

• particle statistics: bosons, fermions, or sometimes boltzmannons(forget indistinquishability)

– A non-relativistic Hamiltonian has no idea if particles arebosons or fermions

• Do we want properties for the ground state, for excited states, or fora finite temperature system ?

• Do you really need a first-principles result? DFT is faster than QMC.


QMC methods

• Variational Monte Carlo (VMC)

– Sample particle positions from a guessed wave function andsimulate the consequences in a system with a certainHamiltonian

– You may improve your guess (for lower energy or variance), butit’s not automatic

• Diffusion Monte Carlo (DMC)

– Usually for sampling particle positions in the ground state

– Numerically improves upon a guessed wave function

• Path Integral Monte Carlo (PIMC)

– Simulate finite temperature quantum systems

– Manifest Feynman’s path integral representation of quantummechanics

– Also a ground-state variant exists


History

1940’s E. Fermi: Schrodinger equation can written as a diffusion equation;Physical principles of Diffusion Monte Carlo (DMC) discovered.

1942 A. A. Frost: evaluation of the energy of few simple molecules bypicking few representative points in the configuration space;Variational Monte Carlo (VMC) is born.

1947 N. Metropolis and S. Ulam: Diffusion Monte Carlo developed into analgorithm; “. . . as suggested by Fermi . . . ”.

1950’s R. Feynman: Path integral formulation of quantum mechanics;Physical principles of Path Integral Monte Carlo (PIMC) discovered

1962 M. H Kalos: Solved the Schrodinger equation in the form of anintegral equation: Green’s function Monte Carlo (GFMC) is born.

1964 H. Conroy: Proposed, that points should be picked at random fromthe configuration space with probability proportional to Ψ2

T , thesquare of the trial wave function. Application to H+

2 , H−, HeH++,He, H2, and Li - the first application to a fermion system!


History

1965 W. L. McMillan: Energy of liquid 4He; Applied the Metropolisalgorithm to sample points with probability proportional to Ψ2

T .

1971 R. C. Grim and R. G. Storer: Importance sampling.

1975 J. B. Anderson: First application of DMC to electronic systems

1980 D. M. Ceperley and B. J. Alder: fixed-node DMC for electron gas

1982 P. J. Reynolds, D. M. Ceperley, B. J. Alder and W. A. Lester, Jr. :fixed-node DMC for molecules

1986 D. M. Ceperley and E. L. Pollock: PIMC calculation of the propertiesof liquid 4He at finite temperatures.

1998 N.V. Prokofev, B. V. Svistunov, and I. S. Tupitsyn: Worm algorithm

2009 G. H. Booth, A. J. W. Thom, and A. Alavi: Fermion MC as a gameof life in Slater determinant space


Recent publications: what’s going onQMC calculations published in October 2018:

I Nonordinary edge criticality of two-dimensional quantum criticalmagnets, L. Weber, F. P. Toldin, S. Wessel

I Simulating quantum annealing via projective quantum Monte Carloalgorithms, E. M. Inack

I Estimates of the Quantum Fisher Information in the S=1Anti-Ferromagnetic Heisenberg Spin Chain with Uniaxial Anisotropy,J. Lambert, E. Sorensen

I Efficient ab initio auxiliary-field quantum Monte Carlo calculations inGaussian bases via low-rank tensor decomposition, M. Motta et al.

I Quantum simulation for thermodynamics of infinite-size many-bodysystems by O(10) sites, S. J. Ran et al.

I Many-body effects in (p, pN) reactions within a unified approach, R.Crespo, A. Arriaga, R. B. Wiringa, E. Cravo ...

I Relative energies and electronic structures of CoO polymorphsthrough diffusion quantum Monte Carlo, K. Saritas, et al.


Computational cost

I Diffusion Monte Carlo (DMC) ∼ 10x Variational Monte Carlo (VMC)I Atomic and molecular physics: fixed-node diffusion Monte Carlo,

coupled cluster method CCSD(T)I N electrons, fixed-node DMC as N4

I CCSD(T) scales as N7 ⇒ small systemsCCSD(T) stands for Coupled Cluster Singles Doubles (Triples),indicating the level of included diagrams; triples are partially included.The method is extremely accurate, but the workload is tremendous.

I fixed-node DMC scales as Z 5.5−6.5 with the atomic number Z : usepseudopotentials?


When is MC effective?The probability that the point (x , y) is inside the circle is

P(hit circle) =Area of a quarter unit circle

Area of a box=π

4. (1)

Very ineffective!

I =

∫ 1

0dxf (x) . (2)

Estimate

I ≈ IN =1

N

N∑i=1

f (xi ) , (3)

Multidimensional integrals: The partition function of N classical atoms is

Z ∝∫

dr1dr2 . . . drN exp [−βV (r1, ..., rN)] , (4)


When is MC effective?

The variance or the square of the standard deviation of independentsamples is

σ2I ≈

1

N

1

N

N∑i=1

f (xi )2 −

(1

N

N∑i=1

f (xi )

)2 , (5)

so

σI ∼1√N. (6)

If one has to evaluate the integrand N times to calculate a d–dimensionalintegral, then for each dimension one has ∼ N1/d points, whose distance ish ∼ N−1/d . The error for every integration cell of volume hd is in thetrapezoid rule ∼ hd+2, so the total error will be N times that, ∼ N−2/d .For high-dimensional integrals this error diminishes very slowly withincreasing number of evaluation points N.


When is MC effective?

Why is MC effective in multidimensional integrations? Heuristically,a 3D cube and a regular grid of N points, take a 2D projection on asquare: you see only N2/3 points⇒ holes in space, significant parts of space never get any attention.

Ways to do better:

I deterministically: quasi-Monte Carlo methods, ”quasi” because thepoints are not random

I completely random points: Monte Carlo


Correlated samples and MC accuracyFor correlated samples

error in mean =

√2τint

Nσ , (7)

where σ is the variance of N independent samples. The integratedautocorrelation time τint describes how badly the samples are correlated.The bad news is that close to a critical point τint diverges. This is knownas critical slowing down, and it tends to make MC calculations slow nearphase transitions.The term ”critical slowing down” can mean two things:

I Decreasing rate of recovery from small perturbations. It signalsapproaching criticality, such as an ecological disaster. Used as anearly-warning mechanism.

I A Monte Carlo method samples the phase space inefficiently due tolong-range correlations near criticality. MC simulation spends longtime in very similar situations. A more clever algorithm does notsuffer from critical slowing down. A typical example is the Ising modelsimulation, single flip vs. cluster algorithms.


Importance samplingProbability distribution function (PDF) w ,∫ 1

0dxw(x) = 1 normalization . (8)

write the integral in an equivalent form

I =

∫ 1

0dx w(x)

f (x)

w(x)(9)

and change variables,

r(x) =

∫ x

0dx ′w(x ′) . (10)

Now we havedr

dx= w(x) , (11)

and r(x = 0) = 0; r(x = 1) = 1, so the integral is

I =

∫ 1

0dr

f (x(r))

w(x(r)). (12)

According to the above strategy this is estimated as the average of theargument evaluated at random points ri ∈ U[0, 1]:

I ≈ IN =1

N

N∑i=1

f (x(ri ))

w(x(ri )). (13)


Importance samplingThe ultimate goal of importance sampling is to make the integrand asconstant as possible, because with a constant a,∫ 1

0dx a ≈ 1

N

N∑i=1

a = a , (14)

one sample (N = 1), say x = 0.02342, gives the exact result. You can’tbeat this. Integrating ∫ 1

0dxf (x) (15)

pick π(x), which you can sample, to be as close to f (x), so that∫ 1

0dxf (x) ≈ 1

N

N∑i=1

f (xi )

w(xi ), xi sampled from π(x), (16)

and f (x)w(x) is nearly constant.


Sampling From a Distribution

Define r(x) as a cumulative function,

r(x) =

∫ x

−∞dx ′w(x ′) , (17)

which gives

dr(x)

dx= w(x) . (18)

Apparently, ∫ 1

0dr =

∫ ∞−∞

dx w(x) . (19)

MC: left integral is evaluated by picking N random points ri ∈ U[0, 1].The right integral is evaluated by picking points xi from distribution w .⇒ values x in Eq. (17) have distribution w .


Using inverse functions

A) Discrete distributionsWe want to sample integers k with 1 ≤ k ≤ N with probability Pk .

1 Construct the cumulative table of probabilities, Fk = Fk−1 + Pk .

2 Sample r ∈ U[0, 1], find k so that Fk−1 < r < Fk . Repeat step 2.

B) Continuous distributions: the mapping method

1 Calculate the cumulative function of w(x),

F (x) =

∫ x

−∞dx ′ w(x ′) (21)

2 Sample r ∈ U[0, 1], solve x from F (x) = r ; the solutions x samplew(x).


Rejection methodSimple case: distribution is limited from above, w(x) < M, x ∈ [a, b].

1 Draw r1 ∈ U[a, b] and r2 ∈ U[0,M]this picks a random point inside the box around w(x), see figure

2 If w(r1) ≥ r2, accept x = r1 as the desired random number, elsediscard both r1 and r2. Go to 1.

a

w(x)

reject

accept

bx

M

0

w(x) =

∫ M

0dr2θ(w(x)− r2)

=

∫ b

adr1

∫ M

0dr2δ(x − r1)θ(w(r1)− r2) . (22)


Rejection method

Interpreting the identity as an algorithm:

θ(w(x)− r2) =

0, if r2 > w(x)

1, if r2 < w(x), (23)

with some fixed x , so integrating r2 from 0 to M gives w(x). Next we useDirac delta to move x out of the argument of w , replacing it with r1. Thepieces can be now interpreted as an algorithm,

w(x) =

∫ b

adr1︸︷︷︸

sample r1∈U[a,b]

∫ M

0dr2︸︷︷︸

sample r1∈U[0,M]

δ(x − r1)︸︷︷︸suggest r1 as x

θ(w(r1)− r2)︸︷︷︸accept, if r2<w(r1)

. (24)

The rejection algorithm is just a Monte Carlo integration of the rhs.


Special case: standard normal distributionA crude algorithmDraw N random numbers ri ∈ U[0, 1] and calculate the sum

x =

√12

N

N∑i=1

(ri −1

2) (25)

As N increases this approaches the standard normal distribution, aconsequence of the Central Limit Theorem (see MC-lecture.pdf for proof):

If you draw N random numbers from any distribution that itselfhas a statistical mean µ and variance σ, their average is normallydistributed with average µ and variance σ/

√N, as N →∞.

Now we draw N random numbers xi = ri − 0.5 ∈ U[− 12, 1

2], a box-shaped distribution f (x) with µ = 〈x〉 = 0 and

σ2 = 〈(x − 〈x〉)2〉 = 〈x2〉 =∫∞−∞ dxx2f (x) =

∫ 12

− 12

dxx2 = 112

, so σ = 1√12

, and hence 1N

∑Ni=1(ri − 1

2) approaches

normal distribution with variance 1√12N

. Multiply with√

12N and you get standard normal distribution (σ = 1).

This gives a rather good Normal(x) already with small N, as N = 6 orN = 12. The large-x tails are wrong, because you can’t get far with afinite number of steps.


Special case: standard normal distribution

Box-Muller algorithm I1

1 Draw r1 and r2 ∈ U[0, 1]

2 Evaluate θ = 2πr1 and R =√−2 ln r2

3 (x1, x2) = (R cos(θ),R sin(θ))

4 Go to step 1.

Proof:∫ 1

0dr1

∫ 1

0dr2δ(x1 −

√−2 ln r1 cos(2πr2)) δ(x2 −

√−2 ln r1 sin(2πr2))

=

∫ 1

0dr1

∫ 1

0dr2δ(r1 − e−(x1+x2)2/2) δ(r2 −

1

2πarctan(

x2

x1))

∣∣∣∣ ∂(r1, r2)

∂(x1, x2)

∣∣∣∣=

1

2πe−(x2

1 +x22 )/2 =

1√2π

e−x21/2 1√

2πe−x

22/2 = Normal(x1) Normal(x2) .

The algorithm simply computes the lhs integrals using MC.1G. Box and M. Muller, Ann. Math. Stat. 38 (1958) 610


Special case: standard normal distribution

Box-Muller algorithm IIUses the rejection method in order to avoid trig. functions.

1 Draw r1 and r2 ∈ U[0, 1]

2 Make R1 = 2r1 − 1 and R2 = 2r2 − 1, they sample ∈ U[−1, 1]

3 If R = R21 + R2

2 > 1 reject the pair and go to step 1

4 Result:

(x1, x2) =

√−2 lnR

R(R1,R2) (27)

5 Go to step 1


Fernandez–Rivero algorithmFernandez–Rivero algorithm2 is a small Maxwell-Boltzmann gassimulation with fictitious energy exchanges (step 5). After thermalization”particle” velocities v have normal distribution (step 8)

1 Initialise vi = 1, i = 1, . . . ,N (N is about 1024) and set j = 0

2 Draw K (about 16) values θk ∈ U[0, 2π].Compute ck = cos(θk) and sk = sin(θk).

3 Set j = (j + 1) mod N, draw i ∈ IU[0,N − 1] so that i 6= j

4 Draw k ∈ IU[0,K − 1]

5 Set tmp = ckvi + skvj and vj = −skvi + ckvj

6 Set vi = tmp

7 If still in thermalization go to step 3

8 Result: vi and vj

9 Go to 3

Here IU[0,N] means random integer between 0 and N.2J. F. Fernandez and J. Rivero, Comp. in Phys. 10 (1996) 83-88.


Markov Chain Monte CarloMarkov chain

Markov Chain is a chain of states, visited with a rule to pick the nextstate xi+1 from the present one xi . If we go on forever, states are visitedwith the weight w(x) (normalized PDF is π(x)).

This Markov chain forms a random walk.

Here π(x) is the asymptotic, stationary, or limiting distribution.

Example 1: Start the chain from x1 = 1. The rule of the next value isxi+1 = (xi + 1)%5 (modulo). The chain is 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, .... Asa result, the weight of each value is w(x) = 1, and the PDF isπ(x) = 1/5.

Example 2: Start from x1 = 0. The rule is xi+1 = xi + ση, where η is arandom number drawn from the standard normal distribution. The weightof each value is w(x) = e−x

2/(2σ2), and the PDFπ(x) = 1/(σ

√2π)e−x

2/(2σ2).


Markov Chain Monte Carlo

The MC estimate with M samples of an integral is computed as a sum,∫ ∞−∞

dx1√2π

e−x2/2f (x) ≈ 1

M

M∑i=1

f (xi ) , xi sampled from1√2π

e−x2/2 .

(28)

We know how to draw independent samples from a standard normaldistribution (Box-Muller, Fernandez-Rivero etc.). On the previous slide westated that the Markov chain with the rule xi+1 = xi + ση hasπ(x) = 1

σ√

2πe−x

2/(2σ2).


Markov chain vs. direct sampling

Think about computing∫ ∞−∞

w(x)f (x) ≈ 1

M

M∑i=1

f (xi ) . (29)

In principle, you can draw the M points xi from distribution π(x) using

a) Direct sampling of π(x): Always do this if you can.

b) Markov chain x1, ..., xM that has the limiting distribution π(x).

Here a) is more effective, because the points xi are uncorrelated . However,we have algorithms only for sampling a few simple distributions, andmany-body wave functions are not among those. Markov chains areinferior in that the points are correlated, but superior in that you cansample an almost arbitrary distribution: All you need is the rule togenerate the next point: finding it is our next topic.


Detailed balance

Master equation: The weight (probability if normalized) of a state S ′

varies as a function of the simulation time (treated as continuous) :

dπ(S ′)

dt=∑S

P(S → S ′)π(S)−∑S

P(S ′ → S)π(S ′) , (30)

Here P(S → S ′) is the rule to change from S to S ′, and we need one that

gives dπ(S ′)dt = 0, because then we get time-independent π(S) as the

limiting, equilibrium distribution. The detailed balance condition3 is asolution:

P(S → S ′)π(S) = P(S ′ → S ′)π(S ′) , (31)

where π(S) is the weight of state S .

The detailed balance condition is a sufficient, but not necessary,condition that the system reaches the correct equilibrium.

3Also known as microscopic reversibility.


Detailed balance

Make the change from state S to S ′ in two stages:

1 Suggest the change S → S ′ with a transition rule (matrix) T (S → S ′)

2 Accept the change S → S ′ with probability A(S → S ′).

so the rule can be written as

P(S → S ′) = T (S → S ′)︸︷︷︸”Try”

A(S → S ′)︸︷︷︸”Accept”

. (32)

This way it’s easier to take into account the fact that you may have anonsymmetric transition rule, T (S → S ′) 6= T (S ′ → S). Requiring thatthe detailed balance condition is satisfied still leaves an infinite number ofpossible rules P(S → S ′). Which one is the most effective? Meaning,which rule gives π(S) fastest? Let’s assume for a while thatT (S → S ′) = T (S ′ → S), so that P(S → S ′) = A(S → S ′), and look atthe Metropolis answer in the quest of the most effective rule.


Metropolis algorithm

Original problem: sample states, so that they obey the Boltzmann PDF,

π(S) =1

Ze−E(S)/T , kB ≡ 1 . (33)

The solution by Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N.Rosenbluth, Augusta H. Teller, and Edward Teller (MR2T 2, 1953):Accept moves from state S to S ′ with probability

A(S → S ′) = min[1, e−∆E/T

]= min

[1,π(S ′)

π(S)

], (34)

where ∆E = E (S ′)− E (S). This Metropolis algorithm has two merits:

I The partition function Z is not involved.At equilibrium the master equation gives

∑S P(S → S′)π(S)−

∑S P(S′ → S)π(S′) = 0, so any constant factors

in π(S) cancels.

I It reaches the equilibrium distribution π(S) fast.


Metropolis-Hastings algorithm

If you try moving right more than left, you should accept left moves moreeasily. The detailed balance condition with nonsymmetric proposalsT (S → S ′) 6= T (S ′ → S):

P(S → S ′)

P(S ′ → S)=

T (S → S ′)A(S → S ′)

T (S ′ → S)A(S ′ → S)=π(S ′)

π(S). (35)

Compensate with acceptance,

A(S → S ′) = min

[1,

T (S ′ → S)

T (S → S ′)e−∆E/T

]= min

[1,

T (S ′ → S)π(S ′)

T (S → S ′)π(S)

](36)


Variational Monte Carlo (VMC)

Energy of the trial state |ϕT 〉 is

ET =〈ϕT |H|ϕT 〉〈ϕT |ϕT 〉

=

∫dRϕ∗T (R)HϕT (R)∫

dR|ϕT (R) |2(37)

=

∫dR|ϕT (R) |2

[ϕT (R)−1 HϕT (R)

]∫dR|ϕT (R) |2

. (38)

The quantity

w(R) =|ϕT (R)|2∫dR|ϕT (R)|2

(39)

is a good PDF ⇒ can be sampled in MC.Use Metropolis (Markov Chain) to draw samples from |ϕT (R)|2Why: only ratios of weights matter ⇒

∫dR|ϕT (R)|2 is irrelevant !


Metropolis VMC algorithmN particles, wf ϕT (R), R = (r1, ..., rN)

1 Pick initial configuration R1 and set step size step.

2 Pick 1:st particle, set i = 1.

3 Draw random displacement vector d = (dx , dy , dz),dx , dy and dz ∈ U[−1

2 ,12 ]

4 Suggest new position r′i = ri + step d

5 Compute ratio = |ϕT (R′)|2/|ϕT (R)|2

6 Metropolis question: Accept the move with probabilitymin [1, ratio]. If ratio > 1 accept move. Else, draw t ∈ U[0, 1]and if ratio > t accept move. If move was accepted set ri = r′i ,else keep old ri .

7 Pick next particle, set i = i + 1. If i < N go to 3.

8 If still in thermalization, go to 3. Adjust step to keep acceptance∈ [40, 60]

9 Accumulate averages, print out progress.

10 Keep collecting data, go to 2Quantum Monte Carlo (QMC) December 14, 2018 30 / 241

Energy optimizationLet’s look at two flavours, energy optimization and variance optimization.Energy optimization relies on the Rayleight-Ritz variational principle:

Among states with the correct symmetry (Bose or Fermi), theground state has the lowest energy.

This ensure, that optimization of ϕT (R;α) wrt. energy,

∂E (α)

∂α= 0 for optimal α . (40)

Caveats:• The average E (α) cannot always be determined exactly – in MC it

has stochastic error⇒ the minimum of the curve 〈E (α)〉 has stochastic error⇒ optimal α has stochastic error

• The many-body wave function is usually not a simple elementaryfunction⇒ optimal α gives optimal ϕT (R;α) only among a class offunctions.


Energy optimization

More serious caveats:• The many-body wave function cannot be exactly represented by a

finite number of samples R.• E (α) is actually evaluated using a finite sample R = R1, ...RN,

so E (α) ≈ 〈E (α)〉R.– There is no natural lower limit to the energy 〈E (α)〉R.⇒ energy optimization may be unstable, stability may require avery big sample R. This usually manifests itself as 〈E (α)〉Rgetting ridiculously low, because the small sample Rhappened to find a low-energy pithole.

This stability problem is one of the reasons why variance optimizationmay be more suitable than energy optimization.


Variance optimizationVariance optimization relies on the zero variance principle:

The variance in the variational energy vanishes for a trial wavefunction, which coincides with any eigenstate of the Hamiltonian.

Proof: Suppose ϕT (R;α) = φi (R), the i :th eigenstate of H. The varianceof energy is σ2 = 〈E (α)2〉 − 〈E (α)〉2 = 0, because

〈E (α)2〉 =

∫dR|ϕT (R;α)|2EL(R;α)2∫

dR|ϕT (R;α)|2

=

∫dR|φi (R)|2[φ−1

i (R)Hφi (R)][φ−1i (R)Hφi (R)]∫

dR|φi (R)|2

=

∫dR|φi (R)|2E 2

i∫dR|φi (R)|2

= E 2i (41)

and

〈E (α)〉2 =

[∫dR|ϕT (R;α)|2EL(R;α)∫

dR|ϕT (R;α)|2

]2

=

[∫dR|φi (R)|2Ei∫dR|φi (R)|2

]2

= E 2i .

(42)Quantum Monte Carlo (QMC) December 14, 2018 33 / 241

Energy vs. variance optimization

Comments on variance optimization:• Variance – unlike energy – has a natural lower limit, σ2 ≥ 0.

• Variance-optimized energy ≥ energy-optimized energy


Energy vs. variance optimization

Energy optimization and variance optimimization are not equivalent.Higher energy result can have lower variance and vice versa:

-2.92

-2.91

-2.90

-2.89

-2.88

-2.87

-2.86

-2.85

-2.84

-2.83

0 10000 20000 30000 40000 50000

En

erg

y [

Ha]

MC steps

He atom:Steps 0-25000: optimized energy, E = −2.878, σ2 = 0.11.Steps 25001- : optimized variance, E = −2.869, σ2 = 0.08.


Objective functions

Minimize an objective function, one possibility is

σ2opt(α) =

∑Nopt

i=1 w(Ri ;α) [EL(Ri )− Eguess]2∑Nopt

i=1 w(Ri ;α), (43)

where the weight function is

w(Ri ;α) =

∣∣∣∣ ϕT (Ri ;α)

ϕT (Ri ;α0)

∣∣∣∣2 , (44)

and Ri are sampled using ϕT (Ri ;α0) with a fixed α0, known ascorrelated sampling, to be explained on the next slide.


Correlated sampling

Direct minimization may be too inaccurate due to statistical fluctuationsof data.

The differences of the correlated averages can bemuch more accurate than the averages themselves.

Energy with parameters α:

〈E (α)〉 ≡∫dR|ϕT (R;α)|2EL(R;α)∫

dR|ϕT (R;α)|2(45)

=

∫dR|ϕT (R;α0)|2

[|ϕT (R;α)|2|ϕT (R;α0)|2EL(R;α)

]∫dR|ϕT (R;α0)|2 |ϕT (R;α)|2

|ϕT (R;α0)|2.

I Sample points R using weight |ϕT (R;α0)|2; parameters are α0

I Evaluate (a) |ϕT (R;α)|2|ϕT (R;α0)|2EL(R;α) (b) |ϕT (R;α)|2

|ϕT (R;α0)|2


Correlated sampling

Observations about correlated sampling:

I Variance of individual values 〈E (α)〉 is larger that sampling from|ϕT (R;α)|2 directly⇒ Correlated sampling is no good for absolute energy values

I Variance of differences 〈E (α1)〉 − 〈E (α2)〉 can be smaller than what itwould be if 〈E (α1)〉 was sampled from |ϕT (R;α1)|2 and 〈E (α2)〉 wassampled from |ϕT (R;α2)|2:Fluctuations ”cancel in the same direction”, the differences

〈E (α1)〉 − 〈E (α2)〉 , (46)

are more accurate.⇒ Correlated sampling is good for hunting energy differences.

This makes correlated sampling suitable for optimization of the trial wavefunction.


Correlated samplingEnergy optimization with correlated samples

-3

-2.8

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

1 1.5 2 2.5 3 0

2

4

6

8

10

12

E (

Ha)

(E-E

guess

)2 (

Ha

2)

alpha

He atom two 1S orbitals; Correlated sampling

(E-Eguess)2

E

The energy-optimized value is 1.6875, which can be found also analytically– see next page.


Optimization on paperWith a 1S product wf the optimization can be done by hand. e.g. He,

〈ϕT (R)|−1

2∇2

i |ϕT (R)〉 =1

2α2 (47)

〈ϕT (R)|− 2

ri|ϕT (R)〉 = −2α (48)

〈ϕT (R)|− 1

rij|ϕT (R)〉 =

5

8α , (49)

so (this is not the local energy, it’s the energy of the trial state)

E (α) = α2 − 4α +5

8α . (50)

This gives optimized value α = 2− 516 . Similarly, the first elements give

elem α E (α) (au) Exact (au)

H 1 -0.5 -0.5He 2− 5

16 =1.6875 -2.8476 -2.90372Li 3− 5

8 =2.375 -8.4609 -7.47806


Virial theorem and full optimizationFor Coulomb potential, the virial theorem states that

〈V〉 = −2〈T 〉 . (51)

A good check, but can’t really tell if one trial wave function is better thanthe other.Why not do full optimization ?Find the normalized wave function of lowest energy:

E [Ψ,Ψ∗, λ] =

∫dRΨ∗(R)HΨ(R)− λ

(∫dR|Ψ(R)|2

)vary Ψ∗ : δE =

∫dRδΨ∗(R)HΨ(R)− λ

∫dRδΨ(R)∗Ψ(R)

=

∫dR[HΨ(R)− λΨ(R)

]δΨ(R)∗ . (52)

Take arbitrary δΨ(R)∗ , and we are back to Schrodinger equation

HΨ(R) = λΨ(R) Harharhar! (53)


Cusp conditions

In the local energy, infinities of the potential energy should be cancelled bythose of the kinetic energy4. These happen if

I two electrons are too close

I electron get too close to nucleus

Concentrate on two electrons, keeping the rest fixed: relative r = r1 − r2

and center-of-mass RCM = 12 (r1 + r2). Since (see lecture notes for details)

∇21 +∇2

2 = 2∇2r +

1

2∇2

RCM, (54)

the Hamiltonian is

H = −(

1

4∇2

RCM+∇2

r

)− 1

2

N∑i=3

∇2i + V (r,Rcm, r3, ..., rN) . (55)

4T. Kato, Comm. Pure Appl. Math. 10, 151 (1957).


Cusp conditionsTry a trial wf of Jastrow type,

Ψ(R) = Ψ(r,Rcm, r3, ..., rN) = e−u(r)f (r)Q(Rcm, r3, ..., rN) , (56)

where

1 f (0) 6= 0, if the spins at r1 and at r2 are anti-parallel

2 f (0) = 0, and f (−r) = −f (r), if the spins at r1 and at r2 are parallel

The potential part diverges as 1/r , so we want to keep this h(r) finite:

h(r) =1

e−u(r)f (r)

(−∇2

r +1

r

)e−u(r)f (r) . (57)

Expand (leave out the arguments r),

h =2u′

r+ u′′ − u′2 +

1

r+ 2u′

1

f

r

r·∇rf −

1

f∇2

r f , (58)

where u′ ≡ du/dr . Look at the two cases separately.


Cusp conditions

Case 1: anti-parallel spins, f (0) 6= 0Only 1. and 4. terms are singular,

h =2u′

r+ u′′ − u′2 +

1

r+ 2u′

1

f

r

r·∇rf −

1

f∇2

r f , (59)

so we set

2u′ + 1

rfinite , (60)

so the cusp condition is

du

dr

∣∣∣∣∣r=0

= −1

2. (61)


Cusp conditions

Case 2: parallel spins, f (0) = 0, and f (−r) = −f (r)Taylor expansion of f (r) around origin is

f (r) = a · r +O(r3) , (62)

which has only odd terms because f (−r) = −f (r). Close to origin we get

h ≈ 2u′

r+ u′′ − u′2 +

1

r+ 2u′

1

a · rr

r· a =

4u′

r+

1

r, (63)

so the cusp condition is

du

dr

∣∣∣∣∣r=0

= −1

4. (64)


Diffusion Monte Carlo (DMC)Schrodinger equation in imaginary time ⇒ diffusion equation

−∂|Ψ(τ)〉∂τ

= H|Ψ(τ)〉 . (65)

formal solution is (if H does not depend on τ)

|Ψ(τ)〉 = e−Hτ |Ψ(0)〉 projection . (66)

The eigenstates |φi 〉 of H are solutions of

H|φi 〉 = Ei |φi 〉, E0 < E1 < E2 < . . . . (67)

We can expand any N-body state in the basis of eigenstates of H,

|Ψ(0)〉 =∑i

ci |φi 〉 , (68)

so

|Ψ(τ)〉 =∑i

cie−Eiτ |φi 〉 . (69)


Diffusion Monte Carlo (DMC)

Stabilise the evolution by subtracting a trial or reference energy ET ≈ E0:

|Ψ(τ)〉 = e−(H−ET )τ |Ψ(0)〉 =∑i

cie−(Ei−ET )τ |φi 〉 . (70)

States with Ei > ET will decay. The projection operator is Green’s function

G(τ) = e−(H−ET )τ , (71)

satisfies the equation

−∂G(τ)

∂τ= (H − ET )G . (72)

Represented by a coordinate-space matrix with elements

〈R′|G(τ)|R〉 = 〈R′|e−(H−ET )τ |R〉 ≡ G (R′,R; τ) . (73)


Short time G (τ): Operator splittingUsually H = T + V, and [T , V] 6= 0 ⇒ we don’t know e−(H−ET )τ exactly.But we can approximate,

G (τ) = e−(H−ET )τ = e−(T +V−ET )τ = e−T τ︸︷︷︸GT (τ)

e−(V−ET )τ︸︷︷︸GV (τ)

+O(τ2) . (74)

The potential part is easy,

GV(R′,R; τ) = e−(V (R)−ET )τδ(R′ − R) . (75)

The kinetic part is a bit trickier (to be derived later),

GT (R′,R; τ) = (4πλτ)−3N/2e−(R−R′)2

4λτ . (76)

Put together, we get the ”primitive approximation”,

G (R′,R; τ) ≈ (4πλτ)−3N/2e−(R−R′)2

4λτ e−(V (R)−ET )τ . (77)


Diffusion GT (τ)Let’s derive the coordinate-space representation of the diffusion Green’sfunction. We use the eigenstate expansion (K is a 3N-dimensionalmomentum space vector),

T φK(R) = (−λ∇2R)φK(R) = λK 2φK(R) (78)

φK(R) = 〈R|K〉 =1√L3N

e−iK·R , (79)

and the states |K〉 are a complete basis. The coordinate-spacerepresentation of GT (τ) is

GT (R′,R; τ) ≡ 〈R′|e−τ T |R〉 =∑

K

〈R′|e−τ T |K〉〈K |R〉 (80)

=∑

K

〈R′|e−τλK2 |K〉φ∗K(R) =∑

K

φK(R′)φ∗K(R)e−τλK2

(81)

=1

L3N

∑K

e−iK·(R′−R)e−τλK2. (82)


Diffusion GT (τ)

The momentum values are 2π/L apart, and for large L the sum can beconverted to an integral. 5

∑K

≈ L3N

(2π)3N

∫dK (83)

1

L3N

∑K

e−iK·(R′−R)e−τλK2 ≈ 1

(2π)3N

∫dKe−iK·(R′−R)e−τλK

2, (84)

where the integral is a Fourier transform of a Gaussian, leading to the niceresult

GT (R′,R; τ) = (4πλτ)−3N/2e−(R−R′)2

4λτ . (85)

This is usually an excellent approximation, so I left out the ”≈”.

5If not, evaluate the sum.


Simulation of GT (τ): DiffusionWe want to compute evolution |Ψ(τ)〉 = GT (τ)|Ψ(0)〉, which is

ψ(R′; τ) =

∫dRGT (R′,R; τ)ψ(R; 0) ≈ 1

M

M∑i=1

ψ(Ri ; 0) , (86)

where Ri is sampled from GT (R′,R; τ) (fixed R′ and τ).GT (R′,R; τ) is a product of 3N Gaussians (in 3D), so all of them can besampled independently ⇒ 3N independent diffusions – but what is σ?Compare the exponents:

2σ2 = 4λτ ⇒ σ =√

2λτ . (87)

Result: we can sample points Ri by a random walk with the rule

Ri+1 = Ri +√

2λτ η , (88)

where η = (ηx , ηy , ηz), ηx ,y ,z sample standard normal distribution.


MC representation of the many-body wave function

In VMC we had an analytical wave function, but in DMC we don’t have ananalytical form for Ψ(R, τ). Instead, DMC represents Ψ(R, τ) in terms ofa finite number of samples R. These configurations are made of”walkers” (they walk a Markov chain).Consequences:

1 Need quite a large sample R to represent Ψ(R, τ)I Use of the order of 100–10000 (depending on the problem)

configurations R, each is one possible configuration in Ψ(R, τ)I Slower than VMC by about the number of configurationsI Many walkers ⇒ consumes memoryI Each configuration is independent ⇒ very good parallelization! Only

now and then collect data and do branching.

2 Too few configurations means Ψ(R, τ) is poorly represented.⇒ The algorithm must keep the number of configurations large – butnot too large, or you’re out of memory.


First DMC algorithm

Pick a large number of configurations R to begin with.

1 Diffusion due to GT (τ): Move from R→ R′ diffusively, in otherwords move each walker according to

r ′i = ri + η√

2λτ ,

where η is a d-dimensional Gaussian random variable with zeromean and unit variance.

2 Branching due to GV(τ): The wave function Ψ(R, 0) is

multiplied by the factor e−(V (R)−ET )τ . So we calculate V (R) foreach configuration and see whether this factor is more or lessthan one, and create/destroy configurations accordingly. Walkersare not moving.

Alas, this is no good if the potential is singular: diffusion takes walkerssooner or later to a near-singular position.


First DMC algorithmThe factor e−(V (R)−ET )τ makes the algorithm problematic:

I Repulsive potential, V (R) 0 if particles come too close. The walkeris killed in branching. Easily lots of killed walkers ⇒ only a few maysurvice ⇒ bad statistics and large fluctuations in number of walkers

I Attractive potential, V (R) 0, if particles come too close. Thewalker branches to very many walkers. ⇒ large jump up in number ofwalkers ⇒ bad statistics and large fluctuations in number of walkers

Bad statistics results if present walkers are descendants of too few walkers.One way to see where problems are is to realize, that the sole energymeasure is the potential energy, so the algorithm would have to producecoordinates R, where

〈V (R)〉R = E0 . (89)

In other words, one evaluates the potential energy in well-chosencoordinates, where it averages to the ground state energy.

Conclusion: we’d better improve the algorithm so that walkers stay awayfrom regions of large |V (R)|. This can be done with importancesampling. Quantum Monte Carlo (QMC) December 14, 2018 54 / 241

Importance sampling in DMCSolve the imaginary-time evolution of the product

f (R; τ) ≡ Ψ(R; τ)ϕT (R) , (90)

knowing that the evolution of Ψ(R; τ) is given by the imaginary-timeSchrodinger equation,

−∂Ψ(R; τ)

∂τ= (H − ET )Ψ(R; τ) . (91)

Insert Ψ(R; τ) = f (R; τ)/ϕT (R) to get

−∂f (R; τ)

∂τ= −λ∇2

Rf (R; τ) + λ∇R · [F(R)f (R; τ)] + [EL(R)− ET ] f (R; τ) .

(92)

drift force (or drift velocity, quantum force)

F(R) ≡ ∇R ln |ϕT (R)|2 = 2∇RϕT (R)

ϕT (R), (93)

and the familiar local energy EL(R) ≡ ϕT (R)−1HϕT (R).Quantum Monte Carlo (QMC) December 14, 2018 55 / 241

Green’s function for importance sampling DMCThe imaginary-time evolution is

−∂f (R; τ)

∂τ= −λ∇2

Rf (R; τ) + λ∇R · [F(R)f (R; τ)] + [EL(R)− ET ] f (R; τ)

≡ HF f (R; τ) , (94)

which defines a new operator, HF . Again the formal solution is

f (R; τ) = e−τHF f (R; 0) , (95)

and the Green’s function is again

G (R,R′; τ) = 〈R′|e−τHF |R〉 . (96)

Once we know what this is, we can write the evolution as an integralequation,

f (R; τ) =

∫dR′G (R,R′; τ)f (R′; 0) . (97)


Green’s function for importance sampling DMCWithout importance sampling we did two things:

1 Deside how accurate Green’s function we want. We settled for themost crude Trotter expansion,

e−τH ≈ e−τ T e−τ V , (98)

which has timestep error O(τ2).

2 Solve GT (R′,R; τ) = 〈R′|e−τ T |R〉 and GV(R′,R; τ) = 〈R′|e−τ V |R〉And we were done. For better accuracy we could improve the Trotterexpansion. Importance sampling adds one complication, namely the driftterm

HF f (R; τ) = −λ∇2Rf (R; τ)︸︷︷︸

diffusion

+λ∇R · [F(R)f (R; τ)]︸︷︷︸drift

+ [EL(R)− ET ] f (R; τ)︸︷︷︸branching

≡ TF f (R; τ) + [EL(R)− ET ] f (R; τ) . (99)

Evolution with only diffusion and drift is known as the Fokker-Planckequation.


Fokker-Planck operator

The drift-diffusion a.k.a Fokker-Planck evolution is

−∂f (R; τ)

∂τ= −λ∇2

Rf (R; τ) + λ∇R · [F(R)f (R; τ)] ≡ TF f (R; τ) (100)

Define the drift operator,

F(R)|R〉 ≡ F(R)|R〉 . (101)

to get

∇R · [F(R)f (R; τ)] = ∇R ·[〈R|F(R)|f (τ)〉

]= ∇R · F(R)f (R; τ) . (102)

so the Fokker-Planck operator is

TF = −λ∇2R + λ∇R · F(R) . (103)


Fokker-Planck operator

The Fokker-Planck operator is usually written in terms of the momentumoperator P = −i∇R,

TF = λP2 + iλP · F(R) . (104)

With timestep error O(τ2),

e−τHF ≈ e−τ TF e−τ EL−ET (105)

where, for shortness, I defined ”local energy operator” EL similar to thepotential operator 〈R′|EL|R〉 = EL(R)δ(R′ − R). That’s simple, theFokker-Planck Green’s function is more involved. We need to evaluate

GTF(R′,R; τ) ≡ 〈R′|e−τ TF |R〉 . (106)

But before we do, let’s look at why did we ”give up” and defined a driftoperator.


Fokker-Planck evolution

Why didn’t we solve the drift-diffusion matrix elements and be happy? Tryto proceed as usual, and turn our attention to the eigenfunction expansion– just to show you how and why this is wrong. Let’s boldly define

TFQQ(R) = qQQQ(R) (not valid !) (107)

QQ(R) = 〈R|Q〉 =∑

K

〈R|K〉〈K|Q〉 =∑

K

aQ,KφK(R) . (108)

Repeat the steps with free diffusion,

GTF(R′,R; τ) =

∑Q

e−τqQQ∗Q(R′)QQ(R) (not valid !) . (109)

and we are done. But hold on, can we solve the eigenproblem for TF ?



Let’s try to solve the Fokker-Planck eigenproblem (non-existent, but wewere curious). First,

TFφK(R) = −λ∇2RφK(R) + λ∇R · [F(R)φK(R)] . (110)

Insert the eigenstates of T , φK(R) = 1√L3N

e−iK·R,

TFφK(R) = λK 2φK(R) + λ∇R · [F(R)φK(R)] . (111)

Using ∇RφK(R) = −iKφK(R) we get

λ∇R · [F(R)φK(R)] = λ [∇R · F(R)]φK(R) + iλK · F(R)φK(R) . (112)



So far we have

TFφK(R) =[λK 2 + λ [∇R · F(R)] + iλK · F(R)

]φK(R) . (113)

The quantity in brackets depends on R, so it’s not an eigenvalue. Theimaginary unit gives us a hint (you can prove it also rigorously) that TF isnon-hermitian, so we conclude that

The Fokker-Planck operator HF is non-hermitian 6,

So what’s the problem? The eigenfunction expansion in Eq. (109) is validonly for hermitian operators, non-hermitian operators are much harder todeal with. Another fine mess importance sampling got us into.

Let’s return to the drawing board and try another route to solve theFokker-Planck (drift-diffusion) problem.

6Non-self-adjoint, to be more precise.


Fokker-Planck evolution – again

Let’s make a new start and look again at the Fokker-Planck equation,

−∂f (R; τ)

∂τ= −λ∇2

Rf (R; τ) + λ∇R · [F(R)f (R; τ)] , (114)

Instead of working with the evolution of the probability density f (R; τ), wecan write an equivalent stochastic PDE for time evolution of theobservable, R(τ). The equation for R(τ) is a Langevin equation. 7

The basic problem is obvious: diffusion and drift don’t commute. We cantry splitting them to two separate evolution processes we can handle. Becareful, you’d wan’t the same accuracy in τ as you had in the originaloperator splitting. A better accuracy may be too costly and a worseaccuracy sets the overall accuracy to lower. If we are happy with O(τ2)

accuracy, then GF (τ) = e−τ(−λ∇2R+λ∇R·F(R)) ≈ e−τ T e−τλ∇R·F(R) is good

enough also here. The first part is diffusion, the second is drift.

7Some mathematicians dislike Langevin equations because of the so-called Ito-Stratonovich dilemma: there are two validinterpretations of how certain stochastic integrals should be evaluated; the values of the integrals disagree.


Drift evolutionConcentrate only on solving only the drift evolution,

−∂f (R; τ)

∂τ= λ∇R · [F(R)f (R; τ)] (only drift) . (115)

The drift Green’s function is propagation of particles along a trajectory,

〈R′|e−τλ∇R·F(R)|R〉 = δ(R′ − R(τ)) , (116)

so let’s ask

What are the trajectories R(τ) of particles under drift?

The drift differential equation (115) is a continuity equation, usuallywritten for density ρ(r) and current j(r) as ∂ρ/∂t = ∇ · j. We caninterpret the function f (R; τ) as density and λF(R)f (R; τ) as current, sowe can pick the solution from literature: the trajectories are solutions of

dR(τ)

dτ= λF(R(τ)) . (117)

The solution that is good enough for present purposes is

R(τ) = R(0) + F(R(0))λτ +O(τ2) . (118)Quantum Monte Carlo (QMC) December 14, 2018 64 / 241

The Langevin equation of Fokker-Planck evolution

Now we know what drift does (correct to O(τ2)),

〈R′|e−τλ∇R·F(R)|R〉 = δ(R′ − R− λτF (R)) , (119)

A random walk with both diffusion and drift is simply

R′ = R +√

2λτη + λτF (R) . (120)

This is the approximate solution of the stochastic Langevin equation,which was equivalent to the Fokker-Planck equation for the probabilitydensity f (R; τ).8

This approximate solution begins to fails because the larger the timestep,the more frequently diffusion takes particles to very large drift. This errormay not be healed in subsequent moves and you begin to get biasedresults.

8It’s sometimes stated that the Langevin equation is analogous to Heisenberg picture, Fokker-Planck equation is analogousto Schrodinger picture. Frankly, I don’t see the analogy.


A working first order DMC algorithm

1 Initialise many (∼ 100− 500) configurations, where walkers aredistributed according to |ϕT (R)|2. We don’t want them all destroyedin the branching process, so we need to generate as diverse a set aspossible. You can pick from a VMC run that samples |ϕT (R)|2positions R now and then to a zeroth generation of configurations.The total number of configurations in a generation is its population.

2 In each configuration of the present generation, move the i th walkerwith

rnewi = ri + λτF(R) + η√

2λτ , (121)

where η is a 3-dimensional Gaussian random vector.

3 Calculate a weight for the move

ratio =|ϕT (R′)|2G (R′,R; τ)

|ϕT (R)|2G (R,R′, τ), (122)

and accept the move R→ R′ with probability min[1, ratio](”Metropolis question”)


A working first order DMC algorithm

4 Take each configuration in turn and make Ncopies copies of it,

Ncopies = int[e−τ(EL(R)−ET ) + ξ

]ξ ∈ U[0, 1] , (123)

5 Estimate the ground-state energy E0 by taking the average of thelocal energies of current generation.

6 Keep the population healthy by adjusting the trial energy ET . Oneway is to use a feedback mechanism,

ET = E0 + κ lnNCtarget

NC, (124)

where E0 is the best guess for the ground-state energy we have so far,NCtarget is the population we’d like to have, and NC is the one wehave now. The positive parameter κ should have the smallest valuethat does the job.

7 go back to step 2


A working first order DMC algorithmComments:• The accept/reject test is a Metropolis-Hastings test.

– The Hamiltonian with drift is non-hermitian; the accept/rejectstep ensures that the detailed balance condition is satisfied ⇒makes sure the points R are really sampled from thedistribution f (R; τ) = Ψ(R; τ)ϕT (R).

– The acceptance must be > 99.9 % – if not, you time step istoo large. At your own discretion you may even omit the testand assume that the acceptance is very high anyhow, but thenyou are on your own.

• Controlling the number of configurations (sets of walkers) should bedone with kid gloves. The more you force it, the more fluctuationsincrease elsewhere. No matter how you control the population youwill be upsetting randomness, so be gentle.

• Use as large a population as you can afford. Something like 5000configurations is common. Debugging can use a smaller set.


Higher order DMC

The order of DMC comes from how we deal with non-commuting parts inthe Hamiltonian H. With importance sampling, we must write e−τHF interms of simpler, albeit approximate, matrix elements:

1 First operator splitting: HF = TF + EL, [TF , EL] 6= 0Separate the Fokker-Planck (drift-diffusion) operator TF and the localenergy operator EL.

2 Second operator splitting: TF = T + λ∇R · F(R),[T , λ∇R · F(R)] 6= 0The Fokker-Planck equation describing probability density evolutionin diffusion under drift is cast to a Langevin equation, an equation ofmotion of a single walker under a velocity field.

To get second or higher order DMC we must improve both splittings inunison, and solve the Langevin equation without loosing accuracy.


What Hamiltonian did we actually use?Let’s ask a question: What ”fake Hamiltonian” H′ has the exact splitting 9

e−τH′

= e−τ T e−τ V ? (125)

The Campbell-Baker-Hausdorff (C-B-H) formula 10 states that

eAeB = eA+B+ 12

[A,B]+ 112

[A−B,[A,B]]+... , (126)

so (drop terms beyond τ3)

eτ T eτ V = eτ(H+ 12τ [T ,V]+ 1

12τ2[T −V,[T ,V]]) ≡ eτH

′, (127)

and read the operator (use [T , V] = [H − V, V] = [H, V]),

H′ = H+1

2τ [H, V]− 1

12τ2[H − 2V, [H, V]] . (128)

9S. A. Chin, Phys. Rev. A42 (1990).10An old theorem (1890,1902), has to do with Lie algebra of a Lie group.


Energy accuracy without importance sampling

Now we can find out how bad mistake we did in energy. After DMCwithout importance sampling has converged, we get the energy

〈E 〉 ≡ 〈EL〉φ0 = 〈φ0|H′|φ0〉 (129)

= E0 +1

2τ〈φ0|[H, V]|φ0〉 −

1

12τ2〈φ0|[H − 2V, [H, V]]|φ0〉 (130)

= E0 −1

12τ2〈φ0|[H − 2V, [H, V]]|φ0〉 , (131)

where the linear term is zero, because |φ0〉 is an eigenstate of H,

〈φ0|[H, V]|φ0〉 = 〈φ0|HV − VH|φ0〉 = 0 . (132)

Without importance sampling the energy converges as τ2

with e−τH ≈ e−τ T e−τ V


Energy accuracy with importance sampling

Importance sampling makes DMC sample the product φ0(R)ϕT (R), andwe evaluate the mixed estimator

〈E 〉 ≡ 〈EL〉φ0ϕT= 〈ϕT |H′|φ0〉 (133)

= E0 +1

2τ〈ϕT |[H, V]|φ0〉 −

1

12τ2〈ϕT |[H − 2V, [H, V]]|φ0〉 ,

which converges only linearly, because 〈ϕT |[H, V]|φ0〉 6= 0.

With importance sampling the energy converges as τ withe−τH ≈ e−τ T e−τ V


Energy accuracy with importance sampling

We saw this linear convergence in the demo program.

-2.9065

-2.906

-2.9055

-2.905

-2.9045

-2.904

-2.9035

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

E (

Hart

ree)

timestep (1/Hartree)

He atom (spins up-down)

DMCExact

-2.90375 -0.362546 timestep


Propagators with second order accuracy

It’s simple to construct a 2nd order splitting, one popular choice is

e−τA+B ≈ e−12τAe−τ Be−

12τA (134)

Next we must also solve the drift trajectories from Eq. (117)

dR(τ)

dτ= λF(R(τ)) (135)

to second order. The drift changes as particles move, so Eq. (118) couldbe improved to (2nd order Runge-Kutta)

R(τ) = R(0) + λτF(R(0) +1

2λτF(R(0))) +O(τ3) . (136)


Propagators with second order accuracy

-2.907

-2.9065

-2.906

-2.9055

-2.905

-2.9045

-2.904

-2.9035

0 0.005 0.01 0.015 0.02 0.025 0.03

E

tau

2nd order resultsExact

-2.90366 -3.46192 x2

Figure: Second order argorithm results for He atom, no accept/reject step.

There are several alternatives (see Chin), which differ in how many driftevaluations (expensive) are needed and what is the range of quadraticbehaviour. I won’t go into details, but push forward to 4th orderalgorithms.


Propagators with fourth order accuracySheng-Suzuki theorem: there is no 4th order splitting

e−τ(A+B) =N∏i=1

e−τai Ae−τbi B (137)

without some ai < 0 and bi < 0. 11

This we cannot have in DMC, because a negative ti means 〈R′|e−τ ti T |R〉is inverse diffusion (backward in time), which cannot be simulated.Something has to be added, and the that’s some commutator(s) of T andV. A convenient one to keep is related to the classical force on particle i ,12

[V, [T , V]] = 2λ|∇RV (R)|2 = 2λN∑i=1

|fi (R)|2 , fi = −∇iV . (138)

11Q. Sheng, IMA J. of Num. Anal. 9, 199 (1989), M. Suzuki, J. Math. Phys. 32, 400 (1991).12The operator [V, [T , V]] acts on some φ(R).


The commutator is evaluated like this, T = −λ∇2R, V := V (R),

[V, [T , V]]φ(R) = [−V 2T − T V 2 + 2V T V ]φ (139)

= λ[V 2∇2φ+∇2(V 2φ)− 2V∇2(Vφ)]

= λ

[V 2∇2φ+ ∇ · [2V (∇V )φ+ V 2∇φ]

− 2V∇ · [(∇V )φ+ V∇φ]

](140)

= λ

[V 2∇2φ+ 2(∇V )2φ+ 2(V∇2V )φ+ 2V (∇V ) ·∇φ

+ 2V∇V ·∇φ+ V 2∇2φ

−2V∇2Vφ−2V∇V ·∇φ−2V∇V ·∇φ−2V 2∇2φ

]coloured terms cancel, only 2nd term survives

= 2λ(∇V )2φ . (141)


Propagators with fourth order accuracy

Caleidoscope of 4th order methods (incomplete list):

I Trotter 13 (divide τ to infinitely many pieces)

e−τH = limM→∞

[e−

τMT e−

τMV]M

, (142)

I Ruth-Forest (Ruth (1983, private channels), Forest and Ruth (1990),re-discovered several times around 1990 by Campostrini and Rossiand by Candy and Rozmus) 14

I Takahashi-Imada (1984)15 and Li-Broughton (1987)16

I Chin (1997)17

13H. F. Trotter, Proc. Am. Math. Soc. 10, 545 (1959)14E. Forest and R. D. Ruth , Physica D. 43, 105 (1990)15M. Takahashi and M. Imada, J. Phys. Soc. Jpn.53, 3765 (1984)16X. P. Li and J. Q. Broughton, J. Chem. Phys.86, 5094 (1987)17S. A. Chin, Phys. Lett. A 226, 344 (1997), and S. A. Chin and C. R. Chen, J. Chem. Phys. 117, 1409 (2002)


Propagators with fourth order accuracy

Forest-Ruth

GFR(τ) = e−v3τ Ve−t3T e−v2Ve−t2τ T e−v1τ Ve−t1τ T e−v0τ V , (143)

with s = 21/3 = 1.25992,

v0 = v3 =1

2

1

2− s(> 0) , v1 = v2 = −1

2

s − 1

2− s(> 0) (144)

t1 = t3 =1

2− s(> 0) , t2 = − s

2− s(< 0!) . (145)

From right to left, the potential will be evaluated at times 0,t1τ ,(t1 + t2)τ (< 0!), and τ . Suzuki was right, there are negativecoefficients. These make the application quite tricky. The good thingabout FR is that it only needs 6 FFT’s to propagate time τ ,


Propagators with fourth order accuracyTakahashi-Imada ”not fully fourth order”

GTI(τ) = e−τ T e−τ V−1

24τ3[V,[T ,V]]e−τ T (146)

Chin Several possible forms, one is

GChin(τ) = e−v0τ Ve−t1τ T e−v1τWe−t2τ T e−v1τWe−t1τ T e−v0τ V (147)

W ≡ V +u0

v1τ2[V, [T , V]] , (148)

v0 =6t1(t1− 1) + 1

12(t1− 1)t1, t2 = 1− 2t1 (149)

v1 =1

2− v0 u0 =

1

48

(1

6t1(1− t1)2− 1

), (150)

and t1 is tunable, 0 ≤ t1 ≤ 1− 1/√

3. Even 5th order errors may cancelwith t1 = 0.35023. All coefficients are positive.


What can a good propagator do?

We have a good approximation for G(τ) = e−τH, such as

GChin(τ) = G(τ) +O(τ5). (151)

Nothing prevents us from overextending the validity, as long as we knowthat we are improving. Take a trial wave function ϕT (R) and propagate it,

〈R|ϕT (τ0)〉 = ϕT (R; τ0) =

∫dR′〈R|GChin(τ0)|R′〉ϕT (R′) . (152)

We can use VMC to see how far in τ0 we can go until the energy begins toincrease: find τ0 were

E (τ0) =〈ϕT (τ0)|H|ϕT (τ0)〉〈ϕT (τ0)|ϕT (τ0)〉

(153)

is at minimum (approximately).18

18In VMC the normalization is irrelevant.


What can a good propagator do?The propagation can now be viewed as a multi-dimensional integral,

ϕT (R; τ0) =

∫dR′dR1dR2e

−v0τ0V (R)〈R|e−t1τ0T |R2〉e−v1τ0W (R2) (154)

〈R2|e−t2τ0T |R1〉e−v1τ0W (R1)〈R1|e−t1τ0T |R′〉e−v0τ0V (R′)ϕT (R′) .

Notice that this applied one GChin for a single, long time step τ0. 19

He4 liquid at equilibrium density

19Rota et al. Physical Review E 81, 016707 (2010); The integrated coordinates can also be thought of as ”shadowcoordinates”, as in the variational shadow wave function by L. Reatto, F. Pederiva et al.


What can a good propagator do?

Large τ means fast projection and less correlated points in DMC: 20

Liquid He4 : Timesteps 0.01(dots), 0.05 (small points) and0.2 (large points). Gettingto the ground state takes 10times more updates with a 2ndorder propagator than with a4th order one.

20H. A. Forbert and S. A. Chin, Phys. Rev. B 63, 144518 (2001)


Fixed-node DMC

Fixed-node DMC (FN-DMC): use antisymmetric trial wave function,

ϕT (...ri , ..., rj ...) = −ϕT (...rj , ..., ri ...) . (155)

A common way to achieve this is to use

ϕT (R) = eU(R)∑k

dkDk [φ](R) (156)

I Jastrow factor eU

I particle-particle correlationsI takes care of particle-particle potentials, especially singularities

I Slater determinants Dk

I antisymmetrized single-particle orbitals φI takes care of Pauli principleI Functional of orbitals, evaluated at R: in my clever notation Dk [φ](R)


Jastrow factor

Often the potential doesn’t depend on symmetry, so we choose aconvenient way to cancel it’s singularities. In electron systems, we havesingular e-n and e-e potentials, so we need at least these correlations(indices i , j for electrons, α for nuclei):

eU(R) = e∑

i,α uen(riα)+∑

i<j uee(rij ) Jastrow factor . (157)

This can be supplemented with an e-e-n factor,

e∑

i<j,α ueen(riα,rjα,rij ) , (158)

which should be enough, because e-e-e-n is rare due to the Pauli principle.

FN-DMC evolution optimizes the correlations to achieve lowest energy


Slater determinantsFor spin-1/2 fermions, one often uses two Slater determinants, 21

ϕT (R) = eU(R)D↑(r1...rN↑)D↓(rN↑+1...rN) . (159)

Here the Slater determinant for N fermions with spins up is

D↑(r1...rN↑ ) =1√N↑!

∣∣∣∣∣∣∣∣∣∣∣

φ1(r1) φ2(r1) ... φN↑ (r1)

φ1(r2) φ2(r2) ... φN↑ (r2)

.

.

.

.

.

. ...

.

.

.φ1(rN↑ ) φ2(rN↑ ) ... φN (rN↑ )

∣∣∣∣∣∣∣∣∣∣∣. (160)

The single-particle orbitals φ can be taken from

I Hartree-Fock (HF) ← more popularI Density functional theory (DFT)

HF has no correlations. This is presumably a good thing, because DMChas the liberty to optimize correlations in the Jastrow factor. DFT tries toadd correlations to orbitals, which, presumably, prevents DMC to do it’sbest in optimizing correlations.

21While not antisymmetric under exchange of spin-up and spin-down fermions, the expectation values of spin-independentoperators are valid. See W. M. C. Foulkes, L. Mitas, R. J. Needs and G. Rajagopal, Rev. Mod. Phys. 73 , 33 (2001).


Slater-type orbitals (STO’s)

First, hydrogen-type orbitals are not very useful. The complete setcontains discrete status plus the continuum, hence ∞ number of states.Too many of them are needed in representation of core electrons.STO’s have the benefit of having only angular node, Rnl is nodeless:

φnlm(r) = Rnl(r)Ylm(θ, φ) (161)

Rnl(r) =(2ζ)3/2

Γ(2n + 1)(2ζr)n−1e−ζr , (162)

with a nuclear-charge-related constant ζ. The notation 1s, 2s, 2p, 3d... isre-used. For more flexibility, the exponent can be chosen to depend on nl ,ζnl .


Gaussian-type orbitals (GTO’s)

The spherical harmonic GTO’s are chosen for their computational merits,rather than physical properties:

φnlm(r) = Rnl(r)Ylm(θ, φ) (163)

Rnl(r) =2(2α)3/4

π1/3

√22n−l−2

(4n − 2l − 3)!!(√

2αr)2n−l−2e−αr2. (164)

Here n!! is a special double factorial,

n!! =

1 n = 0

n(n − 2)(n − 4)...2 n > 0, even

n(n − 2)(n − 4)...1 n > 0, odd1

(n+2)(n+4)...1, n < 0

(165)

Typically, a lot more GTO’s than STO’s are needed, but GTO’s are fast toevaluate. GTO’s have zero cusp and they die out faster in r than STO’s.


Cartesian forms

In cartesian coordinates Slater-type orbitals are

φSTOijk (x , y , z) = x iy jzke−ζr , (166)

and one often uses them as sums of Gaussian primitives,

Xn(x , y , z) = Nnijkx

iy jzke−αnr2, (167)

where the total angular momentum is l = i + j + k , resulting in ones-function (l = 0), three p-functions (l = 1), six d-functions (l = 2) etc..Simple ones are STO-nG, n Gaussians fitted to make an STO.Improvements are double-zeta (the ζ’s in the exponent), two basisfunctions per atomic orbital, triple-zeta etc.Example 3: The ground state of Carbon is 1s22s22p2, and the orbitals1s, 2s, 2px , 2py , 2pz need minimal basis of 5 basis functions, or 10double-zeta basis functions.


Slater-Condon rulesNotation: |Ψpq

mn〉 means from Ψ, change orbital m to p and n to q,(xi := (ri , σi ))

|Ψ〉 := A[ϕ1(x1)ϕ2(x2)...ϕm(xm)ϕn(xn)...ϕN(xN)] (168)

|Ψpm〉 := A[ϕ1(x1)ϕ2(x2)...ϕp(xm)ϕn(xn)...ϕN(xN)] (169)

|Ψpqmn〉 := A[ϕ1(x1)ϕ2(x2)...ϕp(xm)ϕq(xn)...ϕN(xN)] . (170)

The Slater-Condon rules for a one-body operator F =∑N

i=1 fi say, thatthe expectation value and matrix elements are

〈Ψ|F |Ψ〉 =N∑i=1

〈ϕi |f |ϕi 〉 (171)

〈Ψ|F |Ψpm〉 = 〈ϕm|f |ϕp〉 (172)

〈Ψ|F |Ψpqmn〉 = 0 . (173)

For example, the one-body HamiltonianH1 = T + V1 =

∑Ni=1(−∇2

i + V (ri )) can change at most one orbital at atime.


Fixed-node DMC

The exact many-body ground state is a superposition of all possible Slaterdeterminants. Say we have N electrons, scattered on the available orbitals.One way to do this is(collective coordinates xi := (ri , σi ), antisymmetrization A)

I 1st term: all electrons on the lowest HF orbitals

|Ψ〉 = A[φ1(x1)φ2(x2)...φm(xm)φn(xn)...φN(xN)]

I 2nd term: one electron excited to higher orbital p,

|Ψpm〉 = A[ϕ1(x1)ϕ2(x2)...ϕp(xm)ϕn(xn)...ϕN(xN)]

I 3rd term: ...

HF takes a single Slater determinant (here the 1st term), and optimizesφ’s for minimum energy⇒ HF orbitals φ (solve a non-linear, integro-differential equation)⇒ the HF energy – a good start.


Hartree-Fock + DMCHF equations are (simplified to some extent)[

~2

2m∇2 + V (r) + VH(r)

]φi (r)− q2

∫dr′

γ(r, r′)

|r − r′|φi (r) = εiφi (r) , (174)

where

γ(r, r′) ≡∑j

φ∗j (r)φj(r′) (175)

VH(R) = q2

∫dr′∑

j φ∗j (r′)φj(r′)

|r − r′|Hartree potential (176)

Solution: Pick your favorite (constraint: money) from a long list ofsoftware.22 Mike Towler 23 pinpoints the problem in feeding single-particleorbitals to a QMC code:

”Often people find that their HF/DFT code of choice is notsupported, so they give up before they start.”

22wiki DFT software23towler pisa.pdf


https://en.wikipedia.org/wiki/List_of_quantum_chemistry_and_solid-state_physics_software

http://www.tcm.phy.cam.ac.uk/~mdt26/towler_pisa.pdf

Hartree-Fock + DMC

HF energy+ correlation energy = exact energy (definition)

This much of the correlation energy is typically covered:

Hartree-Fock 0 %

optimized Slater-Jastrow ∼ 85 %

FN-DMC ∼ 95 %

Exact 100 %

HF good ⇒ single-determinant Slater-Jastrow wave function OK

atom ground state HF Exp. correlation

He 1s(2) -2.861679993 -2.90338683 -0.041706Li 1s(2)2s(1) -7.432726924 -7.478060323 -0.045333Be 1s(2)2s(2) -14.57302313 -14.6673564 -0.094333B 1s(2)2s(2)2p(1) -24.52906069 -24.541246 -0.012185


Dealing with Slater determinantsMarking ϕT = eUD↑D↓, assuming i ≤ N↑ (it not, swap D↓ and D↑),

∇iϕT = D↑D↓∇ieU + eUD↓∇iD

↑ (177)

Fi = 2∇iϕT

ϕT= e−U∇ie

U +1

D↑∇iD

↑ , (178)

and

∇2i ϕT = D↑D↓∇2

i eU + D↓∇iD

↑ ·∇ieU + eUD↓∇2

i D↑ (179)

−λϕ−1T ∇

2i ϕT = e−U∇2

i eU +

[1

D↑∇iD

↑]·[e−U∇ie

U

]+

1

D↑∇2

i D↑ .

(180)

Furthermore, in moving particles from old to new positions, we need theratio

Ψ(Rnew )

Ψ(Rold)= eU(Rnew )−U(Rold )D

↑(new)

D↑(old). (181)


Dealing with Slater determinantsThe Slater matrix D of spin-up particles has components

Dji = φj(xi ) i ≤ N↑ . (182)

Remember the index order: orbital, particle.I Changing the position of ith particle changes just the ith row of the

Slater matrix:

D↑(r1...rN↑ ) =1√N↑!

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

φ1(r1) φ2(r1) ... φN↑ (r1)

.

.

.

.

.

. ...

.

.

.φ1(ri ) φ2(ri ) ... φN↑ (ri )

.

.

.

.

.

. ...

.

.

.φ1(rN↑ ) φ2(rN↑ ) ... φN (rN↑ )

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣. (183)

I The cofactor of matrix element (i , j), Cij , doesn’t depend on matrixelements on ith row, jth column.

Cij = (−1)i+jMij , Mij is the i, j minor , (184)

and minor Mij leaves out matrix row i and column j .


Dealing with Slater determinants

If the ith particle moves, then24

D := (DT )−1

qi :=D(new)

D(old)=∑j(orb.)

Doldji Dnew

ji =∑j(orb.)

Doldji φj(rnewi ) ,

(185)

(186)

where D = D↑ or D↓. After an accepted move, update(Sherman–Morrison formula)

Djk =

Djk/q, k = i

Djk −Djk

q

∑n(orb.)

Dnkφn(rnewi )q k 6= i .

(187)

The inverses D are computed in the beginning, and only updated asparticles are moved.

24S. Fahy, X. W. Wang, and S. G. Louie, Phys. Rev. B Vol. 42, Nr. 6, 3503 (1990), applied first applied in QMC by Ceperley,Chester, and Kalos (1977).


Dealing with Slater determinantsDetails:Matrix inverse is related to the cofactor matrix,

D := (DT )−1 =C

detD⇒ D−1 =

CT

detD, (188)

and we get the Laplace (cofactor) expansion

detD =∑j

DjiCji , ∀i . (189)

Move the ith particle: the ith row of D changes, but Cij doesn’t, so

qi =detDnew

detDold=

∑j D

newji C old

ji

detDold=

∑j D

newji (detDold Dold

ji )

detDold

=∑j

Doldji Dnew

ji =∑j

Doldji φj(rnewi ) . (190)


Dealing with Slater determinants

Using Laplace expansion again,

∇i detD

detD=

∑j Cji∇iDji

detD=

∑j detDDji∇iDji

detD=∑j

Dji∇iDji , (191)

so∇i detD

detD=∑j

Dji∇iφj(xi ) . (192)


Easy start with basis setsThe Basis Set Exchange (link to portal or link to github):25

# STO-3G EMSL Basis Set Exchange Library 11/12/18 10:13 PM

# Elements References

# -------- ----------

# H - Ne: W.J. Hehre, R.F. Stewart and J.A. Pople, J. Chem. Phys. 2657 (1969).

# Na - Ar: W.J. Hehre, R. Ditchfield, R.F. Stewart, J.A. Pople,

# J. Chem. Phys. 2769 (1970).

# K,Ca - : W.J. Pietro, B.A. Levy, W.J. Hehre and R.F. Stewart,

# Ga - Kr: J. Am. Chem. Soc. 19, 2225 (1980).

# Sc - Zn: W.J. Pietro and W.J. Hehre, J. Comp. Chem. 4, 241 (1983) + Gaussian.

# Y - Cd: W.J. Pietro and W.J. Hehre, J. Comp. Chem. 4, 241 (1983). + Gaussian

#

$basis

*

li STO-3G

*

3 s

16.1195750 0.15432897

2.9362007 0.53532814

0.7946505 0.44463454

3 s

0.6362897 -0.09996723

0.1478601 0.39951283

0.0480887 0.70011547

3 p

0.6362897 0.15591627

0.1478601 0.60768372

0.0480887 0.39195739

*

$end

25Turbomole format.


https://bse.pnl.gov/bse/portal

https://github.com/aoterodelaroza/emsl_basis_set_library

Easy start with basis setsThe STO-3G basis set gives three primitive Gaussians (hence 3G) fit to aSlater-type orbital (hence STO); the s-functions are (always atomic units)

X1(r) = 0.15432897× e−16.1195750r2

X2(r) = 0.53532814× e−2.9362007r2

X3(r) = 0.44463454× e−0.7946505r2. (193)

From the primitives we get contracted Gaussians,

Gn(r) =Nn∑α=1

Xα(r) , (194)

and from contracted Gaussians the orbitals (need normalization),

φi (r) =

Nbasis∑n=1

cinGn(r) . (195)

Obviously STO-3G lacks flexibility. More: basissets.pdf.Quantum Monte Carlo (QMC) December 14, 2018 100 / 241

http://web.uvic.ca/~simmater/Groupweb/Qmodule/basissets.pdf

DMC and chemical accuracy”Chemical accuracy” 1 kcal/mol

1 Hartree = 627.509 kcal/mol

1 kcal/mol = 0.00159360 Hartree

Where are we now:26

26F. R. Petruzielo, J. Toulouse, and C. J. Umrigar, J. Chem. Phys. 136, 124116 (2012).


Path integral Monte Carlo (PIMC)Let’s turn our attention to the statistical physics of equilibrium quantumsystems. In canonical ensemble27 everything can be computed from thecanonical partition function

Z (β) =∑i

e−βEi =∑i

〈i |e−βH|i〉 (196)

where H|i〉 = Ei |i〉 and β = 1/(kBT ), kB is the Boltzmann constant, andT is the temperature. Now Z is the trace (norm) of the density operator

ρ ≡ e−βH , ρ(R′,R;β) ≡ 〈R′|e−βH|R〉 , (197)

where the latter is the density matrix . Trace is basis-independent: Insertposition eigenstates,

Z (β) =

∫dR∑i

〈i |e−βH|R〉〈R|i〉 =

∫dR∑i

〈R|i〉〈i |e−βH|R〉

=

∫dR〈R|e−βH|R〉 =

∫dRρ(R,R;β) . (198)

27Energy and particle number are conserved.Quantum Monte Carlo (QMC) December 14, 2018 102 / 241

Path integral Monte Carlo (PIMC)

In most cases we don’t know 〈R|e−βH|R〉 – except for small β.28 Split theoperator (H obviously commutes with H, so this splitting is exact),

Z (β) =


∫dR〈R|e−

β2He−

β2H|R〉 . (199)

Repeat and insert position states,

Z (β) =

∫dRdR1...dRM−1〈R|e−

βMH|R1〉〈R1|e−

βMH|R2〉...〈RM−1|e−

βMH|R〉 .

(200)

This is a multidimensional integral of a product of high-temperaturedensity matrices

ρ(R′,R; βM

) = 〈R′|ρ ( βM

)|R〉 = 〈R′|e−βMH|R〉 . (201)

28Small β (high T ) in PIMC is equivalent to the small timestep τ in DMC.


Primitive approximationThe high-temperature density matrix can be approximated using ideasdescribed in the DMC section. The splitting e−βH ≈ e−βT e−βV gives theprimitive approximation for the high-T density matrix,

〈R′|e−βH|R〉 ≈ (4πλβ)−3N/2e−(R−R′)2

4λβ e−βV (R) . (202)

In PIMC it’s customary to write the Hamiltonian as

H = −λ∇2R + V (R)

λ ≡ ~2

2m.

(203)

(204)

In DMC it’s also common to mark D ≡ ~2

2m , D for ”diffusion constant”.With M time slices it’s nice to define29

τ ≡ β

M(205)

29Inequal time slices require minor changes.


Quantum-classical analogyDefine a link action

Sm = S(Rm−1,Rm; τ) = − ln ρ(Rm−1,Rm; τ)

ρ(R0,RM ;β) =

∫dR1 . . .RM−1e

−∑M

m=1 S(Rm−1,Rm;τ) .

(206)

(207)

Customarily, one defines

Sm :=3N

2ln(4πλτ) +

(Rm−1 − Rm)2

4λτ︸︷︷︸K(Rm−1,Rm;τ) ”kinetic action”

+U(Rm−1,Rm; τ)︸︷︷︸”inter-action”

(208)

in 3D, where the inter-action U depends on the approximation. Forexample, the primitive action is 30

S(prim.)m =

3N

2ln(4πλτ) +

(Rm−1 − Rm)2

4λτ︸︷︷︸“Spring term”

+ τV (Rm)︸︷︷︸“Potential term”

. (209)

30The potential term can be symmetrized, τ/2(V (Rm−1) + V (Rm)).


Ring polymers

In polymer analogy the spring term holds the polymer together and thepotential term gives the inter-polymer interaction.Example: 5 particles and 4 time slices (not in scale)

Periodic boundary

Periodic boundary

Note: The springs in the figure represent the kinetic , spring action(Rm−1−Rm)2

4λτ , not the potential !


Ring polymers

Another view of 5 particles in 4 time slices:

Only particles on the same time slice interact:Blues interact and reds interact, but blues don’t interact with reds.


Energy estimatorsHaving no trial wave function removes the human bias, but also takesaway one obvious energy measure, the trial energy. There are many energyestimators, with the same average, but with different variance:

I The thermodynamic energy estimator

〈E 〉Th = − 1

Z

∂Z

∂β(210)

I The direct or Hamiltonian energy estimator

〈E 〉dir =Tr[He−βH]

Tr[e−βH](211)

I The virial estimator

〈E 〉vir = no simple form, approximation specific (212)

Here the notation is just to distinguish the three; the averages are thesame, only how the data is collected differently, and also the variancesdiffer. Obviously, you can invent estimators if you feel the need.


Energy estimators

Start from the exact action in 3D, Eq. (207),

ρ(R0,RM ;β) =

∫dR1 . . .RM−1 e−

∑Mm=1 S(Rm−1,Rm;τ)

S(Rm−1,Rm; τ) :=3N

2ln(4πλτ) +

(Rm−1 − Rm)2

4λτ+ Um

Z =

∫dR1 . . .RM e−

∑Mm=1 S(Rm−1,Rm;τ) with R0 ≡ RM .

(213)

(214)

(215)

At high T , Um averages to potential energy, τ〈V 〉. At low T this”inter-action” contains some kinetic terms as well, so it’s really just someunknown term, in need of approximation.


Thermal energy estimator

Remember that τ = β/M, Sm = S(Rm−1,Rm; τ) and usethermodynamics:

〈E〉Th = −1

Z

∂

∂β

∫dR1 . . . RM e−

∑Mm=1 Sm =

1

Z

M∑m=1

∫dR1 . . . RM e−

∑Mm=1 Sm

∂Sm

∂β(216)

=1

Z

M∑m=1

∫dRm−1dRm

∂Sm

∂β

(∫dR1 . . . Rm−2Rm+1RM e−

∑Mm=1 Sm

)︸︷︷︸

ρ(Rm−1,Rm ;τ)

(217)

=M∑

m=1

∫dRm−1dRm

∂Sm

∂β

1

Zρ(Rm−1, Rm ; τ)︸︷︷︸

thermal average

=M∑

m=1

〈∂Sm

∂β〉 =

M∑m=1

1

M〈∂Sm

∂τ〉 , (218)

which gives

〈E 〉Th =1

M

M∑m=1

⟨3N

2τ− (Rm−1 − Rm)2

4λτ2+∂Um

∂τ

⟩. (219)


Thermal energy estimatorThe thermal energy estimator has one serious shortcoming, related to thetwo kinetic terms. The first term,

1

M

M∑m=1

⟨3N

2τ

⟩=

3MN

2β, (220)

is M times the classical kinetic energy. This is the kinetic energy youget, if you start the simulation with M identical time slices. It’s way off,and only running PIMC for a while the second term

1

M

M∑m=1

⟨−(Rm−1 − Rm)2

4λτ2

⟩(221)

cancels the excess kinetic energy. The thermal energy estimator is pooreven for non-interacting particles. Furthermore, at large M (small τ) thetwo partly cancelling kinetic terms fluctuate a lot, making the energyexpectation value slowly converging. 31

31There are, of course, inherent energy fluctuations proportional to the specific heat.


Virial energy estimator

While the thermal kinetic energy fluctuates a lot, the potential energydoesn’t. The quantum virial theorem states, that one can compute thekinetic energy from the potential energy. The energy of a 1D system is

E =1

2x∂V (x)

∂x︸︷︷︸virial kinetic energy

+V (x) . (222)

Herman et al. 32 introduced the virial energy estimator to use this factand get smaller variance.

32M. F. Herman, E. J. Bruskin, and B. J. Berne, J. Chem. Phys. 76, 5150 (1982).


Virial energy estimatorOne way to obtain a better energy estimator is to partially integrate somepath variable and get a kinetic terms with smaller fluctuations. The resultis33

〈E 〉vir =1

M

M∑m=1

⟨3N

2Lτ− (RL+m − Rm) · (Rm+1 − Rm)

4Lτ2λ

− 1

2Fm∆m +

∂Um

∂τ

⟩Fm := −1

τ∇(Um−1 − Um) ∼ classical force

∆m :=1

2L

L−1∑j=−L+1

(Rm − Rm+j) deviation from average ,

(223)

(224)

(225)

and L ∈ [1,M] is a parameter controlling how many time slices areaveraged over.

33Derived in the appendix of Ceperley’s PIMC review (1995).


Virial energy estimatorSpecial cases:

I L = 1, so ∆m = 0 and the virial estimator reduces to the thermal one,

〈E 〉vir =1

M

M∑m=1

⟨3N

2τ− (Rm+1 − Rm)2

4τ2λ+∂Um

∂τ

⟩= 〈E 〉th (226)

I L = M (usual choice; notice: j = 0 term is zero)

∆m =1

2M

M−1∑j=−M+1

(Rm − Rm+j) (227)

〈E 〉vir =1

M

M∑m=1

⟨3N

2β−

with nowindings and exchanges

0︷︸︸︷(Rm+M − Rm) ·(Rm+1 − Rm)

4Mτ2λ

+1

2τ∇(Um−1 − Um)∆m +

∂Um

∂τ

⟩(228)


Virial energy estimator in the primitive approximationThe virial kinetic energy in the primitive approximation is

〈T 〉 =1

2M

M∑m=1

〈Rm ·∇V (Rm)〉 . (229)

This is already quite good, but writing positions wrt. a reference pointreduces the variance further. One choice is to use a centroid , the center ofmass of a polymer ring. Equally well one could use just any point on theparticles path,

〈T 〉 =3N

2β︸︷︷︸classicalkineticenergy

+1

2M

M∑m=1

〈(Rm − RM) ·∇V (Rm)〉︸︷︷︸quantum corrections

. (230)

Alas, the virial energy estimator for each approximate action has to beelaborated separately, because the inter-action term is different.


Virial energy estimatorWhere did the R · V (R) pop up? The virial theorem itself can be derivedusing scaled coordinates,

rs := sr = r + (s − 1)r , (231)

and requiring, that the scaled energy Es is stationary at s = 1,

∂〈Es〉∂s

∣∣∣∣s=1

= − 1

Zs

∂

∂β

[∂Zs

∂s

]∣∣∣∣s=1

= 0 (232)

so∂Zs

∂s

∣∣∣∣s=1

= 0 . (233)

The potential energy can be expanded around s = 1,V (Rs ) = V (r1 + (s − 1)r1, r2 + (s − 1)r2, ..., rN + (s − 1)rN )

= V (R) + (s − 1)R ·∇V (R) +O((s − 1)2) , (234)

so∂V (Rs)

∂s

∣∣∣∣s=1

= R ·∇V (R) . (235)



In a similar fashion we get the scaling of the full action, 34

S(Rm−1;s ,Rm;s ; τ) =3N

2ln(4πλτ) +

s2(Rm − Rm−1)2

4λt+ Um;s , (236)

so∂Sm∂s

∣∣∣∣s=1

= 2(Rm − Rm−1)2

4λt+∂Um;s

∂s

∣∣∣∣s=1

. (237)

34Use(Rs−R′s )2 =

∑Ni=1[(xi,s−x′i,s )2 +(yi,s−y′i,s )2 +(zi,s−z′i,s )2] =

∑Ni=1 s2[(xi−x′i )2 +(yi−y′i )2 +(zi−z′i )2] = s2(R−R′)2.


Virial energy estimatorUsing

dRs = d3r1s ...d3rNs = s3NdR , (238)

the energy stationarity condition is (mark Rk with just k)

∂

∂sZs

∣∣∣∣s=1

=∂

∂s[s3NM

∫d1...dMe−

∑Mm=1 S((m−1)s ,ms ;τ)]

∣∣∣∣s=1

=

∫d1...dMe−

∑Mm=1 S(m−1,m;τ)

M∑m=1

(3N − 2

(Rm−1 − Rm)2

4λτ−∂Ums

∂s

∣∣∣∣s=1

)= 0 . (239)

On the other hand, the energy is ( ∂∂β = 1

M∂∂τ ),

〈E〉 = −1

ZM

∂

∂τZ = −

1

ZM

∂

∂τ

∫d1...dMe−

∑Mm=1 S(m−1,m;τ)

=1

ZM

∫d1...dMe−

∑Mm=1 S(m−1,m;τ)

M∑m=1

[3N

2τ−

(Rm−1 − Rm)2

4λτ2+∂Um

∂τ

]

=1

ZM

∫d1...dMe−

∑Mm=1 S(m−1,m;τ)

[1

2τ

M∑m=1

(3N − 2

(Rm−1 − Rm)2

4λτ−∂Um;s

∂s

∣∣∣∣s=1

)+

M∑m=1

(1

2τ

∂Ums

∂s

∣∣∣∣s=1

+∂Um

∂τ

)].



Use the stationarity condition to eliminate the integral of the blue term,

〈E〉 =1

ZM

∫d1...dMe−

∑Mm=1 S(m−1,m;τ)

[ M∑m=1

(1

2τ

∂Um;s

∂s

∣∣∣∣s=1

+∂Um

∂τ

)], (240)

so from coordinate scaling we get the virial energy estimator

〈E 〉vir =1

M

M∑m=1

⟨1

2τ

∂Um;s

∂s

∣∣∣∣s=1

+∂Um

∂τ

⟩(241)

Notice, how the misbehaving thermal energy terms 3N2τ and − (Rm−1−Rm)2

4λτ2

vanish: The latter came from the the scaled action, ∂∂sSm;s , but the former

came from the scaled differentials, ∂∂s s

3N !


Virial energy estimator for the primitive action

The primitive approximation has Um = τV (Rm), so

1

2τ

∂Um;s

∂s

∣∣∣∣s=1

+∂Um

∂τ=

1

2

∂V (ms)

∂s

∣∣∣∣s=1

+ V (Rm)

=1

2Rm ·∇V (Rm) + V (Rm) , (242)

and we get the result mentioned earlier in Eq. (229),

〈E 〉(prim.)vir =1

M

M∑m=1

⟨1

2Rm ·∇V (Rm) + V (Rm)

⟩. (243)


Virial energy estimator for Chin actionLet’s try scaled coordinates for Chin’s 4th order propagator, Eq. (147).

GChin(τ) = e−v0τ Ve−t1τ T e−v1τWe−t2τ T e−v1τWe−t1τ T e−v0τ V . (244)

Stack to M time slices, kinetic time jumps are (times τ)

[t1, t2, t1], [t1, t2, t1], ..., [t1, t2, t1] , (245)

so

ZChin =

∫d1..dMρ0(1, 2; t1τ)ρ0(2, 3; t2τ)ρ0(3, 4; t1τ)ρ0(4, 5; t1τ)...ρ0(M − 1,M; t2τ)ρ0(M, 1; t1τ)

e−v0τV (1)e−v1τW (2)e−v1τW (3)e−2v0τV (4)...e−2v0τV (M−2)e−v1τW (M−1)e−v1τW (M)e−v0τV (1)

, (246)

and, read the Um of Chin action,

UChin(Rm−1, Rm ; τ) =

2v0τV (m − 1) + v1τW (m) m = 2, 5, 8, ...

v1τW (m − 1) + v1τW (m) m = 3, 6, 9, ...

v1τW (m − 1) + 2v0τV (m) m = 1, 4, 7, ... .

(247)

The modified potential is given in terms of V , but it’s better to scalecoordinates in W directly,

∂W (Rs)

∂s

∣∣∣∣s=1

= R ·∇W (R) . (248)


Virial energy estimator for Chin actionMark

V (R) := R ·∇V (R) (249)

W (R) := R ·∇W (R) , (250)

and we get

1

2τ

∂Um;s

∂s

∣∣∣∣s=1

=

v0Vm−1 + 1

2v1Wm , m = 2, 5, 8, ...12v1Wm−1 + 1

2v1Wm , m = 3, 6, 9, ...12v1Wm−1 + v0Vm , m = 1, 4, 7, ... .

(251)

and, using v1τW (m) = v1τV (m) + 2λu0τ3(∇V (m))2,

∂Um

∂τ=

2v0V (m − 1) + v1V (m) + 6λu0τ

2(∇V (m))2 , m = 2, 5, 8, ...

v1V (m − 1) + 6λu0τ2(∇V (m − 1))2 + v1τW (m) , m = 3, 6, 9, ...

v1V (m − 1) + 6λu0τ2(∇V (m − 1))2 + 2v0V (m) , m = 1, 4, 7, ... .

(252)


Bosons and fermionsWe haven’t touched the subject of particle statistics yet. The Hamiltonian(203) doesn’t know about statistics, we have to force it. The densitymatrix eigenfunction expansion is always valid , 35

ρ(R′,R; τ) = 〈R′|e−βH|R〉 =∑i

e−βEiφi (R′)φi (R) , (253)

so all we need is to require that summed states φi (R) must be(anti)symmetric. We could do this with Slater determinants (fermion) orpermanents (boson), but the notation is simpler with permutations,

φi ,B/F(R) =1

N!

∑P

(±1)Pφi (PR)

+ for bosons (B)

− for fermions (F), (254)

P permutes particle labels and P counts pair permutations;∑

P denotesthe sum over all permutations. For N = 3:

123 : P = 0 213 : P = 1 231 : P = 2

321 : P = 1 132 : P = 1 312 : P = 2 .

35We’ve used is earlier for boltzmannons, distinguishable particles.


Bosons and fermions

Working backwards, we get the Bose/Fermi density matrix 36

ρB/F(R′,R; τ) =∑i

e−βEiφi (R′)

[1

N!

∑P

(±1)Pφi (PR)

](255)

The symmetrized states give the boson density matrix,

ρB(R′,R; τ) =1

N!

∑P

ρ(R′, PR; τ) (256)

and the antisymmetrized states give the fermion density matrix,

ρF(R′,R; τ) =1

N!

∑P

(−1)Pρ(R′, PR; τ) (257)

36It doesn’t matter whether the permutation operator operates on R, on R′, or on both.


Fermion density matrixThe fermion density matrix as determinant of one-body density matricesρ1:

ρF(R′,R; τ) =

∣∣∣∣∣∣∣∣∣ρ1(r1, r′1; τ) ρ1(r2, r′1; τ) ... ρ1(rN , r

′1; τ)

ρ1(r1, r′2; τ) ρ1(r2, r′2; τ) ... ρ1(rN , r′2; τ)

...ρ1(r1, r′N ; τ) ρ1(r2, r′N ; τ) ... ρ1(rN , r

′N ; τ)

∣∣∣∣∣∣∣∣∣ . (258)

The equivalent to fixed node DMC, the fixed-node trial density matrixneeds a reference point R∗. See Militzer: two-particle nodes:

ρT (r1, r2, r∗1, r∗2; τ) =

∣∣∣∣ρ1(r1, r∗1; τ) ρ1(r2, r∗1; τ)ρ1(r1, r∗2; τ) ρ1(r2, r∗2; τ)

∣∣∣∣ , (259)

which for free particles is

ρT (r1, r2, r∗1, r∗2; τ) = (4πλτ)6/2

∣∣∣∣∣∣e− (r1−r∗1 )2

4λτ e−(r2−r∗1 )2

4λτ

e−(r2−r∗1 )2

4λτ e−(r2−r∗2 )2

4λτ

∣∣∣∣∣∣ , (260)

so the nodes are (r1 − r2) · (r∗1 − r∗2) = 0.Quantum Monte Carlo (QMC) December 14, 2018 125 / 241

http://militzer.berkeley.edu/diss/node29.html

Boson superfluidityLet’s go back on the ring polymer picture. How large are the rings? The

spring term is a gaussian e(Rm−1−Rm)2

4λτ with variance σ2 = 2λτ , and M ofthem could cover a circle with radius

r ∼√

2λMτ =

√~2β

m. (261)

Multiply with√

2π and you get the deBroglie wavelength,

λdeBroglie =

√2π~2β

m(262)

Lower T ⇒ larger β ⇒ larger λdeBroglie: at some low T two bosonpolymers may coil together. The inter-polymer spacing in 3D is aboutρ−1/3, so quantum statistics is important below the degeneracytemperature

TD =ρ2/3~2

mkB(3D) . (263)

Remember the periodic boundaries in space.Quantum Monte Carlo (QMC) December 14, 2018 126 / 241

Winding number and superfluidity

I Experimental superfluidity (1938): Kapitza, Allen and Misener

I Superfluid density is directly related to the response of the free energyto a boundary phase twist. 37

I PIMC on liquid 4He: Ceperley and Pollock.38

I Lots of misconceptions and assumptions are about. 39

The winding number is undefined for Hamiltonians that don’tconserve the particle number.

Setup of Andronikashvili experiment: System between two cylinders ofradii R and R + d , and rotate the cylinders slowly ; rotating bucket.PIMC setup: assume d R and put the system between two walls thatare moving with small velocity v wrt. to the liquid.Two-fluid model: The normal component, density ρN , responds to themotion of the walls due to friction. The rest is superfluid , ρs = ρ− ρN .

37M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev. A 8, 1111 (1973)38E. L. Pollock and D. M. Ceperley, Phys. Rev. B 36, 8343 (1987)39V. G. Rousseau, Phys. Rev. B 90, 134503 (2014)



A gas of excitations with energy ε(p) carries momentum (Galilei transf.)

〈P〉v =

∫dr pn(ε− p · v) , (264)

where n(ε) is the distribution function for gas at rest. For small v you canexpand and get 〈P〉v = something × v, the something is the massassociated with excitations - this is the mass of the normal fluid . 40

Follow Pollock and Ceperley and compute the change in free energy. Inthe rest frame walls moving with velocity v

H′ =∑j

(−i~∇j −mv)2

2m+ V , (265)

and the density operator is ρ′ = e−βH′, the same as ρv in the rest frame of

the system. The momentum of the system can now be computed.

40L. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Statistical Physics part II. If a thing has momentum, it has mass.



Using statistical physics,

〈A〉 = Tr[Aρ]/Tr[ρ] =

∑i 〈i |Aρ|i〉∑i 〈i |ρ|i〉

=1

Z

∑i

〈i |Aρ|i〉 , (266)

after some work we find the momentum of the system with moving walls(total density ρ, volume is N/ρ),

〈P〉v =Tr[Pρv ]

Tr[ρv ]≈ normal mass× v = (NNm)v =

(ρN

N

ρm

)v . (267)

Apparently this proportionality can hold only if v is small, but that’s all weneed.



Compute the momentum and expand it to linear order in v . The freeenergy is

Fv = − 1

βlnZv = − 1

βln Tr[ρv ] , (268)

so you finally get (details in Pollock-Ceperley article),

∆FvN

=1

2mv2 ρS

ρ+O(v4) . (269)

In PIMC, we don’t need actual moving walls to measure ∆Fv . The densitymatrix with moving walls, ρv , satisfies the Bloch equation,

−∂ρv (R′,R;β)

∂β= H′ρv (R′,R;β) . (270)

where we use periodic boundary conditions, ri + L = ri .



Define a modified density matrix ρ via a phase factor,

φ(R) ≡ m

~∑j

(rj − r′j) (271)

ρv (R′,R;β) = e iφ(R)ρ(R′,R;β) , (272)

and insert it to the Bloch equation. After some manipulation you get

−∂ρ(R′,R;β)

∂β= Hρ(R′,R;β) , (273)

but the boundary conditions come with a phase factor,

ρ(R′, r1...ri + L, ...rN ;β) = e−imv·L/~ρ(R′,R;β) . (274)

The winding number W counts how many times particle paths windaround periodically. Each winding adds a factor e−imv·L/~.



BTW, the principle how the Galilei transformation

−i~∇j −mv (275)

in the Hamiltonian turns into to a boundary condition for the densitymatrix, phase e−imv·L/~, is widely applicable. Electrons in magnetic fieldhave the Hamiltonian

−i~∇j − eA , (276)

and if they move a full circle round a flux tube with flux Φ, their wavefunction picks up the Aharonov-Bohm phase e−ieΦ/~. 41

41The funny thing is that the circulating electron may never be anywhere where A 6= 0, and still it’s wave function picks upthe phase.


Winding number and superfluidityFrom the winding number we can compute the superfluid fraction, in 3D

ρSρ

=m

~2

〈W 2〉L−1

3ρβ. (277)

On the other hand, the ideal Bose gas Bose-Einstein condensation (BEC)temperature is

Tc =3.31~2

gmkBρ2/3 , (278)

where g is the k-state degeneracy factor. For liquid 4He density this wouldgive Tc ∼ 3.14 K, 42 a bit above the experimental superfluid transitiontemperature at 2.17 K. The fact that the temperatures are so close ledFritz London to propose, that BEC is related to superfluidity, and thatHe-He interactions suppress Tc down to 2.17 K. Let’s see visually whatsuperfluidity requires, next pages contain imaginary time plots of paths.

42A. L. Fetter and J. D. Walecka, Quantum Theory of Many-particle Systems.


Boson superfluidity: not superfluid

R

β



R

β



R

β



R

β



R

β


Boson superfluidity: Start again, not superfluid

R

β


Boson superfluidity: Superfluid!

R

β


Boson superfluidity: Superfluid!

R

β

I Net flux in imaginary time over any boundary in space (such as thegreen boundary, periodic boundaries are not special)

I Four particles in exchange loop, one particle not participatingI Order N particles, not necessarily all particles, is needed for the

exchange loop to reach through the box.Quantum Monte Carlo (QMC) December 14, 2018 141 / 241

Ideal bosons

The simple case of ideal bosons is interesting. We can interpret theparticle exchange at β as a single particle at the lower temperature 2β:

=

This is allowed, since there are no interaction links that would break.


Ideal bosonsWe can now reduce all exchange loops to streched single particle paths,starting from the single-particle partition function Z1 ,

Z1(β) = ”(1)” = Z1(β) (279)

Z2(β) = ”(12) + (21)” =1

2[Z1(β)2 + Z1(2β)] (280)

Z3(β) = ”(123) + (132) + (213) + (321) + (312) + (231)”

=1

3![Z1(β)3 + 3Z1(β)Z1(2β) + 2Z1(3β)] , (281)

and so on. With some combinatorics we find the recursion relation

ZN(β) =1

N

N∑n=1

Z1(nβ)ZN−n(β) . (282)

For example (init recursion with Z1(0) = 1) ,

Z3(β) =1

3

[Z1(β)Z2(β) + Z1(2β)Z1(β) + Z1(3β)

]=

1

3

[Z1(β)

1

2

(Z1(β)2 + Z1(2β)

)+ Z1(2β)Z1(β) + Z1(3β)

]=

1

6

[Z1(β)3 + 3Z1(β)Z1(2β) + 2Z1(3β)

]OK . (283)


Ideal bosons

The (inefficient) bose.py demonstrates how the specific heat peak forms atTc as N increases - and how important it is to use the correct Z1(β). 43

Spoiler: Z1(β) must count the ground state accurately, or you won’t get44

limT→0

CV (T ) = 0 . (284)

One choice is to fix the density , so that more particles occupy largervolume,

ρ =N

Ω= constant⇒ Ω =

N

ρ∝ N . (285)

43See K. Glaum and H. Kleinert, and A Pelster, arXiv 0707.2715v2 (2007).44We need this limit or we can’t define the entropy of a given state, only entropy differences of states.


Ideal bosons

Once Z1(β) is set, the recursion to any N is determined. It doesn’t dependon any spatial quantities, in particular possible periodic boundaries. PBCallows the same permutation, say ”(312)”, to cover more than one spatialinterpretations. Say particles 1 and 2 won’t cross the boundary. Particle 3may end up at the original position of particle 1 (r3(β) = r1(0))

1 without crossing the periodic boundary; r3(τ) is continuous in thebox.

2 crossing the periodic boundary; r3(τ) is discontinuous in the box. Thepath winds (=jumps) from side to side at some τ ′,

r3(τ ′) = (x3(τ ′), y3(τ ′), z3(τ ′)) = (x3(τ ′)− L, y3(τ ′), z3(τ ′)) .

The path is continuous in the extended view of periodic boxes.

Case 1 gives zero winding, but case 2 does wind. From the exactness ofthe recursion we must conclude, that both paths are present in thecomputed Z3(β), if Z1(β), the ”seed”, has PBC .


Ideal bosons in a boxLet’s look at the calculation of Z1(β). For a particle in a box, withoutPBC and V =∞ outside the box, one has45

Z1(β) =

∞∑q=1

e−βq2( π

2λ

Ω2/3)

3

. (286)

Approximately, the sum can be converted to an integral, and taking theground state separately one finds the form used in bose.py.

Z1(β) ≈ 1 +Ω

(4πλ)3/2β−3/2 = 1 +

ρ

(4πλ)3/2β−3/2 , (287)

The reason I’m going through all this is that

As N increases, CV (T ) has the peaked signature of BECtransition at Tc , even though there are no periodic boundariesand no winding paths.

45Again, see K. Glaum and H. Kleinert, and A Pelster, arXiv 0707.2715v2 (2007). I’ve changed notation a bit.


Ideal bosons and superfluidityEarlier we described superfluidity as winding around periodic boundaries,and from the ideal bosons-in-a-box calculation we concluded, that BEChappens regardless of PBC or winding. So what gives? The dilemma is inrelating two phenomena,

I superfluidity as zero viscosity and finite critical velocity.

I Bose-Einstein condensation to the k = 0 state.

For He4, neutron scattering experiments have confirmed, that themomentum distribution shows BEC (about 10 % condensed) at T = 0.Ideal bosons have zero critical velocity, so there’s BEC but nosuperfluidity . As Pethick and Smith puts it,46

However, the connection between Bose-Einstein condensation and superfluidity is asubtle one. A Bose-Einstein condensed system does not necessarily exhibitsuperfluidity, an example being the ideal Bose gas for which the critical velocityvanishes, as demonstrated in Sec. 10.1 below. Also lower-dimensional systems mayexhibit superfluid behavior in the absence of a true condensate, as we shall see inChapter 15.

46C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases Cambridge Univ. Press (2008).


Pair product approximation to the density matrixWe’ve already discussed 4th order propagators, which serve as systemindependent work horses and get us down to low T . One more methoddeserves attention, the pair product approximation. For a hard spherepotential the exact two-body density matrix would obviously be useful,knowing the merits of the pair-product Jastrow wave function. In general,we have two guidelines for density matrices:

I The density matrix satisfies the Bloch equation (see also Eq. (270))

−∂ρ(R,R′;β)

∂β= Hρ(R,R′;β) , (288)

I The Feynman-Kac formula for density matrices,

ρ(R,R′;β) = ρ0(R,R′;β)〈e−∫ β

0 dtV (R(t))〉BRW , (289)

where ρ0 is the free-particle density matrix, and the average is over allBrownian random walk paths.


Pair product approximation to the density matrix

The Feynman-Kac formula is easy to derive, just take the primitiveapproximation and use M − 1 intermediate points,47

ρ(R,R′;β) ≈ 〈R|e−βMT e−

βMV ...e−

βMT e−

βMV |R′〉 (290)

=

∫dR1...RM−1ρ0(R,R1; β

M)...ρ0(RM−1,R

′; βM

)e−βM

∑Mi=1 V (Ri ) (291)

= ρ0(R,R′;β)

∫dR1...RM−1ρ0(R,R1; β

M)...ρ0(RM−1,R

′; βM

)e−βM

∑Mi=1 V (Ri )

ρ0(R,R′;β)︸︷︷︸〈e−

βM

∑Mi=1

V (Ri )〉BRWM

,

with RM = R′. Let M →∞,

〈e−βM

∑Mi=1 V (Ri )〉BRWM

→ 〈e−∫ β

0 dtV (R(t))〉BRW , (292)

and you have the exact Feynman-Kac formula.47The resulting density matrix is not symmetric, but it will be in the limit M →∞.



If the potential in the Feynman-Kac formula is a pair potential, then

e−∫ β

0 dtV (R(t)) =∏i<j

e−∫ β

0 dtV (rij (t)) . (293)

In Jastrow spirit, a good approximation is to assume that particles arecorrelated only in pairs, so that can average each pair independently,

〈∏i<j

e−∫ β

0 dtV (rij (t))〉BRW ≈∏i<j

〈e−∫ β

0 dtV (rij (t))〉BRW

⇒ U(R,R′;β) ≈∏i<j

u(rij , r′ij ;β) .

(294)

(295)

This is the pair product approximation, which ignores three-bodycorrelations and beyond. The point is that u(rij , r

′ij ;β) can be computed

accurately either using matrix squaring or eigenfunction expansion.


Matrix squaring

1 Write the two-body problem in center-of-mass and relative coordinates

2 Use spherical coordinates, angular momentum becomes a parameter(l)

ρ(r , r ′;β) =∑l

ρl(r , r′;β)Pl(cos(θ) (296)

3 Solve the density matrix at high T and iterate, for each l , dow todesired T :

ρl(r , r′′; 2β) =

∫ ∞0

dr ′′ρl(r , r′′, β)ρl(r

′′r ′;β) (297)

4 Compute the final result by summing over l .

The actual numerical implementation needs a few tricks to keep the resultaccurate, but the two-body problem is solved exactly .



Making further approximations,48 the cumulant approximation is∏i<j

〈e−∫ β


e−∫ β

0 dt〈V (rij (t))〉BRW (298)

and, assuming classical paths we get the semi-classical approximation∏i<j

〈e−∫ β


e−∫ β

0 dtV (r(class)ij (t)) , (299)

and for a very small β one recovers the primitive approximation∏i<j

〈e−∫ β


e−β12

[V (rij (0))+V (rij (β))] . (300)

48See D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).


Liquid 4He

Pollock and Ceperley (1987):


Liquid 4He

D. Ceperley, Rev. Mod. Phys. 67, 279 (1995):


Liquid 4He

D. Ceperley, Rev. Mod. Phys. 67, 279 (1995):


Liquid 4HeThe density has a kink at Tλ, so it’s present ”already” in the liquidstructure:

1 1.5 2 2.5 3 3.5 40.019

0.02

0.021

0.022

T (K )

ρ(A−

3) The equilibrium density of liquid He4

as a function of temperature a

aR. J. Donnelly and C. F. Barenghi ,J. Phys. Chem. Ref. Data, Vol. 27, No 6 (1998).

For pair interactions, the potential energy per particle is

〈V 〉/N = N−1∫

dR|ψ(R; T )|2N∑

i<j=1

V (rij )/

∫dR|ψ(R; T )|2

= N−1∫

dr1dr2V (r12)N(N − 1)

2

∫dr3..drN |ψ(R; T )|2/

∫dR|ψ(R; T )|2

= N−1Ω(T )4πρ(T )2∫ ∞

0drr2V (r)g(r ; T ) = 4πρ(T )

∫ ∞0

drr2V (r)g(r ; T ) (301)

so comparing with the PIMC data, most of the T -dependence comes from ρ(T ), the rest is roughly a constant,

∼ −978 KA−3. The pair distribution g(r ; T ) has a very weak temperature dependence. The liquid structure, ρ(T ) and

g(r ; t), is all that’s needed to get the potential energy.Quantum Monte Carlo (QMC) December 14, 2018 156 / 241

Path sampling

We want to sample different terms in Z as effectively as possible, includingpermutations of indistinguishable particles.

I Single bead move: move just one random particle in a random timeslice

I Rigid move: move the whole imaginary time world line of a randomparticle

I Recreate path(s) for multiple time slices, say slices 2, 3, 4, keepingslices 1 and 5 fixedaccept or reject at any stage:

I Bisection: sample 3 as midpoint 1-3-5, sample 2 as midpoint 1-2-3 and4 as midpoint 3-4-5.- Easy permutation sampling, but insufficient for some topologies

I Staging: sample 2 from 1-2-5, then 3 from 2-3-5, then 4 from 3-4-5.

I Worm move: open a path and let the ends move and recombine- Global updates of paths- The ultimate permutation sampler


Path sampling: single-bead move

R

β

Update: two GT ’s and potential energy in the time slice.Ineffective but cheap


Path sampling: rigid move

R

β

Update: potential energy of all time slices. No GT change.Effective but expensive


Path sampling: bisection












Path sampling: Adding permutations to bisection














BisectionComments on the bisection update

I Effective, path changes a lotI If the new midpoint is accepted, it may be far from the original pointI If the new midpoint is rejected (large potential) we don’t waste time on

the restI Does the algorithm early-reject a move that could be finally accepted?

In principle yes, but without any practical consequences 49

I Move to the temporary mid-point (red) is deterministic

I Move from the temporary mid-point to new point (green) can bedone using random Gaussian move. The length can be adjusted foroptimal acceptance.

I Long GT ’s of previous stage can be divided away in the next stage.Intermediate steps can be free-spirited, but in the final step you haveto use the correct weights.

I Adjustable parameters: length of the re-sampled path and amplitudesof random Gaussian moves on each bisection level.

49Llorenc Brualla i Barbera, Thesis (Barcelona, 2002).


Worm

The worm algorithm50 owns to the advances made in the Ising model MCsampling in the 80’s. It solves many problems in PIMC path sampling:

I SOLVED: Global changes are more efficient than local onesI SOLVED: Long permutation loops are easily created with high

acceptanceI other path sampling methods explicitly construct permutationsI acceptance of suggested permutations is very low ⇒ permutation

sampling is very inefficientI on a torus-shaped space you practically never get a path winding

around ⇒ wrong result

I SOLVED: Evaluation of off-diagonal quantities: condensate fractionand imaginary-time correlations, without any ambiquity innormalization.

50N. V. Prokof’ev, B. V. Svistunov, and L. S. Tupitsyn, Physics Letters A 238, 253 (1998); original in JETP. Later, M.

Boninsegni joined Prokof’ev and Svistunov to apply the method on 4He, Phys. Rev. Lett. 96, 070601 (2006), and Phys. Rev. E74, 036701 (2006).


Worm

Start with a configuration that describes one possible term in the partitionfunction Z . This term is in the Z sector . The basic worm updates are:

1 Open: Open a path, this leaves two dangling ends, Ira and Masha.You generated a term that’s not in Z , and entered the G sector .

2 Advance/recede: Ira moves ahead (pulls back) and the worm getslonger (shorter)

3 Insert/remove: Add a completely new piece of path or remove one⇒ Grand Canonical Ensemble (variable N, fixed chemical potential µ)

4 Swap: Ira jumps to another path, close the old path

5 Close: Close the dangling ends of the worm and return to the Zsector. This must be in balance with the open update – detailedbalance!

While in the Z sector, measure anything you would from Z .While in the G sector, measure off-diagonal quantities: Ira and mashacorrespond to the field operators ψ(r; τ) and ψ†(r′; τ ′).


Path sampling: worm open

ψ(r; τI )

ψ†(r; τM)


Path sampling: worm recede


Path sampling: worm advance


Path sampling: worm close


Path sampling: worm insert










Path sampling: worm close


Path sampling: worm swap


Path sampling: worm swap PERMUTATIONS!


Worm partition function

The Z and G sectors are also known with less colourful names: diagonaland off-diagonal sector. Ira and Masha are known as head and tail . Theworm samples an ”extended” partition function, (I ,M are Ira and Masha)

Zworm =∞∑

N=0

ZN

[eβµN + C

∑iI ,iM

∫drIdrMG (rI , rM ; (iI − iM)τ)

],

(302)

where the integrand is the Matsubara Green’s function (thermal average),

G (r, r′; τ − τ ′) = 〈Tτ ψ(r, τ)ψ†(r′; τ ′)〉 . (303)

Here Tτ is the imaginary-time ordering operator and ZN is the canonicalpartition function. The Zworm is sum of the grand canonical partitionfunction and sum over all worms.



I Worm’s natural ensemble is grand canonical, but also a canonicalimplementation has been done. The grand canonical partitionfunction is

ZG = Tr[e−β(H−µN)

]=∞∑

N=0

ZNeβµN , (304)

and we approximate the canonical partition function ZN with themultidimensional integral, (X ≡ R1,R2...RM)

ZG ≈∞∑

N=0

eβµN∫

dXA(X; τ)e−U(X) . (305)

Here A(X; τ) = ρ0(R1,R2; τ)..− ρ0(RM−1,RM ; τ) is the free-particlecontribution and U(X) is the rest, all interaction terms.

I Paths (world lines) are created sampling exactly the free-particledensity matrices ρ0(r1, r2; τ). This can be done using staging .


Staging algorithm

We generate a random walk path between two known endpoints r1 and rm,marked just 1 and m. The imaginary-time difference is τm − τ1 ≡ τ1m.Split a long-leap Gaussian to short-step Gaussians (here in 3D):

ρ0(r1, rm; τ1m) ≡ 〈r1|e−τ1mT |rm〉 =1

(4πλτ1m)3/2e−(r1−rm)2

4λτ1m (306)

=

∫dr2...drm−1〈r1|e−τ12T |r2〉〈r2|e−τ23T |r3〉...〈rm−1|e−τm−1,mT |rm〉

=

∫dr2...drm−1

1

(4πλτ12τ23...τm−1,m)3/2︸︷︷︸N

e−(r1−r2)2

4λτ12 e−(r2−r3)2

4λτ23 ...e−(rm−1−rm)2

4λτm−1,m .

Here the unknown, intermedia coordinates r2, ..., rm−1 are entangled, butwe can re-arrange terms.


Staging algorithm

ρ0(r1, rm; τ1m) = N∫

dr2...rm−1e−(r1−rm)2

4λτ1m[e−(r1−r2)2

4λτ12 e−(r2−rm)2

4λτ2m

e−(r1−rm)2

4λτ1m

][e−(r2−r3)2

4λτ23 e−(r3−rm)2

4λτ3m

e−(r2−rm)2

4λτ2m

][e−(r3−r4)2

4λτ34 e−(r4−rm)2

4λτ4m

e−(r3−rm)2

4λτ3m

]

...

[e−(rm−3−rm−2)2

4λτm−3,m−2 e−(rm−2−rm)2

4λτm−2,m

e−(rm−3−rm)2

4λτm−3,m

][e−(rm−2−rm−1)2

4λτm−2,m−1 e−(rm−1−rm)2

4λτm−1,m

e−(rm−2−rm)2

4λτm−2,m

], (307)

and all extra colored terms cancel. The brackets simplify, for example

e−(r1−r2)2

4λτ12 e−(r2−rm)2

4λτ2m

e−(r1−rm)2

4λτ1m

= e−(r2−r∗2 )2

4λτ12m (308)

τ12m ≡τ12τ2m

τ1mr∗2 ≡

τ2mr1 + τ12rmτ1m

. (309)


Staging algorithmWe now have a formula to generate a random walk between fixed endpoints r1 and rm – don’t forget the blue term! –

ρ0(r1, rm; τ1m) = e−(r1−rm)2

4λτ1m N∫

dr2...drm−1e−(r2−r∗2 )2

4λτ12m ...e−(rm−1−r∗m−1)2

4λτm−2,m−1,m ,

where

τijk ≡τijτjkτik

, τij ≡ τj − τi

r∗j ≡τjkri + τij rk

τik.

(310)

(311)

Usage: compute r∗2 using r1 and rm, and set

r2 = r∗2 +√

2λτ12mη . (312)

Compute r∗3, sample r3, etc. This samples the integrand exactly.Notice that this staging algorithm doesn’t assume that the imaginarytimes are evenly spaced.



The acceptance probabilities of worm updates are given by theMetropolis-Hastings formula:

Pold→new = min

1,

T (new → old)W (new)

T (old → new)W (old)

. (313)

Free-particle paths between fixed r′ and r are created with a random walk,which samples the product

ρ0(r′, r1; τ)ρ0(r1, r2; τ)...ρ0(rk−1, r; τ) . (314)

From the Zworm in Eq. (302) we should be able to compute the updateprobabilities.Let’s go through the worm updates one by one. For more details, see thearticle by Boninsegni et al. from 2006.


Open wormAdjustable parameters are marked with blue. From now on, above meansup, and below means down in imaginary time: imaginary time is periodic,with period β.

Open

1 Set Ira to a random bead, there are Nbeads to choose from.

2 Pick k ∈ U[1,K ] and set Masha k steps above Ira; τIM ≡ τM − τI .3 Remove beads between Ira and Masha.

4 Accept with probability

Popen = min

1,

CKNbeadse−∆U−µτIM

ρ0(rI , rM ; τIM)

. (316)

where

∆U = U(new)− U(old) = U(open)− U(close) .


Worm open, probability detailsLet’s look how this creates the worm term in Eq. (302), repeated here in asimplified form,

Zworm =∞∑

N=0

[ZNe

βµN + C∑iI ,iM

∫drIdrM(ZNe

βµN − path I-M)

].

Ira was picked from Nbeads beads, which is the current value of NM; andthe reverse can be done in 1 way. Masha was picked from K beads,reverse has 1 possible way. Putting together, we have so far

T (new → old)

T (old → new)= KNbead = KNM . (317)

The old weight is eβµN , while the new weight has a factor C and there areN − 1 particles for the duration τIM , eµN(β−τIM)+µ(N−1)τIM , so

W (new)

W (old)= Ce−µτIM . (318)


Open worm, detailsNow things gets interesting. Deleting the path between Ira and Masha alsodeletes the interactions this particle had between Ira and Masha. Thoseare present in ZN , so the weights have changed by the amount of thepotential action; the explicit form depends on the approximation. Using Udefined in Eq. (305), and marking ∆U ≡ U(new)− U(old), we get

W (new)

W (old)= e−∆U . (319)

The free-particle paths are also different,

...ρ0(rI−1, rI ; τ) ρ0(rI , rI+1; τ)...ρ0(rM−1, rM ; τ)︸︷︷︸cut off in open update

ρ0(rM , rM+1 τ)... .

In closing the path we recreate the path segment with staging, which hasthe Ira—Masha factor that’s not taken care of:

W (new)

W (old)=

1

ρ0(rI , rM ; τIM). (320)

Now we have covered all terms in Eq. (316).Quantum Monte Carlo (QMC) December 14, 2018 196 / 241

Open worm, details

I If C = 0 we never open a worm. Fine, because then there is none.

I If C increases, we accept more worms.

I More particles, more worms – unless C cancels N.

I More time slices, more worms – unless C cancels M.

I We accept less worms with Ira and Masha farther apart in a randomwalk.


Close worm

Reverse of open, must satisfy the detailed balance condition.

Close

1 Reject the update, if Masha is zero or more than K steps aboveIra.

2 Generate a path from Ira to Masha.


Pclose = min

1,

ρ0(rI , rM ; τIM)

CKNbeade∆U−µτIM

, (322)

where Nbead is the number of beads after closing, and

∆U = U(new)− U(old) = U(closed)− U(open) ,

hence it’s −∆U in used in the opening acceptance.


Close worm, details

The probability of closing a worm may get very small if the random walkdistance of Ira and Masha is large, i.e., ρ0(rI , rM ; τIM) is very small. Suchworms are rarely created, but they are also hard to close. In order to avoidthese sticky worms Boninsegni et al. used a hard limit, that keepsρ0(rI , rM ; τIM) (actually ln[ρ0(rI , rM ; τIM)]) more reasonable: If

−(rI − rM)2

4λτIM> X (323)

opening and closing are rejected.51 This choice is deterministic, but alsosymmetric, so T (new → old) = T (old → new) and no harm is done tostochasticity.

51Boninsegni et al. used X = 4 in liquid He calculations.


Insert worm

This seeds a new world line strand of length k .

Insert

1 Set Masha to a random r from the simulation volume Ω andrandom time slice

2 Pick k ∈ U[1,K ] generate a k-step path above Masha; Put Irathere.


Pinsert = min

1,CKMΩe∆U−µτIM, (325)


Remove worm

The reverse update of insert.

Remove

1 If the worm length is > K reject remove, the worm is too long.


Premove = min

1,

e∆U−µτIM

CKMΩ

, (327)


Advance

This moves Ira up in imaginary time.

Advance

1 Pick k ∈ U[1,K ]

2 Generate a k-step path from old Ira up in imaginary time and setthe new Ira there.

3 Accept advance with probability

Padvance = min

1, e∆U+µτIM. (329)


Recede

This moves Ira down in imaginary time.

Recede

1 Pick k ∈ U[1,K ]. If the worm has less than k beads reject theupdate.

2 Move Ira k steps down in imaginary time, delete beads on the way.


Precede = min

1, e∆U−µτIM, (331)


SwapSwap

1 Pick k ∈ U[1,K ]. Build a list Si of beads k steps above Ira onanother world line.

2 Build a weight list ρ0(rI , rSi , τISi ). Normalization of the list isΣS =

∑i ρ0(rI , rSi , τISi ).

3 Pick a bead S with a probability of the normalized weigth list.

4 Find the bead T that is k steps down in imaginary time frombead S . If Masha is met along the way, reject the update.

5 Build another weight list ρ0(rS , rTi, τITi

). Normalization of thelist is ΣT =

∑i ρ0(rS , rTi

, τSTi).

6 Generate a free-particle random path from Ira to S .

7 Accept swap and move Ira to T with probability

Pswap = min

1,

ΣS

ΣTe∆U

. (333)


Swap details

The first weight list and sampling bead S from it favors swap attempts tobeads that are not too far in a random walk from Ira. This isT (old → new). Another weight list is needed because the swap is its ownreverse, so we must compute also T (new → old).


Head/tail wiggle


Head/tail wiggle


Head/tail wiggle

Wiggling the head (Ira): In free-particle terms the red coordinates change,

...ρ0(rI−k−2, rI−k−1; τ)ρ0(rI−k−1, rI−k ; τ)...ρ0(rI−2, rI−1; τ)ρ0(rI−1, rI ; τ) .

Here rI−k−1 is fixed, and the new path segment can be sampled exactly,point by point:

r′I−k = rI−k−1 +√

2λτη , samples exactly ρ0(rI−k−1, r′I−k ; τ)

r′I−k+1 = r′I−k +√

2λτη , samples exactly ρ0(r′I−k , r′I−k+1; τ)

...

r′I = r′I−1 +√

2λτη , samples exactly ρ0(r′I−1, r′I ; τ) . (334)

where green coordinates are new. The move is accepted/rejected based onthe potential change in the wiggled time slices. It’s a good idea to add a”hard limit”, and reject all wiggles that take Ira too far from Masha,ρ0(r′I , rM , τIM) < X , so that the wiggled worm can close.


Head/tail wiggle

Another way to sample the new path is to first sample new Ira from thefixed point rl−k , which is a long time jump τI − τk = kτ from Ira,

r′I = rk +√

2λkτη , samples exactly ρ0(rk , r′I ; kτ) , (335)

and fill the path between the new Ira and the fixed point using staging. Becareful to use the minimum image convention properly, so that you don’tsample the new path across the whole system. The wiggle move is it’s own

inverse.


Some PIMC researchers

Just to give you a few pointers to some useful names. In no particularorder:

I D. M. Ceperley, E. L. Pollock, B. Militzer, H. Kleinert

I V. S. Filinov, M. Bonitz, V. E. Fortov, T. Schoof, S. Groth

I J. Boronat, J. Casulleras, F. Mazzanti, S. Giorgini, L. Vranjes Markic,G. E. Astrakharchik

I M. Boninsegni, N. Prokof’ev, B. Svistunov, S. A. Chin, W. Janke, T.Sauer

I D. E. Galli, C.H. Mak, S. Zakharov, M. F. Herman, E. J. Bruskin, B.J. Berne

I Finland (TUT): T. T. Rantala and his students (I. Kylanpaa, M.Leino, J. Tiihonen etc.)

These are just appetizers, many were left out. And, of course, R. P.Feynman.


Fermion sign problem: the Nemesis of QMC.The expectation value of some quantity A is

〈A〉 =Tr[Aρ]

Tr[ρ], (336)

which is evaluated in MC using weights wi ,

〈A〉w =

∑i wiAi∑wi

, (337)

but the weigts wi may be negative, unsuitable for MC. Separate the signas wi = sgn(wi )|wi |,

〈A〉w =

∑i wiAi∑wi

=

∑i sgn(wi )|wi |Ai∑

sgn(wi )|wi |=

∑i |wi |sgn(wi )Ai∑

i |wi |∑i |wi |sgn(wi )∑

i |wi |

. (338)

The factor

pi ≡|wi |∑i |wi |

, (339)

is a good probability, so we should just evalute two averages. Easy, ha?Quantum Monte Carlo (QMC) December 14, 2018 211 / 241

Fermion sign problem: the Nemesis of QMC.Using the probability made of absolute values we get

〈A〉w =

∑i pi sgn(wi )Ai∑i pi sgn(wi )

=〈signA〉|w |〈sign〉|w |

. (340)

In principle, this can be easily done in MC, but there’s a catch. Thedenominator is related to the partition function Z of weights wi , andanother one of weights |wi |,

〈sign〉|w | =∑i

pi sgn(wi ) =

∑i wi∑i |wi |

=Zw

Z|w |=

e−βFw

e−βF|w|= e−β(Fw−F|w|) ,

(341)

where the free energy is F = − 1β lnZ (thermodynamics). As always in

MC, the error of the mean diminishes as 1/√M with M samples, but there

is a factor that grows,

∆(〈sign〉|w |)〈sign〉|w |

=1√M

√〈sign2〉|w | − 〈sign〉2|w |

〈sign〉|w |∼ eβ(Fw−F|w|)

√M

=eβN(fw−f|w|)√M

.

(342)Quantum Monte Carlo (QMC) December 14, 2018 212 / 241

Fermion sign problem: the Nemesis of QMC.

The error behaves like

∆(〈sign〉|w |)〈sign〉|w |

∼ eβN(fw−f|w|)√M

, (343)

where f ’s are free energies per particle – remember F is extensive. Thedifference fw − f|w | is positive, because the boson energy with weights |w |is below the fermion energy with weights w . The factor growsexponentially both with decreasing temperature, and with increasingnumber of fermions N. This is the fermion sign problem.What of the nodes:

I FN-DMC: use fixed nodes that are good at T = 0.

I PIMC: T = 0 nodes and finite-T nodes are not the same. Instead,the density matrix nodes depend on the imaginary time:

ρ(R′,R0; t) = 0 with fixed (reference point) R0 and t . (344)


Fermion sign problem: strategies

I Restricted PIMC (RPIMC): Force nodes via a trial density matrix andkeep paths from crossing to another sign- determinant of free-particle density matrices – accurate at high T- variational trial density matrix (Militzer and Ceperley)- Jellium: RPIMC does not work at high density (rs < 1) 52

I Direct PIMC (DPIMC)

I Configuration PIMC (CPIMC)53

- no timestep error: continuous time QMC- excellent at high density

I Permutation Blocking PIMC (PB-PIMC)54

52RPIMC fails to reproduce the exact ideal fermi gase limit : V. Filinov, J. Phys. A 34, 665 (2001).53Schoof et al. Phys. Rev. Lett. 115, 130402 (2015).54Dornheim at al., New J. Phys. 17, 073017 (2015).


Fermion sign problem; perspectives

Troyer and Wiese 55

Quantum Monte Carlo simulations, while being efficient forbosons, suffer from the “negative sign problem” when applied tofermions–causing an exponential increase of the computing timewith the number of particles. A polynomial time solution to thesign problem is highly desired since it would provide an unbiasedand numerically exact method to simulate correlated quantumsystems. Here we show that such a solution is almost certainlyunattainable by proving that the sign problem is nondeterministicpolynomial (NP) hard, implying that a generic solution of thesign problem would also solve all problems in the complexityclass NP in polynomial time.

55M. Troyer and U-J Wiese, Phys. Rev. Lett. 94, 170201 (2005).


Fermion sign problem; perspectivesTroyer and Wiese study the NP hard problem of determining whether thereexists a state with energy E ≤ E0, for a fixed bound E0, of the IsingHamilton function

H = −∑〈j ,k〉

Jjkσjσk . (345)

What makes this NP hard is the evaluation of the sum over allconfigurations c ,

Z =∑c

e−βE(c) . (346)

The quantum version has σj and σk replaced by Pauli spin matrices σxjand σxk . The bosons problem is easy, it’s a ferromagnet with all couplingsJjk ≥ 0. The fermion problem has a ”random” sign, which can be mappedon random off-diagonal signs in J, so Troyer and Wiese conclude that thesign problem is the origin of the NP hardness. So if NP 6= P (solvable inpolynomial time): tough luck!



Are we out of luck? If we could solve the eigenstates and eigenvalues ofH, H|i〉 = Ei |i〉, we could calculate finite temperature expectation values

from the normalized density operator ρ = 1Z e−βH,

〈A〉 = Tr(ρA) =

∑i e−βEi 〈i |A|i〉∑i e−βEi

. (347)

Here the weights e−βEi are positive, so there is no sign problem. Alas,solving H|i〉 = Ei |i〉 is an exponentially hard problem.A question one should next ask is, just how explicitly do we need to knowthe solution to H|i〉 = Ei |i〉 in order to avoid the sign problem? Is itenough to have the approximate solution hidden in an evolution algorithm?


Fermion sign problem; perspectivesChin 56:

Here we argue that: (1) The sign problem is not intrinsic tosolving a fermion problem; (2) it is only a consequence of a poorapproximation to the exact propagator; and (3) it can beautomatically minimized by using better, higher-orderapproximate propagators. By following up on the last point, thiswork shows that by using optimized fourth-order propagators,accurate results can be obtained for up to 20 spin-polarizedelectrons in a two-dimensional (2D) circular, parabolic quantumdot.

Chin argues that the ”sign problem” is due to the large number ofantisymmetric free-fermion propagators that are needed to extract theground state wave function at large imaginary time.Goal: Get to a large τ with less e−τ T ’s, by using e.g. Eq. (147):

GChin(τ) = e−v0τ Ve−t1τ T e−v1τWe−t2τ T e−v1τWe−t1τ T e−v0τ V . (348)

56S. A. Chin, Phys. Rev. E 91, 031301(R) (2015).



Let’s see how the antisymmetric free-fermion propagator looks like.Define, as usual

GT (R′,R; τ) = 〈R′|eτ T |R〉 . (349)

Antisymmetry comes from the spectral expansion in eigenstates of T withantisymmetric wavefunctions, explicit either as a permutation sumφK(R) =

∑P(−1)Pφbolzmannon

K (PR) or as a determinant. We know theboltzmannon result, just collect them to a determinant:

GT (R′,R; τ) = (4πDτ)−3N/2 detM , (350)

where the matrix M has elements

Mij = e−(ri−r′j )2

4Dτ . (351)

This detM is where the (almost) random signs come from.


Fermion sign problem; perspectivesThe idea is to get the result before the sign problem appears

Fermion sign problem kicks inToo many free propagators

No sign problem – yet


Alternatives with no timestep error: SSE

There are ways to do QMC without any timestep error from splittinge−βH.One is the Stochastic Series Expansion (SSE). Invented by Anders Sandvikand J. Kurkijarvi 57, as an extension to Handscomb’s method. Make aseries expansion

Z = Tr[e−βH] =∑α

∞∑n=0

(−β)n

n!〈α|Hn|α〉 (352)

See O. F. Syljyasen, A. W. Sandvik, Phys. Rev. E 66, 046701 (2002) forSSE with directed loop update. SSE has been mostly used in lattice spinsystems, such as Ising and Hubbard models.

57Physical Review B 43 (7), 5950 (1991), J. Phys A: Math. Gen. 25, 3667 (1992), Physical Review B 59 (22), R14157 (1999)


SSE with the Hubbard modelThe Hubbard Hamiltonian with nearest neighbour jumps t, on-siteinteraction U, and chemical potential µ is

H =∑〈ij〉

[−t(a†i aj + a†j ai )

]+∑i

[−µni +

U

2ni (ni − 1)

](353)

U

t

Figure: Standing waves in crossing laser fields create an optical potential, wherebosons arrange in a lattice and obey the Hubbard Hamiltonian. 59

58Figure: Krissia de Zawadzkia, how-to-draw-an-optical-lattice @stackoverflow59Figure: Krissia de Zawadzkia, how-to-draw-an-optical-lattice @stackoverflow


https://tex.stackexchange.com/questions/54517/how-to-draw-an-optical-lattice

https://tex.stackexchange.com/questions/54517/how-to-draw-an-optical-lattice

SSE with the Hubbard model

Hamiltonian as a sum over bond operators (coordination number Z ),

H = −∑i

Ki (354)

= −∑〈ij〉

t(a†i aj + a†j ai )︸︷︷︸off-diagonal

+µ

Z(ni + nj)−

U

2Z[ni (ni − 1) + nj(nj − 1)]︸︷︷︸diagonal

,

and sample a random product of H terms in the partition function,

Z =∑s

∑s

β∑n=0

βn

n!

∑i1,...,in

〈s|Ki1 ...Kin |s〉 (355)


SSE Diagonal update

I The number of operators is controlled by the temperature.

I Only diagonal operators can be inserted/removed locally

I It’s easier to deal with a constant M operators:use 1 for filling.


SSE Diagonal update

I Start: Pick any reasonable state |s〉 and insert M identity operators1:

I Diagonal update: Change identity ops to diagonal ops or vice versawith acceptance probabilities

Ains(1→ Kd) = min

[〈s|Kd |s〉βNbonds

M − n, 1

](356)

Arem(Kd → 1) = min

[M − n + 1

〈s|Kd |s〉βNbonds

, 1

](357)


SSE Diagonal updateDerivation: Use the detailed balance condition for I → Kd and Kd → I :

1 : M − n + 1 ↔ M − n

Kd : n − 1 ↔ n

P(S → S′)

P(S′ → S)=

W (S′)

W (S)=〈s′|βKd |s′〉〈s|1|s〉

= β〈s′|Kd |s′〉 =

T (S → S′)Ains (1→ Kd )

T (S′ → S)Arem(Kd → 1), (358)

where

T (S → S′) = Prob. of picking a 1 =M − n + 1

Nbonds

(359)

T (S′ → S) = Prob. of picking the Kd = 1 , (360)

so

Ains (1→ Kd )

Arem(Kd → 1)=〈s|Kd |s〉βNbonds

M − n + 1, (361)

and use the Metropolis solution. Here operations are exact reverses,insert is n − 1→ n and remove is n→ n − 1, but in practice n is thecurrent number of Kd ’s, so insert is, instead, n→ n + 1:→ Insert acceptance has M − n, not M − n + 1!


SSE Directed loop update

I Task: find an effective update, such asboson add/remove loop update (+1)/(−1)

I Update changes the type of an operator, but not their total number

I The directed loop update60 is a global, worm update.

60O. F. Syljuasen, and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002).



I Loop update can add/remove bosons ⇒ Grand Canonical Ensemble

I The end state in Z is sampled automatically!

I Knowing the in-leg, how to pick the out-leg so that the Z is sampled ?



Again, it’s up to detailed balance to choose the out-leg. The weights arematrix elements,

W (S) = 〈s1|K|s2〉 , W (S ′) = 〈s ′1|K′|s ′2〉 (362)

and create probability tables, coded P(update, in− leg , out − leg), such as

P(S → S ′) = P(+1; i , j) , P(S → S ′) = P(−1; i , j) or P(+1; j , i) .(363)

One possible normalization is the ”heat bath”,

P(S → S ′) =W (S ′)∑S W (S)

, (364)

which assumes positive weigths.



I Loop update may ”bounce”, up-going adds a boson, down-goingremoves one.

I Loop continues until it reaches the starting point (worm closes). Thisusually happens soon enough.


SSE Forcing positive weigths

Add a constant C to the Hamiltonian,

H = −∑〈ij〉

t(a†i aj + a†j ai )︸︷︷︸off-diagonal

+C +µ

Z(ni + nj)−

U

2Z[ni (ni − 1) + nj(nj − 1)]︸︷︷︸

diagonal

,

I 〈s|off-diagonal|s ′〉 ≥ 0, OK

I 〈s|diagonal|s〉 ≥ 0, if we choose a maximum allowed occupation nmax

and a large enough C to match it.

I If nmax turns out to be too low, start anew with a larger one.


SSE some observables

I Energy, thermodynamical estimator

〈E 〉 = − 1

Z

∂Z

∂β(365)

= − 1

β

1

Z

∑s

∞∑n=0

nβn

n!

∑i1,...,in

〈s|Ki1 . . . Kin |s〉 = −〈n〉β

,

where 〈n〉 is the average number of operators (vertices).I The ground state energy is a finite, fixed number, so 〈n〉 increases withβ ⇒ low-T simulations have more terms and they are slower.

I Energy fluctuates around 〈E 〉, so there exists a finite number of termsM that is reached during any finite time simulation.

I Density and compressibility are trivial to measure.

I Superfluid density ρs is again measured via the winding number, thenet amount of hopping across a boundary.


SSE Bose results

Figure: The average particle number per site in a 2D 8× 8 lattice, at a very lowtemperature, as a function of the hopping parameter t and the chemical potentialµ. The numbered plateaus are Mott insulating phases with the indicated particlenumber per site. Error bars are smaller that the line width. V. Apaja (ca. 2004,unpublished).

Details: See, for example, Boson localization and the superfluid-insulator transition, M. P. A. Fisher et al., Phys. Rev. B 40, 546

(1989). Quantum Monte Carlo (QMC) December 14, 2018 233 / 241

SSE Bose results

Figure: Density ρ, compressibility κ and the superfluid density ρs in a 2D 32× 32lattice, as a function of the chemical potential µ. In Mott insulating phases thedensity is constant, while compressibility and the superfluid density are zero. 62

62O. Syljuasen, PRE 67, 046701 (2003).


Alternatives with no timestep error: Continuous time QMCContinuous time QMC has multiple variants, one way to start is to solveU(β) from

e−βH = e−βT U(β)⇔ U(β) = eβT e−βH (366)

⇒ ∂U(β)

∂β= T eβT e−βH − eβT He−βH

= T eβT e−βH − eβT (T + V)e−βH

= −eβT Ve−βH = −eβT Ve−βT [eβT e−βH]

= −eβT Ve−βT U(β) ≡ −V(β)U(β) , (367)

and integrate,∫ β

0

∂U(t)

∂t= U(β)− U(0)︸︷︷︸

1

= −∫ β

0dtV(t)U(t) (368)


Alternatives with no timestep error: Continuous time QMC

Insert and iterate, Tτ is the imaginary-time ordering operator,

e−βH = e−βT U(β) = e−βT[

1−∫ β

0dτ V(τ)U(τ)

](369)

= e−βT[

1−∫ β

0dτ1V(τ1) +

∫ β

0dτ1

∫ τ1

0dτ2V(τ1)V(τ2)...

]= e−βT

[1−

∫ β

0dτ1V(τ1) +

1

2

∫ β

0dτ1

∫ β

0dτ2Tτ [V(τ1)V(τ2)]...

].

This is often written in the form63

e−βH = e−βT Tτ[e−

∫ β0 dτ V(τ)

], V(τ) ≡ eτ T Ve−τ T . (370)

This can be viewed as the interaction picture representation wrt. V, andthe expanded form does perturbation theory wrt. V .

63Notice the similarity to the Feynman-Kac formula (289).


Alternatives with no timestep error: Continuous time QMCExplicitly,

e−βH = e−βT −∫ β

0dτ1e

−(β−τ1)T Ve−τ1T

+1

2

∫ β

0dτ1dτ2e

−(β−τ1)T Ve(τ1−τ2)T Ve−τ2T ... .

(371)

This gives the canonical partition function, ρ0(R′,R; τ) := 〈R′|e−τ T |R〉,

Z =


∫dRρ0(R,R;β)

−∫ β

0dτ1

∫dRdR1ρ0(R,R1;β − τ1)V (R1)ρ0(R1,R; τ1)

+1

2

∫ β

0dτ1dτ2

∫dRdR1dR2Tτ

[ρ0(R,R1;β − τ1)V (R1)

ρ0(R1,R2; τ1 − τ2)V (R2)ρ0(R2,R; τ2)

]... .

(372)


Alternatives with no timestep error: Continuous time QMC

I This Z is made of DVD-kinks, free diffusion - potential - free diffusion.Free diffusion combined with a singular potential is always bad.

I Terms are positive definite only for an attractive potential – again theHydrogen atom problem works perfectly. MC sampling of Z could bedone by avoiding positive potential regions using domain Green’sfunctions 64 Without any tricks the large |V (R)| occasions overwhelmthe MC sampling.

Eq. (372) uses R basis, where V is diagonal. You could as well choosean occupation number basis, and divide H to diagonal (D) or off-diagonal(Y) contributions in this basis. Next, in Eq. (372), replace T → D andV → Y . This leads to CPIMC formulation by Schoof et al. as in Eqs.(92)—(98) in Introduction to Configuration Path Integral Monte Carlo,discussed on the next slide.

64Schmidt et al., Phys. Rev. E 71, 016707 (2005).


https://arxiv.org/pdf/1408.2011v1.pdf

Configuration Path Integral Monte Carlo

In occupation number basis |n0, n1, ...〉 := |n〉 we have∑ni

|ni〉〈ni| = 1 , Z =∑n

〈n|e−βH|n〉 , (373)

and the perturbation series for H = D + Y, using latter as perturbation, is

Z =∑n〈n|e−βD|n〉 (374)

−∫ β

0dτ1

∑n〈n|e−(β−τ1)D|n〉〈n|Y|n〉〈n|e−τ1D|n〉

+

∫ β0

dτ1

∫ τ1

0dτ2

∑n

∑n1

[〈n|e−(β−τ1)D|n〉〈n|Y|n1〉

〈n1|e−(τ1−τ2)D|n1〉〈n1|Y|n〉〈n|e

−τ2D|n〉]... .

I left out the time ordering operator and factorials, and used explicitordering in time integrals.


Configuration Path Integral Monte CarloUsing n to mark occupation numbers n and marking matrix elementsshortly as

〈n|e−βD|n〉 := e−βD(n)

〈n1|Y|n1〉 := Y (n1, n2) ,

(375)

(376)

one gets (usually Y (n, n) = 0)

Z =∑n

e−βD(n) −∫ β

0dτ1

∑n

e−βD(n)Y (n, n)

+

∫ β

0dτ1

∫ τ1

0dτ2

∑n,n1[

e−(β−τ1)D(n)Y (n, n1)e−(τ1−τ2)D(n1)Y (n1, n)e−τ2D(n)

]... .

(377)

Fermion statistics is built-in in second quantization, but the sign problemprevails, because weights are not as such positive definite.


Things that were not done and things to do

I didn’t talk about

I Path integrals in quantum field theory (Dirac and Majorana fermionsetc.), just path integrals in condensed matter physics. If you want todelve into field theoretical aspects, a good place to start is EdwardWitten’s Fermion Path Integrals And Topological Phases. A youtubevideo is also available here.

I Feynman path integrals (see lecture notes, although they are laggingbehind these slides)

I Fermion PIMC in detail; I recommend Burkhard Militzer’s excellentpresentations on the topic

I Ground state path integrals

Things you should do

I find a way around the sign problem


https://arxiv.org/abs/1508.04715

https://www.youtube.com/watch?v=pOz6e5rmnjQ

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