Density Matrix Quantum Monte Carlo
N.S. Blunt1
T.W. Rogers1 J.S. Spencer1,2
W.M.C. Foulkes1
1 Department of Physics
Imperial College London
2 Department
of Materials
Imperial College London
Quantum Monte Carlo in the Apuan Alps VII
30th July 2012
Outline
1
FCIQMC
2
DMQMC
3
Initial Results
4
Importance Sampling
5
Importance-Sampled Results
Outline
1
FCIQMC
2
DMQMC
3
Initial Results
4
Importance Sampling
5
Importance-Sampled Results
The Full Configuration Interaction Method
Expand the wave function in a basis of Slater determinants
X
|Ψi =
ci |Di i
i
The coefficients ci that minimise
E({ci }) =
hΨ|Ĥ|Ψi
hΨ|Ψi
satisfy the matrix eigevalue problem
Hi j cj = E0FCI ci
where Hi j = hDi |Ĥ|Dj i.
The FCIQMC Algorithm
The lowest eigenvector of the full-CI matrix eigenproblem
Hi j cj = E0FCI ci
can be obtained by solving the imaginary-time Schrödinger
equation
∂ci (t)
= − Hij − Sδij cj (t) = Tij cj (t)
∂t
Sampling Algorithm
∂ci (t)
== Tij cj (t)
∂t
Imagine a population of signed “psips” scattered through
the configuration space.
In one time step, a psip on Dj may spawn children on any
configuration Di for which Tij is non-zero.
The expected number of children spawned on Di is |Tij ∆t|.
If Tij > 0, the children have the same sign as their parent.
If Tij < 0, they have the opposite sign.
Then
hqi in+1 = hqi in + (Ti j ∆t)hqj in
Sampling Algorithm (cont.)
hqi in+1 = hqi in + (Ti j ∆t)hqj in
This is a first-order finite-difference approximation to the
imaginary-time Schrödinger equation
∂ci (t)
= Tij cj (t)
∂t
The FCIQMC psip dynamics solves the
imaginary-time Schrödinger equation.
Psip Cancellation
As with all fermion QMC methods, there is a sign problem.
Psips of both signs appear on the same configurations
and the positive and negative populations almost cancel.
To help control this problem, positive and negative psips on
the same determinant at the same time are cancelled out.
Similar psip cancellation algorithms have been tried many
times in continuum DMC. They do not work very well.
The surprise in Alavi’s work is that, for FCI spaces, psip
cancellation works much better.
No. of psips/108
Energy/t
(a)
2.5
2.0
1.5
1.0
0.5
−14
−15
−16
−17
−18
−19
−20
(b)
0
S(τ )
E(τ )
EFCI
5
10
15
20
Uτ
2D 18-site Hubbard model at half filling, U = 4t.
(Spencer, Blunt, and Foulkes, J. Chem. Phys. 136, 054110 (2012))
Outline
1
FCIQMC
2
DMQMC
3
Initial Results
4
Importance Sampling
5
Importance-Sampled Results
DMQMC
The density operator ρ̂(β) = exp[−β(Ĥ − S Iˆ)] satisfies
the Bloch equation
∂ ρ̂
= −(Ĥ − S Iˆ)ρ̂ = T̂ ρ̂
∂β
DMQMC
The density operator ρ̂(β) = exp[−β(Ĥ − S Iˆ)] satisfies
the Bloch equation
∂ ρ̂
= −(Ĥ − S Iˆ)ρ̂ = T̂ ρ̂
∂β
∂ρij
= Tik ρkj
∂β
DMQMC
The density operator ρ̂(β) = exp[−β(Ĥ − S Iˆ)] satisfies
the Bloch equation
∂ ρ̂
= −(Ĥ − S Iˆ)ρ̂ = T̂ ρ̂
∂β
∂ρij
= Tik ρkj
∂β
This looks rather like the imaginary-time Schrödinger
equation.
∂ci
= Tij cj
∂β
Idea
Can we solve the Bloch equation using an FCIQMC-like
technique that works in the space of operators/matrices
instead of in the space of states/vectors?
Sampling Algorithm
∂ρij
= Tik ρkj
∂β
Imagine a population of two-index psips scattered through
the configuration space of operators |kihj|.
In one time step, a psip on |kihj| may spawn children on
any operator |iihj| for which Tik is non-zero.
The expected number of children spawned on |iihj| is
|Tik ∆t|.
If Tik > 0, the children have the same sign as their parent.
If Tik < 0, they have the opposite sign.
Psips of opposite sign on the same operator cancel.
Sampling Algorithm (cont.)
∂ρij
= Tik ρkj
∂β
Psips spawn along columns of the density matrix only.
However, since [ρ̂, Ĥ] = 0, we can equally well solve the
symmetrized Bloch equation
∂ρij
1
=
Tik ρkj + ρik Tkj
∂β
2
Now psips spawn along row and columns.
ρkj (β = 0) = δkj , so psips are initially scattered at random
along the diagonal of the density matrix.
FCIQMC and DMQMC Compared
The DMQMC equation of motion
∂ρij
∂t
=
1
Tik ρkj + ρil Tlj
2
can be rewritten in the form
∂ρij
∂t
= Lij,kl ρkl
where Lij,kl = Tik δjl + Tlj δik .
FCIQMC finds the large eigenvalue of the matrix Tij .
DMQMC finds the largest eigenvalue of the matrix Lij,kl ,
with ij and kl regarded as composite indices.
Advantages of DMQMC
Finite-temperature properties accessible.
Expectation values of operators that do not commute with
Ĥ easily obtainable:
hÔi =
Oij ρji
ρkk
Disadvantages of DMQMC
Problem 1
The dimension of the space of operators is the square of the
dimension of the space of states.
But . . .
The number of determinants in the FCI space rises
exponentially with the number of particles/sites n, and
ND ∝ eαn
⇒
2
ND2 ∝ (eαn ) = eα(2n)
FCIQMC for an n-site Hubbard model ∼ DMQMC for an
n/2-site Hubbard model. Not so bad.
Problem 2
The simulation goes straight past every inverse temperature β:
you cannot stop to accumulate statistics. Necessary to run
many β-loops and average.
But . . .
Every β-loop provides data at all inverse temperatures
from β = 0 (T = ∞) to β large (T → 0).
Independence of β-loops makes statistical analysis easy.
Outline
1
FCIQMC
2
DMQMC
3
Initial Results
4
Importance Sampling
5
Importance-Sampled Results
The 2D Heisenberg Model
Antiferromagnetic Heisenberg Hamiltonian (J > 0)
Ĥ = J
X
hiji
Ŝi · Ŝj = J
X
hiji
1 + −
− +
Ŝ iz Ŝ jz + (Ŝ i Ŝ j + Ŝ i Ŝ j )
2
2D square lattice is bipartite: no sign problem; no
annihilation.
Ground state and our simulations have Ms = 0.
For 4 × 4 lattice, Ms = 0 Hilbert space has dimension
NC = 16 C8 = 12870. Direct diagonalisation possible.
For 6 × 6 lattice, NC = 9.08 × 109 .
hĤi
0.0
Energy/JN
Exact result
DMQMC result
−0.2
−0.4
−0.6
−0.8
0
1
2
3
4
βJ
4 × 4; 103 psips; 103 β-loops.
5
6
hĤi (cont.)
0.0
Energy/JN
Exact result
DMQMC result
−0.2
−0.4
−0.6
−0.8
0
1
2
3
4
βJ
4 × 4; 105 psips; 103 β-loops.
5
6
Staggered Magnetization
Staggered magnetisation squared
hM̂ 2 i, where M̂ =
1
N
P
i (−1)
xi +yi Ŝ
i
0.30
0.25
0.20
0.15
0.10
Exact ground state = 0.276527
DMQMC result
0.05
0.00
0
1
2
3
4
βJ
4 × 4; 105 psips; 103 β-loops.
5
6
Heat Capacity
Ch = −β 2 dhĤi/dβ = β 2 (hĤ 2 i − hĤi2 )
0.2
Estimator
Spline fit
hCh i/N 2
0.1
0.0
−0.1
−0.2
0.00
1.50
3.00
4.50
βJ
4 × 4; 105 psips; 103 β-loops.
6.00
7.50
Larger Systems
4 × 4: NC = 12870
6 × 6: NC = 9.08 × 109
0.0
Energy/JN
−0.3
−0.6
−0.9
−1.2
0.0
Greens function MC ground state
DMQMC result
0.4
0.8
1.2
βJ
6 × 6; 105 psips; 5 × 103 β-loops.
1.6
2.0
Psip Spreading
Fraction of psips
1.0
Diagonal elements
Single excitations
Double excitations
Triple excitations
0.8
0.6
0.4
0.2
0.0
0
1
2
βJ
3
Psip Spreading (cont.)
As β increases, psips appear further and further from the
diagonal of the density matrix.
Evaluating
hÔi =
Oij ρji
ρkk
becomes more and more difficult.
Outline
1
FCIQMC
2
DMQMC
3
Initial Results
4
Importance Sampling
5
Importance-Sampled Results
Importance Sampling: Nick’s Approach
Idea
Suppress psip population on density matrix elements ρij with
d(i, j) large.
Multiply spawning rate from operators of excitation level d
to operators of excitation level d 0 by wd 0 ,d . Multiply reverse
spawning rate by wd,d 0 = 1/wd 0 ,d .
If d 0 > d, wd 0 ,d < 1:
suppress spawning to operators further from the diagonal
enhance spawning to operators closer to the diagonal
Importance Sampling: Link to Standard Approach
The psips now sample the modified density matrix
ρ̃ij = wd,d−1 wd−1,d−2 . . . w1,0 ρij = ρTij ρij
where d = d(i, j) = d(j, i).
If ρ satisfies
∂ρij
= Lij,kl ρkl
∂β
then
∂(ρTij ρij )
dβ
=
ρTij Lij,kl
1
ρTkl
!
ρTkl ρkl
Importance Sampling: Link to Standard Approach
The psips now sample the modified density matrix
ρ̃ij = wd,d−1 wd−1,d−2 . . . w1,0 ρij = ρTij ρij
where d = d(i, j) = d(j, i).
If ρ satisfies
∂ρij
= Lij,kl ρkl
∂β
then
d ρ̃ij
dβ
=
ρTij Lij,kl
1
ρTkl
!
ρ̃kl
Outline
1
FCIQMC
2
DMQMC
3
Initial Results
4
Importance Sampling
5
Importance-Sampled Results
Importance-Sampled Results
The weights are chosen to make the numbers of psips on
each excitation level similar as β → ∞.
Fraction of psips
1.0
Diagonal elements
Single excitations
Double excitations
Triple excitations
1/9
0.8
0.6
0.4
0.2
0.0
0
1
2
3
4
βJ
It is convenient to switch the weights on gradually as the
simulation progresses (i.e., ρT is β-dependent).
6 × 6 Heisenberg Model: Energy
0.0
Energy/JN
Greens function MC ground state = -0.678871
DMQMC result
−0.2
−0.4
−0.6
0
2
4
6
βJ
6 × 6; 104 psips; 103 β-loops.
(note slight error)
8
10
6 × 6 Heisenberg Model: Energy
0.0
Energy/JN
Greens function MC ground state = -0.678871
DMQMC result
−0.2
−0.4
−0.6
0
2
4
6
βJ
6 × 6; 106 psips; 10 β-loops.
(error removed)
8
10
Staggered magnetisation squared
6 × 6 Heisenberg Model: Staggered Magnetization
0.20
0.15
0.10
0.05
0.00
Greens function MC ground state = 0.2099
DMQMC result
0
2
4
6
βJ
6 × 6; 106 psips; 10 β-loops.
8
10
Larger Systems
Successfully studied an 8 × 8 Heisenberg model
Ms = 0 Hilbert space dimension > 1019
Density matrix has > 1038 elements
using the same psip population (106 ) and only 10–100 times
more β-loops!
Summary
Achievement
Importance-sampled DMQMC works surprisingly well.
Can be applied to large sign-problem-free systems.
Yields full thermodynamics at all temperatures at once.
Yields full density matrix and hence arbitrary expectation
values.
Summary
Achievement
Importance-sampled DMQMC works surprisingly well.
Can be applied to large sign-problem-free systems.
Yields full thermodynamics at all temperatures at once.
Yields full density matrix and hence arbitrary expectation
values.
Not bad for two undergraduate students!
Outlook
Open Questions
Substantial population control errors in large systems need
investigating.
Severity of sign problem? Analogue of initiator approach?
Where is it useful?
The thermal properties of tiny molecules are not very
interesting.
When there is a sign problem, we may not be able to tackle
large enough systems to study phase diagrams.
Entanglement measures such as Tr(ρred ln ρred ) and the
concurrence depend on reduced density matrices.
DMQMC seems able to calculate these better than other
methods.