We propose a simple iterative algorithm to construct the optimal multi-configuration approximation of an N-fermion wave function. That is, M≥N single-particle orbitals are sought iteratively so that the projection of the given wave function in the CNM-dimensional configuration subspace is maximized. The algorithm has a monotonic convergence property and can be easily parallelized. The significance of the algorithm on the study of entanglement in a multi-fermion system and its implication on the multi-configuration time-dependent Hartree-Fock (MCTDHF) are discussed. The ground state and real-time dynamics of spinless fermions with nearest-neighbor interactions are studied using this algorithm, discussing several subtleties.

1. Optimal multi-configuration approximation of an
N-fermion wave function
Jiang-min Zhang
In collaboration with Marcus Kollar
December 12, 2013

2. Outline
A basic (but surprisingly overlooked) problem
How to approximate a given fermionic wave function with Slater
determinants
A simple iterative algorithm
converges monotonically and thus definitely
easily parallelized
Some analytic results
Mathematically interesting and challenging
Multi-configuration time-dependent Hartree Fock
Spinless fermions in 1D

3. A Basic Problem
The simplest type of wave function of a fermionic system is the Slater
wave function:
f(x1, x2, . . . , xN ) =
1
√
N!
φ1(x1) φ1(x2) · · · φ1(xN )
φ2(x1) φ2(x2) · · · φ2(xN )
· · · · · · · · · · · ·
φN (x1) φN (x2) · · · φN (xN )
.
But not every fermionic wave function is in the Slater form:
f(x1, x2) =
1
3
φ1(x1) φ1(x2)
φ2(x1) φ2(x2)
+
1
6
φ3(x1) φ3(x2)
φ4(x1) φ4(x2)
=
1
2
ψ1(x1) ψ1(x2)
ψ2(x1) ψ2(x2)
.
A natural question in the spirit of approximation:
What is the best Slater approximation of a given fermionic wave
function?

4. Significance of the Question
Mathematically a very interesting and very challenging problem
Like the celebrated “N-representability” problem
Geometric measure of entanglement in many-body systems of
identical particles
The most widely used entanglement measure is based on the
Schmidt decomposition
f(x1, x2, . . . , xN ) =
j
λjψj(x1)Ψj(x2, . . . , xN ).
The N indistinguishable particles are split into two parts artificially;
Indistinguishable particles treated as distinguishable!
A slater wave function is an entangled state!
How strong is the correlation between the electrons?
Distance from a free-particle system
Basis of multi-configuration time-dependent Hartree Fock
(MCTDHF).
K. Byczuk, et al., Phys. Rev. Lett. 108, 087004 (2012).
P. Thunstr¨om, et al., Phys. Rev. Lett. 109, 186401 (2012).

5. Mathematical Formulation (the single-configuration case)
N fermions are distributed in L ≥ N orbitals. Given a wave function f,
f(. . . , xp, . . . , xq, . . .) = −f(. . . , xq, . . . , xp, . . .), 1 ≤ xi ≤ L,
find N orthonormal single-particle orbitals φi (1 ≤ i ≤ N) to construct
a Slater determinant wave function
S(x1, . . . , xN ) =
1
√
N!
detAN×N , Aij = φi(xj),
so that the overlap between f and S
I ≡ | f|S |2
= N! dx1 · · · dxN f∗
(x1, · · · , xN )φ1(x1)φ2(x2) · · · φN(xN )
2
is maximized.
A crucial feature: Each orbital appears only once!

6. Mathematical Formulation (the multi-configuration case)
N orbitals might be insufficient! Take M > N orbitals.
Out of {φ1, φ2, . . . , φM }, CN
M Slater determinants can be constructed,
SJ ∝ φj1
∧ φj2
∧ . . . ∧ φjN
,
with J being an N-tuple
J ≡ (j1, j2, . . . , jN ), 1 ≤ j1 < j2 . . . < jN ≤ M.
Maximize the projection of f on the subspace spanned by the SJ ’s,
I =
J
|ηJ |2
,
with
ηJ ≡ f|SJ
=
√
N! dx1 · · · dxN f∗
(x1, · · · , xN )φj1
(x1)φj2
(x2) · · · φjN
(xN ).

7. An “educated” idea
Suppose one needs to maximize function
f(α, β, γ), α, β, γ ∈ R.
An idea based on calculus:
h1(α, β, γ) ≡
∂f
∂α
= 0,
h2(α, β, γ) ≡
∂f
∂β
= 0,
h3(α, β, γ) ≡
∂f
∂γ
= 0.
not object-oriented: only stationary, not maximal
complicated nonlinear equations to solve
even more complicated in case of constraints
K. J. H. Giesbertz, Chemical Physics Letters 591, 220 (2014).

8. A “less-educated” idea (walking upstairs)
A middle-school student’s idea:
fix β and γ to get a function
fβ,γ(α) ≡ f(α, β, γ).
Maximize it with respect to α. ⇒ f ↑.
fix α and γ, maximize f with respect to β. ⇒ f ↑.
fix α and β, maximize f with respect to γ. ⇒ f ↑.
Repeat the procedure above. The value of f ↑ all the way.
Two important factors to take into account:
fβ,γ(α) should be easy to maximize
pitfalls of local maxima (solution: multiple runs with random initial
values)

9. Illustration in the two-fermion case (N = 2)
For a given wave function f(x1, x2) = −f(x2, x1), try to find two
orthonormal single-particle orbitals {φ1, φ2}, so that the Slater
determinant S(x1, x2) = 1√
2
(φ1(x1)φ2(x2) − φ2(x1)φ1(x2))
approximates f best. Equivalently, maximize the absolute value of
I ≡ dx1dx2f∗
(x1, x2)S(x1, x2)
=
√
2 dx1dx2φ1(x1)f∗
(x1, x2)φ2(x2)
= dx1φ1(x1)g∗
1(x1) g1(x1) ≡
√
2 dx2f(x1, x2)φ∗
2(x2)
= dx2φ2(x2)g∗
2(x2) g2(x2) ≡
√
2 dx1f(x1, x2)φ∗
1(x1)
The procedure: Carry out the two steps alternatively
fix φ2 (and calculate g1) and update φ1 as φ1 ∝ g1
fix φ1 (and calculate g1) and update φ2 as φ2 ∝ g2
Luckly, φ1 ⊥ φ2 is satisfied automatically!

10. Trial I: a ring state
Consider such a state with N = 3 fermions in L = 6 orbitals:
f =
1
√
3
(|123 + |345 + |561 ), |ijk ≡ a†
i a†
ja†
k|vac
For the single-configuration approximation (N = M = 3), analytically
Imax = 4/9 = 0.44444 . . .
a transitory plateau at I = 1/3

11. Trial II: another ring state
Consider such a state with N = 4 fermions in L = 9 orbitals:
f =
1
√
2
|1234 +
1
√
3
|4567 +
1
√
6
|7891 ,
For the single-configuration approximation (N = M = 4), analytically
Imax = 1/2 = 0.5
local maxima at I = 1/3 and I = 1/6.

12. Some Analytic Results I
N-fermions in L-orbitals, approximated using M orbitals:
If L = N + 1, the wave function must be a Slater determinant
If M = L − 1, just drop the least occupied natural orbital
If N = 2, for fermions, the wave function has the canonical form
f(x1, x2) =
α
Cα
2
(ψ2α−1(x1)ψ2α(x2) − ψ2α(x1)ψ2α−1(x2)),
with α Cα = 1, and {ψi} being the natural orbitals.
Take the M most occupied natural orbitals
Let λi be the occupation of the ith natural orbital, λi ≥ λi+1,
Imax =
1
N
M
i=1
λi, N = 2,
Imax ≤
1
N
M
i=1
λi, N ≥ 3.

13. Some Analytic Results II (single-configuration)
f = a|12 . . . N + b|N + 1, N + 2, . . . , 2N , N ≥ 2,
Imax = max(|a|2
, |b|2
).
f = a|12 . . . N + b|N, N + 2, . . . , 2N , N ≥ 3,
Imax = max(|a|2
, |b|2
).
A always occupied orbital can be factorized away
Two together-going orbitals allow breaking down the wave function
into two parts
f = 1
2 (|123 + |145 + |256 + |346 ), Imax = 1
2 .
f = 1√
6
(|123 + |234 + |345 + |456 + |561 + |612 ), Imax = 3
4 .
f = a|123 + b|345 + c|561 , Imax ≥ 4
9 . The equality is achieved
when and only when |a|2
= |b|2
= |c|2
= 1
3 .
A conjecture: min Imax = 4/9 for (N, M, L) = (3, 3, 6).

14. 1D Spinless Fermions (e.g., spin-polarized electrons)
N spinless fermions on an L-site 1D lattice, with
nearly-neighbor-interaction, and open boundary condition,
ˆH =
L−1
i=1
−(ˆc†
i ˆci+1 + ˆc†
i+1ˆci) + U ˆniˆni+1.
Ground state
structure of the ground state
Time-evolving state after a quantum quench: Initially the fermions are
confined to the Li sites on the left end and then suddenly released into
the whole lattice.
check the algorithm of Multi-configuration time-dependent Hartree
Fock (MCTDHF)

15. Ground state (repulsive interaction U > 0)
5 10 15 20 25
0.8
0.85
0.9
0.95
1
L
Imax
(a) N = M = 5U=1
U=3
U=5
10 15 20 25
0.8
0.85
0.9
0.95
1
L
Imax
(b) N = M = 6U=1
U=2
U=4
Important features:
L = N and L = N + 1, Imax = 1 irrespective of U.
L → +∞, Imax → 1.
In the large-U limit, a local maximum develops at L = 2N − 1.
charge-density-wave: The N fermions reside every other lattice
site.

16. Ground state (attractive interaction U < 0)
30 60 90 120 150
0.6
0.7
0.8
0.9
1
L
Imax
(a) N = M = 2
U=−1.95
U=−2
U=−2.05
20 40 60 80 100
0.4
0.5
0.6
0.7
0.8
0.9
1
L
Imax
(b) N = M = 3
U=−1.95
U=−2
U=−2.05
Bifurcation at Uc = −2:
|U| < |Uc|, no bound fermionic
pair formation
|U| > |Uc|, bound fermionic
pair formation
Profile of |f(x1, x2)| at U = −3

17. Multi-configuration time-dependent Hartree Fock
Time evolution of a many-body system is difficult!
The conventional approach:
Time-independent basis, chosen a priori
|ψ(t) =
J
CJ (t)eJ ,
with eJ being a many-body basis vector constructed out of
time-independent single-particle orbitals.
Hilbert space exponentially large!
Now a very smart idea:
Adaptively chosen basis,
|ψ(t)
J
CJ (t)eJ (t),
with eJ (t) constructed out of time-dependent single-particle
orbitals.
significantly diminished Hilbert space!

18. Multi-configuration time-dependent Hartree Fock
For N spinless fermions on an L-site lattice,
M L time-dependent single-particle orbitals are taken,
{φ1(t), φ2(t), . . . , φM (t)},
out of which ˜D = M!
N!(M−N)! Slater determinants SJ (t) can be
constructed. The variational wave function is
|ψ(t) =
J
CJ (t)SJ (t).
Evolution of the coefficients CJ (t) and the orbitals φi(t) is
determined by the Dirac-Frenkel variational principle
δ dt i ψ|
∂
∂t
ψ − ψ|H|ψ = 0.
A natural question: can the wave function really be well
approximated by using only M L orbitals?

19. Evolution of Imax
Initially N = 3 fermions are confined to the Li left-most sites (|ψ(t) first
evolved by ED, and then approximated using the algorithm)
Two different cases (U = 1):
0
0.2
0.4
0.6
0.8
1
Imax
(a) N = 3, Li = 3, L = 25
0 5 10 15 20 25 30
0
1
2
t
Eint/U
(c)
0.7
0.8
0.9
1
Imax
(b) N = 3, Li = 5, L = 25
0 50 100 150
0
0.5
t
Eint/U
(d)
From bottom to top, M increases from 3 to 8.
Line M = 3 coincides with line M = 4.
Reduction of Hilbert space: C3
25 = 2300 to C3
8 = 56.

20. Evolution of density distribution—comparison of ED and
MCTDHF
Initially N = 3 fermions are confined to the Li left-most sites (U = 1)
0
0.2
0.4
ˆni
(a1) t = 5
0
0.1
0.2
(a2) t = 10
0 10 20
0
0.1
0.2
0.3
0.4
i
ˆni
(a3) t = 15
0 10 20
0
0.1
0.2
0.3
i
(a4) t = 20
0
0.1
0.2
0.3
0.4
ˆni
(b1) t = 5
0
0.1
0.2
(b2) t = 10
0 10 20
0
0.2
0.4
0.6
iˆni
(b3) t = 15
0 10 20
0
0.1
0.2
0.3
i
(b4) t = 20
Blue line with circles: ED results
Red line with squares: MCTDHF with M = 3 orbitals
Green line with aestrisks: MCTDHF with M = 8 orbitals

21. Conclusions and outlooks
A problem relevant in the MCTDHF context
Numerically, a simple iterative algorithm is proposed
a quantitative approach to geometric measure of entanglement
but the idea is inapplicable to bosons!
Analytically, several scattered nontrivial results have been obtained
A lot of open questions and conjectures
MCTDHF gauged (checked).
An immediate problem: For the Laughlin wave function (zi = xi + iyi)
f(z1, z2, . . . , zN ) =
1≤zi<zj ≤N
(zi − zj)3
N
i=1
exp(−|zi|2
),
how does Imax scale with N (assuming M = N)?
J. M. Zhang and Marcus Kollar, arXiv:1309.1848 (2013).