The Borland-Dennis Discovery and Its Inspiration

Jiang-min Zhang

based on work with Marcus Kollar and Alex Gottlieb

January 2, 2016

Outline

I The N -representability problem and its history

I The Borland-Dennis Discovery

I An elementary proof based on the idea of the best Slaterapproximation of a fermionic wave function

The Quantum N -Body Problem

Generally an extremely difficult problem!

Heavy machines:

I Hartree-Fock

I Density functional theory

I Dynamical mean field theory

I Quantum Monte Carlo

I Explicitly correlated Gaussians

Any N -Body Problem is Actually an Two-Body ProblemThe Hamiltonian decomposed into two-body Hamiltonians:

H =

N∑i=1

H1(i) +∑i<j

H2(i, j) =∑i<j

K2(i, j)

K2(i, j) =∑i<j

[1

N − 1(H1(i) +H1(j)) +H2(ij)

].

The wave function normalized to unity:

1 =

∫dx1dx2 . . . dxN |Ψ(x1, x2, . . . , xN )|2.

The energy:

E = 〈Ψ|H|Ψ〉 = Tr(K2ρ2),

with the two-body reduced density matrix defined as

ρ2(x1, x2;x′1, x′2) ≡ N(N − 1)

2

∫dx3dx3 . . . dxNΨ∗(x′1, x

′2, x3, . . . , xN )

×Ψ(x1, x2, x3, . . . , xN ).

Variational Scheme

Variationally:

Eg ≡ 〈Ψg|H|Ψg〉 = minρ2

Tr(K2ρ2),

with ρ2 satisfying:

I Hermitian

ρ2(x1, x2;x′1, x′2) = ρ∗2(x′1, x

′2;x1, x2)

I Positive-definite

〈ψ(1, 2)|ρ2|ψ(1, 2)〉 ≥ 0

I Trace normalized∫dx1dx2ρ2(x1, x2;x1, x2) = N(N − 1)/2

All can be satisfied readily!

Life is not that easy! Evariatioanl < Eground

“It was while he enjoyed the hospitality of the Summer Research Instituteof the Canadian Mathematical Congress in 1951 that this possibility firstoccurred to the present author. He proceeded to calculate the energy ofthe ground state of Li and was somewhat surprised to obtain a valueabout 30% too low! This shook his naive and unexamined faith in Ritzand Rayleigh but aroused his interest. In that first attempt σ had beenvaried over too large a class of functions.” —A. J. Coleman (1963).

The ρ2 in

Eg = minρ2

Tr(K2ρ2)

needs to be restricted somehow!

A. J. Coleman, “Structure of Fermion Density Matrices”, RMP 35,668 (1963).

The N -Representability Probem

Is a ρ2 representable (or realizable) by an N -fermion wave function?

An inverse problem:

Ψ(x1, x2, . . . , xN )easy

�hard

ρ2(x1, x2;x′1, x′2)

Dependence on N expected!

A simpler problem?

Ψ(x1, x2, . . . , xN )← ρ1(x1;x′1)

More generalizations are possible.

A. J. Coleman, “Structure of Fermion Density Matrices”, RMP 35,668 (1963).

Borland-Dennis’ Approach: Numerical experiments in 1970

Computers are not only a means of

I making calculations faster

I making calculations more precise

I making big tasks possible

but also a means of

I gaining insight and intuition

I discovering new patterns and relationships

I aiding analytic investigation

I ...

Theorists can also do experiments!

Borland & Dennis, “A simple approximation beyond Hartree-Fock”,J. Phys. B 3, 887 (1970).

Fermi, Pasta, & Ulam, “Studies of Nonlinear Problems” (1955).

Borland-Dennis’ Discovery (1972)For N = 3 fermions in M = 6 orbitals (spin degree of freedom included),the eigenvalues of the single-particle reduced density matrix satisfy(λi ≥ λi+1):

λ1 + λ6 = 1,

λ2 + λ5 = 1,

λ3 + λ4 = 1,

λ5 + λ6 ≥ λ4.

These are the necessary and sufficient condition for a single-particledensity matrix to be representable by a wave function!

To be compared with

λ1 + λ2 + λ3 + λ4 + λ5 + λ6 = 3, λi ≥ 0.

Borland & Dennis, “The conditions on the one-matrix for three-bodyfermion wave functions with one-rank equal to six”, J. Phys. B 5, 7(1972).

How Was It Done? The Easy and the Hard Part

λ1 + λ6 = 1?λ2 + λ5 = 1?λ3 + λ4 = 1?

Just print them out:

λ1 0.8402λ2 0.7472λ3 0.6592λ4 0.3408λ5 0.2528λ6 0.1598

Easy to perceive!

λ5 + λ6 ≥ λ4?

Hard, even with a graphic interface!

A Personal Experience: Uniform Sampling of sinc2(x)

Suppose 0 < T ≤ π, then for ∀α ∈ R,

∑n∈Z

sin2(nT + α)

(nT + α)2=π

T.

Zhang and Haque, “ Non-smooth and level-resolved dynamics of aperiodically driven tight binding model”, arXiv:1404.4280.

Back to Borland-Dennis’ Discovery: Why 3-in-6?

If N = 1, the problem is trivial; by the particle-hole transform:N fermions in N + 1 orbitals is always a Slater determinant!

If N = 2, we have the canonical decomposition:

Ψ =√p1φ1 ∧ φ2 +

√p2φ3 ∧ φ4 +

√p3φ5 ∧ φ6 + . . . ,

with φi the natural orbitals. All eigenvalues two-fold denegerate!The M = N + 2 case is then also understood by the particle-holetransform.

3-in-6 is the first nontrivial case!

Lowdin and Shull, “Natural Orbitals in the Quantum Theory ofTwo-Electron Systems”, PR 101, 1730 (1956).

Zhang and Kollar, “Optimal multiconfiguration approximation of anN -fermion wave function”, PRA 89, 012504 (2014).

Previous Proofs of Borland-Dennis’ Discovery

Proofs given by Borland and Dennis themselves, or by Ruskai are bothbased on the numerical finding (8 = 2× 2× 2):

Ψ = A123φ1 ∧ φ2 ∧ φ3+ A124φ1 ∧ φ2 ∧ φ4+ A153φ1 ∧ φ5 ∧ φ3+ A154φ1 ∧ φ5 ∧ φ4+ A624φ6 ∧ φ2 ∧ φ3+ A624φ6 ∧ φ2 ∧ φ4+ A653φ6 ∧ φ5 ∧ φ3+ A654φ6 ∧ φ5 ∧ φ4.

In each Slater determinant, one and only one of φi and φ7−iappears!

Ruskai, “Connecting N -representability to Weyl’s problem: theone-particle density matrix for N = 3 and R = 6”, J. Phys. A 40,F961 (2007).

A Proof of This AssumptionWe need an efficient expansion of Ψ! But where to start?

The best Slater approximation of Ψ!

Let S = f1 ∧ f2 ∧ f3 be the best Slater approximation of Ψ. Extend{f1, f2, f3} into a full orthonormal basis {f1, f2, f3, g1, g2, g3}. Then

Ψ = af1 ∧ f2 ∧ f3 + cg1 ∧ g2 ∧ g3+f1 ∧ (b11g2 ∧ g3 + b12g3 ∧ g1 + b13g1 ∧ g2)

+f2 ∧ (b21g2 ∧ g3 + b22g3 ∧ g1 + b23g1 ∧ g2)

+f3 ∧ (b31g2 ∧ g3 + b32g3 ∧ g1 + b33g1 ∧ g2).

No terms like f1 ∧ f2 ∧ g1! Because

αf1 ∧ f2 ∧ f3 + βf1 ∧ f2 ∧ g1 = f1 ∧ f2 ∧ (αf3 + βg1).

9 of 20 terms have already been killed!

Zhang and Kollar, “Optimal multiconfiguration approximation of anN -fermion wave function”, PRA 89, 012504 (2014).

Schmidt DecompositionDefine (G1, G2, G3) = (g2 ∧ g3, g3 ∧ g1, g1 ∧ g2), then

Ψ = af1 ∧ f2 ∧ f3 + cg1 ∧ g2 ∧ g3 +

3∑i,j=1

bijfi ∧Gj .

Perform a singular value decomposition of b to kill the off-diagonal terms,

fi =

3∑j=1

U∗ikek, Gj =

3∑m=1

VjmHm, U†bV = Λ.

Then

Ψ = af1 ∧ f2 ∧ f3 + cg1 ∧ g2 ∧ g3 +

3∑k=1

Λkkek ∧Hk

with (a 2-in-3 fermionic wave function must be a Slater determinant)

(H1, H2, H3) = (h12 ∧ h13, h23 ∧ h21, h31 ∧ h32),

where hij ∈ Span{g1, g2, g3} and,

〈h12|h13〉 = 〈h21|h23〉 = 〈h31|h32〉 = 0.

A Five-Term Expansion of Ψ

The condition 〈H1|H2〉 = 〈H2|H3〉 = 〈H1|H3〉 = 0 means we can findan orthonormal basis (h1, h2, h3) for span(g1, g2, g3), such that

H1 ∝ h2 ∧ h3, H2 ∝ h3 ∧ h1, H3 ∝ h1 ∧ h2.

Besides that,

f1 ∧ f2 ∧ f3 ∝ e1 ∧ e2 ∧ e3,g1 ∧ g2 ∧ g3 ∝ h1 ∧ h2 ∧ h3.

In the end, we have a five-term expansion of Ψ:

Ψ = Ae1 ∧ e2 ∧ e3 +Be1 ∧ h2 ∧ h3 + Ce2 ∧ h3 ∧ h1+De3 ∧ h1 ∧ h2 + Eh1 ∧ h2 ∧ h3,

with |A|2 + |B|2 + |C|2 + |D|2 + |E|2 = 1.

The Explicit Form of ρ1The 2× 2 block corresponding to {e1, h1} is(

|A|2 + |B|2 E∗BEB∗ |C|2 + |D|2 + |E|2

),

the 2× 2 block corresponding to {e2, h2} is(|A|2 + |C|2 E∗C

EC∗ |B|2 + |D|2 + |E|2),

the 2× 2 block corresponding to {e3, h3} is(|A|2 + |D|2 E∗D

ED∗ |B|2 + |C|2 + |E|2).

All blocks have trace of unity,

|A|2 + |B|2 + |C|2 + |D|2 + |E|2 = 1.

Therefore,

λ1 + λ6 = λ2 + λ5 = λ3 + λ4 = 1.

The Eight-Term ExpansionThe block diagonal form of ρ1 also means

span{e1, h1} = span{φP1, φ7−P1},span{e2, h2} = span{φP2, φ7−P2},span{e3, h3} = span{φP3, φ7−P3},

for some permutation P ∈ S3. That is,

e1 = aφP1 + bφ7−P1, h1 = cφP1 + dφ7−P1.

. . .

Substituting these expressions into

Ψ = Ae1 ∧ e2 ∧ e3 +Be1 ∧ h2 ∧ h3 + Ce2 ∧ h3 ∧ h1+De3 ∧ h1 ∧ h2 + Eh1 ∧ h2 ∧ h3,

We get the eight-term expansion of Ψ. The gap is filled!

Gottlieb, Mauser, and Zhang, “Best Slater approximation of afermionic wave function”, arXiv:1510.05634.

Klyachko’s Heroic Work (2008)

Borland-Dennis’ result generalize!

For example, for the 3-in-7 case,

−4λ1 + 3λ2 + 3λ3 + 3λ4 + 3λ5 − 4λ6 − 4λ7 ≤ 2,

3λ1 − 4λ2 + 3λ3 + 3λ4 − 4λ5 + 3λ6 − 4λ7 ≤ 2,

3λ1 + 3λ2 − 4λ3 − 4λ4 − 3λ5 + 3λ6 − 4λ7 ≤ 2,

3λ1 + 3λ2 − 4λ3 + 3λ4 − 4λ5 − 4λ6 + 3λ7 ≤ 2.

The permissible region is the hollow of a Polyhedron!

Altunbulak and Klyachko, “The Pauli Principle Revisited”, Commun.Math. Phys. 282, 287 (2008).

Summary

I The anti-symmetry condition has deep consequences other than thecommonplace of Pauli’s principle.

I Computers are not only for calculation, but also for exploration.

I Good problems are simple (but not easy) problems understandableto everyone.