New Hohenberg-Kohn theorems Louis Garrigue Singapore, September 24, - - PowerPoint PPT Presentation

New Hohenberg-Kohn theorems

Louis Garrigue Singapore, September 24, 2019 Presentation of : G., Hohenberg-Kohn theorems for interactions, spin and temperature, (arxiv:1906.03191), Journal of Statistical Physics (2019)

Louis Garrigue New Hohenberg-Kohn theorems

Standard Hohenberg-Kohn theorem

HN(v) :=

N

• i=1

−∆i +

• 1i<jN

w(xi − xj) +

N

• i=1

v(xi) Theorem (Standard Hohenberg-Kohn) Let p > max(2d/3, 2), let w, v1, v2 ∈ (Lp + L∞)(Rd) such that HN(v1) and HN(v2) have ground states Ψ1 and Ψ2. If

• Rd(v1 − v2)(ρΨ1 − ρΨ2) = 0,

then v1 = v2 + (E1 − E2)/N. The above assumption enhances the v/ρ duality Extends to more general v’s, and any boundary condition The mathematically technical step relies on a unique continuation property as Lieb realized. It is proved in (G. 19, arXiv:1901.03207) for Coulomb systems.

Louis Garrigue New Hohenberg-Kohn theorems

Hohenberg-Kohn theorem for interactions

HN(v, w) :=

N

• i=1

−∆i +

• 1i<jN

w(xi − xj) +

N

• i=1

v(xi) Theorem (Hohenberg-Kohn for interactions) Let w1, w2, v1, v2 ∈ (Lp + L∞)(Rd). If there are two ground states Ψ1 and Ψ2 of HN(v1, w1) and HN(v2, w2), such that

• Rd(v1 − v2) (ρΨ1 − ρΨ2)

+

• R2d (w1 − w2) (x − y)
• ρ(2)

Ψ1 − ρ(2) Ψ2

• (x, y)dxdy = 0,

then w1 = w2 + c and v1 = v2 + E1−E2

N

− c(N−1)

2

for some c ∈ R. Siedentop and M¨ uller (81’) have a similar statement, but with an incomplete proof Pair correlations contain the information of the interaction

Louis Garrigue New Hohenberg-Kohn theorems

Consequence for the Kohn-Sham setting

Corollary Take potentials v1, v2, w ∈ (Lp + L∞)(Rd), where w is even and not constant, such that HN(v1, w) and HN(v2, 0) have ground states Ψ1 and Ψ2. Then ρ(2)

Ψ1 = ρ(2) Ψ2.

Kohn-Sham effective systems cannot reproduce pair correlations

Louis Garrigue New Hohenberg-Kohn theorems

Interactions and several types of particles

Take N particles of type a and M of type b (bosons or fermions) HN,M(va, vb, wa, wb, wab) :=

N

• i=1

(−∆i+va(xi))+

N+M

• k=N+1

(−α∆k+vb(xk)) +

• 1i<jN

wa(xi−xj)+

• N+1k<lN+M

wb(xk−xl)+

• 1iN

N+1kN+M

wab(xi−xk) Theorem (Hohenberg-Kohn for different particles) Let HN,M(va,i, va,i, wa,i, wb,i, wab,i) have ground states Ψi, i ∈ {1, 2}. If ρ(2)

a,Ψ1 = ρ(2) a,Ψ2, ρ(2) b,Ψ1 = ρ(2) b,Ψ2 and ρ(2) ab,Ψ1 = ρ(2) ab,Ψ2,

then vη,1 − vη,2, wη,1 − wη,2, and wab,1 − wab,2 are constant (for η ∈ {a, b}. Hohenberg-Kohn for interactions is thus very robust

Louis Garrigue New Hohenberg-Kohn theorems

Hohenberg-Kohn for the Zeeman interaction

HN(v, B) =

N

• i=1
• − ∆i + σi · B(xi) + v(xi)
• +
• 1i<jN

w(xi − xj) Counterexample of the (v, B) → (ρ, m) injectivity in Capelle and Vignale (2000). But we have a strong constraint on the external fields. Theorem (Partial Hohenberg-Kohn for Spin DFT) Take HN(v1, B1) and HN(v2, B2) having ground states Ψ1 and Ψ2. If

• R3(v1 − v2)(ρΨ1 − ρΨ2) +
• R3(B1 − B2) · (mΨ1 − mΨ2) = 0, then

|B1 − B2| χ = E1 − E2 N + v2 − v1, where χ is a function taking its values in {−1, −1 + 2

N , −1 + 4 N ,

. . . , 1 − 2

N , 1}.

If we also assume v1 = v2, E1 = E2 and N odd, then B1 = B2.

Louis Garrigue New Hohenberg-Kohn theorems

Counterexample for Matrix DFT

For non local potentials G’s, we define HN(G) =

N

• i=1

−∆i +

• 1i<jN

w(xi − xj) +

N

• i=1

Gi. It is hence natural to ask whether G → γ is injective. But we can find a large class of counterexamples when w = 0. Counterexample Take w = 0, let G1 be such that HN(G1) has a unique ground state Ψ1, isolated in the spectrum. We take G2 = G1 + ǫ |φ φ|, where φ is ⊥ to the N components of γΨ1. For ǫ 0, G2 and G1 have the same (unique) ground state. Open question : prove that it’s true or false for w = |·|−1.

Louis Garrigue New Hohenberg-Kohn theorems

Warm Hohenberg-Kohn

Theorem (Hohenberg-Kohn at positive temperatures) T1, T2 > 0, Γ1, Γ2 the grand canonical Gibbs states corresponding respectively to Ev1,T1 and Ev2,T2. If − (T1 − T2) (SΓ1 − SΓ2) +

• Rd(v1 − v2)(ρΓ1 − ρΓ2) = 0,

then T1 = T2 and v1 = v2. Extends Mermin’s (65’) v → ρ injectivity at fixed T Works also when we only assume that T1, T2 0. Can be extended to (T, v, A, w) → (S, ρ, m, ρ(2)), for non local G → γ, for classical systems, and in the canonical ensemble Conjecture : ρ does not contain the information of both T and v.

Louis Garrigue New Hohenberg-Kohn theorems