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 N N � � � H N ( v ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 Theorem (Standard Hohenberg-Kohn) Let p > max(2 d / 3 , 2) , let w , v 1 , v 2 ∈ ( L p + L ∞ )( R d ) such that H N ( v 1 ) and H N ( v 2 ) have ground states Ψ 1 and Ψ 2 . If � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0 , then v 1 = v 2 + ( E 1 − E 2 ) / 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 N N � � � H N ( v , w ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 Theorem (Hohenberg-Kohn for interactions) Let w 1 , w 2 , v 1 , v 2 ∈ ( L p + L ∞ )( R d ) . If there are two ground states Ψ 1 and Ψ 2 of H N ( v 1 , w 1 ) and H N ( v 2 , w 2 ) , such that � R d ( v 1 − v 2 ) ( ρ Ψ 1 − ρ Ψ 2 ) � ρ (2) Ψ 1 − ρ (2) � � + R 2 d ( w 1 − w 2 ) ( x − y ) ( x , y ) d x d y = 0 , Ψ 2 − c ( N − 1) then w 1 = w 2 + c and v 1 = v 2 + E 1 − E 2 for some c ∈ R . N 2 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 v 1 , v 2 , w ∈ ( L p + L ∞ )( R d ) , where w is even and not constant, such that H N ( v 1 , w ) and H N ( v 2 , 0) have ground states Ψ 1 and Ψ 2 . Then ρ (2) Ψ 1 � = ρ (2) Ψ 2 . 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) N N + M H N , M ( v a , v b , w a , w b , w ab ) := � � ( − ∆ i + v a ( x i ))+ ( − α ∆ k + v b ( x k )) i =1 k = N +1 � � � + w a ( x i − x j )+ w b ( x k − x l )+ w ab ( x i − x k ) 1 � i < j � N N +1 � k < l � N + M 1 � i � N N +1 � k � N + M Theorem (Hohenberg-Kohn for different particles) Let H N , M ( v a , i , v a , i , w a , i , w b , i , w ab , i ) have ground states Ψ i , i ∈ { 1 , 2 } . If ρ (2) a , Ψ 1 = ρ (2) a , Ψ 2 , ρ (2) b , Ψ 1 = ρ (2) b , Ψ 2 and ρ (2) ab , Ψ 1 = ρ (2) ab , Ψ 2 , then v η, 1 − v η, 2 , w η, 1 − w η, 2 , and w ab , 1 − w ab , 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 N � � H N ( v , B ) = � � − ∆ i + σ i · B ( x i ) + v ( x i ) + w ( x i − x j ) i =1 1 � i < j � N 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 H N ( v 1 , B 1 ) and H N ( v 2 , B 2 ) having ground states Ψ 1 and Ψ 2 . � � If R 3 ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) + R 3 ( B 1 − B 2 ) · ( m Ψ 1 − m Ψ 2 ) = 0 , then | B 1 − B 2 | χ = E 1 − E 2 + v 2 − v 1 , N where χ is a function taking its values in {− 1 , − 1 + 2 N , − 1 + 4 N , . . . , 1 − 2 N , 1 } . If we also assume v 1 = v 2 , E 1 = E 2 and N odd, then B 1 = B 2 . Louis Garrigue New Hohenberg-Kohn theorems
Counterexample for Matrix DFT For non local potentials G ’s, we define N N � � � H N ( G ) = − ∆ i + w ( x i − x j ) + G i . i =1 i =1 1 � i < j � N 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 G 1 be such that H N ( G 1 ) has a unique ground state Ψ 1 , isolated in the spectrum. We take G 2 = G 1 + ǫ | φ � � φ | , where φ is ⊥ to the N components of γ Ψ 1 . For ǫ � 0 , G 2 and G 1 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) T 1 , T 2 > 0 , Γ 1 , Γ 2 the grand canonical Gibbs states corresponding respectively to E v 1 , T 1 and E v 2 , T 2 . If � − ( T 1 − T 2 ) ( S Γ 1 − S Γ 2 ) + R d ( v 1 − v 2 )( ρ Γ 1 − ρ Γ 2 ) = 0 , then T 1 = T 2 and v 1 = v 2 . Extends Mermin’s (65’) v �→ ρ injectivity at fixed T Works also when we only assume that T 1 , T 2 � 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
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