Connection between the hopping matrix and the SPDM

In the following we consider the type of particle interaction specified by the parame-tersW_ijkl^σσ0 as fixed, such that the matrixt = {tijσ}of the hopping integrals and energy levels characterizes the system described by the Hamiltonian (3.1). We have already identified theSPDMγ as the fundamental variable inLDFT, and we would now like to formulate a statement with the same fundamental character as the Hohenberg-Kohn (HK)Theorem A.1 in the formulation of conventional DFTin the continuum.

Thus, we would like to establish a unique connection between the hopping matrixt, which characterizes the system, and theSPDMassociated to the ground state. Before we can do so, we need to ascertain under which conditions two hopping matricest andt⁰lead to different ground states. In LemmaA.1we have already shown that ex-ternal potentials which differ by more than a constant must lead to different ground states. It is straight forward to adapt the proof of Lemma A.1 to the case of spin-dependent potentials and to demonstrate that two potentialsvσ(r) andv⁰_σ(r) must lead to different ground states if they differ by more than a possibly spin-dependent constant, i. e., ifv⁰_σ(r) , v_σ(r)+c_σ withc_σ ∈ R.¹ From Eq. (3.3) it is clear, that a spin-dependent shift in the external potentialv_σ⁰(r)=vσ(r)+cσ does not change the hopping integralstijσ,i , j, but leads to a spin-dependent shiftε_iσ⁰ =t_iiσ⁰ =εiσ +cσ of the energy levels. This immediately proves the following

Lemma 3.1. Two hopping matricest = {tijσ} andt⁰ = {t_ijσ⁰ } lead to different ground states of the Hamiltonian(3.1)if they differ by more than a spin-dependent shift in the energy levels, i. e., ift_ijσ⁰ ,t_ijσ +δ_ijc_σ withc_σ ∈R.

We are now prepared to formulate aHK-like theorem and to establish a unique con-nection between the hopping matrixt, which characterizes the system under study,

1Notice, however, that two potentials which differ by a spin-dependent constant notnecessarilylead to the same ground state. For example, the coupling to an external magnetic-field via a Zeeman term corresponds to a spin-dependent shift in the external potential which leads to a polarized ground state if the field is sufficiently strong.

and the correspondingground-state single-particle density matrix (gs-SPDM). We will focus on systems having nondegenerate ground states, however, the inclusion of de-generate ground-states poses no problem, as discussed further below.

Theorem 3.1(Töws and Pastor [81]). The hopping matrixt of the interacting many-particle system described by the Hamiltonian (3.1) is (apart from an irrelevant spin-dependent shift in the energy levels) a functional²of thegs-SPDMγ.

Proof. The proof is carried out in close analogy to the proof of theHKTheoremA.1.

Assume that the ground state|Ψ₀iof the Hamiltonian ˆH with the hopping matrixt is nondegenerate and that there existsanotherhopping matrixt⁰which differs fromt by more than a spin-dependent shift in the energy levels but, nevertheless, leads to thesamegs-SPDMγ as the hopping matrixt. If ˆH⁰and|Ψ₀⁰idenote the Hamiltonian and ground state associated witht⁰, the corresponding ground-state energy is given by

E⁰₀ = hΨ₀⁰|Hˆ⁰|Ψ₀⁰i=X

ijσ

t_ijσ⁰ γijσ +hΨ₀⁰|Wˆ |Ψ₀⁰i. (3.8) Since the hopping matricest andt⁰ differ by more than a spin-dependent shift in the energy levels, the corresponding ground states |Ψ0i and |Ψ₀⁰i must be different according to Lemma3.1. Therefore, from the minimal principle for the ground-state energyE₀ = hΨ₀|Hˆ|Ψ₀iof the Hamiltonian ˆH it follows that

E0 < hΨ₀⁰|Hˆ|Ψ₀⁰i=X

ijσ

tijσγijσ +hΨ₀⁰|Wˆ |Ψ₀⁰i= E⁰₀+X

ijσ

tijσ −t_ijσ⁰

γijσ. (3.9) Notice that the strict inequality holds because we assumed that the ground state as-sociated witht is nondegenerate. By interchanging primed and unprimed quantities one obtains

E⁰₀ ≤ hΨ₀|Hˆ⁰|Ψ₀i=E₀+X

ijσ

t_ijσ⁰ −t_ijσ

γ_ijσ , (3.10)

where no strict inequality holds, since the ground state associated witht⁰ could be degenerate. Adding Eqs. (3.9) and (3.10) the contradiction

E0+E⁰₀ < E⁰₀+E0 (3.11) is obtained. This proves that two hopping matricest andt⁰which differ by more than a spin-dependent shift in the energy levels cannot lead to the samegs-SPDMγ. One concludes thatt is (apart from a spin-dependent shift in the energy levels) a functional

ofγ.

2Here and in the following we will sometimes use the term “functional” for quantities which are ordinary functions in the strict mathematical sense. We choose to do so in order to match the terminology in conventionalDFTand to highlight the corresponding similarities.

Theorem3.1states that the hopping matrixtis, apart from a spin-dependent shift in the energy levels, a functional of thegs-SPDM. A spin-dependent energy shift is, how-ever, of little physical relevance, since it could be absorbed in a spin-dependent chem-ical potential and, most importantly, since it does not change the set of eigenstates of the Hamiltonian. Therefore, we conclude that the physically relevant part of the Hamiltonian ˆH, and thus the full set of corresponding eigenstates as well as all phys-ical properties derived from it, are functionals of the gs-SPDMγ. Notice, however, that the one-to-one correspondence between thegs-SPDMγand the hopping matrixt established by Theorem3.1is no longer valid in the presence of ground-state degen-eracies. Clearly, if the ground state is degenerate thegs-SPDMγ is no longer unique.

Nevertheless, due to the variational principle and the simple fact that all ground states share the same energy, a one-to-one correspondence between the set{γα}formed by the SPDMs of all degenerate ground-states and the hopping matrixt can be estab-lished. Thus, thegs-SPDMof any one of the degenerate ground-states determines the hopping matrixt up to a spin-dependent energy-level shift. Furthermore, in practical applications we will resort to aLevy-Lieb (LL)like formulation ofLDFT, which will be formulated in Section3.1.2 and where degeneracies pose no problem by default.

Returning to the case of a nondegenerate ground-state, where the uniquegs-SPDMγ determines the hopping matrixt up to a spin-dependent energy-level shift and thus all the eigenstates of the Hamiltonian, we may immediately formulate the important Corollary 3.1. The ground state|Ψ₀iof the interacting many-particle system described by the Hamiltonian(3.1)is a functional of thegs-SPDMγ.

Moreover, the converse statement is also true, i. e., thegs-SPDMγijσ = hΨ₀|cˆ_iσ^†cˆjσ|Ψ₀i is a functional of the ground state|Ψ₀i. This establishes a bijective map between the setΨ0 containing all nondegenerate ground states and the corresponding set Γ0 of thegs-SPDMs. Furthermore, from Corollary3.1we obtain the important

Corollary 3.2. The ground-state expectation value of any observableOˆ is a functional of thegs-SPDMγ.

Proof. From Corollary3.1we know that the ground state|Ψ₀i= |Ψ₀[γ]iis a functional of thegs-SPDMγ. Therefore, the ground-state expectation value of any observable ˆO can be obtained from thegs-SPDMas

O[γ]= hΨ₀[γ]|O|ˆ Ψ₀[γ]i. (3.12) In particular, the functional representing the sum of the kinetic and potential energy

K[γ]= hΨ0[γ]|Kˆ|Ψ0[γ]i =X

ijσ

tijσγijσ (3.13)

is explicitly known in the framework ofLDFTand a simple linear form inγ. In the following we will, as it is common practice, refer to ˆKas “kinetic energy”, even though it contains the contribution from the external potentialvσ(r)[see Eqs. (3.1) and (3.3)].

It is one of the major advantages ofLDFTover the conventional formulation ofDFTin the continuum that the functional dependenceK[γ]of the kinetic energy is explicitly known. From Corollary3.2it follows furthermore that the interaction energy

W[γ]= hΨ₀[γ]|Wˆ |Ψ₀[γ]i (3.14) is a functional of thegs-SPDMγ. The interaction-energy functional (3.14) has a uni-versal character in the sense that it does not depend on the hopping integralstijσ, which define the system under study, i. e., the dimensionality and structure of the underlying lattice as well as the range of the hybridizations. The interaction-energy functional (3.14) depends on the interaction integralsW_ijkl^σσ0. Furthermore, it depends on the many-particle Hilbert space under consideration, since the ground state|Ψ₀[γ]i is a state within this Hilbert space. Using Eqs. (3.13) and (3.14), the functional corre-sponding to the ground-state energy is obtained as

E[γ]=X

ijσ

t_ijσγ_ijσ +W[γ]. (3.15) Clearly, for the actualgs-SPDMγ₀ the energy functionalE[γ]assumes its minimum and equals the ground-state energyE₀ associated with the given hopping matrixt. If the interaction energy functionalW[γ]were known, the ground-state energy and SPDMcorresponding to arbitrary hopping matricest could be obtained by minimizing the energy functionalE[γ]. Therefore, the main challenge in practical applications of ground-stateLDFT is to determine the functionalW[γ]. In the next section we will see that already the characterization of the domain of definition ofW[γ]poses some difficulties. We will circumvent these difficulties by introducing a constrained-search method for the interaction-energy functional, which is similar to the LLprocedure discussed in AppendixA.1.2.