Christian Brouder

Gianluca Panati

Gabriel Stoltz

The many-body Green function of degenerate systems

Keywords: numbers: 31.15.am, 71.10.-w, 24.10.Cn

A rigorous non perturbative adiabatic approximation of the evolution operator in the many-body physics of degenerate systems is derived. This approximation is used to solve the long-standing problem of the choice of the initial states of H0 leading to eigenstates of H0 + V for degenerate systems. These initial states are eigenstates of P0V P0, where P0 is the projection onto a degenerate eigenspace of H0. This result is used to give the proper definition of the Green function, the statistical Green function and the non-equilibrium Green function of degenerate systems. The convergence of these Green functions is established.

Non-perturbative Green function methods, such as the GW approximation [1] or the Bethe-Salpeter equation [2,3], have brought remarkable progress in the calculation of the electronic structure and dielectric response of semiconductors. The extension of these methods to transition metals systems faces a serious difficulty: the standard Green function can only be defined when the initial state |0 of the system without interaction is a single Slater determinant. In physical terms, each singleparticle orbital or Bloch state has to be either occupied or unoccupied at zero temperature. However, the physics of transition metals often contradicts this requirement. For the example of a V 3+ ion in an octahedral environment, we do not know a priori how the two 3d electrons are distributed over the six degenerate t 2g orbitals (with up and down spins).

More generally, for a system described by a Hamiltonian H = H 0 + V where the ground state of H 0 is degenerate, we need to determine the parent states, i.e. the initial states of H 0 that evolve into eigenstates of H by adiabatically switching the interaction.

Degenerate systems being ubiquitous in quantum physics, this long-standing problem has been discussed in chemical physics [START_REF] Tolmachev | Correlation effects in atoms and molecules[END_REF]5], nuclear physics [6,7], atomic physics [8,9] and solid state physics [10]. Esterling and Lange [10] summarized the situation as follows: "Since H 0 has degenerate ground states, the choice of the state |0 must be made with care, and this may be considered the key to the problem." This question is also crucial in many-body physics because the Green function of a degenerate system has to be defined from a parent state.

In the present paper, we give a simple method to explicitly determine the parent states and to define the Green function of degenerate systems. Through a nonperturbative analysis of the evolution operator of a degenerate system, we determine the exact form of its singularities. This enables us to derive: (i) an easy and explicit method to determine the parent states; (ii) a nonperturbative proof that the Gell-Mann and Low formula generally converges only for these parent states; (iii) the formula for the Green function of degenerate systems; (iv) the validity of the so-called statistical Green function; (v) the singularity structure of the non-equilibrium Green function of degenerate systems.

Adiabatic switching. Many-body theory [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF][START_REF] Gross | Many-Particle Theory[END_REF] is usually based on the adiabatic switching of the interaction, i.e. the transformation of the time-independent Hamiltonian H = H 0 + V into the time-dependent one H 0 + e -ε|t| V . Adiabatic switching turns the non degenerate ground state |0 of H 0 into an eigenstate |Ψ GML of H first proposed by Gell-Mann and Low [START_REF] Gell-Mann | [END_REF] in 1951

|Ψ GML = lim ε→0 U ε (0, -∞)|0 0|U ε (0, -∞)|0 , (1)
where the evolution operator U ε (t, t ′ ) is the solution of

i ∂U ε (t, t ′ ) ∂t = e iH0t e -ε|t| V e -iH0t U ε (t, t ′ ),
with the initial condition U ε (t ′ , t ′ ) = 1. The wavefunction |Ψ GML is then used to build the Green function of the system [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF][START_REF] Gross | Many-Particle Theory[END_REF]. However, Gell-Mann and Low did not prove that the limit of eq. ( 1) exists [START_REF] Gross | Many-Particle Theory[END_REF]. The convergence of |Ψ GML for nondegenerate systems was first established by Nenciu and Rasche in 1989 [14]. For a degenerate ground state |0 of H 0 , the Gell-Mann and Low formula generally fails to converge when ε → 0, as can be seen even for a trivial two-level system [15]. In the following, we use recent advances in the mathematical analysis of the adiabatic approximation (see [START_REF] Teufel | Adiabatic Perturbation Theory in Quantum Dynamics[END_REF] for a review) to extend the Gell-Mann and Low formula to degenerate systems.

Adiabatic approximation. In this section, we set up the notation and give the theorem that enables us to calculate the parent states and to define the Green functions. We consider t ≤ 0 and we rewrite the time-dependent Hamiltonian as H(s) = H 0 + e s V , where s = εt is the socalled slow variable. The eigenvalues of H(s) are denoted by E j (s) and its eigenprojectors by P j (s). We recall that, if the eigenvalue E j (s) is n j -fold degenerate, the eigenprojector is

P j (s) = nj k=1 |ϕ jk (s) ϕ jk (s)|, where {|ϕ jk (s) } is a set of n j orthonormal eigenstates of H(s)
for the eigenvalue E j (s). For notational convenience, we denote P j (-∞) by P 0 j in the rest of the paper. The model space M is the vector space generated by the eigenstates corresponding to N 0 eigenvalues of H 0 = H(-∞). Each eigenvalue of H 0 can be degenerate and is possibly split by the perturbation V , so that the N 0 eigenvalues of H 0 become N eigenvalues E 1 (s), . . . , E N (s) of H(s), with N ≥ N 0 . Each E j (s) can be degenerate and the eigenvalues are allowed to cross (fig. 1). For an octahedral V 3+ ion, we have N 0 = 1 with degeneracy 15, and there are N = 4 interacting states: 1 A 1g , 1 E g , 1 T 2g and 3 T 1g , with degeneracy n j =1, 2, 3 and 9, respectively.

A key tool of our approach is A(s, s 0 ), the rotating frame operator [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF][START_REF] Messiah | Quantum Mechanics[END_REF], that relates the eigenstates at s 0 and s: A(s, s 0 )|φ jk (s 0 ) = |φ jk (s) , so that A(s, s 0 )P j (s 0 ) = P j (s)A(s, s 0 ).

(

) 2
Using standard technical assumptions [START_REF]Our assumptions hold for a Hamiltonian describing the many-body Coulomb interaction of electrons and nuclei[END_REF], we recently obtained [START_REF] Brouder | [END_REF] a rigorous approximation of the evolution operator projected on each eigenspace:

U ε (0, -∞)P 0 j ≃ e iθj /ε A(0, -∞)P 0 j , (3)
where

θ j = - 0 -∞ (E j (s) -E j (-∞))ds.
In particular, the divergences of the evolution operator are entirely described by the factor e iθj /ε . Construction of the parent states. The parents states are the eigenstates |φ of H 0 such that U ε (0, -∞)|φ tends to an eigenstate |Ψ of H, up to a (divergent) phase. Therefore, the parent states are naturally defined in terms of U ε (-∞, 0)|Ψ and it seems that the interacting states |Ψ are needed to define the parent states [START_REF] Tolmachev | Correlation effects in atoms and molecules[END_REF]9,[START_REF] Kuo | Folded-Diagram Theory of the Effective Interaction in Nuclei, Atoms and Molecules[END_REF]. We now show that the parent states have a more simple and explicit definition as eigenstates of P 0 j and we explain how P 0 j can be calculated by standard time-independent perturbation theory.

For notational convenience, we denote e s by λ and the eigenvalues and eigenprojections are written in terms of λ. We denote by E j (λ) and |ϕ jk (λ) the eigenvalues and eigenstates of H 0 + λV (so that E j (λ) = E j (s)). They can be expanded as [START_REF] Brouder | [END_REF]

E j (λ) = ∞ n=0 λ n E n j , |ϕ jk (λ) = ∞ n=0 λ n |ϕ n jk ,
with the normalization ϕ 0 jk |ϕ jk (λ) = 1. The eigenstates |ϕ jk (λ) are assumed orthonormal only at λ = 0, where P 0 j = nj k=1 |ϕ 0 jk ϕ 0 jk |. The time-independent Schrödinger equation

(H 0 + λV )|ϕ jk (λ) = E j (λ)|ϕ jk (λ) ,
gives, to order 0, H 0 |ϕ 0 jk = E 0 j |ϕ 0 jk , so that |ϕ 0 jk is an eigenstate of H 0 with energy E 0 j . We assume that E 0 j is one of the N 0 eigenvalues of the model space, so that |ϕ 0 jk belongs to the model space. However, the degeneracy of E 0 j as an eigenvalue of H 0 is generally larger than the degeneracy of E j (λ) and we need more information to determine the n j states |ϕ 0 jk . The Schrödinger equation to order λ gives us (H 0 -E 0 j )|ϕ

1 jk = (E 1 j -V )|ϕ 0 jk . This equation can only have a solution if ψ 0 m |(E 1 j -V )|ϕ 0 jk = 0
, where {|ψ 0 m } is a complete set of eigenstates of H 0 with energy E 0 j . Therefore, the initial states |ϕ 0 jk are eigenstates of H 0 with energy E 0 j and eigenstates of P E 0 j V P E 0 j with eigenvalue E 1 j , where P E 0 j is the projection onto the eigenspace of H 0 with eigenvalue E 0 j . In general, the degeneracy is split at this order, in the sense that there are only n j states that are simultaneously eigenstates of H 0 with energy E 0 j and of P E 0 j V P E 0 j with eigenvalue E 1 j . Otherwise, for instance when P E 0 j V P E 0 j is zero by symmetry, the equations coming from higher powers of λ must be taken into account to determine |ϕ 0 jk . In that case, the second order is usually enough [22], but methods have been developed to treat any order [23].

We generally have no a priori knowledge of n j and E 1 j . However, we can calculate all the eigenstates of H 0 and, for each energy E 0 j , we can diagonalize P E 0 j V P E 0 j . Then, each state must be examined to see if it cannot be further split by higher order terms. When degeneracy is due to the symmetry of the Hamiltonian H(s), this can be deduced from the dimension of the irreducible representations to which the states belong. The computational effort required to construct |ϕ 0 jk is small because it is an eigenvalue problem in a vector space whose dimension is the degeneracy of E 0 j , which is small in applications. From the states |ϕ 0 jk we build the projector P 0 j and we define a parent state as a state |φ such that, for some j,

P 0 j |φ = |φ . (4)
In practice, the parent state is one of the |ϕ 0 jk .

Generalized Gell-Mann and Low wavefunction. We show that the parent states previously defined lead to convergent Gell-Mann and Low wavefunctions. For a parent state |φ j such that P 0 j |φ j = |φ j , eq. ( 3) enables us to write U ε (0, -∞)|φ j = U ε (0, -∞)P 0 j |φ j ≃ e iθj /ε A(0, -∞)P 0 j |φ j ≃ e iθj /ε A(0, -∞)|φ j .

Therefore, the following limit exists:

|Ψ GML = lim ε→0 U ε (0, -∞)|φ j φ j |U ε (0, -∞)|φ j (5) = A(0, -∞)|φ j φ j |A(0, -∞)|φ j .
The Gell-Mann and Low wavefunction |Ψ GML is indeed an eigenstate of H 0 + V with energy E j (0) because P j (0)|Ψ GML = |Ψ GML . To show this, we use eq. ( 2):

P j (0)A(0, -∞)|φ j = A(0, -∞)P 0 j |φ j = A(0, -∞)|φ j .
In practice, we are interested in the Gell-Mann and Low wavefunction that is the ground state of H 0 + V . How should we choose the initial state |φ j for this to happen? In the non degenerate case, it is often assumed that the ground state of H 0 leads to the ground state of H 0 + V . When degeneracy is due to the presence of symmetry, band crossing can occur and one should try the |φ j corresponding to the lowest energy E 0 j for each irreducible representation. A typical example of band crossing in the presence of symmetry is given by Tanabe-Sugano diagrams of the multiplet theory [START_REF] Sugano | Multiplets of Transition-Metal Ions in Crystals[END_REF]. For a small crystal field, the ground state has the highest spin value (Hund's rule), but as the crystal field parameter increases, a low spin state can become the ground state.

Green functions. The expression for the Green function is usually derived under the assumption that the ground state of H 0 is non degenerate [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF][START_REF] Gross | Many-Particle Theory[END_REF]. Our results enable us to determine how this expression is extended to the case of degenerate systems. Now we formally extend our previous results to Fock space.

To follow the usual argument [START_REF] Gross | Many-Particle Theory[END_REF], we repeat the calculation by starting from a positively infinite time. In terms of the slow variable s = -ε|t|, the switching function e -ε|t| is the same for positive and negative times. As a result, the rotating frame operator is the same but the divergent phase changes sign:

U ε (0, +∞)|φ j ≃ e -iθj /ε A(0, -∞)|φ j . Therefore, lim ε→0 U ε (0, +∞)|φ j φ j |U ε (0, +∞)|φ j = A(0, -∞)|φ j φ j |A(0, -∞)|φ j .
In other words, the Gell-Mann and Low wavefunctions obtained from positive and negative infinite times are equal. This non-trivial result is due to the fact that the switching function f (t) = e -ε|t| is even. The two-point Green function is defined by [START_REF] Gross | Many-Particle Theory[END_REF] G

(x, y) = Ψ GML |O H |Ψ GML Ψ GML |Ψ GML ,
where x = (r, t), y = (r ′ , t ′ ), O H = T ψ H (x)ψ † H (y) is the time-ordered product of fields in the Heisenberg picture and |Ψ GML is defined by eq. ( 5). Standard manipulations [START_REF] Gross | Many-Particle Theory[END_REF] transform it into

G(x, y) = lim ε→0 φ j |X ε |φ j φ j |U ε (+∞, -∞)|φ j , (6)
where

X ε = U ε (+∞, t)ψ(x)U ε (t, t ′ )ψ † (y)U ε (t ′ , -∞) and X ε = -U ε (+∞, t ′ )ψ † (y)U ε (t ′ , t)ψ(x)U ε (t, -∞) if t > t ′
and t < t ′ , respectively. The expression for the Green function generally converges only when the initial state is a parent state. Indeed, consider a state |φ in the model space and write it as |φ = j |φ j , where |φ j = P 0 j |φ . Thus,

U ε (0, ±∞)|φ ≃ j e ∓iθj /ε A(0, -∞)|φ j .
If there is more than one j in the sum, the phases θ j are generally different (in the absence of eigenvalue crossing, they can be shown to be different). Therefore, the phase factors in the numerator and denominator of eq. ( 6) do not cancel and the expression has no limit for ε → 0. Statistical Green function. The Green function of the previous section has a non-ambiguous meaning when |φ j is the parent state of a non-degenerate interacting state. However, when the interacting state itself is degenerate, there is no reason to choose any particular parent state. To solve that problem, Layzer [START_REF] Layzer | [END_REF] defined the statistical Green function as an equal-weight average over the degenerate states. Such a statistical Green function was advocated, for instance, by Alon and Cederbaum [26]. The statistical Green function can preserve the symmetry of the system: in the example of a spherically symmetric Hamiltonian, the Green function obtained from any state |ℓm with ℓ = 0 gives non spherically symmetric charge density, whereas the statistical Green function obtained from the mixed state m |ℓm (2ℓ + 1) -1 ℓm| gives a spherical charge density [27]. We are now able to prove that the statistical Green function has a welldefined limit when ε → 0.

To define the statistical Green function of a degenerate interacting system with energy E j (0), we use the density matrix ρ = (1/n j ) nj k=1 |ϕ 0 jk ϕ 0 jk |, where the states |ϕ 0 jk are those used to calculate P 0 j . We assume that the degeneracy of E j (0) is n j . Then,

G(x, y) = lim ε→0 Tr(ρX ε ) Tr ρU ε (+∞, -∞) . (7)
If we put |φ j = |ϕ 0 jk in eq. ( 6), the divergences of the numerator and denominator are e 2iθj /ε , which does not depend on k. Therefore, the divergent phases of the numerator and denominator of eq. ( 7) are equal, and the statistical Green function is well defined. For our octahedral V 3+ ion, the Green function is defined with the density matrix built from the nine degenerate states with 3 T 1g symmetry if the ion is high spin.

Non-equilibrium Green function. In the study of nonequilibrium systems, it is often convenient to run the evolution operator over a closed time path instead of taking the limit t → +∞. In this so-called Keldysh approach, the Green function G(x, y) is calculated by a formula involving no denominator [START_REF] Rammer | Quantum field theory of Non-equilibrium States[END_REF]: it is the limit for

ε → 0 of G ε = φ|U ε (-∞, 0)O H U ε (0, -∞)|φ .
As for the standard Green function, this expression generally converges for degenerate systems only when |φ is a parent state. To see this, we expand again a state of the model space over parent states: |φ = j |φ j . Then,

G ε ≃ ij e i(θj -θi)/ε φ i |A(-∞, 0)O H A(0, -∞)|φ j .
This expression converges for ε → 0 when there is a single phase, i.e. when |φ is a parent state. Otherwise, the limit generally does not exist.

Conclusion. The determination of the parent states and the proof of convergence break the last deadlocks in the determination of the Green function of degenerate systems. The main difference with the non-degenerate case is the fact that, for many degenerate systems, the parent state is not a single Slater determinant. To see this, consider a Hamiltonian where H 0 is the restricted Hartree-Fock Hamiltonian of a 3d n transition metal ion and V is the sum of the remaining atomic Coulomb interaction and of an effective potential representing the influence of the surrounding atoms. Then, the parent states are exactly the eigenstates of the crystal-field Hamiltonian and they are generally not single Slater determinants. In that case, the structure of the Green function is more complex because of the so-called initial correlations [START_REF] Hall | [END_REF] coming from the matrix elements between the different Slater determinants. Perturbative [START_REF] Hall | [END_REF] and nonperturbative [START_REF] Brouder | Quantum Field Theory -Competitive Models[END_REF] methods have been developed to tackle initial correlations. Finally, our approach also gives a non-perturbative proof of the convergence of the effective Hamiltonian [START_REF] Weber | [END_REF].

FIG. 1 :

 1 FIG. 1: Exemple of allowed eigenvalue pattern