Hartree--Fock Theory, Lieb's Variational Principle, and their Generalizations

Hartree--Fock theory in quantum mechanics is reviewed, from the proposal of the Hartree--Fock approximation right after quantum mechanics was formulated to its applications in modern physics. This includes the description of traditional Hartree--Fock theory in quantum chemistry, its generalizations of various kinds, and its importance for predicting the presence of symmetry breaking, or the absence thereof.


I Hartree-Fock Theory of Coulomb Systems
One of the biggest triumphes of twentieth century science has been the discovery of quantum mechanics (and quantum field theory) almost one hundred years ago by Heisenberg [60], Born, Heisenberg, and Jordan [19], Schrödinger [91], Dirac [34,35,36], and Pauli [86]. It is a remarkable fact that, while quantum mechanics is of key importance for all technologies discovered in the past century and the complexity of theoretic descriptions of quantum systems has increased by several orders of magnitude, the basic conceptual framework of a complex Hilbert space H of wave functions ψ(t) ∈ H which represent physical states at time t ∈ Ê and evolve according to the (time-dependent) Schrödinger equation iψ(t) = Hψ(t), or observables A(t) which evolve according to the Heisenberg equation of motioṅ A(t) = i[H, A(t)], with H = H * being the self-adjoint Hamiltonian operator, is = remains unchanged until today. In the absence of external fields, the Hamiltonian H is independent of time t. Then the solution of the time-dependent Schrödinger equation can be traced back to determining the spectral resolution, in particular, all eigenvalues E ∈ Ê and all corresponding eigenvectors ψ E ∈ H of H. Eigenvalues and corresponding eigenvectors do not cover all possible cases, and the general task of determining the spectral resolution of H has lead to the mathematical theory of spectral analysis of self-adjoint operators, see [88]. For a Coulomb system, i.e., a nonrelativistic atom (K = 1) or molecule (K ≥ 2), consisting of N ∈ + := {1, 2, 3, . . .} dynamical, mutually repelling electrons revolving about K ∈ + attractive static nuclei of charges Z := (Z 1 , Z 2 , . . . , Z K ) ∈ [Ê + 0 ] K at pairwise distinct positions R := ( R 1 , R 2 , . . . , R K ) ∈ [Ê 3 ] K , the Hilbert space of a single electron is h := L 2 (Ê 3 × {↑, ↓}), and the Hilbert space H (N ) := Ψ ∈ h ⊗N ∀ π ∈ S N : Ψ(x π ) = (−1) π Ψ(x) (I. 1) of the wave function of the system of N dynamical electrons is the space of square-integrable functions of N coordinates x n = ( x n , τ n ) ∈ Ê 3 × {↑, ↓} which are antisymmetric under permutations x π = (x π(1) , . . . , x π(N ) ) of these coordinates (x 1 , . . . , x N ). (Here and henceforth we follow the convention from physics and assume for Hilbert spaces that (ϕ, ψ) → ϕ|ψ is alway antilinear in ϕ and linear in ψ.) The Hamiltonian generating the dynamics of these N electrons is Note that the charges Z and the positions R of the nuclei enter H N as fixed parameters, and H N (Z, R) may be considered the Born-Oppenheimer approximation [20] to lowest order. as the lowest energy expectation value the Hamiltonian H N admits. Instead of solving the Schrödinger equation -which is virtually impossible-one computes the energy expectation value of any normalized trial state Ψ trial ∈ H (N ) . This yields an upper bound Ψ trial |H N Ψ trial ≥ E gs (N) on the ground state energy. If Ψ trial |H N Ψ trial − E gs (N) is small, the trial state Ψ trial is assumed to be a good approximation to (one of) the actual ground state(s) Ψ gs . The mathematical justification for this replacement, e.g., in terms of quantitative error bounds, is a difficult and largely open mathematical problem. The earliest and, perhaps, most natural choice of trial states for Coulomb systems made is known as the Hartree-Fock approximation, which had been originally proposed by Hartree [59] but without an antisymmetry contraint on the wave function. This was followed by improvements by Fock [42] and Slater [95,94], who took the antisymmetry of the trial state correctly into account. It is a variational principle in which the variation in (I. 3) is restricted to Slater determinants, i.e., to wave functions of the form Φ(f ) := f 1 ∧ f 2 ∧ · · · ∧ f N . These are antisymmetrized tensor products of N-tuples f = (f 1 , . . . , f N ) ∈ h N of mutually orthonormal orbitals, i.e., vectors f i ∈ h in the one-particle Hilbert space where the one-particle operator h := −∆ − K k=1 Z k | x − R k | −1 is a secondorder differential operator acting on (a suitable dense domain in) h, and the pair interaction potential V := | x − y| −1 is a multiplication operator on (a dense domain in) h ⊗ h. The energy functional E HF (f ) can be written as a sum of the kinetic energy T (f ) minus the nuclear attraction U(ρ f ), plus the direct term 1 2 D(ρ f ), representing the classical electrostatic energy, minus the exchange term 1 2 X(γ f ), D(ρ) := ρ( x) ρ( y) d 3 x d 3 y | x − y| , X(γ) := |γ( x, y)| 2 d 3 x d 3 y | x − y| , (I. 10) where γ f ( x, y) := τ =↑,↓ N i=1 f i ( x, τ ) f i ( y, τ ) and ρ f ( x) := γ f ( x, x) are the one-particle density matrix and the one-particle density corresponding to Φ(f ), respectively. The explicit and relatively simple forms of these terms are one main reason for the success of the Hartree-Fock approximation.
For large neutral Coulomb systems, i.e., for Z = N ≫ 1, Z = Zz, with z = (z 1 , . . . , z k ) for fixed z k > 0 summing to 1, and nuclear positions R(Z) = R 1 (Z), . . . , R k (Z) not too close to each other, inf Z>0 min k<ℓ {Z 1/3 | R k (Z) − R ℓ (Z)|} > 0, the Hartree-Fock energy is seen [3] to obey E HF (Z) = E TF (Z, Zz, R(Z)) + Z 2 4 K k=1 z 2 k + O Z 5/3 , (I. 11) where the main contribution to leading order in Z is the Thomas-Fermi energy E TF Z, Zz, R(Z) established by Lieb and Simon in [73], which is bounded above and below by universal multiples of Z 7/3 , and followed by the Scott correction Z 2 4 K k=1 z 2 k of order Z 2 derived by Hughes [63] and by Siedentop and Weikard [92,93] for atoms (K = 1), by Ivrii and Sigal [64] for molecules (K ≥ 1) and by Solovej, Sørensen, and Spitzer in the relativistic and nonrelativistic case for both atoms and molecules [98,97]. (See also Siedentop's contribution to this volume.) For our discussion we observe that if f (HF) is a minimizer of E HF (or an approximate minimizer, i.e., E HF (f (HF) ) ≤ E HF (N) + ε, for ε > 0 sufficiently small) under the orthonormality constraint f i |f j h = δ i,j , then there exist universal constants 0 < c < C < ∞ such that, for any choice of z = (z 1 , . . . , z k ) and where ρ HF := ρ f (HF) and γ HF := γ f (HF) . That is, the kinetic energy, the nuclear attraction, and the classical electrostatic energy are all or the order Z 7/3 , while the exchange energy is of order Z 5/3 and hence much smaller in magnitude. The dominance of the three contributions T (f (HF) ), U(ρ HF ), and D(ρ HF ) to the energy compared to the contribution of the exchange term X(γ HF ) can be anticipated from the Cauchy-Schwarz inequality which implies that X(γ f ) ≤ D(ρ f ), for any model with repulsive pair interaction V (x−y) ≥ 0. Note, however, that this takes only total ground state energies of the entire system into account; if we compare energy differences, then the exchange contribution may become the decisive quantity that determines whether a system binds or not.
Furthermore, if f (HF) is a minimizer of E HF under the orthonormality constraint f i |f j h = δ i,j then E HF is stationary at f (HF) = (f Even though the Hartree-Fock equations form a system of nonlinear partial integrodifferential equations in f , the reduction of N dynamical variables to 2 makes it accessible to numerical solution. Concrete numerical algorithms to solve the Hartree-Fock equations have been analyzed mathematically by Cancès and Le Bris in [24]. More recently, the numerical solution of the corresponding self-consistent equation of generalized Hartree-Fock theory described in Section VI, the Bogoliubov-Hartree-Fock equations, has been studied by Lewin and Paul in [70]. Hartree-Fock theory is closely related to density functional theory and Kohn-Sham (KS) theory. These latter two are based on the Hohenberg-Kohn theorem [61] which asserts that the ground state energy of any Coulomb system can be expressed as the infimum of a universal (but unknown) functional of the electron density only. A mathematically precise formulation of the Hohenberg-Kohn theorem was given by Levy [67] and Lieb [74]. We describe the Kohn-Sham theory from the viewpoint of Hartree-Fock theory, although this oversimplifies their physical arguments somewhat. Namely, Kohn and Sham proposed to approximate the exchange term X(γ f ) by a functional G[ρ f ( x)] d 3 x of the one-particle density ρ f only, where G is yet to be determined. A natural candidate for G is G[ρ] = C Dirac ρ 4/3 , which was proposed by Dirac in [33] and whose quality as an approximation to the exchange term was analyzed in [4]. This approximation is known as the local density approximation (LDA) and the KS-LDA theory is widely and successfully used in numerical studies in material science. Its mathematical foundation including a proof of existence of minimizers of the Kohn-Sham energy functional and, hence, of solutions of the corresponding stationarity condition known as the Kohn-Sham equations was given by Anantharaman and Cancès in [1]. An important improvement to the local density approximation defined by a function G[ρ] is the generalized gradient approximation (GGA). It accommodates an additional dependence of the exchange term function G[ρ, ∇ √ ρ] on the gradient of the (square root of the) density, leading to KS-GGA theory. A very successful proposal for the form of G[ρ, ∇ √ ρ] was made by Perdew, Burke, and Ernzerhof in [65] and is known as PBE.
The first mathematically rigorous treatment of the Hartree-Fock approximation and the corresponding Hartree-Fock equations was given by Lieb and Simon in [72]. By applying the so-called direct methods of the calculus of variations, they prove the existence of a minimizer f (HF) of E HF , which then necessarly fulfills the Hartree-Fock equations, under the condition that the number N − 1 of electrons minus one is strictly less than the total nuclear charge Z := K k=1 Z k . This is a natural HVZ-(Hunziker-van Winter-Zhislin-)type condition reflecting the fact that, if one electron is spatially separated far away from the nuclei, it is still attracted by a Coulomb force induced by a net charge Z − N + 1 > 0. This force binds this outer electron to the molecule and prevents its escape to infinity.
An important technical point in [72] is the conversion of the original orthonor- where G(f , g) i,j := f i |g j h denotes the Gram matrix of f = (f 1 , . . . , f N ), g = (g 1 , . . . , g N ) ∈ h N . Lieb and Simon then observe that the minimization over f 's obeying G(f , f ) = 1 can be relaxed to the quadratic form inequality 0 ≤ G(f , f ) ≤ 1 without changing the minimum. This observation foreshadows Lieb's variational principle [71] formulated shortly after. As opposed to the set of f 's obeying G(f , f ) = 1, the set of f 's obeying the weaker constraint 0 ≤ G(f , f ) ≤ 1 is weakly closed (in the appropriate topology) which is necessary for the application of weak lower semicontinuity.
The HVZ-type condition E HF (N) < E HF (N − 1) mentioned above is the key condition for the proof of existence of excited states of Coulomb systems in Hartree-Fock theory, too. As the latter leads to nonlinear Euler-Lagrange equations, the concept of excited state as a higher eigenvalue of a linear operator cannot be applied directly, but there is a natural notion for excited states in variational analysis, namely, stationary points of the functional under consideration for values strictly above the minimum. The first proof that such excited states exist was given by Lions in [76]. Building up on a contribution by Friesecke [45] and an earlier paper [68], Lewin proved in [69] the existence of infinitely many excited states below E HF (N − 1). The essential step is to prove that below E HF (N − 1), the the Hartree-Fock functional for Coulomb systems fulfills a suitable Palais-Smale condition. Moreover, unlike [72] and [76], the proof in [68,69] is entirely given in the space N electrons and does not use any positivity of the pair potential or its Fourier transform.

II Fock Space, Density Matrices, and Second Quantization
Before we turn to Lieb's variational principle we provide a convenient mathematical framework and introduce the second quantization.
Fock Space: We henceforth assume the one-particle Hilbert space h to be a complex separable Hilbert space -not necessarily L 2 Ê 3 × {↑, ↓} as specified in (I.5), although this is a good example to keep in mind. For N ∈ + , the N-particle Hilbert space is denotes norm closure. The fermion Fock space (over h) is defined to be the orthogonal sum where H (0) := · Ω is the one-dimensional vacuum subspace spanned by a unit vector Ω called the vacuum vector. The vacuum subspace represents the physical state of absence of any particle in the quantum system under consideration. The elements of F are sequences Ψ = (ψ 0 , ψ 1 , ψ 2 , . . .) with ψ N ∈ H (N ) . If no confusion is possible, we henceforth consider H (N ) a subspace of F by identifying ψ N ∈ H (N ) with (0, . . . , 0, ψ N , 0, . . .) ∈ F.
Second Quantization: We come to the second quantization of operators. Given N ≥ 2 and three indices i, j, k ∈ {1, . . . , N}, i < j, we define two unitary operators Π Next, given a one-particle operator h on h and a two-particle operator V on h ⊗ h, we define the corresponding N-particle operators h N , V N and furthermore H N on Note that H N agrees with the operator in ( Their second quantizations are the operators acting on finite vectors [defined in (II. 19)]. The question whether À extends to a semibounded quadratic form is subtle, in general. For Coulomb systems, however, Dyson and Lenard [37,38] and Lieb and Thirring [75] have shown stability of matter to hold true, which in our context means that as a quadratic form on F, provided that the chemical potential µ < 0 is chosen sufficiently small, where is the number operator on F. (See also Loss' contribution to this volume.) Abstracting from this situation, in the following we assume the operator h to be essentially self-adjoint on a suitable dense domain s ⊆ h and semibounded, so that h(m) := h + m ≥ 0, for some sufficiently large constant m ∈ Ê. Furthermore, the pair potential is assumed to be an infinitesimal perturbation of h, i.e., V is defined on s and, for any Density Matrices: The energy expectation value ψ N | H N ψ N of a state represented by an N-particle wave function ψ N ∈ H (N ) may be written as Ψ | ÀΨ , where Ψ = (0, . . . , 0, ψ N , 0, . . .) ∈ F has only one non-vanishing component. Allowing for linear combinations, it can be extended to all finite vectors Ψ ∈ F fin of sufficient regularity. For these, we can further rewrite Ψ | ÀΨ = Tr F À|Ψ Ψ| , where |Ψ Ψ| ∈ B(F) denotes the rank-one orthogonal projection onto Ψ.
This suggests we further extend the notion of energy expectation value to all density matrices i.e., all even positive trace-class operators ρ on F of unit trace. (Here and henceforth we use the convention that a ≥ 0 includes the self-adjointness of an operator a.) Evenness of ρ means that φ|ρψ = 0, whenever φ ∈ H (m) and ψ ∈ H (n) with m − n odd. We remark that evenness of density matrices is a natural condition for fermion, but not for boson systems. Since a given density matrix ρ ∈ DM is, in particular, self-adjoint and compact, it can be written in diagonal form as ρ = ∞ ν=1 r ν |Ψ ν Ψ ν |, where r ν ≥ 0 are its nonnegative eigenvalues, which sum to 1, and {Ψ ν } ∞ ν=1 ⊆ F is an orthonormal basis in F of eigenvectors of ρ. If a density matrix ρ ∈ DM obeys ρAE = AEρ = N · ρ, for some N ∈ + , then ρ is called an N-particle density matrix. These are collected in Note that ρ ∈ DM (N ) if, and only if, all its eigenvectors belong to H (N ) . Moreover, the density matrices form a norm-closed convex subset DM ⊆ L 1 (F) in which rank-one orthogonal projections, such as |Ψ Ψ| ∈ DM above, are extremal points called pure. In particular, the orthogonal projection onto ψ N ∈ H (N ) considered above is a pure N-particle density matrix. Density matrices (and, in general, states) are not only natural objects mathematically, but from a physics point of view they are also important conceptually: For any reasonable theoretical framework for the description of a physical system, the scenario that this system is a subsystem of a larger system ("the rest of the universe") ought to be built in. In the latter situation, however, density matrices resulting from projecting onto the subsystem are the natural physical states, not wave functions.
Equipped with the definitions of density matrices and N-particle density matrices above, the Rayleigh-Ritz principle (I.3) assumes the following abstract form: where it is implicity assumed that H N is bounded from below and we denote (II.14) Similarly, for a sufficiently small chemical potential µ < 0 such that À µ is bounded below, we define the total ground state energy Creation and Annihilation Operators: Next, we introduce creation operators.
and extension by linearity. One then easily checks that c * (f ) extends by continuity to a bounded operator on F of norm c * (f ) B(F) = f h , called creation operator c * (f ) ∈ B(F). We observe that of finite vectors in F containing finite linear combinations of finite wedge-products of orbitals in s is a convenient dense domain for second quantizations of one-and two-particle operators on F. The second-quantized operators , Î, À, and AE have a convenient representation in terms of creation and annihilation operators. Assuming that h is infinitedimensional and that {f j } ∞ j=1 ⊆ s ⊆ h is an orthonormal basis of h of sufficiently regular functions lying in a dense subspace s, we have that Reduced Density Matrices: We come to defining reduced density matricesthe central object of this paper. Given a density matrix ρ ∈ DM with finite expectation value AE ρ < ∞ of AE, we define its reduced one-particle density ρ as a linear operator on h ⊗ h by their matrix elements These definitions are meaningful because it turns out that they define trace-class operators, as the following lemma asserts. (II.25) (ii) As quadratic forms, We finally remark that, in case of an N-particle density matrix ρ ∈ DM (N ) , for some N ≥ 2, the corresponding 1-RDM γ (1) ρ can be obtained from its 2-RDM γ (2) ρ by taking a partial trace,

III Lieb's Variational Principle
The formulation of the Hartree-Fock approximation in Section I turns out to be too rigid and inconvenient, mathematically. A more flexible formulation is provided by Lieb's variational principle which uses the reduced one-particle density matrices introduced in (II.23). We first link the 1-RDM γ (1) ρ corresponding to a density matrix ρ ∈ DM to the one-particle density matrix γ f corresponding to an orthonormal family f = (f 1 , . . . , f N ) ∈ h N of N orbitals that enter the Hartree-Fock energy functional E HF (f ) in (I.7). In fact, for pure states we have the following important relation between these operators: Then the following statements hold true.
(ii) The reduced two-particle density matrix of ρ(f ) is twice the orthogonal projection of rank 1 2 The right side of (III.3) is entirely expessed in terms of the rank-N orthogonal projection γ (1) ρ(f ) , and no other property than that enters the functional. That is, Lieb's variational principle [71] asserts that the projection property γ = γ 2 in (III.4) can be relaxed to 0 ≤ γ ≤ 1 h without changing the infimum of the functional.
Proof. We define the auxiliary energy E aux (N) to be the infimium on the right side in (III.6) and observe that, clearly, E aux (N) ≤ E HF (N). We make three simplifying assumptions which are not essential for the validity of Theorem 3 and can be avoided by suitable limiting arguments. The first is the strict positivity of ψ|V ψ > 0, for nonvanishing ψ, as opposed to merely assuming ψ|V ψ ≥ 0. The second simplifying assumption we make is that the infimum E aux (N) is actually a minimum. That is, E aux (N) = E HF (γ 0 ) is attained by a minimizer γ 0 which fulfills 0 ≤ γ 0 ≤ 1 and Tr(γ 0 ) = N. Since γ 0 is compact, there exist an orthonormal basis {f i } ∞ i=1 ⊆ h of eigenvectors of γ 0 with corresponding (not necessarily distinct) eigenvalues λ i ∈ [0, 1] that sum up to N. The third assumption we make is that γ 0 is of finite rank J < ∞, so that λ 1 , . . . , λ J > 0, Before we turn to Lieb's original proof in [71] we sketch the proof that E aux (N) ≥ E HF (N) given in [3], which, however, takes a different perspective. First note that it suffices to show that γ 0 = γ 2 0 is a projection. To this end we assume that γ 0 is not a projection and derive a contradiction from this assumption. If γ 0 is not a projection then there are (at least) two indices p, q ∈ + , p < q such that λ p , λ q ∈ (0, 1) because the sum J j=1 λ j = N is an integer. We set r := min{λ p , 1 − λ p , λ q , 1 − λ q } > 0 and I := [−r, r] and observe that λ p + δ, λ q + δ ∈ [0, 1], for any δ ∈ I. We define Then 0 ≤ γ δ ≤ 1 and Tr(γ δ ) = N, so γ δ is admissible for any δ ∈ I. Moreover which contradicts the assumption that γ 0 is a minimizer of E HF . It follows that γ 0 is a projection, indeed. Note that this proof is constructive in the sense that, fixing the orthonormal orbitals f 1 , . . . , f J , it defines an algorithm to find the γ * of minimal energy E HF (γ * ) among all rank-J operators of the form γ(τ 1 , . . . , We now turn to Lieb's proof of Theorem 3 in [71], starting from (III.7). Its heart is a lemma that, under the assumption that λ 1 , . . . , λ J > 0 and λ 1 +.
. . , J}. We omit its interesting proof. Given these vectors G (1) , . . . , G (N ) , Lieb defines for all n ∈ {1, . . . , N} and any choice θ := (θ j ) J j=1 ∈ [0, 1) J of phases θ 1 , . . . , θ J . Using the orthonormality of {f 1 , . . . , f J } ⊆ h it is easy to check that the set {g N ∈ H (N ) . The energy expectation value of this Slater determinant is This energy expectation value is now averaged over all possible choices of θ by integrating over [0, 1) J . That is, for any integrable function , and this implies that Here, the positivity V ≥ 0 is crucial, see also (IV.2). Adding up (III.11) and half of (III.12), Lieb arrives at Since θ is an average, Eq. (III.13) implies that there is at least one choice of Lieb's variational principle is a formulation of the Hartree-Fock approximation in the natural variable γ. It justifies the introduction of the notion of a oneparticle density matrix as any self-adjoint trace-class operator leaving aside the question whether it is the reduced one-particle density matrix γ = γ (1) ρ corresponding to some density matrix ρ ∈ DM. We come back to this point in the next section. The one-particle density matrices form a norm-closed, and hence weakly closed, convex subset of L 1 (h) which makes them suitable for variational analysis. Lieb's variational principle asserts, briefly speaking, that among one-particle density matrices obeying (III.14) and of trace N the ones with lowest energy are the rank-N projections. Under the assumption of the existence of a minimizer γ HF , this conclusion also follows from the stationarity of the Hartree-Fock functional at E HF at γ HF . In fact, the Hartree-Fock equations (I.14) turn into the following self-consistent equation: with ρ γ ( x) := γ( x, x) being the one-particle density corresponding to γ and a partial trace γ( x, y) = τ =↑,↓ γ( x, τ , y, τ ) as well as a sufficiently regular choice for the integral kernel for γ is understood. Comparing Lieb's variational principle to the original Hartree-Fock approximation, it is interesting to observe that the condition 0 ≤ G(f , f ) ≤ 1 N considered by Lieb and Simon in [72] is actually equivalent to 0 ≤ γ f ≤ 1 h , if we set γ f := N n=1 |f n f n |. Note, however, that γ f is of rank N, at most, and hence that

IV Bogoliubov Transformations and Representability
We begin our discussion of the concept of representability by comparing the two proofs of Theorem 3 given in the previous section. Lieb's original proof seems to be considerably more involved than the one in [3]. One must not overlook, however, that Lieb proves a stronger statement than Eq. (III.6). Namely, the averaging procedure introduced after (III.10) above yields an N-particle density matrix , whose reduced one-particle density matrix equals the minimizing one-particle density matrix γ 0 = γ (1) ρav . Concerning the energy estimate, the key point in Lieb's construction is that which leads to Estimate (III.12), thanks to the positivity V ≥ 0 of the pair interaction potential V . To describe the significance of this observation we introduce some more definitions and notation. We follow the paper [12] by Lieb, Solovej, and the author, Solovej's lecture notes [96], and the papers [9,6,10,8] by Breteaux, Hach, Knörr, Menge, and the author.
Generalized Reduced Density Matrices The Hamiltonian À in (II.8) is a linear which leaves the N-particle Hilbert spaces H (N ) invariant. Thus the variation in the Rayleigh-Ritz principles (II.15) for the total ground state energy E gs and (II.13) for the ground state energy E gs (N) for N particles may both be restricted to density matrices ρ = ∞ N =0 ρ N ∈ DM that are particle-number conserving and even to N-particle density matrices ρ ∈ DM (N ) without changing the infimum.
In general, however, density matrices ρ ∈ DM need not leave the N-particle Hilbert spaces H (N ) invariant, they are only assumed to be even. This can be conveniently formulated with the aid of the self-dual algebra built from creation and annihilation operators which was introduced by Araki [2]. We choose an antiunitary involution j : h → h and define the self-dual field operator All creation and annililation operators can be expessed as self-dual field operators A * (F ) for suitable choices of F . We remark that the antiunitary gives a good illustration of the freedom in the choice of j. Now, suppose that k ∈ + is a positive integer and F 1 , . . . , F 2k ∈ h ⊕ h are generalized orbitals. The evenness of ρ ∈ DM is equivalent to the vanishing of all expectation values of monomials of odd degree in the self-dual field operators. If ρ ∈ DM does not preserve the particle number then expectation values Tr F [ρ A * (F 1 ) · · · A * (F 2k )] of monomials of even degree in the self-dual field operators are, in general, non-vanishing -even if the generalized orbitals are all of the form F j = f j ⊕ 0, for all j = 1, . . . , 2k. While the existence of each of these matrix elements is guaranteed by the boundedness of A * (F ), for any F ∈ h ⊕ h, their summability requires an extra assumption. To formulate this we define the subspace which is a Banach space with respect to the norm ρ AE k := Tr F (AE + 1) k/2 ρ(AE + 1) k/2 . We introduce the subset DM AE k := DM ∩ L 1 AE k (F) of all density matrices ρ ∈ DM for which the expectation AE k ρ < ∞ of the k th power of the particle number operator is finite.
Given k ∈ + and a density matrix ρ ∈ DM AE k , we define its reduced gener- where F 1 , . . . , F k , G 1 , . . . , G k ∈ h ⊕ h. We obtain the reduced k-particle density matrix (k-RDM) γ (k) ρ ∈ B h ⊗k by restricting the matrix elements on vectors of the form g i ⊕ 0 and f j ⊕ 0, that is, We observe that in case ρ preserves particle numbers, i.e., ρAE = AEρ, then Γ (k) ρ is entirely determined by γ The cases k = 1 and k = 2 are obviously of special interest. We first discuss k = 1 and introduce the pairing operator α ρ : h → h corresponding to ρ by noting that α ρ vanishes if ρ preserves particle numbers and further that The pairing operator is convenient for the representation of the 1-gRDM Γ for all F, G ∈ h ⊕ h. Viewed as an operator-valued 2 × 2-matrix acting on vectors F = f 1 ⊕ jf 2 , the generalized 1-RDM Γ (1) ρ appears as where we recall that the 1-RDM γ (1) ρ is given by g| γ (1) Inserting G = F in (IV.10) and using the anticommutation relations, it is easily checked that 0 ≤ Γ (1) ρ ≤ 1 holds true which, in turn, is equivalent to Note that this yields γ Furthermore, if ρ ∈ DM AE has finite particle number expectation then (IV. 13) implies that the pairing operator ρ ) 2 and hence also Γ (1) ρ admits an expansion of the form Γ ρ implies that the eigenvalues and corresponding eigenvectors come in pairs λ ℓ , F ℓ and 1 − λ ℓ , JF ℓ . After changing the order of the eigenvalues, if necessary, we obtain (IV.14) where is an orthonormal basis. If, additionally, ρ is particle-number preserving and so α ρ ≡ 0, then where λ ℓ are the eigenvalues of γ The Bogoliubov linear maps obviously form a subgroup Expressing W as a 2 × 2-matrix of operators, the Bogoliubov linear maps can be alternatively characterized as and, in fact, W → Í W is a bijection Bog h⊕h → Bog F , where ρ of a density matrix ρ ∈ DM AE of finite particle number expectation. These assume the form (IV.11) with nonvanishing pairing operator α ρ unless ρ preserves particle numbers. In an orthonormal basis {F ℓ , JF ℓ } ⊆ h ⊕ h of eigenvectors with eigenvalues λ ℓ and 1 − λ ℓ , respectively, Γ ρ can be represented as in (IV.14). Starting from this one can construct a Bogoliubov linear map W ∈ Bog h⊕h such that Since, for all F, G ∈ h ⊕ h, In other words, the pairing operator α Í * W ρÍ W = 0 of the transformed density matrix Í * W ρÍ W vanishes and γ from which we conclude that the transformed density matrix Í * W ρÍ W has finite particle number expectation, as well, since Inspired by these properties, we define by the set of generalized one-particle density matrices 1-gpdm and by the set of one-particle density matrices (1-pdm).
Representability of 1-gpdm We have just seen that any 1-gRDM of finite particlenumber expectation value necessarily is a 1-gpdm in G (1) . Representability asks for sufficient conditions for this relation. That is, a 1-gpdm Γ (1) ∈ G (1) is called representable, if there exists a density matrix ρ ∈ DM whose reduced generalized one-particle density matrix Γ ρ . The following theorem gives an affirmative answer to question of representability of generalized 1-pdm.
Theorem 4. Every generalized one-particle density matrix Γ (1) ∈ G (1) is representable by a density matrix of finite particle number expectation value.
With these definitions we obtain new characterizations of the total and the N-particle ground state energies as This characterization of the ground state energy was first given by Coleman [29], following a remark by Coulson [31]. It seems to yield a drastic simplification of the task of determining ground state energies and ground states of many-fermion systems, as the number of variables of the problem is reduced from N to 4. This is, however, too optimistic because the problem of restricting the variation in (IV.34) and (IV.35) to representable, respectively N-representable, pairs (γ (1) , γ (2) ) is, perhaps, as difficult as solving the corresponding Schrödinger equation on Fock space altogether.
The requirement that the density matrix from which (γ (1) , γ (2) ) derives is particle number preserving or even an N-particle density matrix adds considerably to the degree of difficulty of the problem, as is seen when comparing the proofs of Theorems 4 and 5 in case that k = 1. A characterization of the representability of (Γ (1) , Γ (2) ) ∈ B(h 2 ) × B(h 2 ⊗ h 2 ) would be great progress.
Nevertheless, we now focus on particle number preserving density matrices ρ for which the reduced generalized 1-pdm (Γ ρ ). The difficulty described above has lead to what is known as the representability problem of quantum chemistry: Specify a condition A : (1) , γ (2) ) = true. The representability problem is still considered open today (at least by those who do not accept tautologies as its solution). It is known to be a hard problem in the sense of QMA complexity in computer science, as demonstrated by Liu, Christandl, and Verstraete in [77]. An overview on questions of reduced density matrices and their representability is given by Coleman and Yukalov in [30].
GPQ Condition and T 1;2 Condition While the representability problem, which is about the specification of a sufficient condition for the representability of a pair (γ (1) , γ (2) ), remains open, research on conditions reduced one-and two-particle density matrices necessarily fulfill has been more successful in the past. Namely, if a condition B : ρ ) = true, for any density matrix ρ ∈ DM then it is immediate that In practise, B is not a single condition but a list of conditions that γ (1) and γ (2) ought to fulfill, and Conditions (i) and (ii) in Lemma 1 are always part of this list. That is, it is understood that γ (1) ∈ g (1) is a 1-pdm and obeys 0 ≤ γ (1) ≤ 1 h and Tr[γ (1) ] < ∞. Theorems 4 and 5 ensure that there are not more conditions on γ (1) alone, that do not involve γ (2) . Almost sixty years ago Coleman [29] and Garrod and Percus [46] specified three conditions, which a representable pair (γ (1) , γ (2) ) of a one-and two-particle density matrix necessarily fulfill. These three conditions were orginally called "G", "P", and "Q", respectively, but we refer to them as a single condition which we call the GPQ condition. We apply the scheme described in (IV.36) and (IV.37) above and introduce Tr[γ (1) observing that E gs (N) ≥ E GPQ (N). In [9], Knörr, Menge, and the author considered self-adjoint, but not necessarily positive, trace-class operators ρ = ρ * ∈ L 1 (F) obeying Tr F |ρ| 1/2 AE 2 |ρ| 1/2 < ∞. It is easy to see that, for these ρ, the operators Γ (2) ρ , given by (IV.6), define trace-class operators on h 2 ⊗ h 2 . In [9], the GPQ condition was proven to be is equivalent to the positivity of Γ Furthermore, it was shown in [9] that the GPQ condition implies the fermion correlation inequality where P = P * = P 2 ∈ B(h) is an arbitrary orthogonal projection. This inequality is the key input for the proof in [3,9] that, for large Coulomb systems, the difference of the Hartree-Fock energy and E GPQ (N) is bounded by o(Z 5/3 ), which implies that the accuracy of the Hartree-Fock approximation is at least as good, A similar inequality was established and then applied to Fermi Jellium (described below) by Graf and Solovej in [51]. Since the exchange term is in magnitude greater than a universal multiple of Z 5/3 , see (I.12)-(I.13), Eq. (IV.40) proves that the accuracy of the Hartree-Fock approximation is better than the smallest contribution to the Hartree-Fock energy. In [40,39], Erdahl found additional representability conditions he called T 1 and T 2 . We refer to these as a single condition, the T 1;2 condition. It arises from observables of the form Q 4 := P * 3 P 3 + P 3 P * 3 , where P 3 is any polynomial in the self-dual field operators of degree three. Obviously, Q 4 is a nonnegative operator. Moreover, while both P * 3 P 3 and P 3 P * 3 are polynomials of degree six, their sum Q 4 is an anticommutator and hence a polynomial of degree four or less. Thus, Tr F [ρ Q 4 ] ≥ 0 yields a condition the pair of reduced generalized 1-pdm and 2pdm (Γ (1) ρ , Γ (2) ρ ) corresponding to ρ necessarily fulfills. We introduce Tr[γ (1) ] = N , (γ (1) , γ (2) ) fulfills GPQ and T 1;2 .
Erdahl's theoretical discovery came into focus of quantum chemists some two decades later, when numerical simulations demonstrated, that, in test cases with small N, the accuracy of E GPQ:T (N) is comparable to the accuray of full CI (configuration interaction) computations, i.e., the full solution of the N electron Schrödinger equation (projected onto a finite dimensional subspace, as part of the Galerkin approximation). These were carried out, e.g., by Mazziotti and the map Bog h⊕h ∋ W → Í W ∈ Bog F is a group isomorphism. Next, we define the second quantization É(T) ∈ B(D(N); F f ) of a bounded which implies that Note that we cannot directly quantize 1 2 T in the sense of (V.2), for if we replace T by 1 2 T in (V.2), we obtain an expression 1 In fact, all Bogoliubov transformations can be written in this form or, at least, approximated in the strong topology. That is, we may identify the Bogoliubov transformations with the family of unitary operators generated by i times selfadjoint quadratic Hamiltonians, where the bar denotes closure in the strong operator topology.
Quasifree Density Matrices: It turns out that quadratic Hamiltonians play an important role not only for Bogoliubov transformations, but also for density matrices. Recall from (IV.30) the definition of the density matrix is the quadratic Hamiltonian corresponding to We now construct an approximation ρ ε ∈ QDM for ρ 0 , such that ρ ε → ρ 0 in DM AE , as ε → 0. For ε > 0, we define noting that τ ℓ (ε) ∈ (0, 1), for all ℓ ∈ + . Next, we set . We further recall from (IV.33) that ρ 0 , after conjugation with the Bogoliubov transformation Í W ∈ Bog F , yields the density matrix ρ 0 := Í W ρ 0 Í * W whose reduced generalized 1-pdm Γ (1) ρ 0 is equal to the prescribed generalized 1-pdm Γ (1) from Theorem 4. Therefore, if T ε = aε bε b * ε 0 , a ε ∈ B(h), a ε ≥ 0, and b ε ∈ L 2 (h), due to the unitarity of Í W and the fact that ρ 0 ∈ DM AE . On the other hand, So, defining the set of quasifree density matrices where the bar indicates closure in DM AE and we conclude from (V.10)-(V.11) that every 1-gpdm is the 1-gRDM of a semigroup generated by a quadratic Hamiltonian or a limit in DM AE thereof.
Note that QDM ⊆ DM. Further note that the closure in the definition (V.12) of quasifree density matrices is important because otherwise the orthog- however, is the Bogoliubov transform Í W Ω of the vacuum vector, with W = P ⊥ jP j P jP ⊥ j , where P = N n=1 |f n f n | is the orthogonal projection onto the subspace spanned by f 1 , . . . , f n ∈ h. Hence, the corresponding rank-one projection ρ = |Í W Ω Í W Ω| is a (pure) quasifree density matrix. The Bogoliubov linear map W ∈ Bog h⊕h is not of the form exp[−iÉ(T )], for any T ∈ qh, but can be obtained as a strong limit of these.
We do not comment on the uniqueness part of Corollary 6 but point out the following important consequence. (V.14) For a density matrix ρ ∈ DM, its image q(ρ) = η ∈ QDM is called its quasifree reduction.
For a quasifree state ρ ∈ QDM the reduced generalized k-pdm can be explicitly computed in terms of its reduced generalized 1-pdm, as the following theorem asserts.
Theorem 8. Let ρ ∈ DM be a density matrix and denote M := Tr F [ρM], for any M ∈ B(F). Then the following statements are equivalent.
(iii) All truncated four-point functions vanish, i.e., for all F 1 , F 2 , F 3 , F 4 ∈ h ⊕ h, Characterization (ii) of quasifree density matrices in the above theorem is often taken as their definition. The somewhat surprising statement for a given density matrix, that the sole vanishing of its truncated four-point functions implies its quasifreeness originates in a theorem of Marcinkiewicz [78] in (classical) probability theory. In the context of quantum physics, it was first proved by Robinson [89] and later generalized in [17,18] for boson systems. The generalization to fermions can be traced back to work of Rajagopal and Sudarshan [87], see also the comment by Titulaer [100]. We refer to Salmhofer [90] for a modern presentation of truncated fermion correlation functions.
The quasifree reduction q : DM → QDM defined in Corollary 7 is a projection, i.e., an idempotent map q 2 = q from the density matrices onto quasifree density matrices. Gottlieb and Mauser [50] observed that the image q(ρ) ∈ QDM of ρ ∈ DM under this projection is the closest element to ρ in QDM in the sense that it minimizes the relative entropy among all quasifree density matrices, as the following Theorem asserts.
Theorem 9. Let ρ ∈ DM be a density matrix and q(ρ) ∈ QDM its quasifree reduction. If the relative entropy exists, then Proof. Let η ∈ QDM be a quasifree density matrix which, for simplicity, is assumed to be given as the exponential since relative entropy is nonnegative.
We note that the existence of the von Neumann entropies S[ρ] := −Tr F {ρ log[ρ]} of ρ and q[ρ] is assumed in the proof of Theorem 9 only for convenience and is not implied by the finiteness of their relative entropy S[ρ, q(ρ)]. Note, however, that if ρ ∈ QDM is quasifree and S[ρ] < ∞ then it assumes the simple form This is not hard to check for a quasifree density matrix ρ 0 ∈ QDM of the form Í W ∈ Bog F to ρ to transform it to ρ 0 . See also [12].

VI Bogoliubov-Hartree-Fock Approximation and Generalizations of Lieb's Variational Principle
Bogoliubov-Hartree-Fock Approximation: Since quasifree density matrices are, in particular, density matrices, we immediately observe that the Bogoliubov-Hartree-Fock energy defines an upper bound E BHF ≥ E gs on the total ground state energy E gs defined in (II.15). For a quasifree density matrix ρ ∈ QDM, the energy expectation value Note that if minimizers Γ BHF exist then they necessarily fulfill a stationarity condition, which in Bogoliubov-Hartree-Fock theory also takes the form of a selfconsistent equation where h BHF [Γ (1) ] is again an effective Hamiltonian on h⊕h and 1 neg (h BHF [Γ BHF and, therefore, also 1 neg (h BHF [Γ (1) BHF ]) necessarily fulfills.
It is possible, however, to use a (further) generalization of Bogoliubov-Hartree-Fock theory to positive temperatures 1/β > 0, which is not reviewed here, and obtain a minimizer Γ BHF by the zero-temperature limit β → ∞ of a family of minimizers Γ (1) β β∈Ê + for inverse temperature 1/β. For fixed β, the minimizer Γ (1) β necessarily fulfills the self-consistent equation is a suitable effective Hamiltonian, itself depending on β, and F β (x) = (1 + e βx ) −1 is the Fermi function.
We remark that the Bogoliubov-Hartree-Fock theory for positive temperature derives from a variational principle, namely the minimization of the Hartree-Fock pressure functional −P β by (VI.6) Lieb, Solovej, and the author have demonstrated in [12] that it fulfills for any generalized 1-pdm Γ (1) ∈ G (1) and, hence, yields a lower bound to the pressure (in the sense of statistical mechanics), in analogy to E BHF (Γ (1) ) being an upper bound to the total ground state energy E gs .
Repulsive Potentials: If h = L 2 (M, dν) for a measure space (M, dν), and V (x, y) ≥ 0 is a repulsive potential, i.e., a nonnegative multiplication operator on h ⊗ h, then In other words: For repulsive pair potentials the pairing operator yields a nonnegative contribution to the energy. Now, if (1) , and by the convexity of is a 1-gpdm, too. Its energy expectation value, however, is It follows that, for repulsive pair potentials, the Bogoliubov-Hartree-Fock energy agrees with the (total) Hartree-Fock energy and does not improve the approximation, where the total Hartree-Fock energy is defined as E HF := inf N ∈ + {E HF (N)}.
We stress that the total Hartree-Fock energy, as a function of the chemical potential µ, is the Legrende transform of the Hartree-Fock energy E HF (N) for N particles.
Attractive ∈ G (1) have a nonvanishing pairing operator α = 0. In this case the Hartree-Fock equations indicating the stationarity of the energy functional at the minimum turn into BCS-type equations for which those found by Bardeen,Cooper,and Schrieffer in [16] for the description of superconductivity are a special case. Because of similarity, the stationarity condition is usually called the BCS equation, and the latter have been systematically analyzed for translation-invariant systems under the additional assumption, or constraint, that only translation-invariant states enter the energy functional [58], see Section VII.
The physical system to which [7] was applied is a star consisting of neutrons, which are spin-1 2 fermions that attract each other by gravity. While it is generally important to prove statements about minimization problems without requiring the actual existence of a minimizer, [7] left this existence question unresolved. Lenzmann and Lewin, however, proved the existence of a minimizer for these neutron stars under natural conditions in [66].

Dirac-Fock Equations:
Shortly after the discovery of nonrelativistic quantum mechanics and the formulation of the Hartree-Fock approximation, a relativistic analogue, the Dirac-Fock (DF) equations, was formulated by Swirles [99]. The proof of existence of solutions the Dirac-Fock equations pose a considerably more difficult problem as compared to proving this for the Hartree-Fock equations, due to the unboundeness of the energy functional from below, direct methods from the calculus of variations do not really apply, as was pointed out by Chaix, Iracane, and Lions in [27,28] who introduced and studied the Bogoliubov- Dirac-Fock (BDF) model. An effective renormalization and then control on the unboundedness below of the BDF model, i.e., of the Hartree-Fock energy functional for electrons and positrons for small particle number and coupling constant was first obtained by Barbaroux, Helffer, Siedentop, and the author in [5]. The existence of solutions for Coulomb systems was shown by Esteban and Séré [41] and by Paturel [85]. In the case of atoms, Barbaroux, Farkas, Helffer, and Siedentop and Barbaroux, Esteban, and Séré related the Dirac-Fock equations to the Hartree-Fock equations of the electron-positron field in [15,14]. For the same model, Hainzl, Lewin, and Séré proved in [54,55] the existence of a minimizer and its uniqueness for the BDF model, but in contrast to [5] Hainzl et al. chose the projection on the free Dirac sea as a reference and extended their results later to atoms and molecules with small particle number and small coupling constants. Huber and Siedentop proved [62], in turn, that the Dirac-Fock equations for atoms possess solutions if, among other smallness conditions on coupling constants, the particle number N is such that the shells of the corresponding hydrogen-like Dirac operator are exactly filled.
Generalization of Lieb's Variational Principle: We come back to Lieb's argument which establishes his variational principle. The N-particle density matrix ρ av in (IV.1) results from averaging the pure quasifree density matrices |Φ(g (θ) ) Φ(g (θ) )| over all possible values of θ. Hence, there exists at least one choice of θ such that the energy expectation value Φ(g (θ) )| ÀΦ(g (Θ) ) of this pure quasifree density matrix is smaller or equal to the energy expectation Tr F [ρ av À] of ρ av .
Derezinski, Napiorkowski, and Solovej [32] and, simultaneously, Breteaux, Knoerr, Menge, and the author generalized this statement in [6] and demonstrated that the Bogoliubov-Hartree-Fock energy E BHF can be approximated by energy expectation values of pure quasifree density matrices to arbitrary accuracy. Moreover, as we additionally point out here, if E BHF is a minimum then there is also a pure quasifree density matrix among the minimizers.
The generalization does not only extend the set of density matrices, over which the Hartree-Fock energy functional is being varied, but also allows for any semibounded self-adjoint Hamiltonian with no additional repulsiveness assumption on the pair potential and not even on the form of the Hamiltonian, that it be a sum of a one-body term and a pair interaction.

VII Symmetries and Restricted
Hartree-Fock Approximation In this final section we discuss symmetries of the quantum system under consideration. In some part, we follow the work of Lieb, Solovej, and the author in [12], and we refer the reader for more details to that paper.
A family S of unitary operators U ∈ S is called a symmetry of À µ Given a symmetry S, we define the restricted Bogoliubov-Hartree-Fock (BHF) energy to be Obviously, E BHF (S) ≥ E BHF , and the approximation made by the restricted BHF energy is not better, and potentially worse, than the one without restriction. The importance of the restricted gHF approxmiation, however, lies in its improved accessability to explicit computation. Translation invariant generalized 1-pdm, for example, can be diagonalized by Fourier transform, or rotationally invariant generalized 1-pdm have a natural decomposition in terms of spherical harmonics.
• If E BHF (S) = E BHF then the symmetry S is called preserved.
• If E BHF (S) > E BHF then the symmetry S is called broken.
It turns out that both cases of preserved symmetry and broken symmetry occur in different models. The reason is the hidden concavity of Hartree-Fock functionals, which is used in the proof of Theorem 3 and which leads to instabilities at the minimum of the restricted functional. We discuss symmetries on various examples of physical interest.
Closed Shell Theorem in Unrestricted Hartree-Fock Theory and Rotation of Atoms: We first discuss rotation symmetry and come back to the Hartree-Fock approximation as originally introduced for atoms. The periodic table of the elements is usually described in terms of angular momentum shells, which contain the electron states. This picture implicitly assumes that the electron orbitals are eigenfunctions of the angular momentum operators L 2 and L z . Indeed, the Hamiltonian and the Hartree-Fock functional of an atom is invariant under rotations about the origin, where the atomic nucleus is located. Its minimizers, however, do generally not possess this rotational symmetry unless we study the restricted theory. Indeed, Griesemer and Hantsch show in [52] that the (unrestricted) HF minimizer becomes rotationally invariant for N electrons that fill up the lowest angular momentum shells (e.g., N = 2, 6, 10, 14, . . .), as Z ≫ N becomes sufficiently large. On the other hand, it is not difficult to see that, without restriction by symmetries, the HF minimizer of an atom with three electrons, say, and a small nuclear charge Z breaks rotational symmetry. This phenomenon is also reflected by the closed shell theorem in [11]: If a HF minimizer γ HF ∈ g (1) for a Coulomb system of N electrons exists, then it is the rank-N orthogonal projection onto the smallest N eigenvalues e 1 ≤ e 2 ≤ . . . The interpretation of this statement is that, in Hartree-Fock approximation, atoms and molecules never possess an open shell because the highest energy level is always fully occupied. In particular, rare earth elements with one loosely bound valence electron in a degenerate high momentum shell do no occur in Hartree-Fock theory. Therefore, the Hartree-Fock approximation for a single Lithium atom, say, does not yield orbitals which are products of a radial function, a spherical harmonic, and a spinor.

Particle Number Conservation:
The strongly continuous one-parameter group N = exp[−itAE] t∈Ê of unitary operators generated by the particle number operator AE is a symmetry of all Hamiltonians À of the form (II.8), as these conserve particle number.
In Section VI it is demonstrated that in case of a repulsive potential, choosing a vanishing pairing operator is always favorable for the energy minimization, and the particle number symmetry is always preserved. For attractive potentials, this is not always the case and, depending on the model, the particle number symmetry is preserved in some cases and broken in others.
Translation Invariance in Ê 3 : Three-dimensional systems are translation invariant, if (the resolvent of) À commutes with all U a ∈ T Ê 3 , where T Ê 3 = {U a | a ∈ Ê 3 } and U a = exp[−i a · Ô] is the translation by a ∈ Ê 3 . These translations are generated by is the second quantization of the momentum operator −i∂ ν in the ν th coordinate direction.
From a physics point of view, it would be desirable to define these translational invariant systems with the (single-fermion) configuration space Ê 3 as described above. This would necessitate general states, rather than density matrices, and ultimately require an operator algebraic framework which we cannot provide here.
Translation Invariance on a large Torus: To circumvent the problem related to the thermodynamic (i.e., infinite volume) limit it is customary to replace the configuration space Ê 3 by a torus Λ := (Ê/L ) 3 of large, but finite, sidelength L ≫ 1. The Hamiltonian À then commutes with translations U a = exp[−i a · Ô] by a ∈ Λ modulo L in T Λ := {U a | a ∈ Λ}. The resulting model is called Fermi Jellium or Fermi gas, and one is interested in the limit L → ∞ and in the energy per unit volume e gs := lim L→∞ {L −3 E gs }. As the ground state energy (at fixed µ) is an extensive quantity, so are the Bogoliubov-Hartree-Fock energy E BHF and the restricted Bogoliubov-Hartree-Fock energy E BHF (T Λ ). For this reason, we define the respective energies e BHF := lim L→∞ {L −3 E BHF } and e BHF (T Λ ) := lim L→∞ {L −3 E BHF (T Λ )} per unit volume.
More than fifty years ago, Overhauser considered the above model with a repulsive interaction, for which the pairing operator vanishes and the Bogoliubov-Hartree-Fock energy agrees with the original total Hartree-Fock energy. At high density, the paramagnetic state represented by a Slater determinant of plane waves occupying for both spin-up and spin-down electrons all momenta k ∈ Λ * below the Fermi energy, i.e., for which the dispersion ω(k) ≤ µ is below the chemical potential, is the natural tranlation invariant HF minimizer and yields e para := e BHF (T Λ ). He demonstrated in [82,83,84], however, that a lower energy e BHF < e BHF (T Λ ) is produced by Slater determinants which are not translation invariant but represent a spin wave. The precise Hartree-Fock minimizer breaking the translation invariance is not known explicitly, but in a recent paper [48] Gontier, Hainzl, and Lewin estimated the difference e BHF (T Λ ) − e BHF > 0 of the energies and proved that it is exponentially small in the interaction coupling. Thus, although the restricted HF energy is higher than the HF energy without restriction, the two terms agree to any order in powers of the coupling constant.
The BCS Model -Spin Invariance: We further introduce global spin transformations which rotate the spin variables 2 at each point in space by the same unitary transformation u ∈ SU (2). As the Hamiltonian is invariant under such global spin rotations, this defines an additional symmetry SU(2) of the system. (One variant of) The BCS model is now defined to be the restricted Bogoliubov-Hartree-Fock energy E BCS (T Λ × SU(2)). In the simplest model case, E BCS (T Λ × SU(2)) can be explicitly computed thanks to the restriction of the variation to translation-invariant generalized 1-pdm which are in the same spin singlet state at any point in Λ. Additionally choosing j to be complex conjugation in Fourier space, the Bogoliubov-Hartree-Fock energy functional is varied only over Γ (1) ∈ G (1) of the form whereγ ∈ L 1 (Λ * ; Ê + 0 ) andα ∈ L 2 (Λ * ) and Λ * = 2π L 3 . Inserting this into the energy functional at zero temperature gives where V , γ, and α are the inverse Fourier transform of V ,γ, andα, respectively. The potential V is assumed to be negative (attractive) for some part of Λ in order not to rule out nonvanishing α = 0 to begin with. This model and its variants, questions of existence and uniqueness of its minimizers, the characterization of the resulting minimizers by the BCS gap equation and the analysis of its solution for zero and positive temperatures have been analyzed and physically interpreted by Hainzl and Seiringer and others in a remarkable series of papers [53,43,57,56,44,22,23], see [58] for a review. Spin symmetry breaking and a phase transition between a ferromagentic and a paramagnetic phase has been recently proved for a Hartree-Fock model like (VII.2) under the additional assumption of the absence α = 0 of pairing, i.e., restriction to conserved particle numbers, by Gontier and Lewin in [49].
The Hubbard Model: The Hubbard model is a translation and spin invariant model on a finite-dimensional one-particle Hilbert space h = ℓ 2 (Λ) ⊗ 2 . We describe the most frequently studied case when Λ := ( /L ) d is the hypercubic d-dimensional periodic lattice and the kinetic energy is represented by a real nearest-neighbour hopping matrix T = t ⊗ 1, where t(x, y) = −1 if x, y ∈ Λ are neighbouring lattice sites, and t(x, y) = 0 otherwise. The pair potential V (x − y) = δ x,y is on-site only and coupled in by a coupling constant λ ≥ 0 which we here assume to be positive, so that BHF minimizers and HF minimizers agree. In spite of its simplicity, the physical properties of the Hubbard model change dramatically, as the model parameters µ and λ vary. A special model situation in which the Hubbard model possesses a rather large symmetry group is given at half-filling, when µ is chosen so that the density in the ground state equals 1, i.e., the number of lattice sites, N = |Λ|. At half-filling, the HF minimizer was explicitly determined by Lieb, Solovej, and the author in [12]. It turns out that both the translation symmetry and the spin symmetry are broken in this case: The HF minimizers γ HF exhibit antiferromagnetic order, i.e., v HF (x) := Tr 2 γ HF (x, x) σ = (−1) x ∆ e, where σ = (σ x , σ y , σ z ) ∈ 2×2 are the Pauli matrices, (−1) x is an alternating sign taking the value 1 on even sites and −1 on odd sites, e ∈ Ê 3 is a(n arbitrary) unit vector, and ∆ > 0 is a self-consistent gap parameter -much like the gap in BCS theory.
For a different choice of the chemical potential µ such that the density 0 < ρ = N/|Λ|≪1 is small and the interaction coupling λ ≫ 1 is large, the Hartree-Fock approximation restricted by the symmetry group S z generated by the spin-z operator S z = x∈Λ σ z was analyzed by Lieb, Travaglia, and the author in [13].
Here, the HF minimizer shows maximal ferromagnetic ordering in the sense that v HF (x) = ±η e 3 with η > 0 taking its maximally possible value. In other words, in the minimizing 1-pdm either all electrons have spin up or all electrons have spin down. In particular, the paramagnetic state is not energetically favorable in this situation.
Periodic structures: If the configuration space Ê 3 is again a torus Λ L := (Ê/L ) 3 for some large integer L ≫ 1 then the Hamiltonian often commutes only with integer translations a contained in the subgroup 3 L = ( /L ) 3 ⊂ Λ, leading to the symmetry S 3 L of À. A typical example is a system of the form À = + 1 2 Î with Î having the full translation symmetry S Λ but having only the smaller symmetry S 3 L due to the presence of a periodic external potential. The closed shell theorem described above does not only hold for Coulomb systems, but for general N fermion systems with a repulsive interaction potential (for which the generalized and the original Hartree-Fock approximation coincide). In general, it may fail, however, in case of restricted Hartree-Fock minimizers. For periodic systems, the existence of minmizer was established by Catto, Le Bris, and Lions in [26] where the 1-pdm are restricted to those which are invariant under (integral) lattice translations. Ghimenti and Lewin have later shown in [47] a kind of closed shell theorem and proved that the minimizer is a projection onto the smallest energies of the corresponding Hartree-Fock effective operator.