Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms
We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea).
We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant . Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to by the formula , where is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge . In the nonrelativistic limit, we obtain that this number is , recovering a result of Lieb.
This work is based on a series of papers by Hainzl, Lewin, Séré and Solovej on the mean-field approximation of no-photon QED.
Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms
Philippe GRAVEJAT, Mathieu LEWIN and Éric SÉRÉ
CEREMADE, UMR 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, FRANCE.
CNRS & Laboratoire de Mathématiques UMR 8088, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, FRANCE.
July 13, 2019
In this paper, we study a model of Quantum Electrodynamics (QED) allowing to describe the behavior of relativistic electrons in an external field and interacting with the virtual electrons of the Dirac sea, in a mean-field type theory. This work should be seen as the continuation of previous papers by Hainzl, Lewin, Séré and Solovej —, in which a more complicated model called Bogoliubov-Dirac-Fock (BDF) is considered. This project was mainly inspired of an important physical paper by Chaix and Iracane [6, 5] in which a model of the same kind was first proposed. We start by summarizing the physical motivation before defining the model properly.
Dirac introduced his operator in 1928  with the purpose to describe the behavior of relativistic electrons. It is defined as
where and are the Dirac matrices . The operator acts on . Contrary to the non-relativistic Hamiltonian , the operator is unbounded from below: . This property is known to be the basic explanation of various peculiar physical phenomena like the possible creation of electron-positron pairs or the polarization of the vacuum. The model that we shall study is a rough approximation of Quantum Electrodynamics but it is able to reproduce many of these physical phenomena. We refer to — for more details.
In QED, one can write a formal Hamiltonian acting on the usual fermionic Fock space, in Coulomb gauge and neglecting photons [15, Eq. (1)]. The mean-field approximation then consists in restricting formally this Hamiltonian to a special subclass of states in the Fock space, called the Hartree-Fock states. Any of these states is uniquely determined by its one-body density matrix which is a self-adjoint operator acting on . Often is an orthogonal projector. The QED energy then becomes a nonlinear functional in the variable , which can be formally written as follows
where for any operator acting on with kernel , is formally defined as . Recall acts on -spinors, i.e. is a complex hermitian matrix. The first term of (2) is the kinetic energy of the particles, whereas the second term describes the interaction with an external electrostatic field created by a smooth distribution of charge (describing for instance a system of classical nuclei). The last two terms account for the interaction between the particles themselves. We have chosen a system of units such that , and also such that the mass of the electron is normalized to 1. The constant (where is the bare charge of an electron) is a small number called the Sommerfeld fine-structure constant.
Expression (2) is purely formal: when is an orthogonal projector on , is never compact and none of the terms above makes sense a priori. However, it is possible to give a meaning to (2) by restricting the system to a box and imposing an ultraviolet cut-off. One can then study the thermodynamic limit, i.e. the behavior of the energy and of the minimizers when the size of the box goes to infinity (but the ultraviolet cut-off is fixed). This approach was the main purpose of .
The last two terms of (2) are respectively called the direct term and the exchange term. In theoretical studies of the Hartree-Fock model, the exchange term is sometimes neglected . The above energy then becomes (formally) convex, a very interesting simplification both from a theoretical and numerical point of view. Refined models exist: in relativistic density functional theory for instance, the exchange term is approximated by a function of the density and its derivatives only, see, e.g., the review . Neglecting the last term, one is led to consider the following reduced formal functional
As usual, one is interested in finding states having lowest energy, possibly in a specific subclass. In QED, a global minimizer in the Fock space is interpreted as being the vacuum, whereas other states (containing a finite number of real electrons for example) are obtained by assuming a charge constraint. When the external field vanishes () and for any values of the coupling constant , one easily proves that has a unique global minimizer which is the negative spectral projector of the free Dirac operator:
The precise mathematical statement is that when the system is restricted to a box of size with an ultraviolet cut-off , the above energy is well-defined; it has a unique minimizer
where is the Dirac operator acting on the box with periodic boundary conditions. The sequence converges (in a weak sense) to which is thus interpreted as the unique global minimizer of . If the exchange term is not neglected, the situation is more complicated and we refer to  where the thermodynamic limit was carried out.
The fact that is found to be the global minimizer of our formal energy is not physically surprising. This corresponds to the usual Dirac picture [7, 8, 9, 10] which consists in assuming that the vacuum should be seen as an infinite system of virtual particles occupying all the negative energy states of the free Dirac operator. Notice however that when the exchange term is taken into account, this picture is no longer valid: does not describe the free vacuum which is instead solution of a complicated translation-invariant nonlinear equation, see .
We want to emphasize the importance of the subtraction of half the identity in all the terms of the above energy (3). Indeed, the kernel of the translation-invariant operator is
If we assume that there is a cut-off in the Fourier domain, i.e. , it is then possible to compute the density
the Dirac matrices being trace-less. We therefore obtain that the free vacuum has no density of charge, which is comforting physically.
When the external field does not vanish, the main idea is then to subtract the (infinite) energy of the free vacuum to (3), in order to obtain a finite quantity. This yields the so-called (formal) reduced-Bogoliubov-Dirac-Fock energy (rBDF) which was already studied in  and is more easily expressed in terms of the difference ,
Note that we have used (4). What we have gained is that can now be a compact operator (it will indeed be Hilbert-Schmidt). We recall that is the density matrix of our Hartree-Fock state, hence it satisfies which translates on as .
A (formal) global minimizer of is interpreted as the polarized vacuum in the presence of the external density . Formally, it solves the self-consistent equation
In order to describe a physical system containing a finite number of real electrons, it is necessary to minimize the above energy not on the full class of states, but rather in a chosen charge sector, i.e. over states satisfying the formal charge constraint . Then a minimizer will satisfy the following equation
where is a Lagrange multiplier due to the charge constraint and interpreted as a chemical potential. The operator is a finite rank operator satisfying and . Notice the number does not need to be an integer as one may want to describe mixed states (in which case ).
We see that in both cases (minimization with or without a charge constraint), a minimizer always corresponds to filling energies of an effective Dirac operator up to some Fermi level . This corresponds to original ideas of Dirac. For the general BDF theory, the idea that one can have a bounded below functional whose minimizer satisfies this kind of equation was first proposed by Chaix and Iracane [6, 5].
In this paper, we shall prove that the range of ’s such that minimizers exist is an interval which contains both the charge of the polarized vacuum (the global minimizer of the energy, solution of (6)) denoted by , and . This proves the existence of neutral molecules and of positively charged molecules the charge of which is not too big, because in this case one has . This extends previous results proved for the BDF theory with the exchange term in : sufficient conditions were given for the existence of minimizers, but these conditions could only be checked in the nonrelativistic or the weak coupling limits. In the present paper, we shall also give interesting properties of a minimizer when it exists, and provide a bound on the maximal number of electrons which can be bound by a nucleus of charge , following ideas of Lieb .
The mathematical formulation and the proofs of the above statements are not straightforward.
The first (and main) difficulty is that we do not expect that a solution of Equations (6) or (7) is a trace-class operator. Indeed our results below will imply that in most cases it cannot be trace-class. This is a big problem as in the energy (3) the first term is expressed as a trace, as well as the total charge of the system which we formally wrote “” in the previous paragraphs. This issue was solved in  where it was proposed to generalize the trace functional and to define the trace counted relatively to the free vacuum as
As we shall see, any minimizer will have a finite so-defined -trace, which does not mean that is trace-class.
If we do not expect to be trace-class, there is a problem in defining the density of charge . Indeed it is known that in QED there are several divergences which need to be removed by means of an ultraviolet cut-off. In previous works —, a sharp cut-off was imposed: the space was replaced by its subspace consisting of functions that have a Fourier transform with support in the ball of radius . This allowed to give a solid mathematical meaning to the energy (5). In [12, 13], it was proved that the energy has a global minimizer , solution of (6). In , sufficient conditions were given on to ensure the existence of a ground state in the charge sector with the exchange term. They could only be checked in the nonrelativistic or the weak coupling limit.
In this paper, we propose other kinds of cut-offs which seem better for obtaining decay properties of the density of charge111A similar remark was made in  in the context of non-relativistic QED.. Essentially, they consist in replacing the Dirac operator by where is a smooth function growing fast enough at infinity. We call these cut-offs smooth in contrast to the previous sharp cut-off. But many of our results will also be valid in the sharp cut-off case.
Even with an ultraviolet cut-off, a minimizer will in general not be trace-class. But we shall be able to prove that anyway its density of charge is an function: . This information can then be used to prove the existence of all atoms and molecules which are either neutral or positively charged and do not have a too strong positive nuclear density. Also we shall prove a formula which relates the integral of and of the form
(see Theorem 4 for a precise statement depending on the chosen cut-off ). When , this proves that , hence cannot be trace-class.
The fact that a minimizer is not trace-class but its density is anyway an function can first be thought of as being a technical issue. But Equation (8) has a relevant physical interpretation. It means that the total observed charge is different from the real charge of the system. Hence the mathematical property that a minimizer is not trace-class is well interpreted physically in terms of charge renormalization. We even recover a standard charge renormalization formula in QED, see [19, Eq. ] and [17, Eq. ], although we use a simple model without photons and within the Hartree-Fock approximation with the exchange term removed.
As announced before, we shall prove in this paper that minimizers exist if and only if , an interval which contains both and the charge of the polarized vacuum. We shall also derive some bounds on and , assuming that the nuclear charge distribution is not too strong. Essentially we prove that is very small and that
In the nonrelativistic limit we recover the usual bound of the reduced Hartree-Fock model which can be obtained by a method of Lieb .
Acknowledgment. M.L. and E.S. acknowledge support from the ANR project “ACCQUAREL” of the French ministry of research.
2 Model and main results
In the whole paper, we denote by the usual Schatten class of operators acting on a Hilbert space and such that . We use the notation for any . A self-adjoint operator acting on is said to be -trace class  if and . We then define its -trace as
The space of -trace class operators on will be denoted by . We refer to  where important properties of this generalization of the trace functional are provided.
2.1 Ultraviolet regularization
It is well-known that in Quantum Electrodynamics a cut-off is mandatory [3, 17]. There are two sources of divergence in the Bogoliubov-Dirac-Fock model. The first is the negative continuous spectrum of the Dirac operator, which is cured by the subtraction of the (infinite) energy of the Dirac sea, as explained above. The second source of divergence is the rather slow growth of the Dirac operator for large momenta: only behaves linearly in at infinity222Notice a model similar to the reduced-BDF theory was recently studied for non-relativistic crystals in the presence of defects , in which case a cut-off is not necessary because of the presence of the Laplacian instead of ..
This can be cured by imposing a sharp cut-off on the space, i.e. by replacing by its subspace
However, when looking at decay properties of the electronic density, it might be more adapted to instead increase the growth of the Dirac operator at infinity. This means we replace by the operator
where grows fast enough at infinity. The operator is self-adjoint on with domain
We remark that the case of the sharp cut-off (9) formally corresponds to
and (or equivalently given by (11));
and satisfies the following properties:
Many of our results will be true under weaker assumptions on but we shall restrict ourselves to (12)–(14) for simplicity. We notice that under these assumptions, the spectrum of is the same as the one of :
Also the negative spectral projector of is the same as the one of :
In the whole paper, we shall consider perturbations of of the form where belongs to the so-called Coulomb space
Notice the dual space of is the Beppo-Levi space
2.2 Definition of the reduced-BDF energy
We recall that or depending on the chosen cut-off. We need to provide a correct setting for the rBDF energy. When , this was done in —. When , this is done similarly to the crystal case studied in . We introduce the following Banach space:
with associated norm
We notice that when and , one has as chosen in —. In the general case, we only have . We recall that is the dual of the space of compact operators acting on . Hence can be endowed with the associated weak- topology where in means that for any compact operator . Together with the fact that is a Hilbert space, this defines a weak topology on .
We also introduce the following convex subset of :
which is the closed convex hull of states of the form where is an orthogonal projector acting on . It is clear that is closed both for the strong and the weak- topology of . As we shall see, the reduced BDF energy will be coercive and weakly lower semi-continuous on .
Besides, the number can be interpreted as the charge of the system measured with respect to that of the unperturbed Dirac sea , see —. Note that the constraint in (21) is indeed equivalent [1, 12] to the inequality
and implies in particular that and for any .
We need to define the density of any state . When , this is easy as any has a smooth kernel (this is because the Fourier transform ). This property was used in – to properly define the density of charge. In the case where and , this is a bit more involved. The following is similar to [14, Lemma 1] and [4, Prop. 1] (we recall that was defined above in (15)):
Proposition 2 (Definition of the density for ).
Let . Then for any . Moreover there exists a constant (independent of and ) such that
Hence, there exists a continuous linear form which satisfies
for any and any . Eventually when , then where is the integral kernel of .
Let us now define the reduced Bogoliubov-Dirac-Fock (rBDF) energy. In the whole paper, we use the notation, for any ,
When , this coincides with the definition of the generalized trace introduced above. The rBDF energy reads:
where we recall that was defined in (16). In (24), is an external density which will be assumed to belong to . We use the notation . The energy is well-defined [12, 14] on the convex set . By (22), we have
this proves both that is bounded from below on ,
and that it is coercive for the topology of .
Since is convex on and weakly lower semi-continuous, it has a global minimizer , interpreted as the polarized vacuum in the presence of the external field created by the density . This was remarked in [13, Theorem 3]. Assuming that where
is the mean field operator, then one can adapt the proof of [13, Theorem 3] to get that is unique and is a solution of the nonlinear equation . The charge of the polarized vacuum is where
When is not too large [13, Eq. (15)], it was proved that . However in general electron-positron pairs can appear, giving rise to a charged vacuum. When , then does not have a unique global minimizer on , but it will be proved that is anyway a uniquely defined quantity.
2.3 Existence of minimizers with a charge constraint
We are interested in the following minimization problem
where the sector of charge is by definition
and is any real number. Of course in Physics but it is convenient to allow any real value. It will be proved below that is a Lipschitz and convex function. Notice that if is a global minimizer of on , then minimizes .
The existence of minimizers to (26) is not obvious: although is convex and weakly lower semi-continuous, and is itself a convex set, the linear form is not weakly continuous. Hence is not closed for the weak topology. Our main result is the following theorem, whose proof is given in Section 3.2 below.
Theorem 1 (Existence of atoms and molecules in the reduced BDF model).
is the largest interval on which is strictly convex. If , then for any . If , then for any ;
the interval contains both and the unique minimizer of ;
if , then has no minimizer in the charge sector ;
if , then has a minimizer in the charge sector . This minimizer is not a priori unique but its associated density is uniquely determined. It is radially symmetric if is radially symmetric. The operator satisfies the self-consistent equation
where is a Lagrange multiplier associated with the charge constraint and interpreted as a chemical potential, and satisfies and . If , then has a finite rank. If , then is trace-class.
Moreover, belongs to and satisfies
if and , and
if and .