In the last thirty years, the study of the quantum many-body problem has made tremendous progress, in particular for weakly interacting regimes where the validity of mean-field theory (or slightly more generally the quasi-free approximation) as an effective theory can be proved. In particular for bosonic systems the mathematical results have been very rich. Just to name some: in the beginning of the 2000s the Gross–Pitaevskii functional for the ground state energy of dilute Bose gases was derived [85, 88]. Later the time-dependent Gross–Pitaevskii equation was derived [46, 47]; bounds on the rate of convergence were obtained by [11, 31]. In 2011 validity of the quasi-free approximation for the excitation spectrum of Bose gases in the mean-field regime was proven [100], thus obtaining also the next-to-leading order of the ground state energy. In contrast, for dilute gases, the quasi-free approximation is not sufficient for obtaining the second order of the energy, although it can be used to derive the leading order with optimal rate of convergence [8, 10, 95]. Very recently, results going beyond the quasi-free approximation were obtained: the excitation spectrum for dilute Bose gases was derived [7, 9]; the Lee–Huang–Yang formula for the second order of the ground state energy was proven [57]; and nonlinear classical Gibbs measures were derived as an approximation at positive temperature [55, 84].
Compared to the development in the theory of bosonic systems, the mathematical progress in the derivation of effective theories for fermionic systems has been lagging behind. For fermions, the mean-field or quasi-free theory leads to the Hartree–Fock approximationFootnote 1 which is widely used in computational physics and chemistry. The validity of the Hartree–Fock approximation was established for the ground state energy of Coulomb systems in a number of seminal works [5, 56, 63]. Rigorous results taking this analysis beyond the quasi-free effective theory have been notably absent, except for a second-order bound [74] on the many-body correction (called correlation energy) to the ground state energy, inspired by [65, 70]. In the present paper we derive an optimal formula for the correlation energy.
Our proof is based on a non-perturbative framework which we started to develop in [23]. The central concept of our approach is that the dominant degrees of freedom are particle-hole pairs which are delocalized over patches on the Fermi surface in momentum space in such a way that they behave approximately as quasi-free bosons. In [23], by means of a trial state, we proved that the formula known as the random phase approximation (RPA) in physics is an upper bound to the correlation energy of a three-dimensional Fermi gas in the mean-field scaling regime (i. e., high density and weak interaction) with a regular interaction potential. In the present paper, we again start from the interacting many-body Hamiltonian and prove the matching lower bound, thus completely validating the random-phase approximation for the ground state energy of the three-dimensional Fermi gas in the mean-field scaling regime.
The problem of calculating corrections to the Hartree–Fock approximation has a long history in theoretical physics. Already in the early days of quantum mechanics the computation of the correlation energy was attempted using second order perturbation theory [69, 104] for a Fermi gas with Coulomb interaction (the electron gas); however, this approach leads to a logarithmically divergent expression due to the long range of the Coulomb potential. It was then noticed [90] that perturbation theory with Coulomb potential becomes even more divergent at higher orders and suggested that a resummation might cure this problem. Then in their seminal work [26], Bohm and Pines developed the RPA: they argued that the Hamiltonian can be partially transformed into normal coordinates which describe collective oscillations screening the long-range of the Coulomb potential, and thus leading to a better behaved perturbative expansion. However, they had to introduce additional bosonic collective degrees of freedom by hand. This was somewhat clarified by [98, 99], who showed that the collective modes can be understood as a superposition of particle-hole pair excitations. The formulation of the RPA due to Sawada et al. has in fact been an important inspiration for our work. Ultimately it was discovered that the RPA can be seen as a systematic partial resummation of perturbation theory; following this line, one even obtains a more precise result [59]. These works have been very influential in the establishment of theoretical condensed matter physics.
The particle-hole pair bosonization of Sawada et al. found application in many settings, for example to describe nuclear rotation and calculate moments of inertia of atomic nuclei [4, 94]. A bosonization method considering only the radial excitations of the Fermi surface was developed by [89]; similar methods applied to systems with square Fermi surface [58, 102]. Later, the bosonization of collective excitations of the Fermi surface became an important tool in the context of renormalization group methods [16, 34, 68, 72, 73]. The collective aspect was further emphasized in the operator-formalism by [33, 35]. In the functional-integral formalism [48, 49, 76,77,78, 80] bosonization was established as a Hubbard–Stratonovich transformation. Despite this popularity, difficulties in judging the quality of the bosonic approximation have been pointed out [79]: “For example, scattering processes that transfer momentum between different boxes on the Fermi surface and non-linear terms in the energy dispersion definitely give rise to corrections to the free-boson approximation for the Hamiltonian. The problem of calculating these corrections within the conventional operator approach seems to be very difficult.” As far as the mean-field scaling regime is concerned, with our result we quantify such corrections as being of subleading order.
A different mathematical approach to the fermionic many-body problem has been developed by employing rigorous renormalization group methods to construct convergent perturbative expansions. This allowed the construction of Gibbs states or ground states for two main classes of interacting fermionic models.
The first class concerns models in the Luttinger liquid universality class (which was first proposed by Haldane [66, 67]), such as interacting fermions or quantum spin chains in one dimension and some two-dimensional models. These models show universal properties agreeing with those of the Tomonaga–Luttinger model which is solvable in one dimension by an exact bosonization method [92]. These predictions of bosonization have been verified rigorously, starting from [16, 17] to [1, 14, 15, 20,21,22, 62, 93]; the proofs however are by detailed analysis of the fermionic theory instead of justifying directly the bosonization. One justification of a bosonization method was achieved by [6], showing equivalence of the massless sine-Gordon model for a special choice of the coupling constant and the massive Thirring model at the free fermion point.
The second class concerns fermions in two or three dimensions at low temperature. In this context, the use of sectors on the Fermi surface, very similar to the construction of patches we use, has been introduced in [53] for the program of proving existence of superconductivity [54]. There, bosonization was implemented as a Hubbard–Stratonovich transformation of sectorized collective excitations. While this ambitious program has not been completed, the sector method was later used to prove Fermi liquid behavior of fermions in two dimensions with uniformly convex Fermi surface at exponentially small positive temperatures (and non-Fermi liquid behavior for fermions with flat Fermi surface) [2, 3, 18, 44, 45, 96]. It furthermore lead to a proof of convergence for the zero-temperature perturbation theory in a special two-dimensional fermionic model with an asymmetry condition of the Fermi surface; this is a series of eleven papers an overview of which is given in [52]. Partial results have been obtained for fermions in three dimensions at positive temperature [43]. We see our approach, while sharing the ‘sectorization’ or ‘patches’ concept, as providing a complementary point of view on related physical problems, based on different, non-perturbative ideas.
Finally, our result should also be contrasted to the study of two-dimensional models that have been constructed to be exactly bosonizable. This goes back to a proposal of [91], who was motivated by high-temperature superconductivity. The analysis and variants of the model were developed by [39,40,41,42, 81, 82]. Furthermore, one may also see similarities (such as the limitation of the number of bosons that can occupy a single bosonic mode) in the bosonization concept to methods such as the Holstein–Primakoff map [12, 36, 37] for spin systems.
Many-body Hamiltonian in the mean-field regime
To describe N spinless fermionic particles on the torus \({\mathbb {T}}^3 := {\mathbb {R}}^3/(2\pi {\mathbb {Z}}^3)\), the Hilbert space is the space of totally antisymmetric \(L^2\)-functions of N variables in \({\mathbb {T}}^3\),
$$\begin{aligned} L^2_\text{ a }({\mathbb {T}}^{3N}) := \{ \psi \in L^2({\mathbb {T}}^{3N}) :&\psi (x_{\sigma (1)}, \ldots , x_{\sigma (N)})\nonumber \\&= {\text {sgn}}(\sigma ) \psi (x_1,\ldots ,x_N)\ \forall \sigma \in {\mathcal {S}}_N \}. \end{aligned}$$
(1.1)
The Hamiltonian is defined as the sum of Laplacians describing the kinetic energyFootnote 2 and a pair interaction, i. e., a multiplication operator defined using a function \(V:{\mathbb {R}}^3 \rightarrow {\mathbb {R}}\),
$$\begin{aligned} H_N := \hbar ^2 \sum _{i=1}^N \left( - \Delta _{x_i} \right) + \lambda \sum _{1 \le i < j \le N} V\left( x_i - x_j \right) . \end{aligned}$$
(1.2)
The positive parameters \(\hbar \) and \(\lambda \) adjust the strength of the kinetic energy and interaction operator, respectively.
In this paper, we assume the interaction potential V to be smooth. Thus the Hamiltonian is bounded from below and its self-adjointness follows from the Kato–Rellich theorem or using the Friedrichs extension. Here we are interested in the infimum of the spectrum (the ground state energy)
$$\begin{aligned} E_N := \inf {\text {spec}}\left( H \right) = \inf \Big \{\langle \psi , H_N \psi \rangle : \psi \in L^2_\text{ a }({\mathbb {T}}^{3N}),\ \Vert \psi \Vert _{L^2}=1 \Big \}. \end{aligned}$$
(1.3)
In full generality, the computation of \(E_N\) is clearly out of reach, simply because the model is too general: it may describe physical systems from superconductors to neutron stars. We thus need to be more specific and consider a particular case of the model, the most accessible case being a mean-field scaling regime: by considering a high density of particles we expect the leading order of the theory to be approximately described by an effective one-particle theory. We thus consider the limit of large particle number on the fixed-size torus. However, kinetic energy and interaction energy in typical states scale differently: the kinetic energy like \(N^{5/3}\) due to the Pauli exclusion principle, the interaction energy like \(N^2\) since there are \(N(N-1)/2\) interacting pairs. To have a chance of obtaining a non-trivial limit we choose to scale the parameters byFootnote 3
$$\begin{aligned} \hbar :=N^{-\frac{1}{3}} \quad \text{ and } \quad \lambda :=N^{-1} \quad \text{ with } N\rightarrow \infty . \end{aligned}$$
(1.4)
With this choice the kinetic energy and the interaction energy in typical states close to the ground state have the same order of magnitude (order N). This scaling regime couples a semiclassical scaling (\(\hbar = N^{-\frac{1}{3}}\rightarrow 0\)) and a mean-field scaling (coupling constant \(\lambda =N^{-1}\)).
If the interaction vanishes, \(V=0\), then the ground state of the system is exactly given by the Slater determinant (i. e., antisymmetrized tensor product) of plane waves
$$\begin{aligned} \psi _\text{ pw } = \bigwedge _{k \in B_\text{ F }} f_k, \qquad f_k(x) = (2\pi )^{-\frac{3}{2}} e^{ik\cdot x}\quad \text{ with } k \in {\mathbb {Z}}^3,\ x \in {\mathbb {T}}^3. \end{aligned}$$
(1.5)
Here the momenta k of the plane waves are chosen such that the expectation value of the kinetic energy operator is minimized. The set of the corresponding momenta \(B_\text{ F }\) is called the Fermi ball. For simplicity we assume that the ball is completely filled, namely we set
$$\begin{aligned} B_\text{ F }:= \{ k \in {\mathbb {Z}}^3: |k|\le k_\text{ F }\}, \end{aligned}$$
(1.6)
and then define the particle number accordingly as \(N := |B_\text{ F }|\). The limit of large particle number is then realized by considering \(k_\text{ F }\rightarrow \infty \). According to Gauss’ classic counting argument we have
$$\begin{aligned} k_\text{ F }= \kappa N^{\frac{1}{3}} \qquad \text{ for } \quad \kappa = \left( 3/4\pi \right) ^{\frac{1}{3}} + {\mathcal {O}}(N^{-1/3}). \end{aligned}$$
If the system is interacting, \(V\not = 0\), the ground state becomes a complicated superposition of Slater determinants. Nevertheless, in Hartree–Fock theory one minimizes only over the set of all Slater determinants. In our setting, the Hartree–Fock energy
$$\begin{aligned} E_N^\text{ HF }:= \inf \Big \{ \langle \psi , H_N \psi \rangle&: \psi =\bigwedge _{i=1}^N u_i \\&\text { with } \{u_i\}_{i=1}^N \text { an orthonormal family in } L^2({\mathbb {T}}^3) \Big \} \end{aligned}$$
is attained by the plane waves as in (1.5) and (1.6); see Appendix A for a proofFootnote 4. Thus in order to gain non-trivial information about the interacting system one must go beyond the Hartree–Fock theory.
Note that by the variational principle, the Hartree–Fock energy \(E_N^\text{ HF }\) is an upper bound to the ground state energy \(E_N\). It follows from the analysis of [5, 63] that Hartree–Fock theory also provides a good lower bound to the ground state energy. In our setting, the approach of [5, 63] shows that
$$\begin{aligned} E_N = E_N^\text{ HF } + o(1) \qquad \text{ as } N\rightarrow \infty . \end{aligned}$$
(1.7)
In particular, both \(E_N\) and \(E_N^\text{ HF }\) contain the Thomas–Fermi energy (in our scaling of order N) and the Dirac correction, also know as the exchange term (in our scaling of order 1).
From the physical point of view, Slater determinants are as uncorrelated as fermionic states (which have to satisfy the Pauli principle) can be, in the sense that they are just antisymmetrized tensor products. Due to the presence of the interaction, the true ground state will contain non-trivial correlations (i. e., it will be a superposition of Slater determinants). Therefore Wigner [104] called the difference
$$\begin{aligned} E_N - E_N^\text{ HF } \end{aligned}$$
the correlation energy. According to (1.7) we know that the correlation energy in our scaling is of size o(1) as \(N \rightarrow \infty \). In the present paper, we are going to determine the leading order of the correlation energy. It is of order \(\hbar = N^{-\frac{1}{3}}\) and given by the explicit formula predicted by the random-phase approximation, as obtained by [59, 90] based on a partial resummation of the perturbation series. We believe that our result is of importance as a rigorous step beyond mean-field theory into the world of interacting quantum systems. Our proof shows that the leading order of the correlation energy can be understood as the ground state energy of an effective quadratic Hamiltonian describing approximately bosonic collective excitations.
Main result
We write the interaction potential via its Fourier coefficients
$$\begin{aligned} V(x) = \sum _{k\in {\mathbb {Z}}^3} {\hat{V}}(k) e^{i k \cdot x}. \end{aligned}$$
Theorem 1.1
(Main Result) There exists a \(v_0 > 0\) such that the following holds true. Assume that \(\hat{V}: {\mathbb {Z}}^3 \rightarrow {\mathbb {R}}\) is compactly supported, non-negative, satisfies \({\hat{V}}(k)={\hat{V}}(-k)\) for all \(k\in {\mathbb {Z}}^3\), and \(\Vert {\hat{V}}\Vert _{\ell ^1}< v_0\). For every \(k_\text{ F }>0\) let the particle number be \(N := |\{ k\in {\mathbb {Z}}^3: |k|\le k_\text{ F }\}|\). Then as \(k_\text{ F }\rightarrow \infty \), the ground state energy of the Hamiltonian \(H_N\) in (1.2) with \(\hbar =N^{-1/3}\) and \(\lambda =N^{-1}\) is
$$\begin{aligned} E_N = E_N^\text{ HF } + E_N^\text{ RPA } + {\mathcal {O}}(\hbar ^{1+\frac{1}{16}}). \end{aligned}$$
(1.8)
Here the correlation energy \(E_N^\text{ RPA }\) is of order \(\hbar \) and, with \(\kappa _0 = \left( \frac{3}{4\pi }\right) ^{\frac{1}{3}}\), given by
$$\begin{aligned} E_N^\text{ RPA }= & {} \hbar \kappa _0 \sum _{k \in {\mathbb {Z}}^3} |k|\left( \frac{1}{\pi }\int _0^\infty \log \left[ 1+2\pi \kappa _0{\hat{V}}(k) \left( 1-\lambda \arctan \left( \lambda ^{-1}\right) \right) \right] {\text{ d }}\lambda \right. \nonumber \\&\qquad \qquad \qquad \quad \ \left. - \frac{\pi }{2}\kappa _0{\hat{V}}(k) \right) . \end{aligned}$$
(1.9)
The upper bound, \(E_N \le E_N^\text{ HF } + E_N^\text{ RPA } + {\mathcal {O}}(\hbar ^{1+\frac{1}{9}})\), was proved in [23], even without smallness condition on the potential. In the present paper we prove the lower bound. The smallness condition is technical, and we expect that the lower bound is also true without this condition.
As already explained in [23], by expanding (1.9) for small \({{\hat{V}}}\), we obtain
$$\begin{aligned} \frac{E_N - E_N^\text{ HF }}{\hbar }= & {} m\pi (1-\log (2)) \sum _{k\in {\mathbb {Z}}^3} |k||{{\hat{V}}}(k)|^2 \left( 1+{\mathcal {O}}({{\hat{V}}}(k))\right) \nonumber \\&\quad + {\mathcal {O}}(\hbar ^{1+\frac{1}{16}}). \end{aligned}$$
(1.10)
Thus we recover the result for the weak-coupling limit of [74]. Moreover, the leading order of the correlation energy of the jellium model as given by Gell-Mann and Brueckner [59, Eq. (19)] (see also [99, Eq. (37)] and [90]) when applied to the case of bounded compactly supported \({\hat{V}}\) agrees with (1.9).
Although some tools from the earlier papers [23, 74] will be useful for us, the proof of Theorem 1.1 requires several important new ingredients. Conceptually, our justification of the random phase approximation is based on the main input that at the energy scale of the correlation energy there are rather few excitations around the Fermi ball. For the upper bound in [23], we consider a trial state whose number of excited particles is of order 1, allowing to control most of error terms easily. However, for the lower bound, the best available estimate for the number of excited particles in a ground state is \({\mathcal {O}}(N^{\frac{1}{3}})\), thanks to a kinetic inequality from [74]. This weaker input breaks most of the error estimates in the upper bound analysis [23], and this is also the reason why a less precise lower bound was obtained in [74]. In fact, using only similar bounds to [23], we can at best show that the error terms are of the same order as the correlation energy. In the present paper, we go beyond that and complete the bosonization approach for the first time.
Let us quickly mention the most important new ingredients of the proof; a more detailed explanation will be given in Sect. 1.3.
-
A refined estimate for the number of bosonic particles. In [23], after removing the Fermi ball by a particle-hole transformation, we control the number of bosonic particles by the fermionic number operator \({\mathcal {N}}\). This is insufficient here, since the bound \(\langle {\mathcal {N}}\rangle \le CN^{\frac{1}{3}}\) mentioned above is too weak. It is natural to try to bound all error terms using the kinetic operator \({\mathbb {H}}_0\), but a serious problem is that \({\mathbb {H}}_0\) is not stable under the Bogoliubov transformation introduced later. Instead, we introduce the gapped number operator \({\mathcal {N}}_\delta \) in (5.6), which takes into account only the fermionic particles far from the Fermi surface and has a much better bound \(\langle {\mathcal {N}}_\delta \rangle \le CN^\delta \) with \(\delta >0\) small. Thus in practice, using \({\mathcal {N}}_\delta \) is as good as using the kinetic operator \({\mathbb {H}}_0\) in many estimates, with the advantage that \({\mathcal {N}}_\delta \) is stable under the Bogoliubov transformation (see Lemma 7.2). Since \({\mathcal {N}}_\delta \) involves the fermionic particles far from the Fermi surface, we have to control separately the contribution from particles close to the Fermi surface, using an improvement of the kinetic inequality in [74] (see Lemma 4.2). The latter issue does not appear in [23] since for an upper bound we can simply take a trial state without any contribution from particles close to the Fermi surface.
-
A refined linearization of the kinetic energy. Similarly to [23], the bosonization approach in the present paper is based on the construction of patches, which allows to linearize the fermionic kinetic operator \({\mathbb {H}}_0\) and relates it to a bosonic operator \({\mathbb {D}}_\text{ B }\). In [23], we prove that the expectation value of \({\mathbb {H}}_0-{\mathbb {D}}_\text{ B }\) against a well-chosen trial state is small, which requires that the number of patches is \(M\gg N^{\frac{1}{3}}\). In the present paper, we only control the commutator of \({\mathbb {H}}_0-{\mathbb {D}}_\text{ B }\) with bosonic pairs operators (see Lemma 8.2). This weaker bound is sufficient to ensure that \({\mathbb {H}}_0-{\mathbb {D}}_\text{ B }\) is essentially invariant under the Bogoliubov transformation (see Lemma 8.1), and importantly it requires only \(M\gg N^{2\delta }\) with \(\delta >0\) small. The possibility of taking a much smaller M is crucial to bound all error terms caused by the Bogoliubov transformation.
-
A refined control on the Bogoliubov kernel. Similarly to [23], we will diagonalize the bosonizable part of the Hamiltonian by a Bogoliubov transformation. In [23] we prove that the kernel of the Bogoliubov transformation is bounded uniformly in the Hilbert–Schmidt topology. This information is sufficient to estimate the error terms when \(\langle {\mathcal {N}}\rangle \sim 1\) (as in the trial state used for the upper bound), but it is insufficient now that there are potentially many excitations. In the present paper, we will derive an optimal bound for the matrix elements of the Bogoliubov kernel (see Lemma 6.1). The new estimate encodes that due to the geometry of the Fermi surface, the interaction energy vanishes at the same rate as the kinetic gap closes. This bound is crucial for improving error estimates involving the Bogoliubov transformation (see Lemma 7.1), especially for controlling the non-bosonizable terms.
-
A subtle analysis of the non-bosonizable terms. As explained in [23], the contribution of the non-bosonizable terms can be controlled by \(N^{-1}\langle {\mathcal {N}}^{2}\rangle \). The trial state in [23] satisfies \(\langle {\mathcal {N}}^{2}\rangle \sim 1\), and hence the non-bosonizable terms are much smaller than the correlation energy. In the present paper, we only know that \(\langle {\mathcal {N}}\rangle \le CN^{\frac{1}{3}}\), which is not enough to rule out the possibility that the non-bosonizable terms are comparable to the correlation energy. It turns out that controlling the non-bosonizable terms is highly nontrivial since these terms couple the bosonic degrees of freedom with the uncontrolled low-energy fermions. Our idea is to bound these terms from below by the kinetic operator. Technically, it is easy to establish the lower bound \(-C \Vert {{\hat{V}}}\Vert _{\ell ^1} {\mathbb {H}}_0\) by completing a square. However, the difficulty here is that we have to validate this bound after implementing the Bogoliubov transformation (see Lemma 9.1). Handling the non-bosonizable terms requires a subtle analysis, using the refined estimate on the Bogoliubov kernel and the smallness assumption on the interaction potential.
-
Analysis of the diagonalized effective Hamiltonian. After implementing the Bogoliubov transformation, we obtain the desired correlation energy plus \({\mathbb {H}}_0-{\mathbb {D}}_\text{ B } + {\mathbb {K}}\) where \({\mathbb {K}} = \sum _{k\in \Gamma ^{\text{ nor }}} \sum _{\alpha ,\beta \in {\mathcal {I}}_{k}} 2\hbar \kappa |k|{\mathfrak {K}}(k)_{\alpha ,\beta } c_\alpha ^*(k) c_\beta (k)\) is the diagonalized effective Hamiltonian. Here \({\mathbb {H}}_0-{\mathbb {D}}_\text{ B }\) remained since it is essentially invariant under the Bogoliubov transformation. For the upper bound in [23], the term \({\mathbb {K}}\) does not cause any problem since its expectation value in the vacuum state is 0. In the present paper, however, we have to bound it from below as an operator (see (10.16)). This task is nontrivial and we have to use again the refined estimate on the Bogoliubov kernel and the smallness assumption on the innteraction potential.
In summary, in the present paper we provide a complete and unified bosonization approach which can handle the states with a lot of low-energy excitations. We believe that our approach is of general interest and could be useful in other contexts.
We also see our result as a possible starting point for further investigations. For example, our bosonization method is general enough to derive a norm approximation on the many-body dynamics [24]. Many questions remain; given the historical context of the problem, maybe most importantly the extension to Coulomb interaction, i. e., the electron gas, at least in some coupled mean-field/large-volume limit, requiring to optimize our bounds for extensivity. Of course, to reach this goal, we would first need to remove the small-potential condition, which at the moment plays a central role. The next key task is to deal with the divergence at small k which appears in the higher orders of perturbation theory. As the small-k singularity is improved to a logarithmic singularity in (1.9), we believe that the bosonization method contains intrinsically the necessary “resummation” that is responsible for this screening of the potential. Of course, hard technical refinements, e. g., in optimizing the k-dependence of our estimates will be necessary. Another question concerns the low-energy spectrum of the Hamiltonian: it is believed that a collective plasmon mode can be isolated from the bosonized excitation spectrum, realizing a theory of electrons dressed by a cloud of excitations and supporting the screening concept. Within the bosonic approximation, the emergence of the plasmon mode has been discussed in [13]. We expect that through a detailed analysis of the spectrum, the screening of the Coulomb potential, and the properties of the approximate ground state, the bosonization method may support the future development of a rigorous, non-perturbative Fermi liquid theory.
Beyond the mean-field scaling regime and the electron gas, there are other systems of physical interest: for example the helium isotope \({}^3\text{ He }\) is fermionic and has short-range isotropic interactions. Furthermore, a high-density limit is particularly important in the description of atomic nuclei; the short-range interactions there are however spin- and isospin-dependent and anisotropic and furthermore have attractive parts. We conjecture that even with attractive potentials the RPA formula for the correlation energy applies as long as the logarithm in \(E^\text{ RPA}_N\) does not become ill-defined. In our scaling, we do not see any contribution from the pairing density related to superconductivity, but one may expect that even if it was non-vanishing, its effect on the energy may be exponentially small. One may speculate that in an appropriate scaling limit the state of a superconductor might be described using a product of a particle-hole pair Bogoliubov transformation as we construct it for the normal phase, times a BCS-type fermionic Bogoliubov transformation.
Sketch of the proof
We will use the Fock space formalism. Recall the fermionic Fock space
$$\begin{aligned} {\mathcal {F}}:= \bigoplus _{n =0 }^\infty L^2_\text{ a }({\mathbb {T}}^{3n}) = {\mathbb {C}} \oplus L^2({\mathbb {T}}^{3}) \oplus L^2_\text{ a }(({\mathbb {T}}^{3})^2) \oplus \cdots \end{aligned}$$
(1.11)
The vector
$$\begin{aligned} \Omega := (1,0,0,\ldots ) \in {\mathcal {F}}\end{aligned}$$
is called the vacuum. For \(\psi = (\psi ^{(0)}, \psi ^{(1)}, \psi ^{(2)}, \ldots ) \in {\mathcal {F}}\) and \(f\in L^2({\mathbb {T}}^3)\) we define the creation operators \(a^*(f)\) and the annihilation operators a(f) by their actions
$$\begin{aligned}&\left( a^*(f) \psi \right) ^{(n)}(x_1,\ldots ,x_n) \\&\quad := \frac{1}{\sqrt{n}} \sum _{j=1}^n (-1)^{j-1} f(x_j) \psi ^{(n-1)} (x_1, \ldots , x_{j-1}, x_{j+1},\ldots ,x_n),\\&\left( a(f)\psi \right) ^{(n)}(x_1,\ldots , x_n) := \sqrt{n+1} \int _{{\mathbb {T}}^3} {\text{ d }}x \overline{f(x)} \psi ^{(n+1)}(x,x_1,\ldots ,x_n). \end{aligned}$$
Since we will work in the discrete momentum space (Fourier space) \({\mathbb {Z}}^3\), it is convenient to write
$$\begin{aligned} a^*_p := a^*(f_p), \quad a_p := a(f_p), \quad \text{ where } f_p(x) = (2\pi )^{-\frac{3}{2}} e^{ip\cdot x} \text{ for } p\in {\mathbb {Z}}^3. \end{aligned}$$
These operators satisfy the canonical anticommutator relations (CAR)
$$\begin{aligned} \{a_p,a^*_{q}\} = \delta _{p,q}, \quad \{a_p,a_q\} =0 = \{a^*_p,a^*_q\} \quad \forall p,q \in {\mathbb {Z}}^3. \end{aligned}$$
(1.12)
The Hamiltonian \(H_N\) in (1.2), originally defined on the N-particle sector \(L^2_\text{ a }(({\mathbb {T}}^{3})^N) \subset {\mathcal {F}}\), can be lifted to an operator on the fermionic Fock space as
$$\begin{aligned} {\mathcal {H}}_N = \hbar ^2 \sum _{p\in {\mathbb {Z}}^3} |p|^2 a^*_p a_p + \frac{1}{2N} \sum _{k,p,q\in {\mathbb {Z}}^3} {{\hat{V}}}(k) a^*_{p+k} a^*_{q-k} a_{q} a_{p}. \end{aligned}$$
(1.13)
Restricted to \(L^2_\text{ a }(({\mathbb {T}}^{3})^N) \subset {\mathcal {F}}\), \({\mathcal {H}}_N\) agrees with the Hamiltonian as given in (1.2).
Correlation Hamiltonian. Now we separate the degrees of freedom described by the Slater determinant of plane waves in (1.5) from non-trivial quantum correlations. Recall the Fermi ball and its complement
$$\begin{aligned} B_\text{ F }:= \{ p \in {\mathbb {Z}}^3: |p|\le k_\text{ F }\}, \quad B_\text{ F}^c:= {\mathbb {Z}}^3 \setminus B_\text{ F }. \end{aligned}$$
We define the particle-hole transformation \(R: {\mathcal {F}}\rightarrow {\mathcal {F}}\) by
$$\begin{aligned} R^* a^*_p R = \left\{ \begin{matrix}{} a^*_p&{}\quad \text{ for } p \in B_\text{ F}^c\\ a_p &{}\quad \text{ for } p \in B_\text{ F }\end{matrix}\right. , \qquad R\Omega := \bigwedge _{p \in B_\text{ F }} f_p. \end{aligned}$$
(1.14)
This map is well-defined since vectors of the form \(\prod _{j}a^*_{k_j} \Omega \) constitute a basis of \({\mathcal {F}}\). Moreover, it is easy to verify that \(R = R^* = R^{-1}\); in particular R is a unitary transformation. (In fact, R is an example of a fermionic Bogoliubov transformation.)
In practice, the action of R on an operator on Fock space is easily computed using the rules (1.14) and the CAR (1.12). For example, consider the particle number operator
$$\begin{aligned} {\mathcal {N}}:= \sum _{p\in {\mathbb {Z}}^3} a^*_p a_p. \end{aligned}$$
For \(\psi = (\psi ^{(0)}, \psi ^{(1)}, \ldots ) \in {\mathcal {F}}\) we have \({\mathcal {N}}\psi = (0, \psi ^{(1)}, 2\psi ^{(2)}, 3 \psi ^{(3)}, \ldots )\); in particular \({\mathcal {N}}\psi = N \psi \) is equivalent to the vector belonging to the N-particle sector of Fock space, \(\psi \in L^2_\text{ a }(({\mathbb {T}}^{3})^N) \subset {\mathcal {F}}\). Now
$$\begin{aligned} \begin{aligned} R^* {\mathcal {N}}R&= \sum _{h \in B_\text{ F }} a_h a^*_h + \sum _{p \in B_\text{ F}^c} a^*_p a_p = \sum _{h \in B_\text{ F }} \left( 1 - a^*_h a_h \right) + \sum _{p \in B_\text{ F}^c} a^*_p a_p\\&= N + \sum _{p \in B_\text{ F}^c} a^*_p a_p - \sum _{h \in B_\text{ F }} a^*_h a_h =: N + {\mathcal {N}}^\text{ p } - {\mathcal {N}}^\text{ h }. \end{aligned} \end{aligned}$$
(1.15)
This identity implies that if \(R\psi \) is a N-particle state, then
$$\begin{aligned} ({\mathcal {N}}^\text{ p } - {\mathcal {N}}^\text{ h}) \psi =0, \end{aligned}$$
(1.16)
namely after the transformation R the number of particles is equal to the number of holes.
The transformed Hamiltonian \(R^* {\mathcal {H}}_N R \) has been computed in [27,28,29,30, 32], in a slightly different way for mixed states in [19], and in the context of the correlation energy in [23, 74]. Let us therefore just give a short sketch of the transformation of the interaction term; the transformation of the kinetic term uses (1.16) but is otherwise very similar to (1.15). We start by using the CAR once to write
$$\begin{aligned}&\frac{1}{2N} \sum _{k,p,q\in {\mathbb {Z}}^3} {\hat{V}}(k) a^*_{p+k} a^*_{q-k} a_{q} a_{p} \nonumber \\&\quad = \frac{1}{2N} \sum _{k \in {\mathbb {Z}}^3} {\hat{V}}(k) \rho (k) \rho (-k) - \frac{1}{2N}\sum _{k \in {\mathbb {Z}}^3} {\hat{V}}(k) {\mathcal {N}}, \end{aligned}$$
(1.17)
where we introduced
$$\begin{aligned} \rho (k) := \sum _{p \in {\mathbb {Z}}^3} a^*_{p+k} a_p . \end{aligned}$$
The second summand of (1.17) equals \(-\frac{1}{2}\sum _{k\in {\mathbb {Z}}^3}{\hat{V}}(k)\), which contributes to the Hartree–Fock energy. For the transformation of the first summand one computes
$$\begin{aligned} R^* \rho (k) R = {\mathfrak {D}}(k)^* + b^*(k) + b(-k) + N \delta _{k,0}, \end{aligned}$$
where we have introduced for any \(k\in {\mathbb {Z}}^3\) the particle-hole pair creation operatorFootnote 5
$$\begin{aligned} b^*(k) := \sum _{p \in B_\text{ F}^c\cap (B_\text{ F }+k)} a^*_p a^*_{p-k} \end{aligned}$$
(1.18)
and the non-bosonizable operator
$$\begin{aligned} {\mathfrak {D}}(k)^* := \sum _{p \in B_\text{ F}^c\cap (B_\text{ F}^c+k)} a^*_{p} a_{p-k} - \sum _{h \in B_\text{ F }\cap (B_\text{ F }-k)} a^*_{h} a_{h+k} . \end{aligned}$$
(1.19)
Note that \({\mathfrak {D}}(k)^* = {\mathfrak {D}}(-k)\) and \({\mathfrak {D}}(0)^* = {\mathcal {N}}^\text{ p } - {\mathcal {N}}^\text{ h }\). Observing that the constant terms (i. e., not containing any creation or annihilation operator) contribute to the Hartree–Fock energy \(E^\text{ HF}_N\) and collecting all quadratic terms in the operator \({\mathbb {X}}\), we arrive at the result
$$\begin{aligned} {\mathcal {H}}_\text{ corr }:= R^* {\mathcal {H}}_N R - E^\text{ HF}_N = {\mathbb {H}}_0 + Q_\text{ B }+ {\mathcal {E}}_1 + {\mathcal {E}}_2 + {\mathbb {X}}\end{aligned}$$
(1.20)
where the summands are given by
$$\begin{aligned} {\mathbb {H}}_0&:= \sum _{k \in {\mathbb {Z}}^3} e(k) a^*_k a_k \quad \text{ with } \text{ dispersion } \text{ relation } e(k) := |\hbar ^2 |k|^2 - \kappa ^2 |, \nonumber \\ Q_\text{ B }&:= \frac{1}{N} \sum _{k \in \Gamma ^{\text{ nor }}} {\hat{V}}(k) \Big [ b^*(k) b(k) + b^*(-k) b(-k) \nonumber \\&\qquad \qquad \qquad \qquad \qquad + b^*(k) b^*(-k) + b(-k) b(k) \Big ], \nonumber \\ {\mathcal {E}}_1&:= \frac{1}{2N} \sum _{k \in \Gamma ^{\text{ nor }}} {\hat{V}}(k) \Big [ {\mathfrak {D}}(k)^* {\mathfrak {D}}(k) + {\mathfrak {D}}(-k)^* {\mathfrak {D}}(-k)\Big ], \nonumber \\ {\mathcal {E}}_2&:= \frac{1}{N} \sum _{k \in \Gamma ^{\text{ nor }}} {\hat{V}}(k) \Big [ {\mathfrak {D}}(k)^* b(k) + {\mathfrak {D}}(-k)^* b(-k) + \text{ h.c. }\Big ], \nonumber \\ {\mathbb {X}}&:= - \frac{1}{2N} \sum _{k \in {\mathbb {Z}}^3} {\hat{V}}(k) \bigg [ \sum _{p \in B_\text{ F}^c\cap (B_\text{ F }+k)} a^*_p a_p + \sum _{h \in B_\text{ F }\cap (B_\text{ F}^c-k)} a^*_{h} a_{h}\bigg ]. \end{aligned}$$
(1.21)
Note that we have introduced the set \(\Gamma ^{\text{ nor }}\) of all momenta \(k=(k_1,k_2,k_3)\) in \({\mathbb {Z}}^3 \cap {\text {supp}}{{\hat{V}}}\) satisfying
$$\begin{aligned} k_3> 0\,\, \text {or}\,\,(k_3=0\,\,\text {and}\,\,k_2>0)\,\,\text {or}\,\,(k_2=k_3=0 \text { and } k_1>0). \end{aligned}$$
This set is chosen such that
$$\begin{aligned} \Gamma ^{\text{ nor }}\cap (- \Gamma ^{\text{ nor }}) =\emptyset , \quad \Gamma ^{\text{ nor }}\cup (- \Gamma ^{\text{ nor }}) =\Big ( {\mathbb {Z}}^3 \cap {\text {supp}}{{\hat{V}}} \Big ) \setminus \{0\}. \end{aligned}$$
The term \(Q_\text{ B }\) is the bosonizable part of the interaction and contains only the pair operators. The term \({\mathcal {E}}_1\) is purely non-bosonizable and \({\mathcal {E}}_2\) couples bosonizable and non-bosonizable excitations. Note that unlike the other terms \({\mathcal {E}}_1\) is not normal-ordered (this choice is made so that we have \({\mathcal {E}}_1 \ge 0\)); for this reason \({\mathbb {X}}\) and \({\mathcal {E}}_1\) differ slightly from the expressions given in [23].
Since \({\mathbb {X}}\) is quadratic in fermionic operators, it can be easily bounded using \({\mathcal {N}}/N\), which will be seen to have expectation value much smaller than the order \(\hbar \) of \(E_N^\text{ RPA }\).
In [23], it was proved that \({\mathbb {H}}_0+Q_\text{ B }\) evaluated in a trial state of quasi-free particle-hole pairs gives rise to \(E_N^\text{ RPA }\) as an upper bound to the correlation energy. Accordingly, an important part of our task will be to prove that the contribution from \({\mathcal {E}}_1+{\mathcal {E}}_2\) is negligible. (Whereas this was easily achieved for the upper bound using the explicit form of the trial state, for the lower bound it actually turns out to be a major challenge.)
The rest of the paper is devoted to the proof of the inequality
$$\begin{aligned} \inf _{\begin{array}{c} \psi \in {\mathcal {F}}:\Vert \psi \Vert =1,\\ ({\mathcal {N}}^\text{ p } - {\mathcal {N}}^\text{ h})\psi =0 \end{array}} \langle \psi , {\mathcal {H}}_\text{ corr } \psi \rangle \ge E_N^\text{ RPA } + {\mathcal {O}}(\hbar ^{1+\frac{1}{16}}) . \end{aligned}$$
(1.22)
Thanks to (1.20) it directly implies the main result, the lower bound in Theorem 1.1.
In the following we explain the key estimates in our proof. We use the symbol C for positive constants that may change from line to line, but are independent of N, \(\hbar \), and M (the number of patches, to be introduced in (1.32)). The constants C may depend on the momentum k, which does not play a role ultimately since we only consider the finitely many \(k \in {\text {supp}}{\hat{V}}\), i. e., we can always take the maximum and so treat all constants as independent of k. We generally absorb any dependence on \({\hat{V}}\) in the constants C; we only write the \({\hat{V}}\)-dependence of estimates explicitly where the smallness condition on \(\Vert {\hat{V}}\Vert _{\ell ^1}\) plays a role.
A priori estimates. Similarly to [23, 74], many approximations used in our approach are based on the idea that the relevant quantum states have only few excitations. For the upper bound in [23], this fact is easily justified by the strong bound \(\langle \Psi _\text{ trial }, {\mathcal {N}}^m \Psi _\text{ trial }\rangle \le C_m\) (for all \(m \in {\mathbb {N}}\)) for the trial state used to compute the expectation value of \({\mathcal {H}}_\text{ corr }\). Compared to that bound, for the ground state we can only derive weaker estimates. In Lemma 2.4 we prove that the particle number operator can be controlled by the kinetic energy (i. e., the kinetic energy operator has a tiny gap, of order \(\hbar ^2\)) by
$$\begin{aligned} {\mathcal {N}}\le 2 N^{\frac{2}{3}} {\mathbb {H}}_0. \end{aligned}$$
(1.23)
To avoid the particle number operator, where possible we bound pair operators directly by the kinetic energy, using an inequality from [74],
$$\begin{aligned} \sum _{p \in B_\text{ F}^c\cap (B_\text{ F }+k)} \Vert a_p a_{p-k} \psi \Vert \le CN^{\frac{1}{2}} \Vert {\mathbb {H}}_0^{1/2} \psi \Vert \quad \forall \psi \in {\mathcal {F}}. \end{aligned}$$
(1.24)
(The idea of directly using the kinetic energy for bounds has appeared already in [65, 70] in the context of rigorous second order perturbation theory.) The bounds (1.24) and (1.23) imply the rough estimates in Lemma 2.1, as in [74]:
$$\begin{aligned} \frac{1}{2} ({\mathbb {H}}_0 +{\mathcal {E}}_1) - \hbar \le {\mathcal {H}}_\text{ corr } \le 2 ({\mathbb {H}}_0 +{\mathcal {E}}_1+ \hbar ). \end{aligned}$$
(1.25)
Together with an upper bound of order \(\hbar \) such as the trivial variational one obtained using the trial state \(\Omega \) (corresponding to the Slater determinant of plane waves before the particle-hole transformation), this implies that the ground state \(\psi _\text{ gs }\) of \({\mathcal {H}}_\text{ corr }\), the minimizer of the expectation value on the left hand side of (1.22), satisfies
$$\begin{aligned} \langle \psi _\text{ gs }, ({\mathbb {H}}_0+{\mathcal {E}}_1) \psi _\text{ gs } \rangle \le C\hbar , \quad \langle \psi _\text{ gs }, {\mathcal {N}}\psi _\text{ gs } \rangle \le CN^{\frac{1}{3}}. \end{aligned}$$
(1.26)
For technical reasons, we will also need to control the expectation of higher powers of \({\mathcal {N}}\), which does not follow from (1.24) and (1.23). To overcome this difficulty, in Lemma 3.1 we replace the ground state \(\psi _\text{ gs }\) by an approximate ground state \(\Psi \) satisfying
$$\begin{aligned} \langle \Psi , ({\mathbb {H}}_0 +{\mathcal {E}}_1) \Psi \rangle \le C\hbar , \quad \Psi = \mathbb {1} ({\mathcal {N}}\le CN^{\frac{1}{3}})\Psi \end{aligned}$$
(1.27)
while its energy is still close to the ground state energy, i. e.,
$$\begin{aligned} \langle \psi _\text{ gs }, {\mathcal {H}}_\text{ corr } \psi _\text{ gs }\rangle \ge \langle \Psi , {\mathcal {H}}_\text{ corr } \Psi \rangle - CN^{-1}. \end{aligned}$$
This is achieved by using the technique of localizing particle number on Fock space, which goes back to Lieb and Solovej [87]. In the proof we will use the formulation from [86, Proposition 6.1]. It is the state \(\Psi \) that most of our subsequent analysis will be applied to.
Approximately bosonic creation operators. When applied to states with few excitations, the pair creation operators behave approximately as bosonic creation operators, namely we have to leading order the canonical commutator relations (CCR)
$$\begin{aligned}{}[b^*(k),b^*(l)] = 0, \quad [b(k),b^*(l)] \simeq \delta _{k,l} \times \text{ const } \quad \forall k,l \in {\mathbb {Z}}^3. \end{aligned}$$
(1.28)
Unfortunately there is no expression for the kinetic energy \({\mathbb {H}}_0\) in terms of the \(b^\natural (k)\)-operatorsFootnote 6. We take inspiration from the solution of the Luttinger model [92]: if the dispersion relation were linear, the \(b^*(k)\) would create eigenvectors of \({\mathbb {H}}_0\). Since the dispersion relation \(\hbar ^2 |k|^2\) is not linear, we will linearize it locally. This is achieved by localizing the creation operators to patches on the Fermi surface. More precisely, we cut the shell of width \(R_{{\hat{V}}} := {\text {diam}}{\text {supp}}{\hat{V}}\) around the Fermi surface into patches \(\{B_\alpha \}_{\alpha =1}^M\). The construction of the patches is recalled in Sect. 4. As discussed in the introduction, under the name of “sectors”, this idea has already been employed in the rigorous renormalization group context.
We consider the pair excitations supported in each patchFootnote 7
$$\begin{aligned} b^*_\alpha (k) := \frac{1}{m_\alpha (k)} \sum _{\begin{array}{c} p:p \in B_\text{ F}^c\cap B_\alpha \\ p-k\in B_\text{ F }\cap B_\alpha \end{array}} a^*_p a^*_{p-k} . \end{aligned}$$
(1.29)
To normalize the constant in the approximate CCR, the normalization constant \(m_\alpha (k)\) should be chosen such that \(\Vert b^*_\alpha (k)\Omega \Vert =1\), namely
$$\begin{aligned} m_\alpha ^2(k) = \sum _{\begin{array}{c} p:p \in B_\text{ F}^c\cap B_\alpha \\ p-k\in B_\text{ F }\cap B_\alpha \end{array}} 1. \end{aligned}$$
(1.30)
This has the meaning of the number of particle-hole pairs \((p,h) \in B_\text{ F}^c\times B_\text{ F }\) inside the patch \(B_\alpha \) with relative momentum \(p-h=k\). However, this number may be zero! In fact, if \(k\cdot {\hat{\omega }}_\alpha <0\) with \({\hat{\omega }}_\alpha \) the unit vector pointing in the direction of the patch \(B_\alpha \), then a simple geometric consideration shows that the summation domain in (1.30) and (1.29) is empty (the condition \(k\cdot {\hat{\omega }}_\alpha <0\) is incompatible with \(p \in B_\text{ F}^c\) and \(p-k \in B_\text{ F }\)). The same problem occurs for \(m_\alpha ^2(-k)=0\) if \(k\cdot {\hat{\omega }}_\alpha >0\).
Furthermore, as suggested by [97, Chapters 8, 9.2.3, and 9.2.4] and [38], bosonization is expected to be a good approximation only if \(m_\alpha (k)\) is large. This cannot be ensured for patches where \(k \cdot {\hat{\omega }}_\alpha \approx 0\) (if we think of the direction of k as defining the north pole of the Fermi ball, these are the patches near the equator). However, the momentum k of such excitations is almost tangential to the Fermi surface and thus their energy is very low. In fact, we will be able to show that their contribution to the ground state energy is small and exclude them from the bosonization. To do so, we introduce a cut-off near the equator by defining the index subset \({\mathcal {I}}_{k}= {\mathcal {I}}_{k}^{+}\cup {\mathcal {I}}_{k}^{-}\) where
$$\begin{aligned} \begin{aligned} {\mathcal {I}}_{k}^{+}&:= \left\{ \alpha \in \{1,2,\ldots , M\} : k \cdot {\hat{\omega }}_\alpha \ge N^{-\delta } \right\} , \\ {\mathcal {I}}_{k}^{-}&:= \left\{ \alpha \in \{1,2,\ldots , M\} : k\cdot {\hat{\omega }}_\alpha \le - N^{-\delta } \right\} . \end{aligned} \end{aligned}$$
(1.31)
We will choose the cut-off parameter \(\delta \) and the number of the patches M such that
$$\begin{aligned} N^{2\delta } \ll M \ll N^{\frac{2}{3}-2\delta }, \quad 0<\delta <\frac{1}{6}. \end{aligned}$$
(1.32)
(Eventually we will choose \(M = N^{4\delta }\) and \(\delta =\frac{1}{24}\).) Note that unlike [23] where we require \(M\gg N^{\frac{1}{3}}\), here we allow a much smaller value of M, which is important to control the error terms due to the Bogoliubov transformation introduced later.
Then by [23, Proposition 3.1], the constant
$$\begin{aligned} n_\alpha (k):=\left\{ \begin{array}{cc} m_\alpha (k) &{}\quad \text{ for } \alpha \in {\mathcal {I}}_{k}^{+}\\ m_\alpha (-k) &{}\quad \text{ for } \alpha \in {\mathcal {I}}_{k}^{-}\end{array}\right. \end{aligned}$$
can be computed to be given by
$$\begin{aligned} n_\alpha (k)^2 = \frac{4\pi k_\text{ F}^2}{M} |k \cdot {{\hat{\omega }}}_\alpha |\left( 1 + o(1) \right) \gg 1. \end{aligned}$$
(1.33)
(Heuristically, the reader may think of the number of particle-hole pairs as given by the surface area of the patch, \(4\pi k_\text{ F}^2/M\), times the depth inside the Fermi ball that can be reached by h, namely \(|k \cdot {{\hat{\omega }}}_\alpha |\). For this counting argument to be justifiable, the diameter of a patch on the Fermi surface may not become too large, requiring \(M \gg N^{2\delta }\).) Consequently, the operators
$$\begin{aligned} c^*_\alpha (k) := \left\{ \begin{array}{cc} b^*_\alpha (k) &{}\quad \text{ for } \alpha \in {\mathcal {I}}_{k}^{+}\\ b^*_\alpha (-k) &{}\quad \text{ for } \alpha \in {\mathcal {I}}_{k}^{-}\end{array}\right. \end{aligned}$$
(1.34)
are well-defined and behave like bosonic creation operators, namely
$$\begin{aligned}&[c_\alpha ^*(k),c_\beta ^*(l)] = 0, \quad [c_\alpha (k),c_\beta ^*(l)] \simeq \delta _{\alpha ,\beta }\delta _{k,l}, \quad \nonumber \\&\quad \forall k,l\in \Gamma ^{\text{ nor }},\ \alpha \in {\mathcal {I}}_{k},\ \beta \in {\mathcal {I}}_{l}. \end{aligned}$$
(1.35)
This is proven in Lemma 5.2, which is a slight extension of [23, Lemma 4.1].
Gapped Number Operator. As we have seen in (1.26) we do not have strong control on the particle number operator, due to the possibility of having many small-energy excitations near the Fermi surface; a problem which in the beginning is avoided by directly using \({\mathbb {H}}_0\) for bounds. However, a serious problem of using \({\mathbb {H}}_0\) is that it is not stable under the Bogoliubov transformation that we will later introduce to approximately diagonalize the effective Hamiltonian. A way of overcoming this problem, and a key improvement compared to [23] is that instead of using the full fermionic number operator \({\mathcal {N}}\) to control error terms, wherever possible we use only the gapped number operator
$$\begin{aligned} {\mathcal {N}}_\delta := \sum _{i \in {\mathbb {Z}}^3:e(i) \ge \frac{1}{4} N^{-\frac{1}{3} -\delta }} a^*_i a_i, \end{aligned}$$
(1.36)
which does not count low-energy excitations. Here we have used the dispersion relation \(e(i)=|\hbar ^2 |i|^2 - \kappa ^2 |\) introduced in (1.21), and due to the artificial gap we obtain
$$\begin{aligned} {\mathcal {N}}_\delta \le N^{\frac{1}{3}+\delta } {\mathbb {H}}_0. \end{aligned}$$
Therefore, (1.27) implies that \(\langle \Psi , {\mathcal {N}}_\delta \Psi \rangle \le CN^\delta \) which is much better than \(\langle \Psi , {\mathcal {N}}\Psi \rangle \le CN^{\frac{1}{3}}\) in (1.26). Thus in practice, controlling error terms by using \({\mathcal {N}}_\delta \) is as good as using the kinetic operator \({\mathbb {H}}_0\). Furthermore, unlike \({\mathbb {H}}_0\), the gapped number operator \({\mathcal {N}}_\delta \) is stable under the Bogoliubov transformation (see Lemma 7.2).
The main instance where \({\mathcal {N}}_\delta \) finds use is Lemma 5.3, where we bound the approximately bosonic number operator by the fermionic gapped number operator,
$$\begin{aligned} \sum _{\alpha \in {\mathcal {I}}_{k}} c_\alpha ^*(k) c_\alpha (k) \le C {\mathcal {N}}_\delta . \end{aligned}$$
(1.37)
This improves [23, Lemma 4.2], where \({\mathcal {N}}\) was used as the bound. The key insight leading to this improvement is that only bosonic pair operators with \(\alpha \in {\mathcal {I}}_{k}\) are needed in the effective Hamiltonian (1.48) and the diagonalizing Bogoliubov transformation (1.49) to obtain the RPA energy (1.53). Since \(\alpha \in {\mathcal {I}}_{k}\) means \(|k\cdot {\hat{\omega }}_\alpha |\ge N^{-\delta }\), the relative momentum k between particles p and holes \(h = p-k\) cannot be tangential to the Fermi surface; i. e., p or h (or both) has to lie above the gap \(e(i) \ge \frac{1}{4}N^{-\frac{1}{3}-\delta }\). This is the reason for the same parameter \(\delta > 0\) appearing both in the gapped number operator and in the equator cut-off (1.31). The new bound allows us to work with the bosonic pairs at the energy scale relevant for the result, while keeping them as much as possible separate from the low-energy excitations on whose number we do not have strong control.
In the next steps, we will write the correlation Hamiltonian \({\mathcal {H}}_\text{ corr }\) as a quadratic Hamiltonian in terms of the approximately bosonic operators \(c_\alpha ^*(k)\) and \(c_\alpha (k)\).
Bosonization of the interaction energy. By decomposing
$$\begin{aligned} b(k) \simeq \sum _{\alpha \in {\mathcal {I}}_{k}^{+}} n_\alpha (k) c_\alpha (k), \quad b(-k) \simeq \sum _{\alpha \in {\mathcal {I}}_{k}^{-}} n_\alpha (k) c_\alpha (k) \end{aligned}$$
(1.38)
we can write the main interaction term as
$$\begin{aligned} Q_\text{ B }\simeq&\frac{1}{N} \sum _{k \in \Gamma ^{\text{ nor }}} {\hat{V}}(k) \Big [ \sum _{\alpha ,\beta \in {\mathcal {I}}_{k}^{+}} n_\alpha (k) n_\beta (k) c^*_\alpha (k) c_\beta (k) \nonumber \\&\qquad \qquad \qquad \qquad + \sum _{\alpha ,\beta \in {\mathcal {I}}_{k}^{-}} n_\alpha (k) n_\beta (k) c^*_\alpha (k) c_\beta (k) \nonumber \\&\qquad \qquad \qquad \qquad + \sum _{\alpha \in {\mathcal {I}}_{k}^{+},\,\beta \in {\mathcal {I}}_{k}^{-}} n_\alpha (k) n_\beta (k) c^*_\alpha (k) c^*_\beta (k) \nonumber \\&\qquad \qquad \qquad \qquad + \sum _{\alpha \in {\mathcal {I}}_{k}^{+},\,\beta \in {\mathcal {I}}_{k}^{-}} n_\alpha (k) n_\beta (k) c_\beta (k) c_\alpha (k) \Big ]. \end{aligned}$$
(1.39)
In the approximation (1.39) we have ignored all excitations outside the patches. It is justified in Lemma 4.1, where we prove that
$$\begin{aligned} Q_\text{ B }+ {\mathcal {E}}_2 - Q_\text{ B}^{{\mathcal {R}}} - {\mathcal {E}}_2^{{\mathcal {R}}} \ge - C\left( N^{ - \frac{\delta }{2}} + C N^{ -\frac{1}{6} + \frac{\delta }{2} } M^{\frac{1}{4}}\right) \big ({\mathbb {H}}_0+{\mathcal {E}}_1+\hbar \big ) \end{aligned}$$
(1.40)
where \(Q_\text{ B}^{{\mathcal {R}}}+{\mathcal {E}}_2^{{\mathcal {R}}}\) is similar to \(Q_\text{ B }+ {\mathcal {E}}_2\) but contains only pair excitations in the patches. The proof of (1.40) requires an improved version of the kinetic inequality (1.24) (see Lemma 4.2). Thanks to (1.27), the error term in (1.40) does not contribute to the leading order of the correlation energy.
Note that the bound (1.40) is not necessary for the upper bound in [23] because the trial state there is constructed to contain only pair excitations inside the patches, so that the expectation value of a pair not belonging completely to relevant patches is identically zero.
Bosonization of the kinetic energy. The bosonization of the fermionic kinetic energy is more complicated. A key observation is that if \(\alpha \in {\mathcal {I}}_{k}^{+}\), then using the CAR (1.12) and linearizing the dispersion relation around \(k_\text{ F }{\hat{\omega }}_\alpha \), we find
$$\begin{aligned} {[}{\mathbb {H}}_0,c^*_{\alpha }(k)]&= \Big [ \sum _{i\in {\mathbb {Z}}^3} e(i) a_i^* a_i, \frac{1}{n_{\alpha }(k)} \sum _{\begin{array}{c} p:p\in B_\text{ F}^c\cap B_\alpha \\ p-k \in B_\text{ F }\cap B_\alpha \end{array}} a^*_p a^*_{p-k} \Big ] \nonumber \\&= \frac{1}{n_{\alpha }(k)} \sum _{\begin{array}{c} p:p\in B_\text{ F}^c\cap B_\alpha \\ p-k \in B_\text{ F }\cap B_\alpha \end{array}} (e(p)+e(p-k))a^*_p a^*_{p-k}\nonumber \\&\simeq 2\hbar \kappa |k\cdot {\hat{\omega }}_\alpha |c_\alpha ^*(k). \end{aligned}$$
(1.41)
For linearizing the dispersion relation we used the fact that for any \(p\in B_\text{ F}^c\cap (B_\text{ F }+k)\cap B_\alpha \), since \({\text {diam}}(B_\alpha )\ll k_\text{ F }/\sqrt{M}\) we have
$$\begin{aligned} e(p)+e(p-k)= \hbar ^2 (2p-k)\cdot k \simeq \hbar ^2 (2 k_\text{ F }{\hat{\omega }}_\alpha ) \cdot k = 2\hbar \kappa |k\cdot {\hat{\omega }}_\alpha |. \end{aligned}$$
(1.42)
Obviously the same holds if \(\alpha \in {\mathcal {I}}_{k}^{-}\). Therefore, within commutators with pair operators, \({\mathbb {H}}_0\) can be approximated as in the Luttinger model [92] by independent modes (i. e., harmonic oscillators) of energies \(\hbar \kappa 2 k\cdot {\hat{\omega }}_\alpha \), namely
$$\begin{aligned} {\mathbb {H}}_0 \simeq 2\kappa \hbar \sum _{k \in \Gamma ^{\text{ nor }}} \sum _{\alpha =1}^M |k\cdot {\hat{\omega }}_\alpha |c^*_{\alpha } (k) c_\alpha (k)=:{\mathbb {D}}_\text{ B }. \end{aligned}$$
(1.43)
A key idea of our analysis is to justify (1.43) not by estimating the difference \({\mathbb {H}}_0-{\mathbb {D}}_\text{ B }\) directly, but rather by proving that it is essentially invariant under the approximate Bogoliubov transformation T which we will introduce below to diagonalize the quadratic bosonized Hamiltonian. More precisely, in Lemma 8.1 we show that with \(\psi := T^* \Psi \) we have
$$\begin{aligned}&\langle \Psi , ({\mathbb {H}}_0 - {\mathbb {D}}_\text{ B}) \Psi \rangle = \langle \psi , ({\mathbb {H}}_0 - {\mathbb {D}}_\text{ B}) \psi \rangle + \text{ error } \end{aligned}$$
(1.44)
where
$$\begin{aligned} |\text{ error }|\le & {} C\hbar \Big [ M^{-\frac{1}{2}} \Vert ({\mathcal {N}}_\delta +1)^{1/2} \psi \Vert ^2 \nonumber \\&\qquad + CM^{\frac{3}{2}}N^{-\frac{2}{3}+\delta } \Vert ({\mathcal {N}}_\delta +1)^{1/2} ({\mathcal {N}}+1) \psi \Vert \Vert ({\mathcal {N}}_\delta +1)^{1/2} \psi \Vert \Big ].\nonumber \\ \end{aligned}$$
(1.45)
Note that only here, in the first error summand, due to the linearization of \({\mathbb {H}}_0\), does M enter in the denominator. With \(\langle \psi , {\mathcal {N}}_\delta \psi \rangle \le C N^\delta \) (this bound is stable under the Bogoliubov transformation), we need to take \(M\gg N^{2\delta }\). We will eventually choose \(M=N^{4\delta }\).
The bound (1.44) is a crucial improvement over the linearization technique in [23] which requires \(M\gg N^{\frac{1}{3}}\), a condition that we cannot fulfill due to the second error summand in (1.45) (recall that in our approximate ground state we only know \({\mathcal {N}}\le C N^{\frac{1}{3}}\)). This improvement is achieved because in [23] we unnecessarily linearized the expectation value of \({\mathbb {H}}_0\), whereas in the present paper we only linearize the necessary commutator with a pair operator \(c^*_\alpha (k)\). In general, this new possibility of choosing a rather small M means that we gain flexibility in the technical steps because we can afford arbitrarily high powers of M as long as there is a negative power of N.
To apply (1.44), prior to using the Bogoliubov transformation, we will decompose
$$\begin{aligned} {\mathbb {H}}_0=({\mathbb {H}}_0-{\mathbb {D}}_\text{ B}) + {\mathbb {D}}_\text{ B }. \end{aligned}$$
(1.46)
Diagonalization of the bosonized Hamiltonian. By combining the approximation (1.39) and the operator \(+{\mathbb {D}}_B\) from (1.46), we find the effective quadratic bosonic Hamiltonian
$$\begin{aligned} {\mathbb {D}}_\text{ B }+ Q_\text{ B}^{\mathcal {R}}= \sum _{k\in \Gamma ^{\text{ nor }}} 2\hbar \kappa |k|h_\text{ eff }(k) \end{aligned}$$
(1.47)
with
$$\begin{aligned} h_\text{ eff }(k)&:= \frac{1}{|k|}\sum _{\alpha \in {\mathcal {I}}_k} |k \cdot {\hat{\omega }}_\alpha |c^*_{\alpha } (k) c_{\alpha } (k) \nonumber \\&\qquad + \frac{{{\hat{V}}}(k)}{2\hbar \kappa |k|N} \Big [ \sum _{\alpha ,\beta \in {\mathcal {I}}_{k}^{+}} n_\alpha (k) n_\beta (k) c^*_\alpha (k) c_\beta (k) \nonumber \\&\qquad \qquad \qquad \quad \qquad + \sum _{\alpha ,\beta \in {\mathcal {I}}_{k}^{-}} n_\alpha (k) n_\beta (k) c^*_\alpha (k) c_\beta (k) \nonumber \\&\qquad \qquad \qquad \qquad \quad + \sum _{\alpha \in {\mathcal {I}}_{k}^{+},\,\beta \in {\mathcal {I}}_{k}^{-}} n_\alpha (k) n_\beta (k) c^*_\alpha (k) c^*_\beta (k) \nonumber \\&\qquad \qquad \qquad \qquad \quad + \sum _{\alpha \in {\mathcal {I}}_{k}^{+},\,\beta \in {\mathcal {I}}_{k}^{-}} n_\alpha (k) n_\beta (k) c_\beta (k) c_\alpha (k) \Big ]. \end{aligned}$$
(1.48)
We have arrived at an effective quadratic Hamiltonian in terms of the approximately bosonic creation and annihilation operators. If the effective Hamiltonian were exactly bosonic, it could be diagonalized by a Bogoliubov transformation [25]. While we do not have this tool available since our operators are not exactly bosonic, we can still use the explicit formula as for a true Bogoliubov transformation and define the unitary map
$$\begin{aligned} T=\exp \Big ( \sum _{k\in \Gamma ^{\text{ nor }}}\frac{1}{2}\sum _{\alpha ,\beta \in {\mathcal {I}}_{k}} K(k)_{\alpha ,\beta } c^*_\alpha (k) c^*_\beta (k) - \text{ h.c. }\Big ) \end{aligned}$$
(1.49)
where the real symmetric matrices K(k) are computed as in the exactly bosonic case. The choice of K(k) is the same as in [23], following the abstract formulation given in [64]. We will quickly recall it in Sect. 6.
Another key aspect of our proof is the observation that the Bogoliubov kernel K(k) satisfies a refined entry-wise bound,
$$\begin{aligned} |K(k)_{\alpha ,\beta }|\le \frac{C}{M} \min \left\{ \frac{n_\alpha (k)}{n_\beta (k)},\frac{n_\beta (k)}{n_\alpha (k)}\right\} \quad \text{ for } \text{ all } k\in \Gamma ^{\text{ nor }}\text { and }\alpha ,\beta \in {\mathcal {I}}_{k}.\nonumber \\ \end{aligned}$$
(1.50)
This is proved in Lemma 6.1. An important role in the proof is played by the fact that due to the geometry of the Fermi surface the normalization factor \(n_\alpha (k)^2\) is proportional to \(|k\cdot {\hat{\omega }}_\alpha |\) (see (1.33)) which is also the linearization of the dispersion relation (see (1.43)), leading to cancellations. This means that as the gap of the kinetic energy closes when we consider particle-hole pairs that are almost tangential to the Fermi surface, the energy gain due to the interaction of such an excitation vanishes at the same rate. While the proof is essentially a detailed computation, it is crucial in controlling the non-bosonizable terms \({\mathcal {E}}_2\), see (9.5).
In Lemma 7.1, we show that T acts approximately as a bosonic Bogoliubov transformation, namely
$$\begin{aligned} T^*_\lambda c_\gamma (l) T_\lambda =&\sum _{\alpha \in {\mathcal {I}}_{l}}\cosh (\lambda K(l))_{\alpha ,\gamma } c_\alpha (l)\nonumber \\&+ \sum _{\alpha \in {\mathcal {I}}_{l}} \sinh (\lambda K(l))_{\alpha ,\gamma } c^*_\alpha (l) + {\mathfrak {E}}_\gamma (\lambda ,l) \end{aligned}$$
(1.51)
where the error operators satisfy
$$\begin{aligned} \sum _{\gamma \in {\mathcal {I}}_{l}} \Vert {\mathfrak {E}}_\gamma (\lambda ,l) \psi \Vert \le C M N^{-\frac{2}{3}+\delta } \Vert ({\mathcal {N}}_\delta +M)^{1/2} ({\mathcal {N}}+1)\psi \Vert \qquad \forall \psi \in {\mathcal {F}}.\nonumber \\ \end{aligned}$$
(1.52)
This is an improvement of [23, Prop 4.4] in that we replaced some \({\mathcal {N}}\) by \({\mathcal {N}}_\delta \). In order to put the error estimate (1.52) in good use, we need also that the particle number operators be stable under the approximate Bogoliubov transformation; this is the content of Lemma 7.2, based on a refinement of the Grönwall argument in [23, 27].
To diagonalize the bosonizable part of the Hamiltonian, we insert (1.51) in \(T^*({\mathbb {D}}_0+Q_\text{ B}^{\mathcal {R}})T\) and write the transformed expression in Wick-normal order (with respect to the approximately bosonic operators). Up to a small error, this produces the ground state energy as desired,
$$\begin{aligned} \inf \text{ spec } \left( \sum _{k\in \Gamma ^{\text{ nor }}} 2\hbar \kappa |k|h_\text{ eff }(k) \right) = E_N^\text{ RPA } + o(\hbar ). \end{aligned}$$
(1.53)
Additionally we obtain the (up to a one-particle unitary) diagonalized quadratic Hamiltonian which in exact Bogoliubov theory would be the excitation spectrum; for some explicit matrix \({\mathfrak {K}}(k)_{\alpha ,\beta }\) it has the form
$$\begin{aligned} \sum _{k\in \Gamma ^{\text{ nor }}} 2\kappa \hbar |k|\sum _{\alpha ,\beta \in {\mathcal {I}}_{k}} {\mathfrak {K}}(k)_{\alpha ,\beta } c^*_\alpha (k)c_\beta (k). \end{aligned}$$
As mentioned before, a further new idea of our proof is that we sacrifice the positive contribution of the excitation spectrum to control the negative term \(-{\mathbb {D}}_\text{ B }\) left from the comparison of the fermionic and bosonic kinetic energy (1.44). In fact, we will prove that (see (10.16))
$$\begin{aligned} \sum _{k\in \Gamma ^{\text{ nor }}} 2\kappa \hbar |k|\sum _{\alpha ,\beta \in {\mathcal {I}}_{k}} {\mathfrak {K}}(k)_{\alpha ,\beta } c^*_\alpha (k)c_\beta (k) \ge {\mathbb {D}}_\text{ B }- C \Vert {{\hat{V}}}\Vert _{\ell ^1} {\mathbb {H}}_0. \end{aligned}$$
(1.54)
The proof of (1.54) is based on an explicit computation of the operator \({\mathfrak {K}}(k)\) and the nice property (1.50) of the Bogoliubov kernel.
When \(\Vert {{\hat{V}}}\Vert _{\ell ^1}\) is small, the error term \(-\Vert {{\hat{V}}}\Vert _{\ell ^1} {\mathbb {H}}_0\) in (1.54) is controlled by the positive term \({\mathbb {H}}_0\) left from the comparison of the fermionic and bosonic kinetic energy (1.44).
Controlling non-bosonizable parts of the Hamiltonian. We still have to show that the non-bosonizable terms \({\mathcal {E}}_1+{\mathcal {E}}_2^{{\mathcal {R}}}\) have only a small effect on the ground state energy. As explained in [23] these error terms can be easily controlled by \({\mathcal {N}}^2/N\). In the trial state in [23], the expectation value of \({\mathcal {N}}^2\) is of order 1, so that \({\mathcal {N}}^2/N\) is a small error. For the lower bound however, in the (approximate) ground state we only know that \({\mathcal {N}}\) is of order \({\mathcal {O}}(N^{\frac{1}{3}})\), so that \({\mathcal {N}}^2/N\) could be of the same order \(\hbar =N^{-\frac{1}{3}}\) as the correlation energy. Another way to see the difficulty in dealing with these terms is to observe that \({\mathcal {E}}_2^{\mathcal {R}}\) couples the “good” bosonic degrees of freedom with “bad” uncontrolled fermions near the Fermi surface (the latter were, by construction, absent in the trial state used for the upper bound).
Thus the non-bosonizable parts require a subtle analysis. The following argument relies on the fact that \({\mathcal {E}}_1\) is non-negative (as \({\hat{V}}(k) \ge 0\)), which helps us in obtaining a lower bound for \({\mathcal {E}}_2^{\mathcal {R}}\). By the Cauchy–Schwarz inequality and the kinetic energy estimate (1.24), it is easy to see that
$$\begin{aligned} {\mathcal {E}}_1+{\mathcal {E}}_2^{\mathcal {R}}\ge - C \Vert {{\hat{V}}}\Vert _{\ell ^1} {\mathbb {H}}_0. \end{aligned}$$
Of course, this bound is useless because \({\mathbb {H}}_0\) is of the same order as \({\mathcal {H}}_\text{ corr }\). However, we are able to rescue this idea by proving a similar lower bound for the transformed operator \(T^*({\mathcal {E}}_1+{\mathcal {E}}_2^{\mathcal {R}})T\). In fact, in Lemma 9.1 we prove that, with \(\psi =T^*\Psi \),
$$\begin{aligned} \langle \Psi , ({\mathcal {E}}_1+ {\mathcal {E}}_2^{\mathcal {R}}) \Psi \rangle&\ge - C \Vert {{\hat{V}}}\Vert _{\ell ^1} \Vert {\mathbb {H}}_0^{1/2} \psi \Vert ^2 - CN^{-\frac{1}{2}} \Vert \psi \Vert \Vert {\mathbb {H}}_0^{1/2} \Psi \Vert \nonumber \\&\quad - C N^{-\frac{5}{3}+2\delta } M \Vert ({\mathcal {N}}_\delta +M)^{1/2} ({\mathcal {N}}+1)\psi \Vert ^2. \end{aligned}$$
(1.55)
The bound (1.55) is one of the most subtle estimates of our analysis and does not have any counterpart in the proof of the upper bound. Note that on the right hand side, once and only once the vector \(\Psi \) appears. The proof of this bound relies on the nice property (1.50) of the Bogoliubov kernel.
The second and third summand on the right hand side of (1.55) are simply bounded by the a-priori estimates (1.27). Unlike \(CN^{-\frac{1}{2}} \Vert \psi \Vert \Vert {\mathbb {H}}_0^{1/2} \Psi \Vert \) with its small pre-factor \(N^{-\frac{1}{2}}\) the expectation value \(-\Vert {{\hat{V}}}\Vert _{\ell ^1} \langle \psi ,{\mathbb {H}}_0 \psi \rangle \) has to be controlled differently: since \(\Vert {{\hat{V}}}\Vert _{\ell ^1}\) is assumed to be small, we can control it using the positive term \({\mathbb {H}}_0\) left after the Bogoliubov transformation of the difference of fermionic and bosonic kinetic energy, see the right hand side of (1.44).
Eventually we will take the parameters \(M=N^{4\delta }\) and \(\delta =\frac{1}{24}\), resulting in the total error \({\mathcal {O}}(\hbar ^{1+\frac{1}{16} })\) to the correlation energy. This completes the sketch of the proof.