On the effective quasi-bosonic Hamiltonian of the electron gas: collective excitations and plasmon modes

We consider an effective quasi-bosonic Hamiltonian of the electron gas which emerges naturally from the random phase approximation and describes the collective excitations of the gas. By a rigorous argument, we explain how the plasmon modes can be interpreted as a special class of approximate eigenstates of this model.

In a series of four seminal papers [8,9,10,19] published in the early 1950s, Bohm and Pines proposed the random phase approximation (RPA) as an effective theory to describe the collective excitations of jellium, a homogeneous high-density electron gas moving in a background of uniform positive charge.In particular, they predicted that the electron gas will be decoupled into quasi-free electrons which emerge from the usual mean-field approximation for independent particles, and collective plasmon excitations which correspond to correlated particle motion.
Although the plasmons were quickly detected by experiments [23,13] after the works of Bohm and Pines, their theoretical explanation remains an important open question in condensed matter and nuclear physics.In 1957, Gell-Mann and Brueckner [15] gave a microscopic derivation of the RPA using a formal summation of a diagrammatic expansion, in which the leading diagrams describe the interaction of pairs of fermions, one from inside and one from outside the Fermi ball.This approach was pushed further by Sawada [21] and Sawada-Brueckner-Fukuda-Brout [22] who interpreted these pairs of electrons as bosons, obtaining an effective Hamiltonian which is quadratic with respect to the bosonic particle pairs.Recently, the bosonization argument in [21,22] has been made rigorous in [18,3,4,2,11,6] for bounded interaction potentials in the mean-field regime, in which the interaction potential is coupled with a small constant such that the interaction energy and the kinetic energy are comparable.In these works, the non-bosonizable terms of the interaction energy are negligible and the rest can be diagonalized by adapting Bogolubov's method [7] to the quasi-bosonic setting.On the mathematical side, the main challenge in this approach is to realize the bosonization structure, which only holds in a very weak sense, making even perturbative results highly nontrivial [18].In the first non-perturbative results in [3,4], the correlation energy was computed exactly to the leading order by using a patching technique (averaging fermionic pairs in patches of the Fermi sphere) to enhance the bosonization structure.This approach has been developed further in [6] to improve the analysis of the ground state energy and in [5] to address the dynamics.In [11] we proposed an alternative approach where the weak bosonization structure was used directly (without relying on the patching technique) to approximately diagonalize the fermionic Hamiltonian.One of the advantages of this approach is that it allows us to derive an effective quasi-bosonic Hamiltonian which describes both the correlation energy and the elementary excitations of the system.In the mean-field regime there are however no approximate eigenstates corresponding to collective plasmon modes.
The aim of the present paper is to give an explanation of the collective plasmon excitations by taking the quasi-bosonic Hamiltonian derived in [11], extrapolating for the In the many-body Schrödinger theory, the system is described by the Hamiltonian which acts on the fermionic space Under our assumption, H N is bounded from below and it can be extended to be a selfadjoint operator on H N with domain D (H N ) = D (H kin ) = N H 2 T 3 .Moreover, H N has compact resolvent and we are interested in the low-lying spectrum of H N when N → ∞.
In general, if V ≡ 0 and N is large, computing the spectrum of H N directly from the microscopic formulation (2.3) is impossible, both analytically and numerically.Consequently one must turn to efficient approximations.One of the most famous approximations for fermions is Hartree-Fock theory, where one restricts the consideration to Slater determinants , which are the least correlated states among all fermionic wave functions.The precision of the Hartree-Fock energy for Coulomb systems can be estimated using general correlation inequalities of Bach [1] and Graf-Solovej [17].Within Hartree-Fock theory, it turns out that the ground state energy can be well approximated by the Fermi state, which is the Slater determinant of the plane waves with momenta inside the Fermi ball B F , namely with for some k F > 0 (the Fermi momentum); see [16] and [4, Appendix A].Here for simplicity we assume that the Fermi ball B F is completely filled by N integer points, which implies that the Fermi state ψ FS is the unique, non-degenerate ground state of the kinetic operator H kin .Without this simplification, the Fermi state is not uniquely defined and the degeneracy of the elementary excitation introduced in the next subsection has to be factored out properly, which complicate the notation but do not improve the physical insight that we want to discuss.
In order to focus on the correlation structure of the interacting system, we need to extract the energy of the Fermi state.For this purpose, it is convenient to write the second-quantized form of the Hamiltonian operator H N in (2.3): are the usual Fermionic creation and annihilation operators associated to the plane wave states u p .Note that although the second-quantized form in (2.7) can be defined on the fermionic Fock space, we will always consider its restriction to the N particle space which coincides with the original Hamiltonian in (2.3).
Using the canonical anticommutation relations (CAR) where {A, B} = AB + BA, it is straightforward to compute the energy of the Fermi state (see e.g.[11,Eq.(1.10) and Eq.(1.20)]) where we define the lune of relative momentum k ∈ Z 3 * by Now we extract the contribution of the Fermi state on the operator level, namely we rewrite the operator in (2.7) as for suitable operators To be precise, we define the localized kinetic operator as and define the localized interaction operator as where We interpret b * k,p as an excitation operator, since it creates a state with momentum p ∈ B c F and annihilates a state with momentum p − k ∈ B F .

The Effective Quasi-Bosonic Hamiltonian
So far, the decomposition of (2.12) is exact, but to proceed further we now make some simplifications.Roughly speaking, the RPA in the physics literature [15,21,22] suggests that the fermionic correlation structure can be described by a bosonic quadratic Hamiltonian.As explained in [11], this bosonic analogy can be summarized in three steps: Step 1.The excitation operators b * k,p , b k,p in (2.16) should be treated as bosonic creation and annihilation operators, where the operators b k,p and b l,q with k = l can be considered as acting on independent Fock spaces.On the mathematical side, we expect the canonical commutation relations (CCR) to hold in an appropriate sense: (2.17) To motivate (2.17), let us consider the simple case k = l where we have the exact relations for all p, q ∈ L k .The last error terms in (2.18) are not small individually (as we only know c * p c p , c p c * p ≤ 1 by Pauli's exclusion principle), but they are small on average.To make it transparent, let us introduce the excitation number operator where the last identity in (2.19) follows from the assumption |B F | = N via the particlehole symmetry 1 .Then it is obvious that while for the low-lying eigenfunctions of H N the excitation number operator N E is expected to be of lower order than p,q∈L k with H k int given in (2.15), and the localized kinetic operator is thought of as (2.23) 1 Namely, the excitation number operator (which counts the number of particles outside the Fermi state) coincides with the hole number operator (which counts the number of holes inside the Fermi state).

The latter approximation (2.23) is motivated by the commutation relations
where the first identity follows from the (exact) CAR (2.9) and the second relation follows from the (approximate) CCR (2.17).
Step 3. If the effective Hamiltonian were an exact bosonic quadratic operator, then it could be diagonalized by a Bogolubov transformation (see e.g.[11, Section 3.2]), resulting in the effective operator (2.26) Here we introduced the correlation energy (2.27) with F (x) = log (1 + x)−x, and for every k ∈ Z 3 * we defined the following real, symmetric operators on ℓ 2 (L k ): (2.28) with (e p ) p∈L k the standard orthonormal basis of ℓ 2 (L k ).However, the quadratic kinetic approximation of (2.23)only holds in the weak sense of (2.24), so the difference is only essentially invariant under the Bogolubov transformation, rather than close to 0 in a direct sense.Therefore, adding (2.29) to (2.26) we obtain the more realistic approximation, up to a unitary transformation, that where we introduced the effective quasi-bosonic Hamiltonian which is an operator on the fermionic space H N = N h.
All in all the bosonization procedure of the random phase approximation thus suggests that (2.30) holds at least for states with few excitations (when N E is not too large).
For regular potentials in the mean-field regime, i.e. when V is replaced by k −1 F W for a fixed potential W satisfying k∈Z 3 * |k|| Ŵ (k)| < ∞, the operator approximation (2.30) has been justified rigorously in [11].To be precise, we proved in [11,Theorem 1] that there exists a unitary operator U : where the error operator satisfies for any fixed ǫ > 0.Moreover, thanks to [11, Theorem 1.2], the bound in (2.33) suffices to show that E U is negligible when applied to low-lying eigenstates while both E corr and Ψ, H eff Ψ are of order k F .
Note that even in the mean-field regime, the Coulomb potential is still excluded in [11].
In this case, when Vk is replaced by gk −1 F |k| −2 , the correlation energy E corr is of order k F log(k F ) instead of k F , and existing techniques seem insufficient to estimate the error terms for the energy lower bound.We refer to the recent work [12] for a rigorous upper bound for the correlation energy.The operator approximation (2.30) for the Coulomb gas in the mean-field regime remains completely open, let alone the corresponding result beyond the mean-field regime.
In the present paper, we will consider the effective operator H eff in more detail, without imposing the mean-field and regularity restrictions on the interaction.In particular, we will focus on the most interesting case of the Coulomb potential Vk = g|k| −2 for which the plasmon modes can be interpreted as a special class of approximate eigenstates of H eff .

Elementary Excitations and the Plasmon Frequency
As explained in [11], since the effective Hamiltonian H eff in (2.31) commutes with N E , we can without loss of generality restrict H eff to the eigenspaces {N E = M } with M = 0, 1, 2, ... The case M = 0 is trivial since the eigenspace {N E = 0} is the one-dimensional space spanned by the Fermi state.In the first non-trivial case, M = 1, the identity (see [11,Eq.(1.55)]) implies that the relation of (2.23) is in fact valid, whence This operator can be diagonalized explicitly on {N E = 1}.More precisely, it was proved in [11,Theorem 1.4] that by introducing the unitary transformation Ũ : where for any we have the identity Consequently, the spectrum of Note that every eigenvalue ǫ of 2 E k = 2(h for a normalized eigenvector w.Therefore, if ǫ is not an eigenvalue of 2h k , we can take the inner product with h k v k and obtain which coincides with [22, Eq. ( 6)].Since the last equality in (2.42) is not obvious, let us add an explanation for the reader's convenience.Using the algebraic identity and the definition and substituting p → p − 1 2 k in the last sum in (2.44) we get where the cancelation comes from the symmetry p → −p.On the other hand, by substituting p → p + k in the first sum in (2.44) we can write where we also transformed p → −p on the second term.Thus (2.42) holds.
In summary, (2.42) characterizes all eigenvalues of 2 E k outside the spectrum of 2h k .In the case of the Coulomb potential Vk = g |k| −2 , with a constant g > 0, the k-dependence in (2.42) is simplified and we obtain (2.47) In this case, among all eigenvalues described in (2.47), the largest one is special as it is proportional to k F while the other eigenvalues are bounded from above by Indeed, note that the function 4  (2.49) is strictly decreasing on (2λ k,max , ∞) and Therefore, the equation f (ǫ) = 1 has a unique solution on (2λ k,max , ∞).Moreover, this solution satisfies (2.52) F , the lower bound in (2.52) implies that and hence (2.52) is asymptotically sharp, namely we have F , while all other eigenvalues of 2 E k , either being characterized by (2.47) or belonging to the spectrum of 2h k , are always bounded by 2λ k,max ≪ k In the physics literature, the largest eigenvalue of 2 E k is often computed in the thermodynamic limit, where we replace Riemann sums by integrals and obtain where is the number density of the system2 .By taking g = 4πe 2 , and also inserting where ω 0 = 4πne 2 m is exactly the plasmon frequency written in [20, Eq. (3-90)] and [14, Eq. (15.16) - (15.18)].
In the present paper, we will study H eff in (2.31) for a general M ≥ 1.In this case, the spectrum of H eff corresponds to not only the elementary excitations but also all of the collective excitations of the system.Unlike the simple case M = 1 discussed above, for M ≥ 2 the operator H eff | N E =M can not be diagonalized explicitly as in (2.40), and hence understanding the spectrum of H eff is both interesting and difficult.We will focus on the part of the spectrum of H eff | N E =M which can be interpreted as describing the collective plasmon modes.

Main results
Consider the effective Hamiltonian H eff in (2.31), i.e.
which is an operator on the fermionic N -particle space As discussed above, for M = 1 the eigenfunctions of H eff are precisely the states of the form b * k (ϕ)ψ FS , where ϕ ∈ ℓ 2 (L k ) is an eigenvector of 2 E k .In the exact bosonic case, the eigenfunctions of H eff | N E =M would be the states of the form where each For the effective Hamiltonian this is generally no longer true when M ≥ 2. However, we will show that in the case that is the eigenvector of the greatest eigenvalue of 2 E k (and so describes the plasmon mode), this is nonetheless approximately correct.
For the specific case of the Coulomb potential, we prove the following: denote the normalized eigenvector corresponding to the greatest eigenvalue, ǫ k , of 2 E k , and define for a constant C > 0 depending only on δ and ε.Furthermore, it holds that Here O(k ) is a quantity that is bounded in absolute value by k F |k| 4 times a constant independent of k F and k.This theorem shows that we can consider ΨM to be an "approximate eigenfunction" of H eff with "approximate eigenvalue" M ǫ k , when M is not too large.Let us give some quick remarks on this theorem: 1.The norm estimate implies both a dynamic and a spectral estimate: Owing to the elementary time evolution estimate (e note that (M ǫ k ) −1 is the characteristic timescale of the oscillation of ΨM , so this is a non-trivial statement for M ≪ (k F |k|) namely the state ΨM is essentially localized in the spectral space This justifies the interpretation that ΨM is an "approximate eigenfunction" of H eff .
2. The condition |k| ≤ k δ F with δ < 1/2 is natural since we need |k| ≪ k F , the plasmon mode merges into the continuum (the interval [0, 2λ k,max ] containing the remaining spectrum of 2 E k ) as argued already by Bohm and Pines.See Figure 1 for a numerical computation of the plasmon frequency and the continuum spectrum of 2 E k when |k| increases.

3.
The estimate for ǫ k is quite precise.Evidently the error term k F .To make connections to the physics literature, we note that replacing the underlying Riemann sums by integrals (and keeping only the leading part of v k , h 3 k v k ), and setting g = 4πe 2 and n = N V = 4π 3 k 3 F we find (with |k| > 0 1 2 3 4 5 6 7 8 9 10 11 12 13 where ω 0 = 4πne 2 m is the plasmon frequency and v F = 2 −1 k F is the Fermi velocity.This describes a plasmon dispersion relation of which is in agreement with [20, Eq. (3.90c)], [19,Eq. (5.19) ] and [14, Eq. (15.60)].See Section 6 for a detailed explanation of (3.6).

4.
In the mean-field regime, where V = k −1 F W with a fixed potential W , the bosonic collective excitations were discussed in [5] on the dynamics and in [2] on the spectrum (see e.g.[2, Eq. (3.38)] for an analogue of (3.6)).In this case, the separation of the plasmon frequency holds in a weak sense: although the largest eigenvalue of 2 E k are within the same order of magnitude of many other eigenvalues, i.e. of order k F , the distance from from the plasmon frequency to the next-highest one is also of order k F while the gaps between other eigenvalues are at most O(|k|) (recall that we are interested in the case |k| ≪ k 1/2 F ).This assertion follows easily from the same argument leading to (2.52).
In contrast, in the present work we focus on the more physical regime where V is independent of k F .As we go beyond the mean-field regime, the largest eigenvalue is much larger than the others, and the genuninely large gap of the spectrum ensures the almostdelocalization of the eigenfunction, which is important for our estimate.

5.
Our analysis can be extended to all potentials satisfying Vk ≥ 0 and we take ǫ k , ϕ k and ΨM as in Theorem 1, then we have the norm estimate (3.8)Note that in the case of the Coulomb potential Vk = g |k| −2 we may explicitly estimate and hence (3.8) boils down to the norm estimate in Theorem 1.We refer to Section 6 for further explanation of (3.8).

Outline of the proof:
The main mathematical difficulty of the proof lies on the fact that H eff is not a bosonic operator.More precisely, the operators b k (ϕ) only satisfies the CCR in a weak sense, and controlling the exchange terms (the error terms from the CCR) requires a careful analysis.In particular, estimating the norm of the approximate eigenstate b * k (ϕ) M ψ FS is already nontrivial, and this will be done in Section 4, together with an analysis of the action of H eff on this state.Until this point, we keep the analysis general and do not use any properties of the one-body operators E k and h k in the definition of H eff .These one-body operators will be analyzed in detail in Section 5. Finally, we conclude the proof of the main theorem in Section 6.

Analysis of the Approximate Eigenstates
Let k ∈ Z 3 * be given and let ϕ ∈ ℓ 2 (L k ) be the normalized eigenstate of 2 E k corresponding to the greatest eigenvalue ǫ k .For M ∈ N 0 we define a state In this section, we estimate the norms of Ψ M and (H eff − M ǫ k )Ψ M / Ψ M ; the main results are stated in Proposition 1 and Proposition 2, respectively.Before going to the two corresponding subsections, let us recall some basic commutator computations.First, we recall that the generalized excitation operators, given by with b k,p = c * p−k c p , obey the commutation relations ϕ, e p e q , ψ δ p,q c q−l c * p−k + δ p−k,q−l c * q c p .(4.4) For use below we calculate the commutator for p ∈ L l and q, r ∈ L k , we find φ, e p ( e p , ϕ e q , ψ + e q , ϕ e p , ψ ) δ p−l,q−k c * q c p−k = − p∈L k ∩L l q∈L k δ p−l,q−k φ, e p ( e p , ϕ e q , ψ + e q , ϕ e p , ψ ) b * 2k−l,q .
In particular

Estimating the Norm of Ψ M
In this subsection we will prove the following: Below we will see that ϕ is "almost completely delocalized", i.e. | e p , ϕ and so the proposition implies that We note the following general estimates: For the second estimate we note that by the quasi-bosonic commutation relations of equation ( 4 whence the second estimate will follow from the first provided k,l∈Z 3 * ε k,l (φ k ; φ l ) ≤ 0. This is indeed the case since by definition φ k , e p e q , φ l δ p,q c q−l c * p−k + δ p−k,q−l c * q c p (4.12) which factorizes as the negative of a sum of squares: Firstly Proof of Proposition 1 (Upper bound): (this is a special case of Lemma 1).In particular, since ϕ ∈ ℓ 2 (L k ) is normalized and Obtaining the lower bound will require some additional work.We note the following: In particular, for Here the third equation in (4.17) is obtained by iterating the second one and commuting the operator b * k (ϕ) to the left.Note that it follows from equation (4.6) that

Proof of Proposition 1 (Lower bound):
We define ϕ (3) ∈ ℓ 2 (L k ) by We then see by Lemma 2 that Ψ M 2 obeys From this we can deduce the desired lower bound by induction.For M = 0, 1 we have equality.Suppose that case M − 1 holds.Then where we recognized that The proof of Proposition 1 is complete.

Action of the Effective Hamiltonian on Ψ M
We now consider the action of on Ψ M , and in doing so prove the following: We start with Lemma 3. We have where Proof: From the first identity of equation (2.24) it follows that implying that For H QB we have by Lemma 2 that (abbreviating In all then ) so as ϕ is an eigenvector of 2 E k with eigenvalue ǫ k , Here the error term on the right-hand side is where we inserted the commutator of equation (4.6).This term can be rewritten as (4.25).
We can now estimate the error term E as follows: Lemma 4. It holds that for brevity, so that E given by equation (4.25) can be written as Then E * E is given by Note that by Lemma 1, the operators B * p,q obey p,q∈L k We estimate the separate terms.For T 1 we can apply the Cauchy-Schwarz inequality and equation (4.35) to bound and T 2 is similarly bounded as the same estimate holding also for T 3 .Finally T 4 is just bounded by so combining the estimates we find Proposition 2 now follows by combining Lemma 3 with Lemma 4 and Proposition 1 to see that

Estimation of One-Body Quantities
To proceed we must now derive some estimates on the one-body quantities involved -we need to verify that ϕ is indeed "almost delocalized" and bound l∈Z 3 . We prove the following: , and

General Estimates
To avoid unnecessary subscripts we consider instead of ℓ 2 (L k ) a general n-dimensional inner product space (V, •, • ), on which a positive symmetric operator h : V → V acts, with diagonalizing basis (e i ) n i=1 and eigenvalues (λ i ) n i=1 , and a fixed v ∈ V such that e i , v > 0 for all 1 ≤ i ≤ n. (5.1) Let ϕ ∈ V be a normalized eigenvector of E with greatest eigenvalue ǫ (> h ) (note that we do not include the factor of 2 in this section), with phase chosen such that h , we have This identity lets us describe the components of ϕ (with respect to (e i ) n i=1 ) in terms of the single unknown ǫ: Taking the inner product with e i yields and now we may note that 2 h 1 2 v, ϕ is simply a constant independent of i.As ϕ is by assumption normalized, we thus have (5.5) Note that by the variational principle , we have So we immediately obtain the following: Proof: We simply estimate and note that by (5.6), (5.8) For the statement of Theorem 1 it is also interesting to bound ǫ from above: We just saw that and the right-hand side is in fact the leading contribution to ǫ 2 : where we estimated as above.Continuing the estimate we then find From the eigenvalue equation for ǫ 2 we can then conclude that (5.12) In [11,Eq. (7.22)] we derived the identity This is asymptotically optimal for "small v", but without the mean-field scaling we also need to consider "large v".While a direct elementwise estimate appears to be more involved in this regime, a good Hilbert-Schmidt estimate is in fact simpler.Covering both regimes, we have the following: Proof: We first note that . Since we may trivially estimate that E ≥ h we can then conclude For the other estimate we simply apply the elementwise estimate:

Proof of Proposition 3
To prove Proposition 3 we now only need to insert the specific one-body operators of our problem; recall that in this case and First, for ϕ 3 6 and ϕ 2 ∞ , we trivially have that and by Lemma 5 we have the estimate Here note that the behavior λ k,max ∼ k F |k| can be deduced easily from (2.48).So the estimate on ϕ ∞ in (5.21) boils down to

.24)
The estimate on ϕ 6 in (5.20) can be simplified using we note that by Lemma 7, when |l| ≤ 2k F ,

Conclusion
We can now conclude the proof of the main result.
Proof of Theorem 1: For the first part of Theorem 1, by Proposition 2 and the estimates of Proposition 3 we have where the assumption |k| ≤ k δ F ensures the applicability of Proposition 3 and the condition M ≤ k ε F ensures that 1 For the second part of Theorem 1, concerning ǫ k , we have by Lemma 6 that (remembering to include a factor of 2) and for the claim that The proof of Theorem 1 is complete.
Further explanation of (3.6) for ǫ k in the Thermodynamic Limit In the thermodynamic limit, in which we replace Riemann sums by the corresponding integrals, we have where L k = p ∈ R 3 | |p − k| ≤ k F < |p| is now the "solid" lune.By integrating along the k • p = constant planes one may reexpress the integral, when |k| ≤ 2k F , as which is the previously mentioned equation (3.6).
Further explanation of (3.8) for general potentials Our analysis can be extended easily to any potential satisfying Vk ≥ 0,  Inserting (6.11), (6.12) and (6.16) in the estimate in Proposition 2, we obtain (3.8).

AUTHOR DECLARATIONS
Conflict of Interest: The authors have no conflicts to disclose.

1 / 2 F
to separate the plasmon frequency from other eigenvalues of 2 E k .When |k| ∼ k 1/2

2 √
a we may then estimate

. 10 ) 1 F and 1 ≤ M ≪ k F |k| 1 2
To be precise, let k ∈ Z 3 * and M ∈ N satisfy Vk ≫ k −, and let ǫ k , ϕ k and ΨM be as in Theorem 1.The proof of Proposition 2 remains unchanged and we only need to generalize slightly the one-body estimates in Proposition 3. We can use exactly (5.21),(5.22)and (5.23), without substituting Vk = g|k| −2 , to get