Abstract
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.
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1 Introduction
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 [13, 23] 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 [2,3,4, 6, 11, 18] 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 Coulomb potential and going beyond the mean-field regime. We hope that our analysis here will provide useful insights towards the ultimate goal of deriving this effective Hamiltonian and understanding the plasmons from first principles.
2 Derivation of the effective Hamiltonian
In this section we give a heuristic derivation of the effective quasi-bosonic Hamiltonian from the microscopic theory, by summarizing the approach in [11].
We consider a system of N (spinless) fermions on the torus \({\mathbb {T}}^{3}=\left[ 0,2\pi \right] ^{3}\) (with periodic boundary conditions), interacting via a repulsive potential \(V:{\mathbb {T}}^{3}\rightarrow {\mathbb {R}}\), which is to say
and which satisfies the square summability condition on the Fourier transform
Here we ignore the contribution of the zero-momentum mode (equivalently we set \({{\hat{V}}}_0=0\)) as it corresponds to a trivial energy shift of the system. (Physically, this is understood to be compensated for by the uniformly charged background.) The reader may keep in mind the typical situation of the Coulomb potential where \({\hat{V}}_{k}=g\left| k\right| ^{-2}\) with \(g>0\), although our analysis applies to a larger class of potentials.
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 self-adjoint operator on \({\mathcal {H}}_{N}\) with domain \(D\left( H_{N}\right) =D\left( H_{\text {kin}}\right) =\bigwedge ^{N}H^{2}\left( {\mathbb {T}}^{3}\right) .\) Moreover, \(H_N\) has compact resolvent and we are interested in the low-lying spectrum of \(H_N\) when \(N\rightarrow \infty \).
In general, if \(V\not \equiv 0\) 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 \(g_{1}\wedge g_{2}\cdots \wedge g_{N}\) with \(\left\{ g_{i}\right\} _{i=1}^{N}\) orthonormal in \(L^{2}\left( {\mathbb {T}}^{3}\right) \), 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 \(\psi _{\textrm{FS}}\) is the unique, non-degenerate ground state of the kinetic operator \(H_{\text {kin}}\). 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):
where
are the usual Fermionic creation and annihilation operators associated with 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 \(\left\{ A,B\right\} =AB+BA\), it is straightforward to compute the energy of the Fermi state (see, e.g., [11, Eqs. (1.10) and (1.20)])
where we define the lune of relative momentum \(k\in {\mathbb {Z}}^3_*\) 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 \(H_{{{\,\textrm{kin}\,}}}^{\prime }, H_{{{\,\textrm{int}\,}}}^{\prime }:D\left( H_{\text {kin}}\right) \subset {\mathcal {H}}_{N}\rightarrow {\mathcal {H}}_{N}\). To be precise, we define the localized kinetic operator as
and define the localized interaction operator as
where
for
We interpret \(b_{k,p}^*\) as an excitation operator, since it creates a state with momentum \(p\in B_{F}^{c}\) and annihilates a state with momentum \(p-k\in B_{F}\).
2.1 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 \(k\ne l\) 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:
To motivate (2.17), let us consider the simple case \(k=l\) where we have the exact relations
for all \(p,q\in L_{k}\). The last error terms in (2.18) are not small individually (as we only know \(c_{p}^{*}c_{p},\,c_{p}c_{p}^{*}\le 1\) 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 particle-hole symmetryFootnote 1. Then it is obvious that
while for the low-lying eigenfunctions of \(H_N\) the excitation number operator \({\mathcal {N}}_{E}\) is expected to be of lower order than
See, e.g., [11, Proposition A.1] for estimates related to (2.21).
Step 2 The full operator in (2.12) is approximated by a quadratic Hamiltonian of \(b_{k,p}^{*}\) and \(b_{k,p}\). Concretely, the non-bosonizable terms, which are the last sum in (2.14), are ignored, so that
with \(H_{\text {int}}^{k}\) given in (2.15), and the localized kinetic operator is thought of as
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, Sect. 3.2]), resulting in the effective operator
Here we introduced the correlation energy
with \(F\left( x\right) =\log \left( 1+x\right) -x\), and for every \(k\in {\mathbb {Z}}_{*}^{3}\) we defined the following real, symmetric operators on \(\ell ^{2}\left( L_{k}\right) \):
with \(\left( e_{p}\right) _{p\in L_{k}}\) the standard orthonormal basis of \(\ell ^{2}\left( L_{k}\right) \). 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 \({\mathcal {H}}_{N}=\bigwedge ^{N}{\mathfrak {h}}\).
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 \({\mathcal {N}}_E\) is not too large).
For regular potentials in the mean-field regime, i.e., when V is replaced by \(k_F^{-1} W\) for a fixed potential W satisfying \(\sum _{k\in {\mathbb {Z}}^3_*} |k| |{{\hat{W}}}(k)| <\infty \), 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 \({\mathcal {U}}: {\mathcal {H}}_N\rightarrow {\mathcal {H}}_N\) such that
where the error operator satisfies
for any fixed \(\epsilon >0\). Moreover, thanks to [11, Theorem 1.2], the bound in (2.33) suffices to show that \({\mathcal {E}}_{{\mathcal {U}}}\) is negligible when applied to low-lying eigenstates \(\Psi \) of \({\mathcal {U}} H_{N} {\mathcal {U}}^*\) satisfying \(\langle \Psi , {\mathcal {U}} H_{N} {\mathcal {U}}^* \Psi \rangle = E_{\textrm{FS}} + E_{\textrm{corr}} + O(k_F)\), namely
while both \(E_{\textrm{corr}}\) and \(\langle \Psi , H_{\textrm{eff}} \Psi \rangle \) 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 \({{\hat{V}}}_k\) is replaced by \(g k_F^{-1} |k|^{-2}\), the correlation energy \(E_{\textrm{corr}}\) 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 \(H_{\textrm{eff}}\) 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 \({{\hat{V}}}_k = g|k|^{-2}\) for which the plasmon modes can be interpreted as a special class of approximate eigenstates of \(H_{\textrm{eff}}\).
2.2 Elementary excitations and the plasmon frequency
As explained in [11], since the effective Hamiltonian \(H_{\textrm{eff}}\) in (2.31) commutes with \({\mathcal {N}}_{E}\), we can without loss of generality restrict \(H_{\text {eff}}\) to the eigenspaces \(\{{\mathcal {N}}_{E}=M\}\) with \(M=0,1,2,\ldots \)
The case \(M=0\) is trivial since the eigenspace \(\{{\mathcal {N}}_E=0\}\) 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 \(\{{\mathcal {N}}_{E}=1\}\). More precisely, it was proved in [11, Theorem 1.4] that by introducing the unitary transformation
where for any \(\varphi \in \ell ^2(L_k)\) the generalized excitation operator \(b_k^*(\varphi )\) is defined by
we have the identity
Consequently, the spectrum of \(\left. H_{\textrm{eff}}\right| _{{\mathcal {N}}_{E}=1}\) is fully determined by the eigenvalues of \(2{\widetilde{E}}_{k}\). Note that every eigenvalue \(\epsilon \) of \(2{\widetilde{E}}_{k}=2(h_{k}^{\frac{1}{2}}\left( h_{k}+2P_{v_{k}}\right) h_{k}^{\frac{1}{2}})^{\frac{1}{2}}\) solves the equation
for a normalized eigenvector w. Therefore, if \(\epsilon \) is not an eigenvalue of \(2h_{k}\), we can take the inner product with \(\langle h_{k}^{\frac{1}{2}}v_{k},w\rangle ^{-1}(\epsilon ^{2} - 4 h_{k}^{2})^{-1}h_{k}^{\frac{1}{2}}v_{k}\) 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 \(L_k = B_F^c \cap (B_F+k) = (B_F + k) \backslash B_F\) we can write
Using \(2\lambda _{k,p}=2k\cdot ( p - \frac{k}{2})\) and substituting \(p\mapsto p-\frac{1}{2}k\) in the last sum in (2.44) we get
where the cancelation comes from the symmetry \(p\mapsto -p\). On the other hand, by substituting \(p\mapsto p+k\) in the first sum in (2.44) we can write
where we also transformed \(p\mapsto -p\) on the second term. Thus (2.42) holds.
In summary, (2.42) characterizes all eigenvalues of \(2 {\widetilde{E}}_{k}\) outside the spectrum of \(2 h_k\). In the case of the Coulomb potential \({\hat{V}}_{k} = g\left| k\right| ^{-2}\), with a constant \(g>0\), the k-dependence in (2.42) is simplified and we obtain
In this case, among all eigenvalues described in (2.47), the largest one is special as it is proportional to \(k_F^{3/2}\) while the other eigenvalues are bounded from above by
Indeed, note that the function
is strictly decreasing on \((2 \lambda _{k, \max },\infty )\) and
Therefore, the equation \(f(\epsilon )=1\) has a unique solution on \((2 \lambda _{k, \max },\infty )\). Moreover, this solution satisfies
namely
When \(|k|\ll k_F^{1/2}\), the lower bound in (2.52) implies that
and hence (2.52) is asymptotically sharp, namely we have
In summary, if \(|k|\ll k_F^{1/2}\), then the largest eigenvalue \(\epsilon \) of \(2 {\widetilde{E}}_{k}\) is proportional to \(k_F^{3/2}\), while all other eigenvalues of \(2 {\widetilde{E}}_{k}\), either being characterized by (2.47) or belonging to the spectrum of \(2 h_k\), are always bounded by \(2 \lambda _{k, \max } \ll k_F^{3/2}\).
In the physics literature, the largest eigenvalue of \(2{\widetilde{E}}_{k}\) is often computed in the thermodynamic limit, where we replace Riemann sums by integrals and obtain
where
is the number density of the systemFootnote 2. By taking \(g=4\pi e^{2}\), and also inserting \(\frac{\hbar ^{2}}{2m}=1\), we find that the largest eigenvalue of \(2{\widetilde{E}}_{k}\) is
where \(\omega _0=\sqrt{\frac{4\pi ne^{2}}{m}}\) is exactly the plasmon frequency written in [20, Eqs. (3–90)] and [14, Eqs. (15.16–15.18)].
In the present paper, we will study \(H_{\textrm{eff}}\) in (2.31) for a general \(M\ge 1\). In this case, the spectrum of \(H_{\textrm{eff}}\) 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 \(M\ge 2\) the operator \(\left. H_{\text {eff}}\right| _{{\mathcal {N}}_{E}=M}\) can not be diagonalized explicitly as in (2.40), and hence understanding the spectrum of \(H_{\text {eff}}\) is both interesting and difficult. We will focus on the part of the spectrum of \(\left. H_{\text {eff}}\right| _{{\mathcal {N}}_{E}=M}\) which can be interpreted as describing the collective plasmon modes.
3 Main results
Consider the effective Hamiltonian \(H_{\textrm{eff}}\) in (2.31), i.e.,
which is an operator on the fermionic N-particle space \({\mathcal {H}}_N= \bigwedge ^{N}L^{2}\left( {\mathbb {T}}^{3}\right) \) with domain \(D(H_N)=\bigwedge ^{N}H^{2}\left( {\mathbb {T}}^{3}\right) \).
As discussed above, for \(M=1\) the eigenfunctions of \(H_\textrm{eff}\) are precisely the states of the form \(b_k^*(\varphi ) \psi _{{{\,\textrm{FS}\,}}}\), where \(\varphi \in \ell ^2(L_k)\) is an eigenvector of \(2{\widetilde{E}}_k\). In the exact bosonic case, the eigenfunctions of \(\left. H_{\mathrm{{eff}}}\right| _{{\mathcal {N}}_{E}=M}\) would be the states of the form
where each \(\varphi _i \in \ell ^2(L_k)\) is an eigenvector of \(2 {\widetilde{E}}_{k_i}\) for \(1 \le i \le M\).
For the effective Hamiltonian this is generally no longer true when \(M \ge 2\). However, we will show that in the case that \(k_1 = \cdots = k_M =: k\), where \(\varphi _1 = \cdots = \varphi _M =: \varphi _k\) is the eigenvector of the greatest eigenvalue of \(2 {\widetilde{E}}_k\) (and so describes the plasmon mode), this is nonetheless approximately correct.
For the specific case of the Coulomb potential, we prove the following:
Theorem 1
Let \(k_F>0\) be a large parameter. Let \({\hat{V}}_{k}=g\left| k\right| ^{-2}\) with a constant \(g>0\). Let \(\delta \in \left( 0,\frac{1}{2}\right) \), \(\varepsilon \in \left( 0,1\right) \), \(\left| k\right| \le k_{F}^{\delta }\) and \(1\le M\le k_{F}^{\varepsilon }\) be given. Let \(\varphi _k \in \ell ^2(L_k)\) denote the normalized eigenvector corresponding to the greatest eigenvalue, \(\epsilon _k\), of \(2 {\widetilde{E}}_k\), and define \(\Psi _M \in \left\{ {\mathcal {N}}_{E}=M\right\} \) by
Then the normalized state \({\hat{\Psi }}_M = \Vert {\Psi }_M \Vert ^{-1} {\Psi }_M\) obeys
for a constant \(C > 0\) depending only on \(\delta \) and \(\varepsilon \). Furthermore, it holds that
Here \(O(k_{F}^{-\frac{1}{2}}\left| k\right| ^{4})\) is a quantity that is bounded in absolute value by \(k_{F}^{-\frac{1}{2}}\left| k\right| ^{4}\) times a constant independent of \(k_F\) and k.
This theorem shows that we can consider \({\hat{\Psi }}_M\) to be an “approximate eigenfunction” of \(H_\textrm{eff}\) with “approximate eigenvalue” \(M \epsilon _k\), 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 \(\Vert (e^{-itH} - e^{-itE}) \psi \Vert \le \Vert (H - E) \psi \Vert t\) this shows that
$$\begin{aligned} \Vert (e^{-itH_\textrm{eff}} - e^{-itM\epsilon _k}) {\hat{\Psi }}_M \Vert \ll 1 \quad \text {for} \quad M \epsilon _k t \ll C \frac{M \epsilon _k}{\vert k \vert ^{-1} \sqrt{k_F} M^\frac{5}{2}} \sim M^{-\frac{3}{2}} k_F \vert k \vert ; \end{aligned}$$(3.3)note that \((M \epsilon _k)^{-1}\) is the characteristic timescale of the oscillation of \({\hat{\Psi }}_M\), so this is a non-trivial statement for \(M \ll (k_F \vert k \vert )^\frac{2}{3}\).
Spectrally, thanks to the operator inequality \(|{\mathbbm {1}}-{\mathbbm {1}}_{[E-\delta , E+\delta ]} (H)| \le \delta ^{-1} |H-E|\), the norm estimate in Theorem 1 implies that
namely the state \({\hat{\Psi }}_M\) is essentially localized in the spectral space \({\mathbbm {1}}_{[M\epsilon _k-\delta , M\epsilon _k+\delta ]} (H_{\textrm{eff}}){\mathcal {H}}_{N}\). This justifies the interpretation that \({\hat{\Psi }}_M\) is an “approximate eigenfunction” of \(H_{\textrm{eff}}\).
-
2.
The condition \(|k| \le k_F^{\delta }\) with \(\delta <1/2\) is natural since we need \(|k|\ll k_F^{1/2}\) to separate the plasmon frequency from other eigenvalues of \(2 {\widetilde{E}}_{k}\). When \(|k|\sim k_F^{1/2}\), the plasmon mode merges into the continuum (the interval \([0, 2 \lambda _{k, \max }]\) containing the remaining spectrum of \(2 {\widetilde{E}}_k\)) as argued already by Bohm and Pines. See Fig. 1 for a numerical computation of the plasmon frequency and the continuum spectrum of \(2 {\widetilde{E}}_{k}\) when |k| increases.
-
3.
The estimate for \(\epsilon _k\) is quite precise. Evidently the error term \(k_F^{-1/2} |k|^4\) is much smaller than \(k_F^{3/2}\) when \(|k|\ll k_F^{1/2}\); moreover
$$\begin{aligned} \epsilon _{k}=\sqrt{8\left\langle v_{k},h_{k}v_{k}\right\rangle +4\frac{\left\langle v_{k},h_{k}^{3}v_{k}\right\rangle }{\left\langle v_{k},h_{k}v_{k}\right\rangle }}+o\left( 1\right) \end{aligned}$$(3.5)for \(\left| k\right| \ll k_{F}^{\frac{1}{8}}\). To make connections to the physics literature, we note that replacing the underlying Riemann sums by integrals (and keeping only the leading part of \(\left\langle v_{k},h_{k}^{3}v_{k}\right\rangle \)), and setting \(g = 4 \pi e^2\) and \(n=\frac{N}{{\mathcal {V}}} = \frac{4\pi }{3}k_F^3\) we find (with \(\frac{\hbar ^2}{2 m} = 1\))
$$\begin{aligned} \epsilon _{k}\approx \sqrt{\frac{4 \pi e^2}{3\pi ^2}k_{F}^{3}+\frac{12}{5}k_{F}^{2}\left| k\right| ^{2}} \approx \hbar \omega _0 + \frac{\frac{6}{5} k_F^2 \vert k \vert ^2}{\hbar \omega _0} = \hbar \left( \omega _0 + \frac{3}{10} \frac{v_F^2}{\omega _0} \vert k \vert ^2 \right) , \end{aligned}$$(3.6)where \(\omega _{0}= \sqrt{\frac{4\pi ne^{2}}{m}}\) is the plasmon frequency and \(v_{F}= 2 \hbar ^{-1} k_F\) is the Fermi velocity. This describes a plasmon dispersion relation of
$$\begin{aligned} \omega (k) \approx \omega _0 + \frac{3}{10} \frac{v_F^2}{\omega _0} \vert k \vert ^2, \end{aligned}$$(3.7)which is in agreement with [20, Eq. (3.90c)], [19, Eq. (5.19) ] and [14, Eq. (15.60)]. See Sect. 6 for a detailed explanation of (3.6).
-
4.
In the mean-field regime, where \(V=k_F^{-1}W\) 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 \(2{\widetilde{E}}_{k}\) are within the same order of magnitude of many other eigenvalues, i.e., of order \(k_F\), the distance 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 \(|k|\ll k_F^{1/2}\)). This assertion follows easily from the same argument leading to (2.52).
The spectrum of \(2 {\widetilde{E}}_{k}\) with \({{\hat{V}}}_k=4\pi |k|^{-2}\) and \(k_F=500\)
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 genuinely large gap of the spectrum ensures the almost-delocalization of the eigenfunction, which is important for our estimate.
-
5.
Our analysis can be extended to all potentials satisfying \({{\hat{V}}}_k \ge 0\) and \(\sum _{k\in {\mathbb {Z}}^{3}_*} {{\hat{V}}}_k^2 <\infty \). To be precise, for any \(k\in {\mathbb {Z}}_{*}^{3}\) and \(M\in {\mathbb {N}}\) such that \({\hat{V}}_{k}\gg k_{F}^{-1}\) and \(1\le M\ll k_{F}\left| k\right| ^{\frac{1}{2}}\), if we take \(\epsilon _k, \varphi _k\) and \({{\hat{\Psi }}}_M\) as in Theorem 1, then we have the norm estimate
$$\begin{aligned} \left\| \left( H_{\textrm{eff}}-M\epsilon _{k}\right) {{\hat{\Psi }}}_M\right\| \le \frac{C}{\sqrt{k_{F}}\left| k\right| }\sqrt{\sum _{l\in 2B_{F}}\min \left\{ 1,k_{F}{\hat{V}}_{l}\right\} {\hat{V}}_{l}\left| l\right| ^{2}+Ck_{F}^{3}\sum _{l\in {\mathbb {Z}}^{3}\backslash 2B_{F}}{\hat{V}}_{l}^{2}}M^{\frac{5}{2}}. \end{aligned}$$(3.8)Note that in the case of the Coulomb potential \({\hat{V}}_{k}=g\left| k\right| ^{-2}\) we may explicitly estimate
$$\begin{aligned} \sum _{l\in 2B_{F}}\min \left\{ 1,k_{F}{\hat{V}}_{l}\right\} {\hat{V}}_{l}\left| l\right| ^{2}&\sim \int _{0}^{\sqrt{k_{F}}}r^{2}dr+k_{F}\int _{\sqrt{k_{F}}}^{2k_{F}}\frac{1}{r^{2}}r^{2}dr \le k_{F}^{\frac{3}{2}}+k_{F}k_{F}\le k_{F}^{2},\nonumber \\ k_{F}^{3}\sum _{l\in {\mathbb {Z}}^{3}\backslash 2B_{F}}{\hat{V}}_{l}^{2}&\sim k_{F}^{3}\int _{2k_{F}}^{\infty }\frac{1}{r^{4}}r^{2}dr\le k_{F}^{3}k_{F}^{-1}=k_{F}^{2}, \end{aligned}$$(3.9)and hence (3.8) boils down to the norm estimate in Theorem 1. We refer to Sect. 6 for further explanation of (3.8).
Outline of the proof The main mathematical difficulty of the proof lies on the fact that \(H_{\textrm{eff}}\) is not a bosonic operator. More precisely, the operators \(b_{k}\left( \varphi \right) \) 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 \(b_{k}^{*}\left( \varphi \right) ^{M}\psi _{{{\,\textrm{FS}\,}}}\) is already nontrivial, and this will be done in Sect. 4, together with an analysis of the action of \(H_{\textrm{eff}}\) on this state. Until this point, we keep the analysis general and do not use any properties of the one-body operators \({\widetilde{E}}_k\) and \(h_k\) in the definition of \(H_{\textrm{eff}}\). These one-body operators will be analyzed in detail in Sect. 5. Finally, we conclude the proof of the main theorem in Sect. 6.
4 Analysis of the approximate eigenstates
Let \(k\in {\mathbb {Z}}_{*}^{3}\) be given and let \(\varphi \in \ell ^{2}\left( L_{k}\right) \) be the normalized eigenstate of \(2{\widetilde{E}}_{k}\) corresponding to the greatest eigenvalue \(\epsilon _{k}\). For \(M\in {\mathbb {N}}_{0}\) we define a state \(\Psi _{M}\in \left\{ {\mathcal {N}}_{E}=M\right\} \) by
In this section, we estimate the norms of \(\Psi _M\) and \((H_{\textrm{eff}}-M \epsilon _k) \Psi _M/\Vert \Psi _M\Vert \); the main results are stated in Propositions 1 and 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
with
For use below we calculate the commutator \(\left[ \varepsilon _{l,k}\left( \phi ;\varphi \right) ,b_{k}^{*}\left( \psi \right) \right] \): As
for \(p\in L_{l}\) and \(q,r\in L_{k}\), we find
In particular
4.1 Estimating the norm of \(\Psi _{M}\)
In this subsection we will prove the following:
Proposition 1
It holds that
Below we will see that \(\varphi \) is “almost completely delocalized,” i.e., \(\left| \left\langle e_{p},\varphi \right\rangle \right| \sim \dim \left( \ell ^{2}\left( L_{k}\right) \right) ^{-\frac{1}{2}}=\left| L_{k}\right| ^{-\frac{1}{2}}\), whence
and so the proposition implies that
We note the following general estimates:
Lemma 1
Let \(\left( \phi _{k}\right) _{k\in {\mathbb {Z}}_{*}^{3}}\) be a collection of vectors \(\phi _{k}\in \ell ^{2}\left( L_{k}\right) \). Then for any \(\Psi \in {\mathcal {H}}_{N}\)
Proof
For \(\sum _{k\in {\mathbb {Z}}_{*}^{3}}b_{k}\left( \phi _{k}\right) \Psi \) we estimate
by the usual fermionic estimate \(\left\| \sum _{p}a_{p}c_{p}^{*}\right\| _{\textrm{Op}}\le \sqrt{\sum _{p}\left| a_{p}\right| ^{2}}\). For the second estimate we note that by the quasi-bosonic commutation relations of Eq. (4.3)
whence the second estimate will follow from the first provided \(\sum _{k,l\in {\mathbb {Z}}_{*}^{3}}\varepsilon _{k,l}\left( \phi _{k};\phi _{l}\right) \le 0\). This is indeed the case since by definition
which factorizes as the negative of a sum of squares: Firstly
Similarly
\(\square \)
Proof of Proposition 1 (Upper bound):
For any \(\Psi \in {\mathcal {H}}_{N}\) and \(\phi \in \ell ^{2}\left( L_{k}\right) \) it holds that
(this is a special case of Lemma 1). In particular, since \(\varphi \in \ell ^{2}\left( L_{k}\right) \) is normalized and \(\Psi _{M}\in \left\{ {\mathcal {N}}_{E}=M\right\} \) we have
whence \(\left\| \Psi _{M}\right\| ^{2}\le M!\left\| \psi _{{{\,\textrm{FS}\,}}}\right\| ^{2}=M!\). \(\square \)
Obtaining the lower bound will require some additional work. We note the following:
Lemma 2
For any \(l\in {\mathbb {Z}}_{*}^{3}\) and \(\phi \in \ell ^{2}\left( L_{l}\right) \) it holds that
In particular, for \(k=l\),
Proof
We calculate
Here the third equation in (4.17) is obtained by iterating the second one and commuting the operator \(b_k^*(\varphi )\) to the left. Note that it follows from Eq. (4.6) that \(\left[ \varepsilon _{l,k}\left( \phi ;\varphi \right) ,b_{k}^{*}\left( \varphi \right) \right] \) commutes with \(b_{k}^{*}\left( \varphi \right) \). As \(b_{l}\left( \phi \right) \psi _{{{\,\textrm{FS}\,}}}=0=\varepsilon _{l,k}\left( \phi ;\varphi \right) \psi _{{{\,\textrm{FS}\,}}}\) we thus find by applying (4.17) to \(\psi _{\textrm{FS}}\) that
The \(k=l\) case follows by inserting Eq. (4.7). \(\square \)
Proof of Proposition 1 (Lower bound):
We define \(\varphi ^{\left( 3\right) }\in \ell ^{2}\left( L_{k}\right) \) by
We then see by Lemma 2 that \(\left\| \Psi _{M}\right\| ^{2}\) 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. \(\square \)
4.2 Action of the effective Hamiltonian on \(\Psi _{M}\)
We now consider the action of
on \(\Psi _{M}\), and in doing so prove the following:
Proposition 2
For \({\hat{\Psi }}_{M}=\left\| \Psi _{M}\right\| ^{-1}\Psi _{M}\) it holds that
We start with
Lemma 3
We have
where
with \(A_{l}=2\left( {\widetilde{E}}_{l}-h_{l}\right) \).
Proof
From the first identity of Eq. (2.24) it follows that
whence
implying that
For \(H_{\textrm{QB}}\) we have by Lemma 2 that (abbreviating \(A_{l}=2\left( {\widetilde{E}}_{l}-h_{l}\right) \))
In all then
so as \(\varphi \) is an eigenvector of \(2{\widetilde{E}}_{k}\) with eigenvalue \(\epsilon _{k}\),
Here the error term on the right-hand side is
where we inserted the commutator of Eq. (4.6). This term can be rewritten as (4.25).
\(\square \)
We can now estimate the error term \({\mathcal {E}}\) as follows:
Lemma 4
It holds that
Proof
Write
for brevity, so that \({\mathcal {E}}\) given by Eq. (4.25) can be written as
Then \({\mathcal {E}}^{*}{\mathcal {E}}\) is given by
Note that by Lemma 1, the operators \(B_{p,q}^{*}\) obey
Since \(\left[ c_{p-k}^{*}c_{q},B_{p,q}\right] =0\) it holds that
so using also that \(\left[ c_{p-k}^{*}c_{q},c_{q'}^{*}c_{p'-k}\right] =\delta _{p,p'}\delta _{q,q'}-\delta _{p,p'}c_{q'}^{*}c_{q} -\delta _{q,q'}c_{p'-k}c_{p-k}^{*}\) we find
We estimate the separate terms. For \(T_{1}\) we can apply the Cauchy–Schwarz inequality and Eq. (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
\(\square \)
Proposition 2 now follows by combining Lemma 3 with Lemma 4 and Proposition 1 to see that
5 Estimation of one-body quantities
To proceed we must now derive some estimates on the one-body quantities involved - we need to verify that \(\varphi \) is indeed “almost delocalized” and bound \(\sum _{l\in {\mathbb {Z}}_{*}^{3}}\left\| {\widetilde{E}}_{l}-h_{l}\right\| _{\textrm{HS}}^{2}\). We prove the following:
Proposition 3
For \(\left| k\right| \ll \sqrt{k_{F}}\), it holds that
and
for \(C>0\) independent of k and \(k_{F}\).
Note that the condition \(\left| k\right| \ll \sqrt{k_{F}}\), namely \(\left| k\right| / \sqrt{k_{F}} \rightarrow 0\) as \(k_F\rightarrow \infty \), holds if \(|k| \le k_F^{\delta }\) with \(\delta <1/2\).
5.1 General estimates
To avoid unnecessary subscripts we consider instead of \(\ell ^{2}\left( L_{k}\right) \) a general n-dimensional inner product space \(\left( V,\left\langle \cdot ,\cdot \right\rangle \right) \), on which a positive symmetric operator \(h:V\rightarrow V\) acts, with diagonalizing basis \(\left( e_{i}\right) _{i=1}^{n}\) and eigenvalues \(\left( \lambda _{i}\right) _{i=1}^{n}\), and a fixed \(v\in V\) such that \(\left\langle e_{i},v\right\rangle >0\) for all \(1\le i\le n\).
Define \({\widetilde{E}}:V\rightarrow V\) by
Let \(\varphi \in V\) be a normalized eigenvector of \({\widetilde{E}}\) with greatest eigenvalue \(\epsilon \) (\(>\left\| h\right\| \)) (note that we do not include the factor of 2 in this section), with phase chosen such that \(\left\langle h^{\frac{1}{2}}v,\varphi \right\rangle \ge 0\). Then as \({\widetilde{E}}\) squares to \(h^{2}+2P_{h^{\frac{1}{2}}v}\), we have
whence
This identity lets us describe the components of \(\varphi \) (with respect to \(\left( e_{i}\right) _{i=1}^{n}\)) in terms of the single unknown \(\epsilon \): Taking the inner product with \(e_{i}\) yields
and now we may note that \(2\left\langle h^{\frac{1}{2}}v,\varphi \right\rangle \) is simply a constant independent of i. As \(\varphi \) is by assumption normalized, we thus have
Note that by the variational principle \(\epsilon ^{2} \ge {\tilde{E}}^2 = h^{2}+2P_{h^{\frac{1}{2}}v}\), we have
So we immediately obtain the following:
Lemma 5
Let \(\lambda _{\max }=\max _{1\le i\le n}\lambda _{i}\). Then provided \(2\left\langle v,hv\right\rangle >\lambda _{\max }^{2}\) it holds that
Proof
We simply estimate
and note that by (5.6),
\(\square \)
For the statement of Theorem 1 it is also interesting to bound \(\epsilon \) from above: We just saw that
and the right-hand side is in fact the leading contribution to \(\epsilon ^{2}\):
Lemma 6
Provided \(2\left\langle v,hv\right\rangle >\lambda _{\max }^{2}\) it holds that
Proof
By the identity \(\varphi =\left\| \left( \epsilon ^{2}-h^{2}\right) ^{-1}h^{\frac{1}{2}}v\right\| ^{-1}\left( \epsilon ^{2}-h^{2}\right) ^{-1}h^{\frac{1}{2}}v\) we have that
where we estimated as above. Continuing the estimate we then find
From the eigenvalue equation for \(\epsilon ^{2}\) we can then conclude that
\(\square \)
5.2 Estimating \(\left\| {\widetilde{E}}-h\right\| _{\textrm{HS}}\)
Finally we consider \(\left\| {\widetilde{E}}-h\right\| _{\textrm{HS}}=\left\| \left( h^{2}+2P_{h^{\frac{1}{2}}v}\right) ^{\frac{1}{2}}-h\right\| _{\textrm{HS}}\). In [11, Eq. (7.22)] we derived the identity
from which it follows that
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:
Lemma 7
It holds that
Proof
We first note that
as \({\widetilde{E}}^{2}=h^{2}+2P_{h^{\frac{1}{2}}v}\). Since we may trivially estimate that \({\widetilde{E}}\ge h\) we can then conclude
For the other estimate we simply apply the elementwise estimate:
\(\square \)
5.3 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 \(\left\| \varphi \right\| _{6}^{3}\) and \(\left\| \varphi \right\| _{\infty }^{2}\), we trivially have that
and by Lemma 5 we have the estimate
provided \(2\left\langle v_{k},h_{k}v_{k}\right\rangle -\lambda _{k,\max }^{2}>0\). Moreover, under the condition that \(\left| k\right| \ll \sqrt{k_{F}}\), the k-dependent quantities behave as (recall that \({\hat{V}}_{k}=g\left| k\right| ^{-2}\))
and
Here note that the behavior \(\lambda _{k,\max }\sim k_{F}\left| k\right| \) can be deduced easily from (2.48). So the estimate on \(\left\| \varphi \right\| _{\infty }\) in (5.21) boils down to
The estimate on \(\left\| \varphi \right\| _{6}\) in (5.20) can be simplified using \(\sqrt{\left| L_{k}\right| }\sim \sqrt{k_{F}^{2}\left| k\right| }\). It follows that
when \(\left| k\right| \ll \sqrt{k_{F}}\), as claimed.
For \(\sum _{l\in {\mathbb {Z}}_{*}^{3}}\left\| {\widetilde{E}}_{l}-h\right\| _{\textrm{HS}}^{2}\) we note that by Lemma 7, when \(\left| l\right| \le 2k_{F}\),
and when \(\left| l\right| >2k_{F}\)
whence
as claimed. \(\square \)
6 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 \(\left| k\right| \le k_{F}^{\delta }\) ensures the applicability of Proposition 3 and the condition \(M\le k_{F}^{\varepsilon }\) ensures that \(\left( 1-\frac{M^{2}}{k_{F}^{2}\left| k\right| }\right) ^{-\frac{1}{2}}\le C\).
For the second part of Theorem 1, concerning \(\epsilon _{k}\), we have by Lemma 6 that (remembering to include a factor of 2)
and
As \(\sqrt{a+b}-\sqrt{a}\le \frac{b}{2\sqrt{a}}\) we may then estimate
for the claim that
The proof of Theorem 1 is complete. \(\square \)
6.1 Further explanation of (3.6) for \(\epsilon _{k}\) in the Thermodynamic Limit
In the thermodynamic limit, in which we replace Riemann sums by the corresponding integrals, we have
where \({\mathcal {L}}_{k}=\left\{ p\in {\mathbb {R}}^{3}\mid \left| p-k\right| \le k_{F}<\left| p\right| \right\} \) is now the “solid” lune. By integrating along the \(k\cdot p=\text {constant}\) planes one may reexpress the integral, when \(\left| k\right| \le 2k_{F}\), as
with \(f(x) = x^\beta .\) It follows that for Coulomb
whence
which is the previously mentioned Eq. (3.6).
6.2 Further explanation of (3.8) for general potentials
Our analysis can be extended easily to any potential satisfying
To be precise, let \(k\in {\mathbb {Z}}_{*}^{3}\) and \(M\in {\mathbb {N}}\) satisfy \({\hat{V}}_{k}\gg k_{F}^{-1}\) and \(1\le M\ll k_{F}\left| k\right| ^{\frac{1}{2}}\), and let \(\epsilon _k, \varphi _k\) and \({{\hat{\Psi }}}_M\) 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 \({{\hat{V}}}_k= g |k|^{-2}\), to get
under the condition that \({\hat{V}}_{k}\gg k_{F}^{-1}\). Using again (5.20) and \(\sqrt{\left| L_{k}\right| }\sim \sqrt{k_{F}^{2}\left| k\right| }\), we also have
Moreover, we can split
and estimate by Lemma 7 that for \(\left| l\right| \le 2k_{F}\)
For \(\left| l\right| >2k_{F}\) we simply estimate
Thus
Inserting (6.11), (6.12) and (6.16) in the estimate in Proposition 2, we obtain (3.8).
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Notes
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).
Here we consider the spinless fermions for simplicity. If we include a factor of q for the electron spin states (e.g., \(q=2\) for electrons), the equality (2.55) is still correct provided that \(n=\frac{q}{6\pi ^{2}}k_{F}^{3}\).
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We thank the referees for helpful suggestions. MRC and PTN acknowledge the support from the Deutsche Forschungsgemeinschaft (DFG project Nr. 426365943).
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Christiansen, M.R., Hainzl, C. & Nam, P.T. On the effective quasi-bosonic Hamiltonian of the electron gas: collective excitations and plasmon modes. Lett Math Phys 112, 114 (2022). https://doi.org/10.1007/s11005-022-01607-1
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DOI: https://doi.org/10.1007/s11005-022-01607-1