Effective mass of the Polaron: a lower bound

We show that the effective mass of the Fr\"ohlich Polaron is bounded below by $c\alpha^{2/5}$ for some constant $c>0$ and for all coupling constants $\alpha$. The proof uses the point process representation of the path measure of the Fr\"ohlich Polaron.


Introduction and results
The Polaron models the slow movement of an electron in a polar crystal. The electron drags a cloud of polarization along and thus appears heavier, it has an "effective mass" larger then its mass without the interaction. In the Fröhlich model of the Polaron, the Hamiltonian describing the interaction of the electron with the lattice vibrations is the operator given by 1 |k| e ik·x a k + e −ik·x a * k dk that acts on L 2 (R 3 ) ⊗ F (where F is the bosonic Fock space over L 2 (R 3 )). Here x and p are the position and momentum operators of the electron, N ≡ R 3 a * k a k dk is the number operator, the creation and annihilation operators a * k and a k satisfy the canonical commutation relations [a * k , a k ′ ] = δ(k − k ′ ) and α > 0 is the coupling constant. The Hamiltonian commutes with the total momentum operator p ⊗ 1 + 1 ⊗ P f , where P f ≡ R 3 ka * k a k dk is the momentum operator of the field, and has a fiber decomposition in terms of the Hamiltonians at fixed total momentum P (which act on F). Of particular interest is the energymomentum relation E α (P ) := inf spec(H α (P )). It is known that E α is rotationally symmetric and has a global minimum at P = 0, which is believed to be unique [DS20]. The effective mass m eff (α) is defined as the inverse curvature at P = 0 i.e.
While the asympotics of the ground state energy E α (0) in the strong coupling limit α → ∞ are known, the asymptotics of the effective mass m eff (α) still remain an important open problem. For the ground state energy, Donsker and Varadhan [DV83] gave a probabilistic proof (using the path integral formulation) of Pekars conjecture [Pek49] that states lim α→∞ E α (0) α 2 = min dxdy |ψ(x)ψ(y)| 2 |x − y| . (1.1) There also exists a functional analytic proof by Lieb and Thomas [LT97], that additionally contains explicit error bounds. While it has been conjetured by Landau and Pekar [LP48] that lim α→∞ m eff (α) α 4 = 16 √ 2π 3 R 3 dx |ψ(x)| 4 , ψ minimizer of (1.1), (1.2) so far the only progress on the asympotics of the effective mass has been achieved by Lieb and Seiringer [LS20], who showed that lim α→∞ m eff (α) = ∞ by estimating with suitably chosen trial states φ α,P . For models with stronger regularity assumptions (excluding the Fröhlich Polaron), quantitative estimates on the effective mass were recently obtained in [MS21]. Our goal is to show that there exists some constant c > 0 such that for all α > 0 and thereby giving a first quantitative growth bound for the effective mass of the Fröhlich Polaron.
We give a short introduction into the probabilistic representation of the effective mass.
We refer the reader to [DS20] for details in this representation, and to [Moe06] for a review covering functional analytic properties of the Polaron. An application of the Feynman-Kac formula to the semigroup (e −T Hα ) T 0 leads to the path measure in finite volume T > 0, where W is the distribution of three dimensional Brownian motion and the partition function Z α,T is a normalization constant. In particular, the partition function can be expressed as the matrix element Z α,T = Ω, e −T Hα(0) Ω , where Ω is the Fock vacuum. This leads to E α (0) = − lim T →∞ log(Z α,T )/T , which was applied (up to boundary conditions) in the aforementioned proof of Pekars conjecture by Donsker and Varadhan. Spohn conjectured [Spo87] that the path measure (in infinite volume) converges under diffusive rescaling to Brownian motion with some diffusion constant σ 2 (α), and showed that the effective mass then has the representation m eff (α) = (σ 2 (α)) −1 .
The existence of the infinite volume measure and the validity of the central limit theorem were shown by Mukherjee and Varadhan [MV19] for a restricted range of coupling parameters. The proof relies on a representation of the measures (1.4) as a mixture of Gaussian measures, the mixing measure being the distribution of a point process on {(s, t) ∈ R 2 : s < t} × (0, ∞) which has a renewal structure. This approach was extended in [BP21] to a broader class of path measures and a functional central limit theorem, and, for the Fröhlich Polaron, to all coupling parameters (relying on known spectral properties of H α (0)). We also refer the reader to [MV21], [BMPV22], where proofs are given that do not need the spectral properties of H α (0). We use this point process representation and derive a "variational like" formula for the effective mass (see Proposition 3.2). We obtain the estimate (1.3) by minimizing over sub-processes of the full point process.
Without additional effort, we can allow for a bit more generality and look at the probability where d 2, v(x) = 1/|x| γ with γ ∈ [1, 2) and g : [0, ∞) → (0, ∞) is a probability density with finite first moment satisfying sup t 0 (1 + t)g(t) < ∞ and x s,t := x t − x s for x ∈ C([0, ∞), R d ) and s, t 0. By the results in [BMPV22], these conditions are sufficient for the existence of an infinite volume measure and the validity of a functional central limit theorem in infinite volume. By the proof of the central limit theorem given in [MV19] 1) , the respective diffusion constant σ 2 (α) can also be obtained by taking the "diagonal limit", that is Here X t (x) := x t for any x ∈ C([0, ∞), R d ) and t 0.
Theorem 1.1. There exists a constant C > 0 (depending on γ, g and d) such that for all α > 0. Consequently, there exists a constant c > 0 such that the effective mass of the Fröhlich Polaron satisfies m eff (α) cα 2/5 for all α > 0.
As we will point out in Remark 2.1, the results obtained in [BMPV22] imply for the Fröhlich Polaron with minor additional effort the existence of somec > 0 such that σ 2 (α) e −cα 2 for reasonably large α.

Point process representation
By using the identity 1 |x| = 2 π ∞ 0 du e −u 2 |x| 2 /2 and expanding the exponential into a series, Mukherjee and Varadhan [MV19] represented the path measure of the Fröhlich Polaron as a mixture of Gaussian measures P ξ,u , the mixing measure Θ α,T being the distribution of a suitable point process on {(s, t) ∈ R 2 : s < t} × [0, ∞). In the following, we give an introduction to the point process representation in the form given in [BP21]. We will additionally use the invariance of (1.5) under replacing v by v ε (x) := v(x) + ε for some ε 0. While this transformation does not change the measure P α,T , we will obtain different point process representations depending on the choice of ε. Let Γ α,T be the distribution of a Poisson point process on △ := {(s, t) ∈ R 2 : s < t} with intensity measure µ α,T (dsdt) := αg(t − s)1 {0 s<t T } dsdt. By expanding the exponential into a series and interchanging the order of summation and integration, we obtain for any A ∈ B(C([0, ∞), R d )) and any ε 0 it is generalizable to the measures above, see Remark 4.14 in [BP21] where N f (△) is the set of finite integer valued measures on △. We will view ξ = The measure Γ α,T can be interpreted in the following way: Consider a M/G/∞-queue started empty at time 0 where the arrival process ∞ n=0 δ sn is a homogeneous Poisson process with rate α, and the service times (τ n ) n are iid with density g and are independent of the arrival process. Consider the process η = ∞ n=1 δ (sn,tn) where t n := s n + τ n is for n ∈ N the departure of customer n. Then the process off all customers in η that arrive and depart before T has distribution Γ α, Then Equation (2.1) yields the representation P α,T (·) = Γ ε α,T (dξ)P ε ξ (·) of P α,T as a mixture of the pertubed path measures The function F ε depends in an intricate manner on the configuration ξ, depending on number, length and relative position of the intervals. Nevertheless, the perturbed measure Γ ε α,T can still be interpreted as a queuing process by identifying (s, t) ∈ supp(ξ) with a service interval [s, t] as indicated above. Under the given assumptions on g and v, Γ ε α,T can then be expressed in terms of an iid sequence of "clusters" of overlapping intervals, separated by exponentially distributed dormant periods not covered by any interval. The processes of increments can be drawn independently along these dormant periods and clusters (according to the kernel (ξ, A) → P ε ξ (A)). This yields the existence of the infinite volume limit as well as the functional central limit theorem [BP21], [BMPV22]. In order to make the connection to the representation used in [MV19], we notice that for all It will turn out to be useful to make the atoms of out point process distinguishable. We will abuse notation and identify probability measures on N f (△) with symmetric probability measures on △ ∪ := ∞ n=0 △ n . With Equation (2.1), we obtain where N := ∞ n=0 n · 1 △ n . We define for ξ = ((s i , t i )) 1 i n ∈ △ n and u ∈ [0, ∞) n the Gaussian measures and obtain We will view (s, t, u) ∈ △ × [0, ∞) as an interval [s, t] equipped with the mark u. In contrast to the representation in [MV19], we draw a sample of the mixing measure Θ α,T (which is the distribution of a point process on △ × [0, ∞)) in two steps: First, draw a sample ξ according to Γ 0 α,T and then draw marks u according to the kernel κ 0 (ξ, ·). This added flexibility will be useful later. If we define we get with the previous considerations Taking the limit T → ∞ in (2.3) then yields the diffusion constant i.e. the inverse of the effective mass. One can also express the pertubed measure Γ ε α,T in terms of the path measures P α,T (a similar calculation was already made in [MV21, (4.18)]). The Laplace transform of Γ ε α,T evaluated at the measurable function f : We introduced the parameter ε in order to add a component to Γ ε α,T which is far easier to understand than the whole process: notice that we can write Γ ε α,T as the distribution of the sum η α,T + η ε α,T where η α,T ∼ Γ 0 α,T and where the Poisson point process η ε α,T ∼ Γ εα,T is independent of η α,T . Notice also that the distribution of marks of η ε α,T under κ ε still depends on the whole process. We will prove Theorem 1.1 in three steps: 2) By finiteness of the partition function, we have Fε(ξ) < ∞ for Γα,T almost all ξ 3) I.e. Γ ε α,T is the distribution of a point process that is conditionally on (Xt)t ∼ Pα,T a Poisson point process with intensity measure µα,T (dsdt)vε(Xs,t) (1) We identify configurations (ξ, u) for which σ 2 T (ξ, u) is small. We will see that σ 2 T (ξ, u) is small if there exists some subconfiguration (ξ ′ , u ′ ) of (ξ, u) (obtained by deletion of marked intervals) such that ξ ′ consists of pairwise disjoint intervals which cover large parts of [0, T ] and have lengths exceeding 1 and marks exceeding some large C(α). While not necessary 4) for the proof of Theorem 1.1, we give a characterization of σ 2 T (ξ, u) in terms of a L 2 -distance which emphasizes the variational type flavor of our approach and might be useful for further refining the method.
(2) For some C(α) (to be determined later), we estimate the kernel κ ε by a product kernel that marks intervals independent of each other with a mark in {0, C(α)}.
In particular, the product kernel allows us to mark η ε α,T without knowledge of the full process η α,T + η ε α,T . (3) We mark η ε α,T with this product kernel and obtain a Poisson point process on △ × [0, ∞). Thinning this process by only considering all marked intervals whose length exceeds 1 and that have the mark C(α) = α 1/(2+d) yields Theorem 1.1.
3. Properties of σ 2 T (ξ, u) We derive a few properties of σ 2 T (ξ, u). First, we will show that σ 2 T (ξ, ·) is decreasing with respect to the partial order on [0, ∞) N (ξ) defined by u ũ iff u i ũ i for all 1 i N (ξ). We will then express σ 2 T (ξ, u) in terms of a L 2 -distance and calculate σ 2 T (ξ, u) in case that ξ consists of pairwise disjoint intervals.

4) Lemma 3.3 later in the text can also be shown directly by using independence of Brownian increments
Lemma 3.1. For fixed ξ ∈ △ ∪ , the function σ 2 T (ξ, ·) is decreasing on [0, ∞) N (ξ) . Proof. Let ξ = ((s i , t i )) 1 i n ∈ △ n . Then By Isserlis' theorem (and as the coordinate processes X 1 , . . . , X d are iid under the centered Gaussian measure P ξ,u ) In particular, σ 2 T (ξ, u) is increasing under deletion of a marked interval (s i , t i , u i ) (set u i = 0). As already used in [MV19], the normalization constant φ(ξ, u) can also be expressed in terms of a determinant of the covariance matrix of a suitable Gaussian vector. Let (B t ) t 0 be an one dimensional Brownian motion and (Z n ) n be an iid sequence of N (0, 1) distributed random variables, independent of (B t ) t 0 . Then, by Lemma 6.1 of the appendix, we have φ(ξ, u) = 1 det(C(ξ, u)) d/2 where C(ξ, u) is the covariance matrix of the Gaussian vector (u 1 B s 1 ,t 1 +Z 1 , . . . , u n B sn,tn + Z n ).
Finally, by Lemma 6.2 which concludes the proof.
In the situation above, 1 − 1 where λ denotes the Lebesgue measure. Assume that τ i 1 and u i C for all 1 i n where C > 0. Then For a general configuration (ξ, u), the previous considerations allow us to obtain estimates on σ 2 T (ξ, u) by considering subconfigurations (ξ ′ , u ′ ) of pairwise disjoint marked intervals with intervals lengths exceeding 1 and marks exceeding C. In combination with an application of renewal theory, this implies the following technical Lemma: Proof.
In case that f = 0 a.e. the statement is trivial, so assume We consider the Poisson point process η = ∞ n=1 δ (sn,tn) of a M/G/∞-queue where the arrival process ∞ n=1 δ sn is a homogeous Poisson point process with intensity β, the service times (τ n ) n are iid, have density ρ and are independent of the arrival process and t n := s n + τ n is for n ∈ N the departure of customer n. Then the restriction of η to the process of all customers that arrive and depart before T has distribution Ξ T . We inductively define i 0 := 0, t 0 := 0 and i n+1 := inf{j > i n : s j > t in } for n 0, i.e. customer i n+1 is for n 1 the first customer that arrives after the departure of customer i n . The waiting times (s in − t i n−1 ) n 1 are iid Exp(β) distributed and are independent of the iid service times (t in − s in ) n 1 which have density ρ. Notice that (t in ) n 0 defines a renewal process. Let N T := inf{n : t in T }. By the considerations after Lemma 3.3, we have where t 0 := 0 and B T := T − t i N T −1 is the backward recurrence time of the renewal process (t in ) n at time T . By renewal theory, (B T ) T 0 converges in distribution as T → ∞. Let (T k ) k be a sequence in [0, ∞) that converges to infinity. Then B T k /T k → 0 in probability as k → ∞. Hence, there exists a subsequence (T k j ) j such that B T k j /T k j → 0 almost surely as j → ∞. By dominated convergence, E[B T k j /T k j ] → 0 as j → ∞. As (T k ) k was arbitrary, we have E[B T /T ] → 0 as T → ∞. By renewal theory almost surely as T → ∞ and by the law of large numbers we thus have almost surely as T → ∞. By the dominated convergence theorem, we have convergence in L 1 as well and the statement follows.

Estimation of the kernel κ ε
While the component η ε α,T ∼ Γ εα,T is far easier to understand than the whole process, the distribution of marks of η ε α,T under κ ε still depends on the whole process. We get rid of this problem by estimating κ ε by a suitable product kernel that marks the intervals independent of each other. For t, C > 0 we define and p ε (C) := 2 −γ/2 (1 + 4C 2 ) d/2 1 − ε/h ε (2) .

Appendix: facts about Gaussian measures
In the following, we collect a few facts about Gaussian measures (which were in a similar form already applied in [MV19]) in order to be self contained.