Weak coupling limit for the ground state energy of the 2D Fermi polaron
aa r X i v : . [ m a t h - ph ] D ec Weak coupling limit for the ground state energyof the 2D Fermi polaron
David MitrouskasDecember 18, 2020
Abstract
We analyze the ground state energy for N fermions in a two-dimensional box in-teracting with an impurity particle via two-body point interactions. We allow for massratios M > .
225 between the impurity mass and the mass of a fermion and considerarbitrarily large box sizes while keeping the Fermi energy fixed. Our main result showsthat the ground state energy in the limit of weak coupling is given by the polaron en-ergy. The polaron energy is an energy estimate based on trial states up to first order inparticle-hole expansion, which was proposed by Chevy in the physics literature. For theproof we apply a Birman–Schwinger principle that was recently obtained by Griesemerand Linden. One main new ingredient is a suitable localization of the polaron energy.
It is a universal challenge in quamtum theory to understand the physics of few particlesimmersed into a complex enviroment in terms of properties of quasi-particles. A famousexample of a quasi-particle is the Fr¨ohlich polaron developed in a series of influential worksby Landau, Pekar and Fr¨ohlich [Lan33, LP48, Pek54, Fr¨o54]. They suggested to describethe motion of an electron through a polarizable crystal in terms of a polaron, that is, aquasi-particle composed of an electron dressed by a local deformation of the crystal. Thispicture leads to a drastic simplification as the complex many-body problem is replaced by aself-consistent non-linear one-body model, which is much more accessible to computations.While the Fr¨ohlich polaron is certainly the most prominent example for a polaron model,the concept of quasi-particles and polarons has turned out very useful far beyond its originalapplication in the theory of electrons moving through crystals. For instance the experimentalrealization of impurities immersed into ultracold atomic gases during the last two decades hastriggered the invention and analysis of many new models such as the Fermi polaron [Che06],1he Bose polaron [GD15] and the angulon [SL15]. In the present work we are interestedin the two-dimensional Fermi polaron which is a popular model in theoretical physics todescribe strongly population imbalanced Fermi gases at low temperature confined to the two-dimensional plane. In case of extreme imbalance there is only a single particle interacting witha gas of non-interacting fermions via a two-body short range interaction.We consider N identical fermions and an additional distinguished particle, called impurityparticle, in a two-dimensional box Ω = [ − L/ , L/ with periodic boundary conditions.The underlying Hilbert space is L (Ω) ⊗ H N where H N = V N L (Ω) denotes the space ofanti-symmetric N -particle wave functions. For a short-range potential, the Pauli principlesuppresses the interaction among the fermions which is therefore neglected. The Hamiltonianof the system is formally described by − M ∆ y − N X i =1 ∆ x i − g N X i =1 δ ( x i − y ) , (1.1)where y represents the coordinate of the impurity, ∆ is the Laplace operator and M denotesthe ratio between the mass of the impurity particle and the mass of a fermion. The interactionis given by a Dirac-delta-potantial δ ( x ) with coupling strength g >
0. This model is known asthe 2D Fermi polaron and has been analyzed to a great extent in the physics literature, seee.g. [Che06, CG08, PS08, CM09, PDZ09, BM10, Par11, SEPD12, PL13]. The Fermi polaronis of interest, among other reasons, because of the occurrence of a pairing mechanism some-what analogous to the famous BCS–BEC crossover. In two space dimensions, one expects atransition of the ground state as a function of the coupling strength. While for weak coupling,the impurity particle is expected to be surrounded by a cloud of particle-hole excitations,in the strong coupling regime it is predicted that the impurity is closely bound by a singlefermion forming a molecular state.Here we provide a rigorous analysis of the ground state energy in the limit of weak couplingby which we confirm its asymptotic form conjectured in the physics literature. From themathematical point of view, our work is a continuation of recent articles by Griesemer andLinden [Lin17, GL18, GL19] in which they provide a definition of the self-adjoint Hamiltonian H associated with the formal expression (1.1), derive a Birman-Schwinger type principle forthis Hamiltonian and prove stability of the Fermi polaron at zero density. The Birman–Schwinger priciple characterizes the low energy spectrum by means of an operator φ ( λ ) withspectral parameter λ . Compared to H the operator φ ( λ ) is given more explicitly and thusprovides a suitable tool for the analysis of the low energies, in particular for upper and lowerbounds for the ground state energy inf σ ( H ). Two such upper bounds, called polaron andmolecule energy, respectively, were discussed in [GL19]. Motivated by the derivation of theseupper bounds, we shall provide a matching lower bound for inf σ ( H ) in the limit of weak2oupling. A possible approach to define the Fermi polaron is to start with a regularized version of thepoint interaction and then remove the regularization in a suitable sense [GL19]. Since thislays the foundation for our work, we provide a short summary.For reasons of convenience we describe the fermions in the formalism of second quanti-zation. This means that we think of H N as the N -particle sector of F = L ∞ n =0 V n L (Ω),the fermionic Fock space over L (Ω). We denote the vacuum state with zero particles by | i = (1 , , , ... ) and define creation and annihilation operators a ∗ k , a k : F → F of planewaves ϕ k ( x ) = L − e ikx , k ∈ (2 π/L ) Z ,( a k Ψ) ( n ) = √ n + 1 Z Ω d x n +1 ϕ k ( x n +1 )Ψ ( n +1) ( x , ..., x n +1 ) , (1.2)( a ∗ k Ψ) ( n ) = 1 √ n n X j =1 ( − j ϕ k ( x j )Ψ ( n − ( x , ..., x j − , x j +1 , ..., x n ) (1.3)for Ψ = (Ψ ( n ) ) n ≥ ∈ F . The creation and annihilation operators satisfy the usual canonicalanti-commutation relations (CAR), a k a ∗ l + a ∗ l a k = δ kl , a k a l + a l a k = 0 (1.4)for all pairs k, l ∈ (2 π/L ) Z .For any number E B < g − n = X k ≤ n M ) k − E B (1.5)and define the sequence of regularized Hamiltonians ( H n ) n ∈ N , acting on L (Ω) ⊗ H N , by H n = − M ∆ y + X k k a ∗ k a k − g n X k ,l ≤ n e i ( k − l ) y a ∗ l a k . (1.6)If not stated otherwise, sums run over the two-dimensional momentum lattice (2 π/L ) Z withpossible restrictions indicated, e.g., as k ≤ n .The following statement proves the existence of the self-adjoint Hamiltonian describingthe 2D Fermi polaron. Proposition 1.1. (see [GL19, Theorem 6])
For given
L > , N ≥ , M > and E B < there exists a self-adjoint Hamiltonian H : D ( H ) ⊆ L (Ω) ⊗ H N → L (Ω) ⊗ H N such that n → H in strong resolvent sense as n → ∞ . H is bounded from below. Remarks. a From Proposition 5.1 [GL19] we know that the spectrum σ ( H ) is purely discrete. The choice of g n ensures the following renormalization condition (note that g n has alogarithmic divergence as n → ∞ ): For N = 1 the Hamiltonian H has exactly one negativeeigenvalue which coincides with E B <
0. Hence the number E B corresponds to the bindingenergy of the 1 + 1-particle model and can be used as a suitable coupling parameter of thepoint interaction.The goal of this work is to derive an asymptotic formula for the ground state energymin σ ( H ) in the limit of weak coupling E B ր N L − → ∞ (the two limits turn out to be closely connected). Instead of working with the particle number N as a free parameter, it is more convenient to fix a chemical potential µ > N = N ( µ ) with N ( µ ) = (cid:12)(cid:12)(cid:8) k ∈ κ Z : k ≤ µ (cid:9)(cid:12)(cid:12) , κ = 2 πL . (1.7)Since the number of fermions now coincides with the number of eigenvalues of − ∆ that are lessor equal than µ , counting multiplicities, the parameter µ plays the role of the Fermi energy.We write the ground state energy of H as a function of µ and E B as E ( µ, E B ) = min σ ( H )and introduce the energy of N ( µ ) non-interacting fermions inside the box Ω, E ( µ ) = X k ≤ µ k . (1.8)Our main result, Theorem 1.2, shows that the energy difference E ( µ, E B ) − E ( µ ) is givenat leading order by the polaron energy e P ( µ, E B ), that is E ( µ, E B ) − E ( µ ) e P ( µ, E B ) = 1 + o (1) as µ | E B | → ∞ . (1.9)The polaron energy e P ( µ, E B ) < e P ( µ, E B ) = − L X k ≤ µ G ( k, − k − e P ( µ, E B )) , (1.10) The energies E ( µ, E B ) and E ( µ ) depend of course also on L but we omit this in our notation. G ( q, τ ) is defined for τ > − µ and q ∈ κ Z by G ( q, τ ) = 1 L X k (cid:18) M ) k − E B − χ ( µ, ∞ ) ( k ) M ( q − k ) + k + τ (cid:19) . (1.11)Here χ ( µ, ∞ ) ( s ) denotes the characteristic function χ ( µ, ∞ ) ( s ) = 1 for s > µ and χ ( µ, ∞ ) ( s ) = 0otherwise. That (1.10) admits a lowest negative solution was shown in [GL19, Proposition7.1]. Let us mention that our main result (1.9) holds in particular in the thermodynamiclimit, i.e. after taking the limit L → ∞ .The polaron equation (1.10) was proposed in [Che06] based on a formal variational calcu-lation with trial states w P ∈ L (Ω) ⊗ H N ( µ ) of the form w P = α ϕ ⊗ | FS µ i + X k ≤ µ X l >µ α k,l ϕ k − l ⊗ a ∗ l a k | FS µ i (1.12)where α , α k,l ∈ C , ϕ k ( y ) = L − e iky and | FS µ i = Y k ≤ µ a ∗ k | i (1.13)denotes the ground state of the kinetic operator P k k a ∗ k a k ↾ H N ( µ ) (called the Fermi sea). Arigorous proof of the upper bound E ( µ, E B ) ≤ E ( µ ) + e P ( µ, E B ) was given in [GL19] utilizinga generalized Birman–Schwinger principle for the Hamiltonian H (see Section 2).In the physics literature the polaron energy is considered to be a good approximation in theweak coupling limit E B ր µ → ∞ [Che06, CG08, Par11].In the regime of strong coupling E B → −∞ , it is expected that the ground state undergoes atransition to states in which the impurity is tightly bound by a single fermion. This behavioris represented by the so-called molecule or dimer ansatz [CM09, PDZ09, Par11]. In contrastto the latter, the polaron state (1.12) is interpreted as an impurity that is surrounded by weakdensity fluctuations in the Fermi sea. The two classes of trial states were investigated exten-sively in the physics literature leading to indications for the anticipated difference betweenthe shape of the ground state in the weak and strong coupling limits (see, e.g., the literaturequoted in the previous section). For this reason the Fermi polaron is also discussed in thecontext of the BCS–BEC crossover. Most results in the physics literature, however, are basedon variational estimates using suitable classes of trial states. We remark that this can onlyjustify upper bounds for the ground state energy, whereas here we provide a correspondinglower bound.The Fermi polaron has been studied also in three dimensions. The problem of defining asemi bounded self-adjoint Hamiltonian in this case was solved in [Min11, CDFMT12, MS17].Contrary to the 2D model, it is known that the Hamiltonian is semi-bounded in three dimen-5ions only if M ≥ M ∗ for some critial mass ratio M ∗ >
0. Rigorous results concerning theground state energy mostly addressed the question of stability and the existence of a lowerbound that is uniform in the particle number N . In [MS17] it was shown that at zero densitythere is such a uniform lower bound under the condition that M > .
36. In a more recentwork, Moser and Seiringer generalized their findings to the positive density setup by provingthat the energy shift caused by the impurity particle depends only on the average density andthe interaction strength but not on the size of the system [MS19]. The question whether thepolaron energy describes the correct asymptotic form of the ground state energy similar to(1.9) is still open for the three-dimensional model.Quantum models with N + 1 particles interacting via two-body point interactions havebeen studied in the mathematical literature from various points of views. Besides the works al-ready quoted, we refer to [DFT94, DR04, AGHH05, Min11, CDFMT12, MO18] and referencestherein. We are now ready to state our main result which provides an asymptotic estimate for theground state energy min σ ( H ) of the 2D Fermi polaron. Theorem 1.2.
Set
M > . and for L > and µ > , fix the number of particles N ( µ ) by (1.7) . Moreover, let the Hamiltonian H be the limit operator of ( H n ) n ∈ N as stated inProposition 1.1. Then the ground state energy E ( µ, E B ) = min σ ( H ) and the lowest solu-tion e P ( µ, E B ) < of the polaron equation (1.10) satisfy the following property. There existconstants c , C > (possibly depending on M ) such that (cid:12)(cid:12) E ( µ, E B ) − E ( µ ) − e P ( µ, E B ) (cid:12)(cid:12) ≤ C | e P ( µ, E B ) | log( µ/ | E B | ) (1.14) for all L > , µ > and E B < with L | E B | ≥ and µ/ | E B | ≥ c . Remarks. a In Lemma 3.3 we show that e P ( µ, E B ) = O ( µ/ log( µ/ | E B | )) as µ/ | E B | → ∞ . The condition L | E B | ≥ E B is at least of the order of the minimal kinetic excitation energy whichequals (2 π/L ) . In this sense our analysis is beyond the perturbative regime. Since the constant on the right side of (1.14) does not depend on
L >
0, we can directlyinfer a statement about the ground state energy in the thermodynamic limit,lim sup L →∞ (cid:12)(cid:12)(cid:12)(cid:12) E ( µ, E B ) − E ( µ ) e P ( µ, E B ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C log( µ/ | E B | ) (1.15)6or all µ/ | E B | ≥ c . The condition
M > .
225 is related to the problem of stability (of second kind), thatis, to find a uniform lower bound for the ground state energy in the thermodynamic limit L → ∞ . While it is known that H is bounded from below for all M > M ≤ . The upper bound in (1.14) was proven in [GL19]. For the convenience of the reader, wegive a brief sketch of the argument in Section 2.2. The novel contribution of the present workis the derivation of the lower bound.
A similar result was obtained in [LM19] for the case of an infinitely heavy impurity,formally corresponding to M = ∞ . In this case the N fermions interact with an externaldelta potential which simplifies the analysis significantly.The rest of the article is organized as follows. In the next section we introduce theBirman–Schwinger operator φ ( λ ) associated to the Hamiltonian H and state the correspond-ing Birman–Schwinger principle. Upper and lower bounds for the ground state energy followfrom suitable bounds for φ ( λ ). In Section 2.2 we recall how to obtain the upper bound in(1.14). Sections 3–6 are about the matching lower bound. They account for the main partof this work. In Section 3 we derive a localization of the polaron energy inside a suitablesubspace of the Hilbert space. In the two subsequent sections we provide lower bounds for theBirman–Schwinger operator on the localization subspace and its orthogonal complement. Onthe localization subspace, we obtain a perturbed polaron equation whose solution we compareto the polaron energy, see Section 4. In Section 5 we analyze the Birman–Schwinger operatoron the orthogonal complement of the localization subspace. The lower bound on this subspacecan be understood as a proof of stability of the Fermi polaron at positive density which gen-eralizes analogous findings for the zero density model [GL19]. In Section 5.1 we combine theobtained results to conclude the proof of Theorem 1.2. The last section contains the proof ofa technical lemma that is used several times throughout the article. In this section we discuss the Birman–Schwinger principle for the Hamiltonian H which pro-vides a suitable tool for the analysis of upper and lower bounds for E ( µ, E B ) = min σ ( H ). φ ( λ ) Our starting point for the proof of Theorem 1.2 is a Birman–Schwinger type principle for theoperator H . This is the second result from [GL19] which is important for our analysis. For7he precise statement, let us introduce the resolvent set ρ ( H ) ⊂ C of the non-interactingHamiltonian H = (cid:16) − M ∆ y + T (cid:17) ↾ L (Ω) ⊗ H N ( µ ) , (2.1)with T = P k k a ∗ k a k the kinetic energy operator on the fermionic Fock space. Proposition 2.1. (See [GL19, Sections 5 and 6])
There exists a family of operators φ ( λ ) , λ ∈ ρ ( H ) , acting on L (Ω) ⊗ H N ( µ ) − with λ -independent domain D , such that for all real-valued λ , φ ( λ ) is essentially self-adjoint and its closure (denoted again by φ ( λ ) ) satisfies inf σ ( φ ( λ )) ≤ ⇔ E ( µ, E B ) ≤ λ, (2.2) with equality on one side implying equality on both sides. Moreover the φ ( λ ) form an analyticfamily of type (A) and for λ ∈ R − ∪ ( C \ R ) ⊂ ρ ( H ) they are given explicitly by φ ( λ ) = F ( i ∇ y , T − λ ) + 1 L X k,l a ∗ l e iky − M ∆ y + T + k + l − λ e − ily a k (2.3) where F ( q, τ ) = 1 L X k (cid:18) mk − E B − M q + k + τ (cid:19) , m = M + 1 M . (2.4)
Remarks.2.1
Note that while H is defined on the Hilbert space L (Ω) ⊗ H N ( µ ) , the Birman–Schwingeroperator φ ( λ ) acts on L (Ω) ⊗ H N ( µ ) − . We also remark that the domain D is given bythe set of all finite linear combinations of states of the form ϕ q ⊗ ϕ k ∧ . . . ∧ ϕ k N ( µ ) − with q, k , ..., k N ( µ ) − ∈ κ Z and ϕ k the normalized plane waves in L (Ω). The operator defined in (2.3) coincides with the Birman–Schwinger operator φ ( z ) from[GL19, Lemma 6.3] up to a multiplicative factor L − . Apart from renaming z into λ , we writethe impurity degree of freedom in first quantization whereas in [GL19], all degrees of freedomare expressed in second quantization. Proposition 2.1 is a direct consequence of the statementsfrom [GL19, Section 5 and Lemma 6.3].For explicit computations it is useful to invert the normal order of creation and annihilationoperators in (2.3) when k , l ≤ µ . With G ( q, τ ) defined in (1.11), this leads for λ ∈ R − ∪ ( C \ R )to φ ( λ ) = G ( i ∇ y , T − λ ) − L X k ,l ≤ µ a k e iky − M ∆ y + T − λ e − ily a ∗ l (cid:18) L X k ≤ µl >µ e iky a k a ∗ l − M ∆ y + T + l − λ e − ily + h.c. (cid:19) + 1 L X k ,l >µ a ∗ l e iky − M ∆ y + T + k + l − λ e − ily a k (2.5)understood as an operator on L (Ω) ⊗ H N ( µ ) − . Through analytic continuation the aboveidentity extends to λ < E ( µ ). This explicit expression of φ ( λ ) will be the main object to beanalyzed.To arrive at (2.5) we made use of the CAR and the pull-through formula, which for suitablefunctions f : κ Z × R → C reads a k f ( P f , T ) = f ( P f + k, T + k ) a k , a ∗ k f ( P f , T ) = f ( P f − k, T − k ) a ∗ k . (2.6)Here P f = P k ka ∗ k a k denotes the momentum operator of the fermions. We show how to use Proposition 2.1 to obtain an upper bound for E ( µ, E B ). This resembles theanalysis performed in the first part of Section 7 [GL19]. For an upper bound, it is sufficientto find a trial state w and a suitable λ that satisfy h w, φ ( λ ) w i ≤
0. As such we choose λ = E ( µ ) + e P ( µ, E B ) and the wave function w = X k ≤ µ G ( k, − k − e P ( µ, E B )) ϕ k ⊗ a k | FS µ i . (2.7)With the aid of (2.5), a straightforward computation leads to (cid:10) w, φ ( E ( µ ) + e P ( µ, E B )) w (cid:11) = X k ≤ µ G ( k, − k − e P ( µ, E B )) (cid:20) L X k ≤ µ G ( k, − k − e P ( µ, E B )) · e P ( µ, E B ) (cid:21) , (2.8)which is identically zero because of (1.10). By Proposition 2.1 this implies the upper bound E ( µ, E B ) ≤ E ( µ ) + e P ( µ, E B ) . (2.9) φ ( λ ) For the analysis of the lower bound it is convenient to make use of the translational invarianceof the model, in particular, that φ ( λ ) commutes with the total momentum operator P tot =9 i ∇ y + P f with P f = P k ka ∗ k a k ↾ H N ( µ ) − . This guarantees a total momentum decompositionof φ ( λ ), meaning that there is a unitary map V : L (Ω) ⊗ H N ( µ ) − → M p ∈ κ Z H N ( µ ) − (2.10)that diagonalizes P tot by eliminating the y coordinate in favor of the total momentum p ∈ κ Z .This unitary is given by ( V w ) p = ( h ϕ p | ⊗ H N ( µ ) − ) e iP f y w where h ϕ p | ⊗ H N ( µ ) − shall indicateto take the scalar product in the coordinate y with the plane wave ϕ p ∈ L (Ω). To see thatthe parameter p describes the total momentum, use ( V P tot w ) p = p ( V w ) p to verify (cid:10) w, P tot w (cid:11) = X p ∈ κ Z p (cid:10) ( V w ) p , ( V w ) p (cid:11) . (2.11)The map V is called Lee–Low–Pines transformation [LLP53] and its inverse is given by V ∗ ( w p ) = e − iP f y ( ϕ p ⊗ w p ).From this definition it is not difficult to check that φ ( λ ) in (2.5) transforms into V φ ( λ ) V ∗ = P p ∈ κ Z φ p ( λ ) where φ p ( λ ) = G ( p − P f , T − λ ) − H p ( λ ) − X p ( λ ) + P p ( λ ) (2.12)is defined as an operator on H N ( µ ) − with H p ( λ ) = a ( η ) 1 M ( p − P f ) + T − λ a ∗ ( η ) , (2.13) X p ( λ ) = a ( η ) A ∗ p ( λ ) + A p ( λ ) a ∗ ( η ) , (2.14) P p ( λ ) = 1 L X k ,l >µ a ∗ l M ( p − P f − k − l ) + T + k + l − λ a k , (2.15)and a ( η ) = 1 L X k ≤ µ a k , A p ( λ ) = 1 L X k >µ M ( p − P f − k ) + T + k − λ a k . (2.16)Note that the first summand in φ p ( λ ) defines an unbounded operator whereas the three otherterms can be shown to be bounded operators. The domain of essential self-adjointness of φ p ( λ ) is the dense subspace consisting of all finite linear combinations of states of the form ϕ k ∧ . . . ∧ ϕ k N ( µ ) − with k , ..., k N ( µ ) − ∈ κ Z .10 Localization of the polaron energy
By proposition 2.1 the lower bound E ( µ, E B ) ≥ λ is equivalent to φ ( λ ) ≥
0. The next foursections are therefore devoted to the analysis of the condition φ p ( λ ) ≥ p ∈ κ Z . In view of the upper bound (2.2) it is sufficient to consider λ ≤ E ( µ )+ e P ( µ, E B )from now on.To prepare our first main statement we need to introduce a suitable orthogonal projector inthe Hilbert space H N ( µ ) − . For its definition let us give names to the subsets of the momentumlattice κ Z that correspond to hole and particle momenta w.r.t. the Fermi sea,Λ h = (cid:8) k ∈ κ Z : k ≤ µ (cid:9) , Λ p = (cid:8) k ∈ κ Z : k > µ (cid:9) . (3.1)Moreover for ε > ≤ p ,ε = (cid:26) k ∈ Λ p : µ < k ≤ (cid:18) ε log e µ (cid:19) µ (cid:27) (3.2)and define the orthogonal projectors Π ε and Π ⊥ ε = I − Π ε throughRan(Π ε ) = lin (cid:8) a ∗ l ...a ∗ l m − a k ...a k m | FS µ i : m ≥ , k , ..., k m ∈ Λ h , l , ...., l m − ∈ Λ ≤ p ,ε (cid:9) . (3.3)Here lin V stands for the closure in H N ( µ ) − of the linear hull of the subset V ⊆ H N ( µ ) − . Fora better understanding of Ran(Π ε ) and its orthogonal complement, let us recall that the setof all anti-symmetric products of N ( µ ) − D = (cid:8) a ∗ l ...a ∗ l m − a k ...a k m | FS µ i : m ≥ , k , ..., k m ∈ Λ h , l , ..., l m − ∈ Λ p (cid:9) , (3.4)is a total set of the Hilbert space H N ( µ ) − , i.e. lin D = H N ( µ ) − . A comparison with (3.3)shows that Π ε projects on all states in H N ( µ ) − that have particle modes occupied solely inthe momentum lattice region Λ ≤ p ,ε (this includes all states with zero particle modes occupied),whereas the range of Π ⊥ ε consists of states that have at least one mode occupied in Λ > p ,ε =Λ p \ Λ ≤ p ,ε .In the next proposition we provide a lower bound for φ p ( λ ) in terms of two operators thatact only on Ran(Π ε ) and Ran(Π ⊥ ε ), respectively. The physical meaning of the two subspacesis the following: On Ran(Π ⊥ ε ) it is not clear how to obtain a suitable L -independent boundfor the operator P p ( λ ) which is one of the main obstacles in the analysis. On this subspacewe estimate the negative part of P p ( λ ) in terms of G ( p − P f , T − λ ). This is closely connectedto the problem of obtaining a lower bound of H uniformly in the system size L → ∞ . Sucha bound, though necessary for the proof of Theorem 1.2, is however not much related to theasymptotic form of E ( µ, E B ) − E ( µ ). The latter will be determined on Ran(Π ε ) on which the11perator P p ( λ ) is easily estimated with a suitable uniform bound. Hence on this subspace allof G ( p − P f , T − λ ) is available (and needed) for the analysis of the correct energy asymptotics.This explains the motivation behind the following decomposition of φ p ( λ ). Since the operator G ( p − P f , T − λ ) is needed on both subspaces separately, it is an important step in ourargument. Proposition 3.1.
There are constants c , ε > such that for all p ∈ κ Z , L | E B | ≥ , µ/ | E B | ≥ c and ε ∈ (0 , ε ) , it holds that φ p ( λ ) ≥ Φ p ( λ, ε ) + Ψ p ( λ, ε ) , with Φ p ( λ, ε ) = Π ε (cid:0) G ( p − P f , T − λ ) − H p ( λ ) − X p ( λ ) − ε − (cid:1) Π ε , (3.5)Ψ p ( λ, ε ) = Π ⊥ ε (cid:0) G ( p − P f , T − λ ) + P p ( λ ) − K ( ε, e µ ) (cid:1) Π ⊥ ε , (3.6) and K ( ε, e µ ) = ε − / ( ε − / + p log e µ + ε log e µ ) . We use the notation e µ = µ/ | E B | . Remark 3.1.
From the discussion above it is clear that the ε - and e µ -dependent errors in(3.5) and (3.6) have physically different meanings. Eventually only the error in (3.5) entersthe constant on the right side of (1.14). For that reason, we do not optimize the error termsas e µ → ∞ , and always consider ε sufficiently small but fixed w.r.t. e µ and L > G ( q, τ ) and e P ( µ, E B ). G ( q, τ ) and e P ( µ, E B ) As the following bound will be used several times, we note that it follows easily with the aidof Lemma A.1, (cid:12)(cid:12)(cid:12)(cid:12) L X aµ ≤ k a ≥ G ( q, τ ) by thecorresponding integral which can be evaluated explicitly. Lemma 3.2.
There are constant c , C > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ( q, τ ) − πm log q M +1 + mµ + τ | E B | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) µ ( µ + τ ) log( µ/ | E B | ) (cid:19) (3.8) for all q ∈ R , τ > − µ , L | E B | ≥ and µ/ | E B | ≥ c . Recall m = 1 + 1 /M . e P ( µ, E B ) as µ/ | E B | → ∞ . The precise statement is Lemma 3.3.
There are constants c , C > such that the polaron energy satisfies (cid:12)(cid:12)(cid:12)(cid:12) e P ( µ, E B ) + (cid:0) M (cid:1) µ log( µ/ | E B | ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C µ (log( µ/ | E B | )) (3.9) for all L | E B | ≥ and µ/ | E B | ≥ c .Proof. Let us set z P = | e P ( µ, E B ) | and e µ = µ/ | E B | . To prove suitable upper and lower boundsfor z P we first show z P ≤ µ for all µ/ | E B | ≥ c given that the constant c is chosen largeenough.Consider the set of parameters for which z P exceeds the value µ , M c = (cid:8) ( L, µ, E B ) : z P > µ ≥ c | E B | and L | E B | ≥ (cid:9) ⊆ R + × R + × R − . (3.10)By monotonicity of G ( q, τ ) in the τ variable, we have G ( k, − k + z p ) ≥ G ( k,
0) for all k ≤ µ and ( L, µ, E B ) ∈ M c . By Lemma 3.2 this implies G ( k, ≥ πm log( mc ) − C (cid:18) c (cid:19) ≥ πm log( mc ) . (3.11)Inserting this into the polaron equation (1.10) and employing (3.7) leads to z P ≤ mµ log( mc ) (cid:18) √ c + 24 πc (cid:19) , (3.12)which implies z P ≤ µ for c large enough. Hence M c is empty and we can assume z P ≤ µ .Utilizing again monotonicity of G ( q, τ ), we get G ( k, − k + z P ) ≤ G ( k, µ ). By (3.8) we havefor all k ≤ µ , G ( k, µ ) ≤ πm log µM +1 + mµ + µ | E B | ! + C ≤ πm log e µ + C (3.13)for two constants C , C >
0. Using (1.10) together with (3.7), we obtain the lower bound z P ≥ µm − log e µ + C (cid:18) − C pe µ (cid:19) ≥ mµ log e µ − C/ (log e µ ) . (3.14)13ith the lower bound (3.14) we can estimate for 0 ≤ k ≤ µ , G ( k, − k + z P ) ≥ πm log( m e µ − e µ ) − C (cid:18) µz P log e µ (cid:19) ≥ πm log e µ − C . (3.15)Similarly as above, using the polaron equation and (3.7), one finds z P ≤ mµ log e µ (cid:18) − C / log e µ (cid:19)(cid:18) C pe µ (cid:19) , (3.16)which gives the desired upper bound. Remark 3.2.
For λ ≤ E ( µ ) + e P ( µ, E B ) it follows from T ↾ H N ( µ ) − ≥ E ( µ ) − µ that thereare constants c , C > ± G ( p − P f , T − λ ) − πm log ( p − P f ) M +1 + mµ + T − λ | E B | !! ≤ C (3.17)as operator inequalities on H N ( µ ) − for all L | E B | ≥ µ/ | E B | ≥ c . A useful implicationof this bound is G ( p − P f , T − λ ) ↾ H N ( µ ) − ≥ πm log( µ/ | E B | ) − C. (3.18) Since G ( p − P f , T − λ ) and H p ( λ ) both commute with the projector Π ε , we have φ p ( λ ) = Π ε φ p ( λ ) Π ε + Π ⊥ ε φ p ( λ ) Π ⊥ ε + (cid:16) Π ε (cid:0) − X p ( λ ) + P p ( λ ) (cid:1) Π ⊥ ε + h.c. (cid:17) . (3.19)The statement of Proposition 3.1 is a consequence of the following estimates. Note that fornotational convenience we estimate the constant C from above by ε − / . Lemma 3.4.
There are constants c , ε , C > such that for all p ∈ κ Z , L | E B | ≥ , e µ = µ/ | E B | ≥ c and ε ∈ (0 , ε ) , Π ε P p ( λ )Π ε ≥ − C √ ε Π ε , (3.20)Π ⊥ ε H p ( λ ) Π ⊥ ε ≤ Cε log e µ Π ⊥ ε , (3.21)Π ⊥ ε X p ( λ ) Π ⊥ ε ≤ C p log e µ Π ⊥ ε , (3.22)14 nd Π ε X p ( λ ) Π ⊥ ε + h.c. ≤ Cε / log e µ Π ⊥ ε + Cε − / Π ε , (3.23)Π ε P p ( λ ) Π ⊥ ε + h.c. ≥ − C √ ε . (3.24) Proof.
Line (3.20). Using a k Π ε = 0 for all k ∈ Λ p \ Λ ≤ p ,ε together with the pull-throughformula (2.6), we obtainΠ ε P p ( λ ) Π ε = Π ε L X k,l ∈ Λ ≤ p ,ε a ∗ l a k M ( p − P f − l ) + T + l − λ ! Π ε . (3.25)By means of the commutation relations a k a ∗ l + a ∗ l a k = δ kl and1 L X l ∈ Λ ≤ p ,ε M ( p − P f − l ) + T + l − λ ≥ H N ( µ ) − , we further get(3.25) ≥ Π ε − L X k,l ∈ Λ ≤ p ,ε a k M ( p − P f ) + T − λ a ∗ l ! Π ε . (3.27)For f ( µ, E B ) > − f ( µ, E B )2 L X k,l ∈ Λ ≤ p ,ε a k a ∗ l ! − f ( µ, E B ) L X k,l ∈ Λ ≤ p ,ε a k M ( p − P f ) + T − λ ) a ∗ l ! (3.28)acting on Π ε H N ( µ ) − . In the first summand, we use the CAR together with (3.7) for a = µ , b = µ (1 + ε log e µ ), and further employ L | E B | ≥ e µ ≥ c . This gives1 L X k,l ∈ Λ ≤ p,ε a k a ∗ l ≤ L X k ∈ Λ ≤ p,ε ≤ Cµε log e µ . (3.29)In the second summand in (3.28), we use T − λ > H N ( µ ) in order to neglect the positiveoperator M ( p − P f ) in the denominator. Then we use again the pull-through formula and theCAR to obtain 1 L X k,l ∈ Λ ≤ p ,ε a k M ( p − P f ) + T − λ ) a ∗ l ≤ L X k,l ∈ Λ ≤ p ,ε a k T − λ ) a ∗ l L X k ∈ Λ ≤ p ,ε T + k − λ ) . (3.30)Note that in the last step, we applied the inequality1 L X k,l ∈ Λ ≤ p ,ε a ∗ l T + k + l − λ ) a k ≥ , (3.31)which is verified by writing1( T + k + l − λ ) = Z ∞ exp (cid:0) − t ( T − λ + k + l ) (cid:1) d t (3.32)and estimatingexp (cid:0) − t ( T − λ + k + l ) (cid:1) ≥ exp (cid:0) − t l (cid:1) exp (cid:0) − t T − λ ) (cid:1) exp (cid:0) − t k (cid:1) . (3.33)With T ≥ E ( µ ) − µ on H N ( µ ) − and E ( µ ) − λ ≥ | e P ( µ, E B ) | we next get(3.30) ≤ L X k >µ k − µ + | e P ( µ, E B ) | ) . (3.34)By Lemma A.1 and the estimate ∞ Z √ µ d t ( t − µ + | e P ( µ, E B ) | ) ≤ ∞ Z √ µ d t (( t − √ µ ) + | e P ( µ, E B ) | ) = π | e P ( µ, E B ) | / , (3.35)one finds the upper bound(3.30) ≤ π | e P ( µ, E B ) | + 1 L | e P ( µ, E B ) | / + (cid:18) √ µL + 6 L (cid:19) | e P ( µ, E B ) | ≤ C log e µµ . (3.36)Hence, Π ε P ( λ ) Π ε ≥ − C (cid:18) f ( µ, E B ) µε log e µ + log e µf ( µ, E B ) µ (cid:19) Π ε ≥ − C √ ε Π ε , (3.37)if we choose f ( µ, E B ) = ( √ ε log e µ ) /µ . Line (3.21). Since states of the form e w = a ∗ ( η ) w ∈ H N ( µ ) with w ∈ Ran(Π ⊥ ε ) have at least16ne momentum mode occupied in Λ ≥ p ,ε , it follows thatΠ ⊥ ε H p ( λ ) Π ⊥ ε ≤ Π ⊥ ε (cid:18) a ( η ) a ∗ ( η ) | e P ( µ, E B ) | + µ/ ( ε log e µ ) (cid:19) Π ⊥ ε . (3.38)The remaining expression is estimated using a ( η ) a ∗ ( η ) ≤ L X k ≤ µ ≤ Cµ (3.39)which follows from the CAR in combination with (3.7). Lines (3.22) and (3.23). It is straightforward to verify that for any two orthogonal projectors Q, e Q acting on H N ( µ ) − and for any f ( µ, E B ) , g ( µ, E B ) > QX p ( λ ) e Q + h.c. ≤ Q (cid:18) f ( µ, E B ) A p ( λ ) A ∗ p ( λ ) + a ( η ) a ∗ ( η ) g ( µ, E B ) (cid:19) Q + e Q (cid:18) g ( µ, E B ) A p ( λ ) A ∗ p ( λ ) + a ( η ) a ∗ ( η ) f ( µ, E B ) (cid:19) e Q. (3.40)Similar as in the analysis of (3.25), one further shows A p ( λ ) A ∗ p ( λ ) ↾ H N ( µ ) − ≤ C log e µµ . (3.41)Together with (3.39) and (3.40) this leads to Q X p ( λ ) e Q + h.c. ≤ C (cid:18) f ( µ, E B ) log e µµ + µg ( µ, E B ) (cid:19) Q + C (cid:18) g ( µ, E B ) log e µµ + µf ( µ, E B ) (cid:19) e Q. (3.42)For f ( µ, E B ) = g ( µ, E B ) = µ/ p log e µ , this shows (3.22), whereas the inequality in (3.23)follows from f ( µ, E B ) = √ εµ and g ( µ, E B ) = µ/ ( √ ε log e µ ). (In the latter case we set Q = Π ⊥ ε and e Q = Π ε .) Line (3.24). Using a l Π ε = 0 for l ∈ Λ p \ Λ ≤ p ,ε and the pull-through formula together with theCAR, we find Π ε P p ( λ ) Π ⊥ ε = Π ε − L X l ∈ Λ ≤ p ,ε k >µ a k M ( p − P f ) + T − λ a ∗ l ! Π ⊥ ε , (3.43)17here we made use ofΠ ε M ( p − P f − l ) + T + l − λ Π ⊥ ε = 1 M ( p − P f − l ) + T + l − λ Π ε Π ⊥ ε = 0 . (3.44)From here the proof works the same way as for (3.28) (with the difference that we end upwith an identity on the right side). We obtainΠ ε P p ( λ ) Π ⊥ ε + h.c. ≥ − C √ ε , (3.45)which completes the proof of the lemma and thus also the proof of Lemma 3.1. Φ p ( λ, ε ) : perturbed polaron equation In this section we show that the condition Φ p ( λ, ε ) ≥ λ and then provide a suitable estimate for the solution of this equation.In order to obtain the presumably optimal asymptotics of the error in (1.14), we introduceanother orthogonal projector Π ε, with Ran(Π ε, ) ⊆ Ran(Π ε ) defined as the closed subspaceof all states containing exactly one unoccupied momentum mode (a hole) in the lattice regionΛ ≤ h ,ε = (cid:26) k ∈ κ Z : k ≤ (cid:18) − ε log e µ (cid:19) µ (cid:27) ⊂ Λ h . (4.1)More precisely we set for n ≥
0, Ran(Π ε,n ) = lin V ε,n with V ε,n ⊂ H N ( µ ) − the subset V ε,n = n w = a ∗ l ...a ∗ l m − a k ...a k m | FS µ i (cid:12)(cid:12) m ≥ , k , ..., k m ∈ Λ h , l , ...., l m − ∈ Λ ≤ p ,ε , and X k ∈ Λ ≤ h ,ε a k a ∗ k w = nw o . (4.2)Clearly Ran(Π ε ) = L n ≥ Ran(Π ε,n ). (Note that the operator P k ∈ Λ ≤ h ,ε a k a ∗ k counts the numberof holes in Λ ≤ h ,ε .)The next lemma provides a more accurate localization of the polaron energy inside thesubspace Ran(Π ε, ). Lemma 4.1.
There are constants c , ε > such that for all p ∈ κ Z , L | E B | ≥ , µ/ | E B | ≥ c and ε ∈ (0 , ε ) , we have Φ p ( λ, ε ) ≥ Π ε, (cid:0) pol ( λ ) − ε − (cid:1) Π ε, (4.3) where pol ( λ ) = G (0 , T − λ ) − a ( η )( T − λ ) − a ∗ ( η ) . roof. We write Π ε = Π ε, + Π ε, + Π ε, with Π ε, = P n ≥ Π ε,n . Below we prove theinequality Φ p ( λ, ε ) ≥ Π ε, (cid:0) G ( p − P f , T − λ ) − H p ( λ ) − Cε − (cid:1) Π ε, + (Π ε, + Π ε, ) (cid:0) G ( p − P f , T − λ ) − Cε log e µ (cid:1) (Π ε, + Π ε, ) (4.4)from which the statement of the lemma follows by G ( p − P f , T − λ ) − H p ( λ ) ≥ G (0 , T − λ ) − a ( η )( T − λ ) − a ∗ ( η ) − C (4.5)together with inequality (3.18). By choosing ε small enough the second line in (4.4) is positivefor all e µ ≥ c . The bound in (4.5) is a direct consequence of (3.17) and the fact that T − λ ≥| e P ( µ, E B ) | on H N ( µ ) .The derivation of (4.4) occupies the remainder of this proof. To this end, noteΠ ε G ( p − P f , T − λ )Π ε = Π ε, G ( p − P f , T − λ )Π ε, + (Π ε, + Π ε, ) G ( p − P f , T − λ )(Π ε, + Π ε, ) . (4.6) • Introducing Λ > h ,ε = Λ h \ Λ ≤ h ,ε and a ( η >ε ) = P k ∈ Λ > h ,ε a k , we can start withΠ ε, H p ( λ ) Π ε, ≤ Π ε, (cid:18) a ( η >ε ) a ∗ ( η >ε ) | e P ( µ, E B ) | (cid:19) Π ε, ≤ Cε − Π ε, (4.7)which follows from (cid:18) M ( p − P f ) + T − λ (cid:19) ↾ H N ( µ ) ≥ | e P ( µ, E B ) | ≥ Cµ log e µ , (4.8) a ∗ ( η )Π ( ε ) = a ∗ ( η >ε )Π ( ε ), and a ( η >ε ) a ∗ ( η >ε ) ≤ L X k ∈ Λ >h,ε ≤ Cµε log e µ . (4.9)The latter is obtained via (3.7). • Next we considerΠ ε, H p ( λ ) Π ε, + h.c.= Π ε, a ( η >ε ) 1 M ( p − P f ) + T − λ a ∗ ( η ) Π ε, + h.c.19 ( ε log e µ ) µ Π ε, a ( η >ε ) a ∗ ( η >ε ) Π ε, + µ ( ε log e µ ) Π ε, a ( η ) 1( T − λ ) a ∗ ( η ) Π ε, ≤ C (cid:0) ε log e µ Π ε, + ε − Π ε, (cid:1) , (4.10)where we made use of (3.39), (4.8) and (4.9). • The contribution Π ε, H p ( λ ) Π ε, + h.c. = 0 (4.11)vanishes identically since a ( η ) 1 M ( p − P f ) + T − λ a ∗ ( η )Π ε, w ∈ Ran(Π ε, ) ⊕ Ran(Π ε, ) (4.12)for any w ∈ H N ( µ ) − and Π ε, Π ε, = Π ε, Π ε, = 0. (The operator a ∗ ( η ) can reduce thenumber of unoccupied modes at most by one.) • We proceed withΠ ε, H p ( λ ) Π ε, + h.c. = Π ε, a ( η >ε ) 1 M ( p − P f ) + T − λ a ∗ ( η ) Π ε, + h.c. (4.13)which holds because of( a ( η ) − a ( η >ε )) 1 M ( p − P f ) + T − λ a ∗ ( η ) Π ε, w ∈ Ran(Π ε, ) (4.14)and Π ε, Π ε, = 0. (Note that a ( η ) − a ( η >ε ) adds an unoccupied mode in Λ ≤ h ,ε .) We estimatethe r.h.s. of (4.13) from above by ε log e µµ Π ε, a ( η >ε ) a ∗ ( η >ε ) Π ε, + µε log e µ Π ε, a ( η ) 1( M ( p − P f ) + T − λ ) a ∗ ( η ) Π ε, ≤ C (cid:0) Π ε, + ε log e µ Π ε, (cid:1) , (4.15)where we used another time that states of the form ψ = a ∗ ( η )Π ε, w ∈ H N ( µ ) are either zeroor have at least one unoccupied mode in Λ ≤ h,ε . The latter implies (cid:10) ψ, ( T − λ ) − s ψ (cid:11) ≤ h ψ, ψ i ( ε log e µ ) s µ s ( s > . (4.16)20 In the bound for Π ε, H p ( λ ) Π ε, we use (4.16) with s = 1 to getΠ ε, H p ( λ ) Π ε, ≤ Cε log e µµ Π ε, a ( η ) a ∗ ( η )Π ε, ≤ Cε log e µ Π ε, (4.17)by means of (3.39).So far we have shownΠ ε H p ( λ ) Π ε − Π ε, H p ( λ ) Π ε, ≥ Cε − Π ε, + Cε log e µ (cid:0) Π ε, + Π ε, (cid:1) . (4.18)For the bounds involving X p ( λ ), we recall (3.42). • With f ( µ, E B ) = g ( µ, E B ) = µ/ p log e µ , we obtain(Π ε, + Π ε, ) X p ( λ ) (Π ε, + Π ε, ) ≤ C p log e µ (cid:0) Π ε, + Π ε, (cid:1) . (4.19) • Choosing f ( µ, E B ) = µ/ ( ε log e µ ) and g ( µ, E B ) = εµ leads toΠ ε, X ( λ ) (Π ε, + Π ε, ) + h.c. ≤ Cε − Π ε, + Cε log e µ (Π ε, + Π ε, ) . (4.20) • For the term with Π ε, on both sides, the estimate in (3.42) is not good enough (for obtainingan error of order one w.r.t. e µ ). A possible improvement, however, is readily obtained fromΠ ε, X p ( λ ) Π ε, = Π ε, ( A p ( λ ) a ∗ ( η >ε ) + h.c.) Π ε, (4.21)which is true since A p ( λ )( a ∗ ( η ) − a ∗ ( η >ε ))Π ε, w ∈ Ran(Π ε, ) and Π ε, Π ε, = 0. Following nowthe same steps that led to (3.42) and using in addition (4.9), we obtainΠ ε, X p ( λ ) Π ε, ≤ Π ε, (cid:18) f ( µ, E B ) A p ( λ ) A ∗ p ( λ ) + a ( η >ε ) a ∗ ( η >ε ) g ( µ, E B ) (cid:19) Π ε, ≤ C (cid:18) f ( µ, E B ) log e µµ + µg ( µ, E B ) ε log e µ (cid:19) Π ε, . (4.22)With f ( µ, E B ) = µ/ ( √ ε log e µ ) and g ( µ, E B ) = √ εµ/ log e µ , this provides Π ε, X p ( λ ) Π ε, ≤ Cε − / Π ε, .Bringing the above bounds together proves (4.4).The goal of the next lemma is to analyze the condition pol ( λ ) − r ≥ r ≥
0. To see for which λ such a bound may hold, we use the fact that this operator is givenby an expression of the form K − V ∗ V with K = G − r and V = ( T − λ ) − / a ∗ ( η ). If K is21elf-adjoint and K ≥ c for some number c >
0, it follows easily that K − V ∗ V = ( K − V ∗ V ) K − ( K − V ∗ V ) + V ∗ (1 − V K − V ∗ ) V ≥ V ∗ (1 − V K − V ∗ ) V. (4.23)This is a key argument in the proof of the following proposition. Proposition 4.2.
For any fixed r > there exists a constant c > such that for all L | E B | ≥ and µ/ | E B | ≥ c the following implication holds: pol ( λ ) ≥ r if λ satisfies E ( µ ) − λ − L X k ≤ µ G (0 , E ( µ ) − λ − k ) − r = 0 . (4.24) Remark 4.1.
We call (4.24) the perturbed polaron equation.
Proof.
Since T ≥ E ( µ ) − µ on H N ( µ ) − and λ ≤ E ( µ ) + e P ( µ, E B ), it follows by (3.18) that G (0 , T − λ ) exceeds the value of r for e µ large enough. For such e µ we can use (4.23) to find pol ( λ ) − r ≥ a ( η ) 1 T − λ F ( T − λ, r ) 1 T − λ a ∗ ( η ) ↾ H N ( µ ) − (4.25)where F ( T − λ, r ) = (cid:18) T − λ − a ∗ ( η ) 1 G (0 , T − λ ) − r a ( η ) (cid:19) ↾ H N ( µ ) . (4.26)From here it follows similarly as in the proof of [LM19, Lemma 4.2] that F ( T − λ, r ) ≥ λ satisfies the inequality E ( µ ) − λ − L X k G (0 , E ( µ ) − k − λ ) − r ≥ . (4.27)For convenience of the reader we provide the proof of the last statement in Appendix B.Next we prove the existence of a unique solution to the perturbed polaron equation (4.24)in the interval ( −∞ , E ( µ ) + e P ( µ, E B )] and provide a suitable estimate for the difference ofthis solution and E ( µ ) + e P ( µ, E B ). Lemma 4.3.
For any fixed r > there exists a constant c > such that for all L | E B | ≥ and µ/ | E B | ≥ c , the perturbed polaron equation (4.24) admits a unique solution in the interval ( −∞ , E ( µ ) + e P ( µ, E B )] . We call this solution λ ( µ, E B ) . Moreover there exists a constant The omission of the r dependence of λ ( µ, E B ) is justified by (4.28). > such that E ( µ ) + e P ( µ, E B ) − λ ( µ, E B ) ≤ C (1 + r ) | e P ( µ, E B ) | log( µ/ | E B | ) (4.28) for all L | E B | ≥ and µ/ | E B | ≥ c .Proof. To prove the existence of a solution we write (4.24) as E ( µ ) − λ = f ( λ ) with f ( λ ) = 1 L X k ≤ µ G (0 , E ( µ ) − λ − k ) − r (4.29)a continuous monotonically increasing function f ( λ ) : ( −∞ , E ( µ ) + e P ( µ, E B )] → R . Bydefinition of G ( q, τ ) we have f ( λ ) → λ → −∞ . Next consider f (Λ( µ, E B )) withΛ( µ, E B ) = E ( µ ) + e P ( µ, E B ). With the help of (1.10), f (Λ( µ, E B )) = − e P ( µ, E B )+ 1 L X k ≤ µ r + G ( k, − e P ( µ, E B ) − k ) − G (0 , − e P ( µ, E B ) − k )( G (0 , − e P ( µ, E B ) − k ) − r ) G ( k, − e P ( µ, E B ) − k ) , (4.30)and by way of Lemma 3.2, G ( k, − e P ( µ, E B ) − k ) − G (0 , − e P ( µ, E B ) − k ) ≥ − C , we seethat the second line in (4.30) is bounded from below by a constant times − µ/ (log e µ ) . Hencefor all e µ large enough, we infer f (Λ( µ, E B )) ≥ | e P ( µ, E B ) | >
0. These observations implythat there is a unique λ ( µ, E B ) ∈ ( −∞ , Λ( µ, E B )) such that f ( λ ( µ, E B )) = E ( µ ) − λ ( µ, E B ).The difference between Λ( µ, E B ) and λ ( µ, E B ) is estimated byΛ( µ, E B ) − λ ( µ, E B )= 1 L X k ≤ µ (cid:18) r + G ( k, − e P ( µ, E B ) − k ) − G (0 , E ( µ ) − k − λ ( µ, E B )) G ( k, − e P ( µ, E B ) − k )( G (0 , E ( µ ) − k − λ ( µ, E B )) − r ) (cid:19) ≤ C (1 + r )(log e µ ) (cid:18) µ + 1 L X k ≤ µ (cid:0) G ( k , − e P ( µ, E B ) − k ) − G (0 , E ( µ ) − k − λ ( µ, E B )) (cid:1)(cid:19) , (4.31)where we used Lemma 3.2 to estimate the denominator from below by a constant times(log e µ ) . In the remainder we show1 L X k ≤ µ (cid:0) G ( k, − e P ( µ, E B ) − k ) − G (0 , E ( µ ) − k − λ ( µ, E B )) (cid:1) ≤ Cµ (4.32)which proves that the left side of (4.28) is bounded from above by C (1 + r ) µ/ (log( µ/ | E B | )) .23o verify (4.32) we use λ ( µ, E B ) ≤ E ( µ ) + e P ( µ, E B ) and again Lemma 3.2 to estimatethe expression inside the brackets from above by14 πm log k M +1 − e P ( µ, E B ) + mµ − k − e P ( µ, E B ) + mµ − k ! + 2 C. (4.33)With 0 ≤ k ≤ µ , 0 ≤ − e P ( µ, E B ) ≤ µ and | e P ( µ, E B ) | = O ( µ/ log e µ ), one further verifiesthat the logarithm is bounded from above by log((2 M + 4 M + 1) / ( M + 1)) ≤ C .Let us summarize the result of this section. Corollary 4.4.
For any fixed ε > there exist constants c , C > such that λ ( µ, E B ) ≤ E ( µ ) + e P ( µ, E B ) , the unique solution of the perturbed polaron equation (4.24) with r = ε − ,satisfies the following two properties: Φ p ( λ ( µ, E B )) ≥ and λ ( µ, E B ) − E ( µ ) − e P ( µ, E B ) ≥ − C (1 + ε − ) | e P ( µ, E B ) | log( µ/ | E B | ) (4.34) for all p ∈ κ Z , L | E B | ≥ and µ/ | E B | ≥ c . In the next section we show that Ψ p ( λ, ε ) ≥ λ ≤ E ( µ ) + e P ( µ, E B ) provided that M > .
225 and ε is sufficiently small. Ψ p ( λ, ε ) : stability condition On the subspace Ran(Π ⊥ ε ) it is not clear how to obtain a suitable L -independent bound forthe operator P p ( λ ). A possible solution to this difficulty is to estimate its negative part interms of G ( p − P f , T − λ ). Such a bound was derived in [Lin17, GL18] in the context of the 2DFermi polaron at zero density (there the model is defined on R instead of the box Ω and thekinetic energy E ( µ = 0) is zero). The strategy of our proof follows the one developed there,but several new obstacles need to be dealt with in the present case. The new obstacles aredue to µ > P p ( λ ) = P p ( λ ) − e P p ( λ, ε ) + e P p ( λ, ε ) where e P p ( λ, ε ) = 1 L X k ,l >µ/ε a ∗ l M ( p − P f − k − l ) + T + k + l − λ a k . (5.1)The operator P p ( λ ) − e P p ( λ, ε ) is the easy part and can be estimated by the following lemma. Lemma 5.1.
There are constants c , ε , C > such that P p ( λ ) − e P p ( λ, ε ) ≥ − C p ε − log( µ/ | E B | ) (5.2)24 n H N ( µ ) − for all p ∈ κ Z , L | E B | ≥ , µ/ | E B | ≥ c and ε ∈ (0 , ε ) .Proof. Write P p ( λ ) − e P p ( λ, ε ) = 1 L X µ
1] be given by β ( u, ε ) = min ( , ( M + 1 − u )( M + 2) (cid:0) − (1 + M +1 − uM ( M +2) ) √ ε (cid:1) ( M + 1 − u )(1 − √ ε ) + M ( M + 2)(1 − √ ε ) ) (5.6)and set α ( M, ε ) = 12 (cid:18) M (1 − √ ε ) + 1 + Z β ( u, ε )( M (1 − √ ε ) + 1 − u ) d u (cid:19) . (5.7) Proposition 5.2.
There are constants c , ε , C > such that e P p ( λ, ε ) ≥ − − ε (cid:18) α ( M, ε )4 π log (cid:18) T − λ + 2 µµ (cid:19) + C p µ/ | E B | (cid:19) (5.8) on H N ( µ ) − for all p ∈ κ Z , L | E B | ≥ , µ/ | E B | ≥ c and ε ∈ (0 , ε ) . To prove Proposition 5.2 we combine the next two lemmas.
Lemma 5.3.
Let T >µ/ε = P k >µ/ε k a ∗ k a k . For any ε ∈ (0 , , it holds that T >µ/ε T − λ + 2 µ ↾ H N ( µ ) − ≤ − ε . (5.9)25 emma 5.4. There are constant c , ε , C > such that e P p ( λ, ε ) ≥ − T >µ/ε T − λ + 2 µ (cid:18) α ( M, ε )4 π log (cid:18) T − λ + 2 µµ (cid:19) + C p µ/ | E B | (cid:19) (5.10) on H N ( µ ) − for all p ∈ κ Z , L | E B | ≥ , µ/ | E B | ≥ c and ε ∈ (0 , ε ) .Proof of Proposition 5.2. Since T − λ ↾ H N ( µ ) − ≥ − µ , the operatorlog (cid:18) T − λ + 2 µµ (cid:19) ↾ H N ( µ ) − ≥ log(2) (5.11)is positive. Since T and T >µ/ε commute, we can use Lemma 5.3 to prove Proposition 5.2 withthe aid of (5.10). Proof of Lemma 5.3.
Using again T − λ ↾ H N ( µ ) − ≥ − µ in combination with 0 ≤ T >µ/ε ≤ T ,the operator T >µ/ε T − λ + 2 µ ≤ λ − µT − λ + 2 µ ≤ E ( µ ) − µµ (5.12)is bounded when restricted to H N ( µ ) − . Hence it is sufficient to show (5.9) on the densesubspace lin D ⊆ H N ( µ ) − given by all finite linear combinations of anti-symmetric productsof plane waves, see (3.4). Since the states in lin D are linear combinations of simultaneouseigenstates of T ↾ H N ( µ ) − and T >µ/ε ↾ H N ( µ ) − , we can restrict the argument further to theset D itself. This becomes particularly useful when writing D = S n ≥ W n ( ε ) with W n ( ε ) = n w ∈ D : (cid:16) X k >µ/ε a ∗ k a k (cid:17) w = nw o (5.13)the set of anti-symmetric products of plane waves with exactly n momentum modes occupiedin { k ∈ κ Z : k > µ/ε } .Since T >µ/ε w = 0 for w ∈ W ( ε ), we consider w ∈ W n ( ε ), n ≥
1, with || w || = 1. We call β ( w ) the eigenvalue of T and γ ( w ) the eigenvalue of T >µ/ε . It follows that γ ( w ) = h w, T >µ/ε w i > n µε − (5.14)as well as β ( w ) − γ ( w ) = h w, ( T − T >µ/ε ) w i ≥ E ( µ ) − ( n + 1) µ. (5.15)To derive the last inequality we denote the eigenvalues of − ∆ by λ i ( − ∆) ( i ≥
1, numbered26ith increasing order and counting multiplicities) and use h w, ( T − T >µ/ε ) w i = h w, ( P k ≤ µ/ε k a ∗ k a k ) w i ≥ E ( µ ) − N ( µ ) X i = N ( µ ) − n λ i ( − ∆) . (5.16)From λ i ( − ∆) ≤ µ for i ≤ N ( µ ), we obtain (5.15). The latter together with λ ≤ E ( µ ) implies β ( w ) − λ ≥ γ ( w ) − ( n + 1) µ, (5.17)and combining this with (5.14), we get h w, T >µ/ε T − λ +2 µ w i = γ ( w ) β ( w ) − λ + 2 µ ≤ γ ( w ) γ ( w ) − ( n − µ ≤ n µε − n µε − − ( n − µ . (5.18)Since the expression on the right does not exceed the value − ε , we have proven the statement. Proof of Proposition 5.2.
We start by introducing the abbreviations b k = k + 1 M + 2 ( p − P f ) , b l = l + 1 M + 2 ( p − P f ) (5.19)by which one writes the denominator in the expression defining e P ( λ, ε ) as m ( b k + b l ) + 2 M b k · b l + 1 M + 2 ( p − P f ) + T − λ. (5.20)For w ∈ H N ( µ ) − we define e w ∈ L ( κ Z ; H N ( µ ) − ) by e w ( k ) = a k w . Moreover we define theunitary operator U ∈ L ( L ( κ Z ; H N ( µ ) − )) by ( U ϕ )( k ; k , ..., k N ( µ ) − ) = ϕ (cid:0) k + 1 M + 2 (cid:0) p − N ( µ ) − X i =1 k i (cid:1) ; k , ..., k N ( µ ) − (cid:1) , (5.21)where we use the notation ( U ϕ )( k ; k , ..., k N ( µ ) − ) for the Fourier space representation of( U ϕ )( k ) ∈ H N ( µ ) − . With these definitions at hand, it is not difficult to compute h w, e P p ( λ, ε ) w i = 1 L X k,l h ( χ µ/ε e w )( k ) , U σ ( k, l ) U ∗ ( χ µ/ε e w )( l ) i (5.22) Note that we omit the p -dependence of the unitary operator U . χ ( µ/ε, ∞ ) stands for the characteristic function k χ ( µ/ε, ∞ ) ( k ) and where σ ( k, l ) = 1 L m ( k + l ) + M k · l + M +2 ( p − P f ) + T − λ . (5.23)Denoting the scalar product on L ( κ Z ; H N ( µ ) − ) by hh· , ·ii , (5.22) is rewritten as h w, e P p ( λ, ε ) w i = hh χ ( µ/ε, ∞ ) e w, U σ U ∗ χ ( µ/ε, ∞ ) e w ii (5.24)where σ is the operator on L ( κ Z ; H N ( µ ) − ) with operator-valued kernel σ ( k, l ). Next, weshow that the negative part of σ has the kernel σ − ( k, l ) = ( σ ( − k, l ) − σ ( k, l )). To this end,consider the reflection operator R defined by ( R e w )( k ) = e w ( − k ) for any e w ∈ L ( κ Z ; H N ( µ ) − ).It is straightforward to verify Rσ = σR . Moreover, Rσ is a positive operator, which can beseen as follows. The integral kernel of Rσ is given by ( Rσ )( k, l ) = σ ( − k, l ) and has the integralrepresentation σ ( − k, l ) = 1 L Z ∞ e − tk (cid:16) e − t ( k − l ) /M e − t ( M +2 ( p − P f ) + T − λ ) (cid:17) e − tl d t. (5.25)Then use the following identity for ψ ∈ L (Ω) and its Fourier transform b ψ ∈ ℓ ( κ Z ),1 L X k,l b ψ ( k ) e − t ( k − l ) /M b ψ ( l ) = Z Ω | ψ ( x ) | X k e ikx e − tk /M d x. (5.26)This together with Poisson’s summation formula (see e.g. [Gra14, Section 3.2]) and the factthat the Fourier transform of a Gaussian is a positive function implies that Rσ is a positiveoperator. Consequently, we have Rσ = | σ | since Rσ is positive and σ = ( Rσ )( Rσ ). Thepositive and negative parts of σ are thus given by σ ± = ± ( σ ± Rσ ) / σ ± ( k, l ) = ± ( σ ( k, l ) ± σ ( − k, l )) / σ − ( k, l ) = R − ddu σ ( − uk, l )d u ,and hence σ − ( k, l ) = M k · lL Z − M + 1)( k + l ) − uk · l + B ] d u, (5.27)where B = MM +2 ( p − P f ) + M ( T − λ ). Using this in combination with (5.24), we get e P p ( λ, ε ) ≥ − ML X k ,l >µ/ε a ∗ k Z − b k · b l (cid:2) ( M + 1)( b k + b l ) − u b k · b l + B (cid:3) d u a l . (5.28)28o the expression on the right we apply the following inequality which is a version of theSchur test and is easily proven by applying the Cauchy-Schwarz inequality two times, X k ,l >µ/ε a ∗ k J ( k, l ) a l ≤ X k >µ/ε k a ∗ k X l >µ/ε | J ( k, l ) | l a k (5.29)for any family of bounded operators ( J ( k, l )) k,l ∈ κ Z on F satisfying J ( k, l ) ∗ = J ( k, l ). Thisprovides e P p ( λ, ε ) ≥ − M X k >µ/ε k a ∗ k L X l >µ/ε Z − | b k · b l | l (cid:2) ( M + 1)( b k + b l ) − u b k · b l + B (cid:3) d u | {z } = f ( k ,p − P f ,T ) a k (5.30)as an operator inequality on H N ( µ ) − . Our next goal is to find a suitable function g such thatfor k > µ/ε , we have f ( k , p − P f , T ) ≤ g ( T + k ) on H N ( µ ) − . For such a function we have e P p ( λ, ε ) ≥ − M X k >µ/ε k a ∗ k g ( T + k ) a k ≥ − M T >µ/ε g ( T ) (5.31)since g ( T + k ) a k = a k g ( T ) and T >µ/ε = P k >µ/ε k a ∗ k a k .To find a suitable function g , it is helpful to check that the expression inside the squarebrackets in the denominator in (5.30) is positive for ε small enough. To see this we use b k ≥ √ εk − √ ε − √ ε ( p − P f ) ( M + 2) , (5.32)and similarly for b l , together with k , l > µ/ε and T − λ ≥ − µ on H N ( µ ) − to find( M + 1)( b k + b l ) − u b k · b l + B ≥ M ( ε − / − µ + (cid:18) − √ ε − √ ε M + 2 (cid:19) MM + 2 ( p − P f ) . (5.33)Next we use that − u b k · b l ≥ u ∈ [ − ,
0] or for u ∈ [0 , f larger and also independent of u on the respective interval. On the otherinterval, we employ 0 ≥ − u b k · b l ≥ −| u | ( b k + b l ). In both cases this leads to Z − | b k · b l | l (cid:2) ( M + 1)( b k + b l ) − u b k · b l + B (cid:3) d u ≤ | b k · b l | l (cid:2) ( M + 1)( b k + b l ) + B (cid:3) (5.34a)29 Z | b k · b l | l (cid:2) ( M + 1 − u )( b k + b l ) + B (cid:3) d u. (5.34b)In the denominators we proceed with the bound( M + 1 − u )( b k + b l ) + B ≥ | b k · b l | (cid:0) M (1 − √ ε ) + 1 − u (cid:1) . (5.35)The latter is verified by( M + 1 − u )( b k + b l ) + B ≥ (cid:18)b k + b l + ( p − P f ) M ( M + 1)( M + 2) (cid:19)(cid:18) M + 1 − u − µM b k + b l + ( p − P f ) ( M +1)( M +2) (cid:19) (5.36)on H N ( µ ) − in combination with b k + b l + ( p − P f ) M ( M + 1)( M + 2) > µ √ ε (5.37)which, in turn, follows from (5.32) and k + l ≥ µ/ε . Putting the different steps together,one obtains f ( k , p − P f , T ) ≤ e f ( k , p − P f , T,
0) + Z e f ( k , p − P f , T, u )d u (5.38)with e f ( k , p − P f , T, u ) = 1 L X l >µ/ε l ( M (1 − √ ε ) + 1 − u ) (cid:2) ( M + 1 − u )( b k + b l ) + B (cid:3) . (5.39)In the expression inside the square brackets we estimate b k and b l by (5.32) to get the lowerbound (cid:2) ... (cid:3) ≥ ( M + 1 − u ) (cid:18) √ εl + √ δk − µM ( M + 1 − u ) (cid:19) + M ( T − λ + 2 µ )+ (cid:18) M ( M + 2) − √ ε − √ ε ( M + 1 − u ) − √ δ − √ δ ( M + 1 − u ) (cid:19) ( p − P f ) ( M + 2) . (5.40)Requiring that the second line vanishes implies √ δ = M ( M + 2)(1 − √ ε ) − √ ε ( M + 1 − u ) M ( M + 2)(1 − √ ε ) + ( M + 1 − u )(1 − √ ε ) . (5.41)30ence we can bound the expression in square brackets by( M + 1 − u )( b k + b l ) + B ≥ √ ε l ( M + 1 − u ) + M β ( u, ε )( T + k − λ + 2 µ ) (5.42)with β ( u, ε ) = min { , √ δ ( M + 1 − u ) /M } . (5.43)Note that for ε small enough β (0 , ε ) = 1. Applying this to (5.39), we obtain e f ( k , p − P f , T, u ) ≤ M (1 − √ ε ) + 1 − u ) 1 L X l >µ/ε l − √ ε l ( M + 1 − u ) + M β ( u, ε )( T + k − λ + 2 µ ) . (5.44)Here we sum a non-negative and monotonically decreasing function of l so that we can applyLemma A.1. To follow the next steps with more ease, let us write(5.44) = 1 X (cid:18) L X l >µ/ε l (1 + Y l ) (cid:19) (5.45)with (all understood as operator on H N ( µ ) − ) X = 2( M (1 − √ ε ) + 1 − u ) Z, Y = √ ε ( M + 1 − u )2 Z , (5.46)and Z = M β ( u, ε )( T + k − λ + 2 µ ). Since for b > ∞ Z √ µ/ε s (1 + bs ) d s = 12 log (cid:18) εµ b (cid:19) , ∞ Z √ µ/ε s (1 + bs ) d s ≤ p µ/ε + π √ b , (5.47)we obtain the bound1 L X l >µ/ε l (1 + Y l ) ≤ π log (cid:18) εµ Y (cid:19) + 2 πL (cid:18) p µ/ε + π √ Y (cid:19) + (cid:18) p µ/επL + 6 L (cid:19) µε (1 + µε Y ) . (5.48)Using T − λ ≥ − µ on H N ( µ ) − , k ≥ µ/ε and L | E B | ≥
1, the second line is easily seen to31e bounded by a constant times e µ − / . In the first line, we estimatelog (cid:18) √ εM β ( u, ε )( T + k − λ + 2 µ )( M + 1 − u ) µ (cid:19) ≤ log (cid:18) T + k − λ + 2 µµ (cid:19) (5.49)by choosing ε sufficiently small. This together with (5.45) leads to e f ( k , p − P f , T, u ) ≤ T + k − λ + 2 µ (cid:18) log (cid:0) T + k − λ +2 µµ (cid:1) πM ( M (1 − √ ε ) + 1 − u ) β ( u, ε ) + C pe µ (cid:19) . (5.50)Recalling definition (5.7) for α ( M, ε ), we set g ( T ) = 1 T − λ + 2 µ (cid:18) α ( M, ε )4 πM log (cid:18) T − λ + 2 µµ (cid:19) + C pe µ (cid:19) (5.51)for some suitable constant C . In view of (5.38) and (5.50), it follows that f ( k , p − P f , T ) ≤ g ( T + k ) as desired. With the aid of (5.31) this leads to e P p ( λ, ε ) ≥ − T >µ/ε T − λ + 2 µ α ( M, ε )4 π log (cid:18) T − λ − µµ (cid:19) + C pe µ ! (5.52)for some constant C >
M > .
225 enters as a technical assumption and is not expected to be optimal.
Corollary 5.5.
Let
M > . and λ ≤ E ( µ ) + e P ( µ, E B ) . There exist constants c , ε > such that Ψ p ( λ, ε ) ≥ for all p ∈ κ Z , L | E B | ≥ , µ/ | E B | ≥ c and ε ∈ (0 , ε ) .Proof. Recalling the definition of Ψ p ( λ, ε ) in (3.6), we writeΨ p ( λ, ε ) = Π ⊥ ( ε )(Ψ p, ( λ, ε ) + Ψ p, ( λ, ε ))Π ⊥ ( ε ) (5.53)with Ψ p, ( λ, ε ) = ε / G ( p − P f , T − λ ) + P p ( λ ) − e P p ( λ, ε ) − K ( ε, e µ ) − d, (5.54)Ψ p, ( λ, ε ) = (1 − ε / ) G ( p − P f , T − λ ) + e P p ( λ, ε ) + d, (5.55)where K ( ε, e µ ) = ε − + ε − / p log e µ + ε / log e µ and d > p, ( λ, ε ) ≥ ε / πm log e µ − C (cid:0) ε / + d + p ε − log e µ + K ( ε, e µ ) (cid:1) (5.56)for some ( ε -independent) C >
0. With ε > e µ large enough.In line (5.55) we apply (3.17) to get(1 − ε / ) G ( p − P f , T − λ ) + d ≥ − ε / πm log (cid:18) T − λ + mµ | E B | (cid:19) (5.57)on H N ( µ ) − . Since m = 1 + M , µ/ | E B | ≥ c ≥ M as well as ( T − λ + µ ) ↾ H N ( µ ) − ≥
0, wecan estimate the logarithm further bylog (cid:18) T − λ + mµ | E B | (cid:19) ≥ log (cid:18) T − λ + µ + c µ/Mµ (cid:19) ≥ log (cid:18) T − λ + 3 µµ (cid:19) . (5.58)Proposition 5.2 gives a bound for the second term in (5.55), e P p ( λ, ε ) + d ≥ − − ε α ( M, ε )4 π log (cid:18) T − λ + 2 µµ (cid:19) . (5.59)Adding both estimates together leads toΨ p, ( λ, ε ) ≥ (cid:18) − ε / πm − − ε α ( M, ε )4 π (cid:19) log (cid:18) T − λ + 2 µµ (cid:19) (5.60)on H N ( µ ) − .The condition Ψ p ( λ, ε ) ≥ − ε / ) MM + 1 − − ε α ( M, ε ) ≥ . (5.61)This is similar to the stability condition at zero density that was derived in [GL18]. There itwas shown that the Fermi polaron defined on R is stable if MM + 1 − α ( M, ≥ M > .
225 [GL18, Theorem 1]. Since α ( M, ε ) dependscontinuously on ε , we can conclude that (5.61) holds for any given M > .
225 if we choose ε sufficiently small. This completes the proof of the corollary.33 .1 Proof of Theorem 1.2 The lower bound in (1.14) is a direct consequence of the Birman–Schwinger principle (2.2)together with Corollaries 4.4 and 5.5. As the upper bound was already discussed in Section2.1, we have completed the proof of Theorem 1.2.
As a first step we replace G ( q, τ ) by e G ( q, τ ) = 1 L X k (cid:18) mk − E B − ξ µ ( k ) M ( q − k ) + k + τ (cid:19) , (6.1)where ξ µ ( s ) = s ≤ µ ) cos (cid:16) π ( s − µ ) log e µµ (cid:17) + ( µ ≤ s ≤ µ + µ/ log e µ )1 ( s ≥ µ + µ/ log e µ ) . (6.2)Compared to G ( q, τ ) we have replaced the characteristic function χ ( µ, ∞ ) ( k ) with a smoothercutoff described by ξ µ ( k ). The error for this can be controlled by a crude estimate like | G ( q, τ ) − e G ( q, τ ) | ≤ L X k ≥ µ χ ( µ,µ + µ/ log e µ ) ( k ) M ( q − k ) + k + τ ≤ C µ ( µ + τ ) log e µ , (6.3)which is easily justified by means of (3.7). Next we write e G ( q, τ ) = L − P k g ( k ) with g ( k ) = 1 mk + | E B | − ξ µ ( k ) M ( q − k ) + k + τ , (6.4)and apply Poisson’s summation formula (see e.g. [Gra14, Section 3.2]) to find e G ( q, τ ) − π Z R g ( k )d k = 14 π (cid:18) (cid:18) πL (cid:19) X k ∈ κ Z g ( k ) − (2 π ) b g (0) (cid:19) = 12 π X z ∈ L Z z =0 b g ( z ) . (6.5)To compute b g (0) we replace ξ µ ( s ) again with χ ( µ, ∞ ) ( k ) and estimate the difference (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 π ) b g (0) − Z R (cid:18) mk + | E B | − χ ( µ, ∞ ) ( k ) M ( q − k ) + k + τ (cid:19) d k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C µ ( µ + τ ) log e µ . (6.6)34he integral can be evaluated explicitly, Z R (cid:18) mk + | E B | − χ ( µ, ∞ ) ( k ) M ( q − k ) + k + τ (cid:19) d k = πm log M +1 q + τ + mµ | E B | ! + πm log (cid:18) − F ( q , τ )2 (cid:19) (6.7)where F ( s, τ ) = M s + τ + mµ M +1 s + τ + mµ − s − s µM (cid:0) M s + τ + mµ (cid:1) ! . (6.8)The last expression is not larger than 1 + 1 /M such that for M >
1, we have (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) − F ( q , τ )2 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. (6.9)It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 π ) b g (0) − πm log M +1 q + τ + mµ | E B | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) µ ( µ + τ ) log e µ (cid:19) . (6.10)Next we need to estimate the right side in (6.5). To this end, write g ( k ) = g ( k ) + g ( k )with g ( k ) = 1 mk + | E B | , g ( k ) = ξ µ ( k ) M ( q − k ) + k + τ , (6.11)and use rotational symmetry, i.e. b g i ( z ) = b g i ( | z | e u ), i = 1 ,
2, where e u denotes the first unitvector in the ( k u , k v ) plane. We can then use integration by parts to compute the Fouriertransform for z = 0, b g ( z ) = m − ∞ Z −∞ d k u e ik u | z | ∞ Z −∞ d k v k + | E B | /m = 1 m ( i | z | ) ∞ Z −∞ d k u (cid:18) ∂ ∂k u e ik u | z | (cid:19) ∞ Z −∞ d k v k + | E B | /m = 1 im | z | ∞ Z −∞ d k u e ik u | z | ∞ Z −∞ d k v k u ( k v + | E B | /m ) − k u ( k + | E B | /m ) . (6.12)35f the last expression we estimate the absolute value to get | b g ( z ) | ≤ m | z | Z d k (cid:18) | k | ( k + | E B | /m ) + 24 | k | ( k + | E B | /m ) (cid:19) ≤ C | z | | E B | / . (6.13)The bound for | b g ( z ) | works similarly but is slightly more cumbersome. We start again with b g ( z ) = 1 im | z | | E B | / ∞ Z −∞ d k u e ik u | z | ∞ Z −∞ d k v ∂ ∂k u (cid:18) ξ µ ( k ) | E B | / M ( q − k ) + k + τ (cid:19) (6.14)for which we need to compute the different derivatives. A straightforward computation shows (cid:12)(cid:12)(cid:12) ∂ n ∂k nu ξ µ ( k ) (cid:12)(cid:12)(cid:12) ≤ C (log e µ ) n µ n/ χ ( µ,µ + µ/ log e µ ) ( k ) , n ∈ { , , } . (6.15)Abbreviating the denominator as D ( k ) = M ( q − k ) + k + τ it is not difficult to verify χ ( µ,µ + µ/ log e µ ) ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂k u D ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) µ / ( k + τ ) + µ / ( k + τ ) (cid:19) , (6.16) χ ( µ,µ + µ/ log e µ ) ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂k u D ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) k + τ ) + µ ( k + τ ) (cid:19) , (6.17) χ ( µ,µ + µ/ log e µ ) ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂k u D ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) µ / ( k + τ ) + µ / ( k + τ ) + µ / ( k + τ ) (cid:19) . (6.18)To illustrate this for the first line, we compute (cid:12)(cid:12)(cid:12)(cid:12) ∂∂k u D ( k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) D ( k ) (cid:18) M ( k u − q u ) + 2 k u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) | k − q | D ( k ) + | k | D ( k ) (cid:19) (6.19)and use D ( k ) ≥ M ( k − q ) and k ≤ µ in combination with a balanced Cauchy-Schwarzestimate. This leads to (cid:12)(cid:12)(cid:12)(cid:12) ∂∂k u D ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) D ( k ) / + √ µD ( k ) (cid:19) ≤ C (cid:18) √ µD ( k ) + √ µD ( k ) (cid:19) (6.20)from which the bound in (6.16) follows by D ( k ) ≥ k + τ . The other two lines are obtainedin close analogy.Summing up the different combinations we obtain (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂k u (cid:18) ξ µ ( k ) | E B | / M ( p − k ) + k + τ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) C (log e µ ) e µ / (cid:18) χ ( µ,µ + µ/ log e µ ) ( k ) 1 k + τ + χ ( µ, ∞ ) ( k ) X j =2 µ j − ( k + τ ) j (cid:19) (6.21)by which we can estimate the integral Z R (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂k u (cid:18) ξ µ ( k ) | E B | / M ( q − k ) + k + τ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d k ≤ C (cid:18) µ ( µ + τ ) log e µ (cid:19) . (6.22)Together with (6.13) and (6.14), this gives | b g ( z ) | ≤ C | z | | E B | / (cid:18) µ ( µ + τ ) log e µ (cid:19) . (6.23)The remaining series can be bounded as X z ∈ L Z z =0 | z | − | E B | − / = ( L | E B | / ) − X z ∈ Z z =0 | z | − ≤ C (6.24)because of L | E B | ≥ X z ∈ Z z =0 | z | = X n,m ≥ n + m ) / + X n ≥ n ≤ ∞ Z (cid:18) πs + 4 s (cid:19) d s ≤ C (6.25)by the integral test of convergence.We conclude that the absolute value of the right side in (6.5) is bounded from above by12 π X z ∈ L Z z =0 | b g ( z ) | ≤ C (cid:18) µ ( µ + τ ) log e µ (cid:19) . (6.26)Hence the proof of the lemma is complete. A Replacing sums by integrals
For a short proof of the following lemma, see [LM19, Appendix B].
Lemma A.1. (a) Let f : [0 , ∞ ) → [0 , ∞ ) be monotonically decreasing. Then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X k f ( k ) − π ∞ Z f ( t ) t d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ πL ∞ Z f ( t )d t + 3 f (0) L . (A.1)37 b) Let m ≥ and f : [ m, ∞ ) → [0 , ∞ ) be monotonically decreasing. Then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X k ≥ m f ( k ) − π ∞ Z √ m f ( t ) t d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ πL ∞ Z √ m f ( t )d t + (cid:16) √ mπL + 6 L (cid:17) f ( m ) . (A.2) B Completing the proof of Lemma 4.2
It remains to analyze the condition F ( T − λ, r ) ≥
0. For that we approximate F ( T − λ, r ) by F ( n ) ( T − λ, r ) where the operator F ( n ) ( T − λ, r ) arises by replacing the function G (0 , τ ) in(4.26) by G ( n ) (0 , τ ) = 1 L X k ≤ n (cid:18) k − E B − χ ( µ, ∞ ) ( k ) k + τ (cid:19) . (B.1)Note that G ( n ) (0 , τ ) → G (0 , τ ) as n → ∞ for every τ > − µ . Thus, G ( n ) (0 , T − λ ) ψ → G µ (0 , T − λ ) ψ as n → ∞ for every ψ ∈ D (recall that D ⊂ H N ( µ ) − is the set of all anti-symmetric product states, see (3.4)). Since D forms a total set of eigenstates of G (0 , T − λ )on H N ( µ ) − , its linear hull lin D ⊆ H N ( µ ) − is a domain of essential self-adjointness for thisoperator. Furthermore, G ( n ) (0 , T − λ ) ≥ G ( n ) (0 , − µ − e P ( µ, E B )) and thus, by the convergenceof G ( n ) (0 , τ ) and the fact that G (0 , − µ − e P ( µ )) ≥ C log e µ , it follows that there is a c > G ( n ) (0 , T − λ ) − r > c for n large enough. Hence as n → ∞ , ( G ( n ) (0 , T − λ ) − r ) − → ( G (0 , T − λ ) − r ) − and F ( n ) ( T − λ, r ) → F ( T − λ, r ) strongly.Using the pull-through formula (2.6), we can write F ( n ) ( T − λ, r ) = T − λ − L X k ≤ µ ( G ( n ) (0 , T − k − λ ) − r ) − + 1 L X k ,l ≤ µ a k ( G ( n ) (0 , T − k − l − λ ) − r ) − a ∗ l (B.2)on H N ( µ ) . Assuming that the last term in (B.2), which we call P ( n ) ( T − λ, r ) in the following,is a positive operator on H N ( µ ) , we obtain F ( n ) ( T − λ, r ) ≥ E ( µ ) − λ − L X k ≤ µ ( G ( n ) (0 , E ( µ ) − k − λ ) − r ) − , (B.3)since T ≥ E ( µ ) on H N ( µ ) . In view of (4.25), this completes the proof of Lemma 4.2.38t remains to show P ( n ) ( T − λ, r ) ≥
0. For ψ ∈ H N ( µ ) , L h ψ, P ( n ) ( T − λ, r ) ψ i = ∞ Z X k ,l ≤ µ h ψ, a k exp( − t [ G ( n ) (0 , T − k − l − λ ) − r ) a ∗ l ψ i d t = ∞ Z exp( − t [ L − X p ≤ n p − E B − r ]) I ( n ) ( t ) d t (B.4)with I ( n ) ( t ) = X k ,l ≤ µ h ψ, a k Q µ m -sum as X k ,l ≤ µ Y q ∈ A n m ( q ) =0 t m ( q ) m ( q )! c m ( q ) ∞ Z d s q h ψ, a k Q p ∈ A n m ( p ) =0 e − ( p + T − k − l − λ ) s /m ( p ) p a ∗ l ψ i Y q ∈ A n m ( q ) =0 t m ( q ) m ( q )! c m ( q ) ∞ Z d s q (cid:12)(cid:12)(cid:12)(cid:12) X k ≤ µ Y p ∈ A n m ( p ) =0 e − ( p + T − k − λ ) s /m ( p ) p a ∗ k ψ (cid:12)(cid:12)(cid:12)(cid:12) . (B.9)This yields I ( n ) ( t ) ≥ h ψ, P ( n ) ( T − λ, r ) ψ i ≥ n ∈ N . Acknowledgements.
I am very grateful to Ulrich Linden for introducing me to the Fermi polaron as well as for hiscontributions to this project in its early stage.
References [AGHH05] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden. Solvable models in quantum mechan-ics.
AMS Chelsea Publishing, 2nd edition . (2005)[BM10] G.M. Bruun and P. Massignan. Decay of polarons and molecules in a strongly polarized Fermigas.
Phys. Rev. Lett. 105:020403 . (2010)[Che06] F. Chevy. Universal phase diagram of a strongly interacting Fermi gas with unbalanced spinpopulations.
Phys. Rev. A 74, 063628 . (2006)[CM09] F. Chevy and C. Mora. Ground state of a tightly bound composite dimer immersed in a Fermisea.
Phys. Rev. A 80, 033607 . (2009)[CG08] R. Combescot and S. Giraud. Normal state of highly polarized Fermi gases: Full many-bodytreatment.
Phys. Rev. Lett. 101, 050404 . (2008)[CDFMT12] M. Correggi, G.F. Dell’Antonio, D. Finco, A. Michelangeli and A. Teta. Stability for a systemof N fermions plus a different particle with zero-range interactions. Rev. Math. Phys. 24(7),1250017, 32 . (2012)[DFT94] G.F. Dell’Antonio, R. Figari and A. Teta. Hamiltonians for systems of N particles interactingthrough point interactions Ann. Inst. H. Poincar´e Phys. Th´eor.
J. Phys. A 37(39), 9157–9173 . (2004)[Fr¨o54] H. Fr¨ohlich. Electrons in lattice fields.
Adv. Phys 3(11):325 . (1954)[Gra14] L. Grafakos. Classical Fourier Analysis. Graduate texts in mathematics.
Springer, 3rd edition. (2014)[GL18] M. Griesemer and U. Linden. Stability of the two-dimensional Fermi polaron.
Lett. Math. Phys.108(8), 1837–1849 . (2018) GL19] M. Griesemer and U. Linden. Spectral Theory of the Fermi Polaron.
Ann. Henri Poincar´e 20(6),1931–1967 . (2019)[GD15] F. Grusdt and E. Demler. New theoretical approaches to Bose polarons.
Preprint . (2015)arXiv:1510.04934[Lan33] L.D. Landau. ¨Uber die Bewegung der Elektronen in Kristallgitter.
Phys. Z. Sowjetunion. 3,644–645 . (1933)[LP48] L.D. Landau and S.I. Pekar.
Zh. Eksp. Teor. Fiz. 18, 419 . (1948)[LLP53] T.D. Lee, F.E. Low and D. Pines. The motion of slow electrons in a polar crystal.
Phys. Rev.90(2), 297–302 . (1953)[Lin17] U. Linden. Energy estimates for the two-dimensional Fermi polaron. PhD thesis.
Universit¨atStuttgart . (2017)[LM19] U. Linden and D. Mitrouskas. High density limit of the Fermi polaron with infinite mass.
Lett.Math. Phys. 109, 1805–1825 . (2019)[MO18] A. Michelangeli and A. Ottolini. Multiplicity of self-adjoint realisations of the (2 + 1)-fermionicmodel of Ter-Martirosyan-Skornyakov type.
Rep. Math. Phys. 81(1):1–38 . (2018)[Min11] R. Minlos. On point-like interaction between n fermions and another particle. Mosc. Math. J.11(1), 113–127, 182 . (2011)[MS17] T. Moser and R. Seiringer. Stability of a fermionic N + 1 particle system with point interactions. Commun. Math. Phys. 356(1), 329–355 . (2017)[MS19] T. Moser and R. Seiringer. Energy contribution of a point interacting impurity in a Fermi gas.
Ann. Henri Poincar´e 20, 1325–1365 . (2019)[Par11] M.M. Parish. Polaron-molecule transitions in a two-dimensional Fermi gas.
Phys. Rev. A 83,051603 . (2011)[PL13] M.M. Parish and J. Levinsen. Highly polarized Fermi gases in two dimensions.
Phys. Rev. A 87,033616 . (2013)[Pek54] S.I. Pekar. Untersuchung ¨uber die Elektronentheorie der Kristalle.
Berlin, Akad. Verlag . (1954)[PS08] N. Prokof’ev and B. Svistunov. Fermi-polaron problem: Diagrammatic monte carlo method fordivergent sign-alternating series.
Phys. Rev. B 77, 020408 . (2008)[PDZ09] M. Punk, P.T. Dumitrescu and W. Zwerger. Polaron-to-molecule transition in a strongly imbal-anced fermi gas.
Phys. Rev. A 80, 053605 . (2009)[SEPD12] R. Schmidt, T. Enss, V. Pietil¨a and E. Demler. Fermi polarons in two dimensions.
Phys. Rev. A85, 0216029 . (2012)[SL15] R. Schmidt and M. Lemeshko. Rotation of quantum impurities in the presence of a many-bodyenvironment.
Phys. Rev. Lett. 114, 20300 . (2015) -mail: [email protected]@mathematik.uni-stuttgart.de