On the impact of surface defects on a condensate of electron pairs in a quantum wire
aa r X i v : . [ m a t h - ph ] J un On the impact of surface defects on acondensate of electron pairs in a quantumwire
Joachim Kerner Department of Mathematics and Computer ScienceFernUniversit¨at in Hagen58084 HagenGermany,
Abstract
In this paper we are interested in understanding the impact of surface defectson a condensate of electron pairs in a quantum wire. Based on previous results weestablish a simple mathematical model in order to account for such surface effects.For a system of non-interacting pairs, we will prove the destruction of the condensatein the bulk. Finally, taking repulsive interactions between the pairs into account, wewill show that the condensate is recovered for pair densities larger than a critical onegiven the number of the surface defects is not too large. E-mail address:
Introduction
In this paper we are interested in establishing a mathematical model which allows us tounderstand the impact of surface defects on a (Bose-Einstein) condensate of electron pairsin a simple quantum wire, namely the half-line R + = [0 , ∞ ). It is motivated by theseminal work of Bardeen, Schrieffer and Cooper [Coo56, BCS57] which demonstrated thatthe superconducting phase in (type-I) superconductors results from a coherent behaviour ofpairs of electrons (Cooper pairs) similar to the one occurring in Bose gases (Bose-Einsteincondensation) [MR04]. Discovered first by Onnes, the most striking experimental feature ofsuperconductors is a vanishing of the electrical resistance below some critical temperature[Onn91].In a superconductor, the formation of a Cooper pair is a result of the interaction oftwo electrons with the lattice constituting the solid (electron-phonon-electron interaction).Due to the negative binding energy of each formed pair, the many-particle ground stateof the superconductor (which itself is formed of pairs only) is separated from the excitedstates by a finite energy gap ∆ > We consider the quantum wire which is modelled by the half-line R + = [0 , ∞ ). On thisquantum wire we place, as in [Ker], a system of two interacting electrons (with same spin)whose Hamiltonian shall formally be given by H p = − ∂ ∂x − ∂ ∂y + v b ( | x − y | ) (2.1)2ith a binding-potential v b : R + → R + defined as v b ( x ) := ( ≤ x ≤ d , ∞ else . (2.2)Due to the binding potential, the two electrons form a pair whose spatial extension ischaracterised by the parameter d >
0. We refer to [Ker] where a mathematically rigorousrealisation of (2.1) was obtained via the construction of a suitable quadratic form on L a (Ω) := { ϕ ∈ L (Ω) | ϕ ( x, y ) = − ϕ ( y, x ) } with Ω := { ( x, y ) ∈ R | | x − y | ≤ d } being the two-particle configuration space.Now, in order to incorporate (localised) surface effects we extend our Hilbert space.More explicitly, we shall be working on the direct sum H = L a (Ω) ⊕ ℓ ( N ) (2.3)which means that we couple the (continuous) quantum wire to a discrete graph which issupposed to model surface defects. From a physical point of view this seems reasonable ina regime where the surface defects are relatively small compared to the bulk.Furthermore, the Hamiltonian of a free pair (meaning without surface-bulk interactions)shall be given by H = H p ⊕ L ( γ ) , (2.4) L ( γ ) being the (weighted) graph Laplacian, i.e., f ∈ ℓ ( N ),( L ( γ ) f )( n ) := X m γ nm ( f ( m ) − f ( n )) (2.5)where γ := ( γ n,m ∈ R + ) = γ T is the associated edge weight matrix [Chu97]. Since ourgraph is actually assumed to be a path graph (or chain graph), one sets γ mn = δ | n − m | , e n with ( e n ) n ∈ N ⊂ R + . In order to study the effect of the surface defects on a condensate of electron pairs we shallinvestigate the condensation phenomenon similar to [Ker]. Of course, given one wishesto describe the dynamics of a pair one would like to add a non-diagonal interaction termto (2.4) which describes the coupling between the bulk and the surface. However, sincewe are interested in quantum statistical properties only, we will simplify the discussion inthis paper by modelling the interaction as a diagonal operator. The coupling between thesurface and the bulk is then realised through the “heat bath” [Rue69]. More explicitly, weconsider the one-pair operator H α ( γ ) := H p ⊕ ( L ( γ ) − α ) (3.1)3ith α ≥ R + is replaced by the interval [0 , L ], L >
0, and oneconsiders the restriction H Lα ( γ ) := H p | L (Ω L ) ⊕ ( L ( γ ) − α ) | C n ( L ) (3.2)of (3.1) defined on H L = L (Ω L ) ⊕ C n ( L ) (3.3)where Ω L := { ( x, y ) ∈ Ω | ≤ x, y ≤ L } and n ( L ) ∈ N refers to the number of surfacedefects up to length L of the wire.Since H Lα ( γ ) is a direct sum of two operators, one has σ ( H Lα ( γ )) = σ ( H p | L (Ω L ) ) ∪ σ (( L ( γ ) − α ) | C n ( L ) ). As a consequence, H Lα ( γ ) has purely discrete spectrum (see [Ker]for a discussion of H p | L (Ω L ) ). In the following, the eigenvalues of H Lp := H p | L (Ω L ) shallbe denoted by E n ( L ) and the corresponding eigenfunctions by ϕ n , n ∈ N . Similarly,the eigenvalues of ( L ( γ ) − α ) | C n ( L ) by λ j ( L ) and the associated eigenfunctions by f j , j = 1 , ..., n ( L ). In both cases, the eigenvalues are counted with multiplicity.Now, as a first result we establish the following. Proposition 3.1.
For all sequences of edge weights ( e n ) n ∈ N ⊂ R + one has inf σ ( H Lα ( γ )) = − α . (3.4) Furthermore, E ( L ) ≥ E := π d and lim L →∞ E ( L ) = E . (3.5) Proof.
The first equation follows directly from the fact that zero is the lowest eigenvalue tothe discrete Laplacian associated with the constant eigenfunction (1 , , , ..., T ∈ C n ( L ) .The second part of the statement was proved in [Lemma 3.1, [Ker]].In order to investigate condensation of pairs we will work, as customary in statisticalmechanics, in the grand canonical ensemble [Ver11, Rue69]. The associated Gibbs state is ω Lβ,µ L ( · ) := Tr F b ( e − β (Γ( H Lα ( γ )) − µ L N) [ · ]) Z ( β, µ L ) , (3.6)where β = T ∈ (0 , ∞ ) is the inverse temperature, µ L ∈ ( −∞ , µ max ( L )) the chemicalpotential (with µ max ( L ) specified later) and Z ( β, µ L ) = Tr F b ( e − β (Γ( H Lα ) − µ L N) ) the partitionfunction. Furthermore, F b is the bosonic Fock space over H L ,N = n ( L ) X j =1 a ∗ j a j + ∞ X n =0 a ∗ n a n (3.7)4he number operator andΓ( H Lα ( γ )) = n ( L ) X j =1 ( λ j ( L ) − α ) a ∗ j a j + ∞ X n =0 E n ( L ) a ∗ n a n (3.8)the second quantisation of H Lα ( γ ), see [MR04, BHE08] for more details. Note here that { a ∗ j , a j } are the creation and annihilation operators corresponding to the states { ⊕ f j } n ( L ) j =1 and { a ∗ n , a n } the ones corresponding to the states { ϕ n ⊕ } ∞ n =0 .Most importantly, in the grand-canonical ensemble there is an explicit formula for thenumber of pairs occupying a given eigenstate [Rue69]: for every state ϕ n ⊕ n ϕ n := a ∗ n a n , the number of pairs occupying this state is ω Lβ,µ L ( n ϕ n ) = 1 e β ( E n ( L ) − µ L ) − . (3.9)An equivalent formula applies to any element of the form 0 ⊕ f j , setting n f j := a ∗ j a j .For the Hamiltonian (3.8), the thermodynamic limit shall then be realised as the limit L → ∞ such that ρ = 1 L n ( L ) X j =1 ω Lβ,µ L ( n f j ) + ∞ X n =0 ω Lβ,µ L ( n ϕ n ) (3.10)holds for all values of L with µ L denoting the sequence of the chemical potentials and ρ > ϕ n ⊕ n ∈ N , is macroscopicallyoccupied in the thermodynamic limit iflim sup L →∞ e β ( E n ( L ) − µ L ) − > ϕ is macroscopically occupied in thethermodynamic limit given the underlying Hilbert space is L (Ω L ) only. In contrast tothis, we obtain the following result when working on H L , i.e., when including the surfacedefects. Theorem 3.2 (Destruction of the condensate in the bulk I) . Assume that H Lα ( γ ) is givenwith an arbitrary sequence of edge weigths ( e n ) n ∈ N ⊂ R + . Then, for the associated Gibbsstate and all pair densities ρ > , no bulk state ϕ n ⊕ , n ∈ N , is macroscopically occupiedin the thermodynamic limit. Actually, one has lim L →∞ L e β ( E n ( L ) − µ L ) − , ∀ n ∈ N . Proof.
Since µ max ( L ) = inf σ ( H Lα ( γ )) in the non-interacting case [Rue69], Proposition 3.1implies that µ L ∈ ( −∞ , − α ). The result then readily follows from (3.11) taking Proposi-tion 3.1 into account. 5 On the condensate in the bulk in the presence ofsurface pair interactions
In the previous section we have seen, by Theorem 3.2, that the condensate of electron pairsin the bulk is destroyed through the presence of surface defects. However, since the defectsare imagined relatively small when compared to the bulk, (repulsive) interactions betweenthe pairs should be taken into account for large pair surface densities.In order to account for those interactions, we pursue a (quasi) mean-field approach.More explicitly, the first term on the right-hand side of (3.8) (i.e., the free Hamiltonianassociated with the discrete graph) shall be replaced by n ( L ) X j =1 ( λ j ( L ) − α + λρ s ( µ L , L )) a ∗ j a j := h L ( α, λ ) , (4.1)where ρ s ( µ L , L ) ≥ C n ( L ) ; see eq. (4.4)below. Furthermore, λ > Remark 4.1.
Note that the interaction term in the standard mean-field approach is λ N V with V the associated volume, see [MV99, Ver11].Since we can write h L ( α, λ ) as in (4.1) we conclude that the eigenvalues λ j ( L ) areeffectively only shifted by λρ s ( µ, L ) − α . Accordingly, the problem thereby reduces to aneffective non-interacting “particle” model and one has µ L < min { λρ s ( µ L , L ) − α, E ( L ) } (4.2)for the sequence of chemical potentials µ L [Rue69], taking into account that the lowesteigenvalue of the Laplacian is zero. In particular, µ max ( L ) = min { λρ s ( µ L , L ) − α, E ( L ) } .Furthermore, µ L and the surface pair density ρ s ( µ L , L ) shall be chosen in a way such that ρ = lim k →∞ L k n ( L k ) X j =1 e β [( λ j ( L k ) − α + λρ s ( µ Lk ,L k ) ) − µ Lk ] − ∞ X n =0 e β ( E n ( L k ) − µ Lk ) − , (4.3)for a subsequence µ L k together with ρ s ( µ L , L ) = 1 n ( L ) n ( L ) X j =1 e β [( λ j ( L ) − α + λρ s ( µ L ,L )) − µ L ] − . (4.4)In the rest of the section we shall assume that L/n ( L ) is bounded from above and that µ L k converges to a (possibly negative infinite) limit value µ ≤ E (from eq. (4.3) and eq. (4.4)we indeed conclude that there are values and, in particular, arbitrarily large/small valuesof ρ > heorem 4.2. Let µ L ∈ ( −∞ , E ( L )) be a corresponding sequence of chemical potentialswith limit value µ ≤ E . Then lim L →∞ (cid:18) n ( L ) L ρ s ( µ L , L ) + ρ ( µ L , L ) (cid:19) = ρ − √ π ∞ X n =1 Z ∞ e β π n d e β ( x − µ ) − x , (4.5) where ρ ( µ L , L ) := ω Lβ,µ L ( n ϕ ) /L .Proof. Starting from (4.3), the statement follows directly from formula (3.4) of [Ker] setting ~ = 1, m e = 1 / d by d/ √ L →∞ Ln ( L ) =: δ with 0 ≤ δ < ∞ then, setting ρ := lim L →∞ ρ ( µ L , L ),lim L →∞ ρ s ( µ L , L ) = δ ρ − √ π ∞ X n =1 Z ∞ e β π n d e β ( x − µ ) − x ! − δρ =: ˜ ρ ( µ, δ ) − δρ , (4.6)for a limit value µ ≤ E . From a physical point of view it is also interesting to write˜ ρ ( µ, δ ) = δ ( ρ − ρ exc ) , (4.7)where ρ exc = ρ exc ( β, µ ) equals the second term within the brackets in eq. (4.6). Note that ρ exc is the density of pairs occupying all excited (eigen-)states (i.e., ϕ n ⊕ n ≥
1) inthe bulk in the thermodynamic limit.
Lemma 4.3 (Destruction of the condensate in the bulk II) . Assume that λ > and ˜ ρ ( µ, δ ) < E + ανλ (4.8) for some ν > and µ the limit point of µ L . Then lim L →∞ L e β ( E n ( L ) − µ L ) − , ∀ n ∈ N . (4.9) Proof.
By relation (4.2) and the assumptions we conclude that µ < E − ε (4.10)for the limit point of µ L and for some constant ε >
0. Consequently, by Proposition 3.1we get lim L →∞ L e β ( E ( L ) − µ L ) − , ∀ n ∈ N , (4.11)which yields the statement since the bulk ground state is occupied the most.7e immediately obtain the following corollary which is particularly interesting from aphysical point of view. Corollary 4.4.
Assume that δ = 0 . Then, for all values λ > , (4.9) holds for the states ϕ n ⊕ , n ∈ N .More generally, if δ · ρ < E + ανλ (4.12) for some ν > and λ > , then (4.9) holds for the states ϕ n ⊕ , n ∈ N . Remark 4.5.
Corollary 4.4 implies that the condensate of electron pairs in the bulk (whichexists due to [Theorem 3.3, [Ker]] whenever no surface defects are present) is destroyed forarbitrarily large repulsive (quasi) mean-field interactions if the number of surface defectsis large, i.e., of order larger than L .In addition, Lemma 4.3 implies that the condensate in the bulk is destroyed for arbi-trarily large pair densities ρ > λ > α ≥ Theorem 4.6 (Reconstruction of the condensate) . If δ, λ > then there is a critical pairdensity ρ crit = ρ crit ( β, δ, α, λ ) > such that for all pair densities ρ > ρ crit one has lim L →∞ L e β ( E ( L ) − µ L ) − > . Proof.
Assume to the contrary that such a critical pair density doesn’t exist. Then thereexist arbitrarily large ρ > ρ = 0.Now, in a first step pick such a ρ , for given values β, δ, α, λ >
0, so large that˜ ρ ( E , δ ) > E + αλ . (4.13)Using the same reasoning as in the proof of Lemma 4.3 one then concludes that µ ≤ E (4.14)for the limit point of the associated sequence of chemical potentials, taking Proposition 3.1into account.In a second step we use (4.13) and (4.14) in (4.4) to conclude, for such ρ , the existenceof a constant C > | ρ s ( µ L , L ) | < C (4.15)for all L ≥ L , L large enough. Finally, increasing ρ even more then yields a contradictionwith (4.6) and consequently the statement. Remark 4.7.
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