On Existence of Ground States in the Spin Boson Model
aa r X i v : . [ m a t h - ph ] F e b On Existence of Ground Statesin the Spin Boson Model
David Hasler ∗ Benjamin Hinrichs † Oliver Siebert ‡ Friedrich-Schiller-University Jena
Department of MathematicsErnst-Abbe-Platz 207743 JenaGermany
March 1, 2021
Abstract
We show the existence of ground states in the massless spin boson model without any infraredregularization. Our proof is non-perturbative and relies on a compactness argument. It worksfor arbitrary values of the coupling constant under the hypothesis that the second derivativeof the ground state energy as a function of a constant external magnetic field is bounded.
The spin boson model describes a quantum mechanical two-level system which is linearly coupledto a quantized field of bosons. If the bosons are relativistic and massless, the model is used asa simplified caricature describing an atom, coarsely approximated by two states, coupled to thequantized electromagnetic field. Although the model has been extensively investigated, see forexample [Spo98, AH97, Gér00] and references therein, it is still an active area of research, cf.[BBKM17, DM20].If this system has a ground state, i.e., if the infimum of the spectrum is an eigenvalue, thisphysically means that it exhibits binding. Furthermore, ground states are a necessary ingredient tostudy scattering theory in quantum field theories. In the case of massless bosons or photons in R d ,we have the dispersion relation ω ( k ) = | k | . As a consequence, the infimum of the spectrum is notisolated from the rest of the spectrum and establishing existence of a ground state is non-trivial.If one imposes a mild infrared regularization of the interaction function f , such that the quotient f /ω is square-integrable, e.g., in the case d = 3 if we have δ > − / such that f ( k ) ∼ | k | δ forsmall photon momentum | k | , then existence of ground states has been shown [Spo89, Spo98, AH95,AH97, BFS98b, BFS98a, Gér00, LMS02] and its analytic dependence on coupling parameters hasbeen established [GH09]. However, in models of physical interest where d = 3 , the coupling functiontypically has the behavior f ( k ) ∼ | k | − / and f /ω is no longer square-integrable. In such a situation ∗ [email protected] † [email protected] ‡ [email protected] . Hasler, B. Hinrichs, O. Siebert 2 the model is infrared-critical in the sense that an infrared problem may occur and a ground stateceases to exist. Such a behavior was most prominently observed for translation invariant models in[Frö73, HH08, DH19], see also references therein. Moreover, the absence of ground states was shownfor the Nelson model [LMS02] as well as for generalized spin boson models [AHH99], provided anonvanishing expectation condition is satisfied. However, it may also happen in the infrared-criticalcase that the infrared divergences cancel and a ground state exists. Heuristically, the reason behindthis cancellation is an underlying symmetry of the model. In particular, existence of ground stateshave been shown for models of non-relativistic quantum electrodynamics [BFS99, GLL01, HH11a,HH11c, BCFS07, HS20]. Due to the absence of diagonal entries in the coupling matrix, Herbst andthe first author [HH11b] proved that the spin boson model does actually exhibit a ground stateeven in the infrared-critical case, see also [BBKM17] for a recent alternative proof providing newinsight.In this paper, we consider couplings which are more singular than in [HH11b, BBKM17] andprove the existence of a ground state in the spin boson model, e.g., in d = 3 for any coupling f ( k ) ∼ | k | δ for | k | → with δ > − , provided an energy bound is satisfied. We note, this result isoptimal in the sense that for δ = − the field operator is no longer bounded in terms of the freefield energy. In contrast to previous results, our result is non-perturbative and holds for all valuesof the coupling constant as long as the energy inequality holds. Let us be more precise on thestatement. Denote by ω : R d → [0 , ∞ ) the boson dispersion relation and f : R d → R the interactionof the quantum field and the two-level system. Then, the lower-bounded and self-adjoint Hamiltonoperator describing the spin boson model acts on the Hilbert space C ⊗ F , with F being the usualFock space on L ( R d ) , and is given as H ( ω, f ) = σ z ⊗ + ⊗ d Γ( ω ) + Z R d f ( k ) σ x ⊗ ( a † k + a k ) d k. (1.1)Here, d Γ( ω ) denotes the second quantization of the operator of multiplication by ω , moreover a k , a † k are the distributions describing annihilation and creation operators, respectively, and σ x and σ z denote the Pauli matrices. A more rigorous definition can be found in Section 2. For the energyinequality we consider e n ( µ ) = inf σ ( H ( ω n , f ) + µ ( σ x ⊗ )) , where the sequence ( ω n ) n ∈ N convergesto ω uniformly and is chosen, such that f /ω n is square-integrable. The parameter µ ∈ R can herebyby interpreted as an external magnetic field. Explicitly, we assume that the second derivative of e n ( µ ) exists at µ = 0 and is bounded as n → ∞ , for our result to hold. This assumption isrelated to a bound on the magnetic susceptibility of a continuous Ising model. We note that sucha bound has been shown in our situation for the case d = 3 and δ = − / [Spo89]. For d = 3 and δ ∈ ( − / , − , the bound has been shown to hold for the discrete Ising model, cf. [Dys69]and references therein. We believe that a proof of our assumption can be obtained by taking acontinuum limit of the discrete Ising model. As a consequence of our result, non-existence of aground state for large coupling would imply the divergence of the magnetic susceptibility in thecorresponding Ising model.For the proof of our result, we utilize that the existence of ground states for the infrared regularsituation has been established using a variety of techniques. Hence, if we consider H ( ω n , f ) asabove, then a ground state ψ n exists. We then prove these ground states lie in a compact set andhence there exists a strongly convergent subsequence ( ψ n k ) k ∈ N . The limit of this sequence will bethe ground state of H ( ω, f ) . On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 3
Throughout this paper we assume d ∈ N and write h = L ( R d ) for the state space of a single boson.Then, let F be the bosonic Fock space defined by F = C ⊕ ∞ M n =1 F ( n ) with F ( n ) = L ( R nd ) , (2.1)where we symmetrize over the n R d -variables in each component. We write an element ψ ∈ F as ψ = ( ψ ( n ) ) n ∈ N and define the vacuum Ω = (1 , , , . . . ) .For a measurable function ω : R d → R , we define d Γ( ω ) = 0 ⊕ ∞ M n =1 ω ( n ) with ω ( n ) ( k , . . . , k n ) = n X i =1 ω ( k i ) (2.2)as operators on F . Further, for f ∈ h , we define the annihilation operator a ( f ) and creationoperator a † ( f ) using a ( f )Ω = 0 , a † ( f )Ω = f and for g ∈ F ( n ) ( a ( f ) g )( k , . . . , k n − ) = √ n Z f ( k ) g ( k, k , . . . , k n − ) d k ∈ F ( n − , (2.3) ( a † ( f ) g )( k , . . . , k n , k n +1 ) = 1 √ n + 1 n +1 X i =1 f ( k i ) g ( k , . . . , b k i , . . . , k n +1 ) ∈ F ( n +1) , (2.4)where b k i means that k i is omitted from the argument. One can show that these operators can beextended to closed operators on F that satisfy ( a ( f )) ∗ = a † ( f ) . From the creation and annihilationoperator, we define the field operator ϕ ( f ) = a ( f ) + a † ( f ) . (2.5)The following properties are well-known and can for example be found in [RS75, Ara18]. Lemma 2.1.
Let ω, ω ′ : R d → R and f ∈ h . Then(i) d Γ( ω ) and ϕ ( f ) are self-adjoint.(ii) If ω ′ ≥ ω , then d Γ( ω ′ ) ≥ d Γ( ω ) . Especially, if ω ≥ , then d Γ( ω ) ≥ .(iii) Assume ω > almost everywhere and ω − f ∈ h . Then ϕ ( f ) and a ( f ) are d Γ( ω ) / -boundedand for ψ ∈ D ( d Γ( ω ) / ) we have k a ( f ) ψ k ≤ k ω − f kk d Γ( ω ) ψ k and k ϕ ( f ) ψ k ≤ k ( ω − + 1) g kk ( d Γ( ω ) + 1) ψ k . In particular, ϕ ( g ) is infinitesimally d Γ( ω ) -bounded. Now, let H = C ⊗ F ∼ = F ⊕ F , (2.6) On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 4 where the unitary equivalence is implemented by ( v , v ) ⊗ ψ v ψ ⊕ v ψ . Furthermore, let σ x and σ z be the usual × Pauli-matrices σ x = (cid:18) (cid:19) and σ z = (cid:18) − (cid:19) . (2.7)For a measurable function ω : R d → R and f ∈ h , we define the spin boson Hamiltonian on H as H ( ω, f ) = σ z ⊗ + ⊗ d Γ( ω ) + σ x ⊗ ϕ ( f ) . (2.8) Lemma 2.2.
Assume that ω : R d → R is measurable and almost everywhere positive, f ∈ h , ω − / f ∈ h , and µ ∈ R . Then H ( ω, f ) + µσ x ⊗ defines a lower-bounded self-adjoint operator onthe domain D ( ⊗ d Γ( ω )) . Any core for ⊗ d Γ( ω ) is a core for H ( ω, f ) + µσ x ⊗ .Proof. The operator K = σ z ⊗ + ⊗ d Γ( ω ) + µσ x ⊗ is self-adjoint as a sum of a self-adjointoperator with bounded self-adjoint operators and has domain D ( K ) = D (1 ⊗ d Γ( ω )) . Moreover,it is bounded from below since d Γ( ω ) is non-negative by Lemma 2.1 and σ x , σ z are bounded. Thesymmetric operator σ x ⊗ ϕ ( f ) is infinitesimally K -bounded by Lemma 2.1. Hence, the statementfollows from the Kato-Rellich theorem (cf. [RS75, Theorem X.12]).From now on, we fix a measurable non-negative function ω : R d → R and an f ∈ h and work underthe following assumptions. Hypothesis A. (i) ω takes positive values almost everywhere,(ii) ω ( k ) | k |→∞ −−−−→ ∞ .(iii) There exists α > , such that ω is locally α -Hölder continuous.(iv) There exists ǫ > , such that ω − / f ∈ L ( R d ) ∩ L ǫ ( R d ) .(v) There exists α > , such that sup | p |≤ Z R d | f ( k + p ) − f ( k ) | p ω ( k ) | p | α d k < ∞ . (vi) We have sup | p |≤ Z R d | f ( k ) | p ω ( k ) ω ( k + p ) d k < ∞ . Example . In d = 3 dimensions elementary estimates show that the assumptions of Hypothesis Ahold for the choices ω ( k ) = | k | , f ( k ) = gκ ( k ) | k | δ (2.9)for any δ > − , g ∈ R , and a cutoff function κ of the form κ ( k ) = 1 | k |≤ Λ , for some Λ > , or κ ( k ) = exp( − ck ) , for some c > . The number g will be referred to as the coupling parameter. Example . More generally as in Example 2.3, we consider, for k ∈ R d , the functions ω ( k ) = | k | α , f ( k ) = gκ ( k ) | k | β , with some α > , β ∈ R , g ∈ R , and κ a cutoff function as in Example 2.3.Then Hypothesis A holds under the condition d > max { α − β, α − β, α − β } . On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 5
The second assumption we need is a differentiability condition for the infimum of the spectrumwith respect to the constant magnetic field µ . For this, we need to ensure that ω can be approximatedby a family of functions which are bounded from below by some positive constant. Hypothesis B.
There exists a decreasing sequence ( ω n ) n ∈ N of nonnegative measurable functions ω n : R d → R converging uniformly to ω , with the following properties.(i) There exists α > , such that ω n is locally α -Hölder continuous for all n ∈ N .(ii) inf k ∈ R d ω n ( k ) > .(iii) The function e n ( µ ) = inf σ ( H ( ω n , f ) + µσ x ⊗ ) is twice differentiable at zero and C χ := sup n ∈ N ( − e ′′ n (0)) < ∞ . (2.10) Remark . We note that (i) and (ii) of Hypothesis B are satisfied for the typical choice of a massivephoton dispersion relation ω n = p m n + ω , (2.11)or also ω n = ω + m n , where ( m n ) n ∈ N is any sequence of positive numbers decreasing monotonicallyto zero. The constant m n can be understood to be a photon mass. The result we prove is, however,independent of the specific choice of ω n . Remark . The differentiability assumption in Hypothesis B (iii) can be shown to hold by regularanalytic perturbation theory provided (ii) holds, since (ii) implies that the ground state energyis separated from the rest of the spectrum (cf. [AH95, AH97] or Proposition 3.2). However, theuniform bound on the second derivative is nontrivial to establish. We note that this assumptionwill be translated into a bound on the resolvent by means of second order perturbation theory, seeLemma 4.3. In fact, this bound on the resolvent is what we need in the proof of the main result,i.e., Theorem 2.8 holds if one replaces (iii) by the bound in Lemma 4.3.
Remark . Let us discuss (2.10) in d = 3 dimensions for the case given in (2.9) and (2.11). If δ > − / then (2.10) follows, e.g., from the estimates in [GH09]. In the case δ = − / , (2.9) hasbeen shown for small values of the coupling constant in [Spo89], using that ground state properties ofthe spin boson Hamiltonian are related to the correlation functions of a continuous one dimensionalIsing model with long range interaction, cf. [SD85] and [AN86]. In particular, (2.10) translatesto the corresponding Ising model having finite magnetic suszeptibility. For δ ∈ ( − / , − thefiniteness of the magnetic suszeptibility has been shown for the discrete Ising model, [Dys69]. Infact, for δ ∈ [ − / , − the bound (2.10) does not hold anymore for large values of the couplingconstant. This follows from the relation to the Ising model and the phase transition for Ising modelswith coupling decaying quadratically in the distance, cf. [Spo89, AN86, IN88]. On the other hand,for small couplings, using the relation to a continuous Ising model, it has been shown in [Abd11]that g inf σ ( H ( | · | , gf )) is analytic in a neighborhood of zero if f, f | · | − / ∈ h , which correspondsto δ > − . Note that the infimum of the spectrum can be analytic although there does not exista ground state (cf. [AH12]). Thus, it is not unreasonable to suspect that (2.10) might in fact holdfor δ > − provided the coupling is sufficiently small.Our main result now is the following. Theorem 2.8.
Assume Hypotheses A and B hold. Then inf σ ( H ( ω, f )) is an eigenvalue of H ( ω, f ) . On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 6
Remark . This result has been proven for infrared regular models, e.g., under the additionalassumption inf k ∈ R d ω ( k ) > in [AH95] (see [AH97] for a generalization of this result) and under theassumption ω − f ∈ h in [Gér00], which in d = 3 corresponds to δ > − / in (2.9). Specifically for d = 3 the existence has been shown in situations where ω − f / ∈ h in [HH11b, BBKM17]. The resultsin these papers include the case (2.9) with δ = − / provided the coupling g is sufficiently small.The results are perturbative in nature and were obtained using operator theoretic renormalizationand iterated perturbation theory, respectively. In particular, [HH11b] not only shows existence,but also analyticity of the ground state and the ground state energy in the coupling constant.Concerning existence, the result of Theorem 2.8 goes beyond. It shows existence for any δ > − and arbitrary coupling, as long as the derivative bound (2.10) is finite. Remark . In [Spo89] finite temperature KMS states of the spin boson Hamiltonian where in-vestigated for R R d f ( k ) e − ω ( k ) | t | d k ∼ = t − for large t . For d = 3 this corresponds to δ = − / in(2.9). Using results about the one dimensional continuous Ising model, it was established that theKMS states have a weak limit as the temperature drops to zero. In particular, it was shown thatthere exists a critical coupling such that the expectation of the number of bosons is finite below andinfinite at and above the critical coupling strength. We note that for the proof of the main theoremwe use a similar bound on the number of photons, see Lemma 4.5 (i) (which is in fact weaker thanthe one in [Spo89]). Remark . We note that our result gives a physically explicit bound on the coupling constantvia (2.10), where the left hand side of (2.10) is proportional to the magnetic suszeptibility of thecorresponding Ising model. As a consequence of Theorem 2.8 the absence of a ground state impliesthat the magnetic suszeptibility must diverge. Given the existence results in [HH11b, BBKM17], incase δ = − / in (2.9), the absence of a ground state for large coupling could provide an alternativemethod of proof for phase transitions in continuous long range Ising models. To the best of ourknowledge the absence of a ground state in the spin boson model with µ = 0 for δ ∈ ( − , − / andlarge coupling has not yet been shown. Nevertheless, we refer the reader to results [Spo89, DM20]where the large coupling limit has been investigated.The method of proof we use is based on the proof in [GLL01]. It was applied to the infrared-critical model of non-relativistic quantum electrodynamics by two of the authors in [HS20].For the proof of Theorem 2.8, we denote by ψ n the ground state of H ( ω n , f ) , which exists dueto the assumption inf k ∈ R d ω n > . We then prove, that all of them lie in a compact set K ⊂ C ⊗ F .Hence, there exists a subsequence ( ψ n j ) j ∈ N converging strongly to some ψ ∈ K . It then remains toshow that ψ = 0 actually is a ground state of H ( ω, f ) .The rest of this paper is organized as follows. In Section 3, we show some simple properties ofthe states ψ n and the corresponding ground state energies. In Section 4, we then derive necessaryupper bounds with respect to the photon number to construct the compact set K in Section 5. In this section, we derive some simple properties of the ground state energy of the infrared regu-lar spin boson Hamiltonian. Throughout this section we will assume Hypothesis A and that thesequence ( ω n ) n ∈ N is chosen as in Hypothesis B (ii). We set H = H ( ω, f ) and H n = H ( ω n , f ) , (3.1)as well as E = inf σ ( H ) and E n = inf σ ( H n ) for all n ∈ N . On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 7
Lemma 3.1.
We have(i) H ≤ H n ′ ≤ H n for n ≤ n ′ ,(ii) lim n →∞ E n = E .Proof. (i) follows from the monotonicity of ( ω n ) and Lemma 2.1. We set N = ⊗ d Γ(1) . Then,due to the uniform convergence of ( ω n ) , there is a sequence ( C n ) ⊂ R + satisfying C n n →∞ −−−→ and ω n ≤ ω + C n . Hence, d Γ( ω n ) ≤ d Γ( ω ) + C n d Γ(1) , which implies H n ≤ H + C n N. On the other hand let ε > and fix ϕ ε ∈ D ( N ) ∩ D ( H ) with k ϕ ε k = 1 , such that h ϕ ε , Hϕ ε i ≤ E + ε. This is possible, since D ( N ) ∩ D ( H ) is a core for ⊗ d Γ( ω ) and hence for H , by Lemma 2.2.Together with (i), we obtain E ≤ E n ≤ h ϕ ε , H n ϕ ε i ≤ h ϕ ε , Hϕ ε i + C n h ϕ ε , N ϕ ε i ≤ E + ε + C n h ϕ ε , N ϕ ε i n →∞ −−−→ E + ε. Now (ii) follows in the limit ε → .As mentioned above the bound Hypothesis B (ii) implies the existence of a ground state, which isthe content of the following lemma. Proposition 3.2.
For all n ∈ N , E n is a simple eigenvalue of H n .Further, [ E n , E n + inf k ∈ R d ω n ( k )) ∩ σ ess ( H n ) = ∅ .Proof. The existence has been shown in [AH95] and the uniqueness for arbitrary couplings has beenshown in [HH11b], see also [Frö73].Let ψ n be a normalized eigenvector of H n to the eigenvalue E n . A main ingredient of our proofthen is the following proposition. Proposition 3.3.
The sequence ( ψ n ) n ∈ N is minimizing for H , i.e., ≤ h ψ n , ( H − E ) ψ n i n →∞ −−−→ . Proof.
We use Lemma 3.1 and find ≤ h ψ n , ( H − E ) ψ n i ≤ h ψ n , ( H n − E ) ψ n i = E n − E → . In this section we derive essential bounds on the ground states ψ n , which are uniform in n ∈ N .Throughout this section we will assume that Hypotheses A and B hold. We recall the followingdefinition in Hypothesis B e n ( µ ) = inf σ ( H ( ω n , f ) + µσ x ⊗ ) for n ∈ N and µ ∈ R . (4.1)Note that by the definitions in (3.1), we have E n = e n (0) . The next lemma is a simple symmetryargument. On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 8
Lemma 4.1.
We have e n ( µ ) = e n ( − µ ) for all n ∈ N .Proof. We define the unitary operator U = e i π σ z ⊗ ( − d Γ(1) . It easily follows from the definitionsthat e i π σ z σ x e − i π σ z = − σ x and ( − d Γ(1) ϕ ( f )( − d Γ(1) = ϕ ( − f ) . Now, using that d Γ -operatorscommute, we obtain U ( H ( ω n , f ) + µσ x ⊗ ) U ∗ = H ( ω n , f ) − µσ x ⊗ , which proves the claim.Now, for k ∈ R d , we define the pointwise annihilation operator a k acting on ψ ( ℓ +1) ∈ F ( ℓ +1) by ( a k ψ ( ℓ +1) )( k , . . . , k ℓ ) = √ ℓ + 1 ψ ( ℓ +1) ( k, k , . . . , k ℓ ) . (4.2)Note that by the Fubini-Tonelli theorem ( a k ψ ( ℓ +1) ) ∈ F ( ℓ ) for almost every k . Further, for n ∈ N ,we define the operator R n ( k ) = ( H n − E n + ω n ( k )) − for k ∈ R d , (4.3)which is bounded by Hypothesis B, and the spectral theorem directly yields k R n ( k ) k ≤ ω n ( k ) . (4.4)The next statement is well-known and can be found under the name pull-through formula through-out the literature, cf. [BFS98a, Gér00]. In the statement we write ψ n = ( ψ n, , ψ n, ) in the sense of(2.6) and denote a k ψ n = ( a k ψ n, , a k ψ n, ) and σ x ψ n = ( σ x ⊗ ) ψ n = ( ψ n, , ψ n, ) . (4.5) Lemma 4.2.
Let n ∈ N . Then, for almost every k ∈ R d , the vector a k ψ n ∈ H and a k ψ n = − f ( k ) R n ( k ) σ x ψ n . The infrared bounds we want to obtain in this section are bounds on R n ( k ) σ x ψ n . To that end, westart by translating Hypothesis B into a resolvent bound. Lemma 4.3.
For all n ∈ N , we have h σ x ψ n , ψ n i = 0 and ≤ h σ x ψ n , ( H n − E n ) − σ x ψ n i = − e ′′ n (0) . Proof.
The proof uses analytic perturbation theory, for details see [Kat80, RS78]. The operatorvalued function η H n ( η ) := H n + ησ x ⊗ defines an analytic family of type (A) for η ∈ C (cf.[Kat80, Theorem 2.6]). By Proposition 3.2, we know that e n (0) is a non-degenerate eigenvalue of H n (0) isolated from the essential spectrum. First order perturbation theory now yields e ′ n (0) = h ψ n , σ x ψ n i . Hence, Lemma 4.1 implies h ψ n , σ x ψ n i = 0 . Then, by second order perturbation theory,we obtain the second equality.This gives us the required infrared bound. Lemma 4.4.
We have k R n ( k ) σ x ψ m k ≤ s − e ′′ n (0) ω n ( k ) for all n ∈ N . On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 9
Proof.
By the product inequality, we have k R n ( k ) σ x ψ n k ≤ k R n ( k )( H n − E n ) / kk ( H n − E n ) − / σ x ψ n k . (4.6)By Lemma 4.3, the second factor on the right hand side can be estimated using k ( H n − E n ) − / σ x ψ n k ≤ p − e ′′ n (0) . It remains to estimate the first factor in (4.6). Using (cid:13)(cid:13) R n ( k ) / ( H n − E n ) / (cid:13)(cid:13) ≤ we find with(4.4) (cid:13)(cid:13) R n ( k )( H n − E n ) / (cid:13)(cid:13) ≤ (cid:13)(cid:13) R n ( k ) / (cid:13)(cid:13) ≤ p ω n ( k ) . We combine this result with the pull-through formula.
Lemma 4.5.
Let B = { x ∈ R d : | x | ≤ } .(i) For all n ∈ N and almost all k ∈ R d , we have k a k ψ n k ≤ | f ( k ) | p ω ( k ) C / χ .(ii) There exist an α > and a measurable function h : B × R d → [0 , ∞ ) with sup p ∈ B k h ( p, · ) k < ∞ , such that for all n ∈ N and almost all p ∈ B and k ∈ R d k a k + p ψ n − a k ψ n k ≤ | p | α h ( p, k ) . Proof. (i) follows directly from Lemmas 4.2 and 4.4, and from the monotonicity of ( ω n ) n ∈ N .Let α be the minimum of the values from Hypothesis A (iii) and Hypothesis B (i) and let α be as in Hypothesis A (v). Then, we set α = min { α , α } and ˜ h ( p, k ) = max ( | f ( k + p ) − f ( k ) || p | α p ω ( k ) , | f ( k + p ) | ω ( k ) p ω ( k + p ) ) . Then, by Hypothesis A, ˜ h satisfies the above statements on h . Further, using the resolvent identityand Lemma 4.2, we obtain a k + p ψ n − a k ψ n = f ( k ) R n ( k ) σ x ψ n − f ( k + p ) R n ( k + p ) σ x ψ n = ( f ( k ) − f ( k + p )) R n ( k ) σ x ψ n + f ( k + p )( R n ( k ) − R n ( k + p )) σ x ψ n = ( f ( k ) − f ( k + p )) R n ( k ) σ x ψ n (4.7) + f ( k + p ) R n ( k )( ω n ( k + p ) − ω n ( k )) R n ( k + p ) σ x ψ n . (4.8)By Lemma 4.4 and Hypothesis B, we find | (4.7) | ≤ C / χ | f ( k + p ) − f ( k ) | p ω ( k ) ≤ C / χ | p | α ˜ h ( p, k ) . Further, the local α -Hölder continuity of ω n yields there is C > , such that | (4.8) | ≤ C | p | α ˜ h ( p, k ) . This proves the statement for the function h = ( C / χ + C )˜ h . On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 10
We use the above infrared bounds to derive an upper bound on the photon number operator andthe free field energy N = ⊗ d Γ(1) and H f = ⊗ d Γ( ω ) (4.9)acting on the ground states ψ n . The proof uses the following well-known representation of thequadratic form associated with second quantization operators in terms of pointwise annihilationoperators. Lemma 4.6.
Assume A : R d → [0 , ∞ ) is measurable and ψ ∈ F . Then the map k
7→ k A ( k ) / a k ψ k is in L ( R d ) if and only if ψ ∈ D ( d Γ( A ) / ) . Further, for any φ , φ ∈ D ( d Γ( A ) / ) we have h d Γ( A ) / φ , d Γ( A ) / φ i = Z R d A ( k ) h a k φ , a k φ i d k. Proof.
The statement is standard in the literature, see for example [RS75].The next lemma will provide a photon number bound.
Lemma 4.7.
For all n ∈ N we have ψ n ∈ D ( N / ) ∩ D ( H f ) and the inequalities h N / ψ n , N / ψ n i ≤ C χ k ω − / f k and h ψ n , H f ψ n i ≤ C χ k f k .Proof. The property ψ n ∈ D ( H f ) was proven in Lemma 2.2. The remaining statements follow fromcombining the upper bound in Lemma 4.5 (i) and Lemma 4.6. In this section, we construct a compact set K ⊂ H , such that ( ψ n ) n ∈ N ⊂ K . We then use thecompactness of K to prove Theorem 2.8. Throughout this section, we assume that Hypotheses Aand B hold.Let us begin with the definition of K . To that end, assume y i for i = 1 , . . . , ℓ is the positionoperator acting on ψ ( ℓ ) ∈ F ( ℓ ) as [ y i ψ ( ℓ ) ( x , . . . , x ℓ ) = x i d ψ ( ℓ ) ( x , . . . , x ℓ ) , (5.1)where b · denotes the Fourier transform. For δ > , we now define a closed quadratic form q δ actingon φ = ( φ , φ ) ∈ Q ( q δ ) ⊂ H with natural domain as q δ ( φ ) = h N / φ, N / φ i + X ℓ ∈ N s ∈{ , } ℓ ℓ X i =1 (cid:10) φ ( ℓ ) s , | y i | δ φ ( ℓ ) s (cid:11) + h H / φ, H / φ i , (5.2)where N and H f are defined as in (4.9). Now define K δ,C := { φ ∈ Q ( q δ ) : k φ k ≤ , q δ ( φ ) ≤ C } for C > . (5.3) Lemma 5.1.
For all δ, C > the set K δ,C ⊂ H is compact.Proof. By Lemma 2.1, q δ is nonnegative. Hence, there exists a self-adjoint nonnegative operator T associated to q δ . By the general characterization of operators with compact resolvent (cf. [RS78,Theorem XIII.64]), K C is compact iff T has compact resolvent iff the i -th eigenvalues of T obtainedby the min-max principle µ i ( T ) tend to infinity, i.e., lim i →∞ µ i ( T ) = ∞ . On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 11
To that end, we observe T preserves the ℓ photon sectors C ⊗F ( ℓ ) and denote T ℓ = T ↾ C ⊗F ( n ) .Now, since ( ω + 1) ( ℓ ) ( K ) → ∞ as K → ∞ by Hypothesis A (ii), we can apply Rellich’s criterion (cf.[RS78, Theorem XIII.65]) and hence T ℓ has compact resolvent for all ℓ ∈ N . As argued above, wehave lim i →∞ µ i ( T ℓ ) = ∞ . Further, since T ℓ ≥ ℓ , we have µ i ( T ℓ ) ≥ ℓ and therefore lim i →∞ µ i ( T ) = ∞ .We now need to prove the following proposition, where ψ n are the normalized ground states of H n as defined in Section 3. Proposition 5.2.
There are δ, C > , such that ψ n ∈ K δ,C for all n ∈ N . For the proof the following lemma is essential. Hereby, for n ∈ N , s ∈ { , } and y, k ∈ R d , weintroduce the notation d ψ ( ℓ ) n,s ( y ) : ( y , . . . , y ℓ − ) ψ ( ℓ ) n,s ( y, y , . . . , y ℓ − ) ,ψ ( ℓ ) n,s ( k ) : ( k , . . . , k ℓ − ) ψ ( ℓ ) n,s ( k, k , . . . , k ℓ − ) . (5.4)Due to the Fubini-Tonelli theorem, we have d ψ ( ℓ ) n,s ( y ) , ψ ( ℓ ) n,s ( k ) ∈ L ( R ( ℓ − d ) for almost every k, y ∈ R d .Further, comparing with the definition (4.2), we observe ψ ( ℓ ) n,s ( k ) = 1 √ ℓ + 1 ( a k ψ n,s ) ( ℓ ) . (5.5) Lemma 5.3.
There exist δ > and C > , such that for all p ∈ R d and n, ℓ ∈ N , s ∈ { , } Z R d | − e − ipy | (cid:13)(cid:13)(cid:13)(cid:13) d ψ ( ℓ ) n,s ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L ( R ( ℓ − d ) d y ≤ Cℓ + 1 min n , | p | δ o . (5.6) We note that δ can be chosen as δ = ǫα ǫ , where α > and ǫ > are as in Lemma 4.5 (ii) andHypothesis A (iv), respectively.Proof. That the left hand side of (5.6) is bounded by a constant C , uniformly in p , follows easily dueto the Fock space definition, since the Fourier transform preserves the L -norm. Now lets consider | p | ≤ . Note that Z R d | − e − ipy | (cid:13)(cid:13)(cid:13)(cid:13) d ψ ( ℓ ) n,s ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L ( R ( ℓ − d ) d y = Z R d (cid:13)(cid:13) ψ ( ℓ ) n,s ( k + p ) − ψ ( ℓ ) n,s ( k ) (cid:13)(cid:13) d k = 1 ℓ + 1 Z R d (cid:13)(cid:13) ( a k + p ψ n,s ) ( ℓ ) − ( a k ψ n,s ) ( ℓ ) (cid:13)(cid:13) d k, where we used (5.5). Let θ ∈ (0 , . By Lemma 4.5, we have some C > such that (cid:13)(cid:13) ( a k + p ψ n,s ) ( ℓ ) − ( a k ψ n,s ) ( ℓ ) (cid:13)(cid:13) ≤ C | p | θα h ( p, k ) θ | f ( k ) | p ω ( k ) ! − θ . For r, s > with r + s = 1 , we now use Young’s inequality bc ≤ b r /r + c s /s to obtain a constant C r,s > with (cid:13)(cid:13) ( a k + p ψ n,s ) ( ℓ ) − ( a k ψ n,s ) ( ℓ ) (cid:13)(cid:13) ≤ C r,s | p | θα h ( p, k ) θr + | f ( k ) | p ω ( k ) ! − θ ) s . (5.7) On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 12
Set r = θ . Then, the first summand in (5.7) is integrable in k due to Lemma 4.5. Further, theexponent of the second summand equals − θ ) s = 2(1 − θ ) (cid:18) − r (cid:19) − = 2(1 − θ )1 − θ . Hence, we can choose θ > such that − θ )1 − θ = 2 + ǫ . By Hypothesis A (iv), it follows that (5.7)is integrable in k and the proof is complete.From here, we can prove an upper bound for the Fourier term in (5.2). Lemma 5.4.
Let δ > be as in Lemma 5.3. Then there exists C > such that for all n, ℓ ∈ N and s ∈ { , } Z R d · ℓ ℓ X i =1 | x i | δ/ (cid:12)(cid:12)(cid:12)(cid:12) d ψ ( ℓ ) n,s ( x , . . . , x ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( x , . . . , x ℓ ) ≤ C. Proof.
From Lemma 5.3, we know that there exists a finite constant C such that Z R d Z R d | − e − ipy | k d ψ ( ℓ ) n,s ( y ) k | p | δ/ d y d p | p | d ≤ Cℓ + 1 . After interchanging the order of integration and a change of integration variables q = | y | p , we find Cℓ + 1 ≥ Z R d k d ψ ( ℓ ) n,s ( y ) k Z R d | − e − ipy | | p | δ/ d p | p | d d y = Z R d k d ψ ( ℓ ) n,s ( y ) k | y | δ/ Z R d | − e − iqy/ | y | | | q | δ/ d q | q | d | {z } =: c d y , where c is nonzero and does not depend on y .We can now conclude. Proof of Proposition 5.2 . Combine Lemmas 4.7 and 5.4.
Proof of Theorem 2.8.
By Lemma 5.1 and Proposition 5.2, we know there exists a subsequence ( ψ n k ) k ∈ N , which converges to a normalized vector ψ ∞ . By the lower semicontinuity of non-negativequadratic forms, we see from Proposition 3.3 that h ψ ∞ , ( H − E ) ψ ∞ i ≤ lim inf k →∞ h ψ n k , ( H − E ) ψ n k i = 0 . Acknowledgements
D.H. wants to thank Ira Herbst for valuable discussions on the subject.
On Existence of Ground States in the Spin Boson Model . Hasler, B. Hinrichs, O. Siebert 13
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