FKN Formula and Ground State Energy for the Spin Boson Model with External Magnetic Field

We consider the spin boson model with external magnetic field. We prove a path integral formula for the heat kernel, known as Feynman-Kac-Nelson (FKN) formula. We use this path integral representation to express the ground state energy as a stochastic integral. Based on this connection, we determine the expansion coefficients of the ground state energy with respect to the magnetic field strength and express them in terms of correlation functions of a continuous Ising model. From a recently proven correlation inequality, we can then deduce that the second order derivative is finite. As an application, we show existence of ground states in infrared-singular situations.


Introduction
In this paper, we study the spin boson model with external magnetic field. This model describes the interaction of a two-level quantum mechanical system with a boson field in presence of a constant external magnetic field. We derive a Feynman-Kac-Nelson (FKN) formula for this model, which relates expectation values of the semigroup generated by the Hamilton operator to the expectation value of a Poisson-driven jump process and a Gaussian random process indexed by a real Hilbert space obtained by an Euclidean extension of the dispersion relation of the bosons. Especially, when calculating expectation values with respect to the ground state of the free Hamiltonian, one can explicitly integrate out the boson field and obtain expectations only with respect to the jump process. This allows us to express the ground state energy and its derivatives in terms of correlation functions of a continuous Ising model, provided a gap assumption is satisfied. As an application, we show the existence of ground states for the spin boson model in the case of massless bosons for infrared singular interactions, using a recent correlation bound and a regularization procedure.
The history of FKN-type theorems dates back to the work of Feynman and Kac [Fey05,Kac51]. Such functional integral respresentations were used to study the spectral properties of models in quantum field theory by Nelson [Nel73]. Since then, many authors have used this approach to study models of non-relativistic quantum field theory, see for example [GJ85,GJ87,Spo87,FFG97,Hir97, BHL + 02, BS05, HL08, BH09] and references therein. The spin boson model without an external magnetic field has been investigated using this approach in [SD85,FN88,Abd11] and recently in [HHL14]. In [Spo89] path measures for the spin boson model with magnetic field were studied by means of Gibbs measures. In this paper, we extend the FKN formula for the spin boson model to external magnetic fields.
This paper is structured as follows. Section 2 is devoted to the definition of the spin boson model and the statement of our main results. We start out with a rigorous definition of the spin boson Hamiltonian with external magnetic field as a selfadjoint lower-semibounded operator in Section 2.1. In Section 2.2, we then describe its probabilistic description through the FKN formula stated in Theorem 2.4 and reduce the degrees of freedom to study expectation values with respect to the ground state of the free operator as expectation values of a continuous Ising model in Corollary 2.6. In Section 2.3, we then use the well-known connection between expectation values of the semigroup and the ground state energy to express the derivatives of the ground state energy with respect to the magnetic field strength as correlation functions of this continuous Ising model, under the assumption of massive bosons. The proofs of the results presented in Section 2 are given in Section 3.
In Section 4, we then apply our results and prove Theorem 4.1. Explicitly, we use the recent result from [HHS21b] to prove the existence of ground states of the spin boson Hamiltonian with vanishing external magnetic field. Our proof especially includes the case of massless bosons with infrared-singular coupling.
The article is accompanied by a series of appendices. In Appendices A to C, we present some essential technical requirements for our proofs, including standard Fock space properties in Appendix C.1 and a construction of the so-called Q-space in Appendix C.2. In Appendix D, we give a proof for the existence of ground states at arbitrary external magnetic field in the case of massive bosons, a case which to our knowledge is not covered in the literature.

General Notation
L 2 -spaces: For a measure space (M, dµ) and a real or complex Hilbert space h, we denote by L 2 (M, dµ; h) the real or complex Hilbert space of square-integrable h-valued measurable functions on M, respectively. If h = C, we write for simplicity L 2 (M, dµ) = L 2 (M, dµ; C). Further, we assume R d for any d ∈ N to be equipped with the Lebesgue measure without further mention.
Characteristic functions: For A ⊂ X, we define the function 1 A : X → R with 1 A (x) = 1, if x ∈ A, and 1 A (x) = 0, if x / ∈ A.

Spin Boson Model with External Magnetic Field
In this section, we give a precise definition of the spin boson Hamiltonian with external magnetic field and prove that it defines a selfadjoint lower-semibounded operator. Let us recall the standard Fock space construction from the Hilbert space perspective. Textbook expositions on the topic can, for example, be found in [RS75,Par92,BR97,Ara18].
Throughout, we assume h to be a complex Hilbert space. Then, we define the bosonic Fock space over h as where ⊗ n s h denotes the n-fold symmetric tensor product of Hilbert spaces. We write Fock space vectors as sequences ψ = ψ (n) n∈N 0 with ψ (0) ∈ C and ψ (n) ∈ ⊗ n sym h. Especially, we define the Fock space vacuum Ω = (1, 0, 0, . . .).
For a self-adjoint operator A, let the (differential) second quantization operator dΓ(A) on F (h) be the operator where (·) denotes the operator closure. Next, if v is another complex Hilbert space and B : h → v is a contraction operator (i.e., B ≤ 1), the second quantization operator Γ(B) : Furthermore, for f ∈ h, we define the creation and annihilation operators a † (f ) and a(f ) as the closed linear operators acting on pure tensors as a(f )g 1 ⊗ s · · · ⊗ s g n = 1 √ n n k=1 f, g k g 1 ⊗ s · · · g k · · · ⊗ s g n , a † (f )g 1 ⊗ s · · · ⊗ s g n = √ n + 1f ⊗ s g 1 ⊗ s · · · ⊗ s g n , where ⊗ s denotes the symmetric tensor product and · in the first line means that the corresponding entry is omitted. Note that a † (f ) is the adjoint of a(f ). We introduce the field operator as In Appendix C.1, we provide a variety of well-known properties of the operators defined above, which will be used throughout this article. From now on, we also write F = F (L 2 (R d )).
To define the spin boson Hamiltonian with external magnetic field, let σ x , σ y , σ z denote the 2 × 2-Pauli matrices We consider the Hamilton operator To prove that the expression (2.7) defines a selfadjoint lower-semibounded operator, we need the following assumptions.
Hypothesis A.
is measurable and has positive values almost everywhere.

Feynman-Kac-Nelson Formula
In this section, we move to a probabilistic description of the spin boson model. Except for Lemma 2.2, all statements are proved in Section 3.1. The spin part can be described by a jump process, which we construct here explicitly. To that end, let (N t ) t≥0 be a Poisson process with unit intensity, i.e., a stochastic process with state space N 0 , stationary independent increments, and satisfying realized on some measurable space Ω. We refer the reader to [Bil99] for a concrete realization of Ω. Moreover, we can choose Ω such that N t (ω) is right-continuous for all ω ∈ Ω, see for example [Bil12,Section 23]. Further, let B be a Bernoulli random variable with P[B = 1] = P[B = −1] = 1 2 , which we realize on the space {−1, 1}. Then, we define the jump process ( X t ) t≥0 on the product space Ω × {−1, 1} (equipped with the product measure) by (2.9) To fix a suitable measure space to work with, we use the law of the process ( X t ) t≥0 . That is, we realize the stochastic process on the space where we equip D with the σ-algebra generated by the projections π t (x) = x t , t ≥ 0. The measure, µ X , on D is then given by the pushforward with respect to the map for which it is straightforward to see that it is measurable. We define the process X t (x) = x t for x ∈ D, t ≥ 0. It follows by construction that the stochastic processes X t and X t are equivalent, in the sense that they have the same finite-dimensional distributions. For random variables Y on the measure space (D, µ X ), we define We note that by the construction (2.8), the paths of X µ X -almost surely have only finitely many jumps in any compact interval. We denote the set of all such paths by D f . The property µ X (D f ) = 1 can alternatively also be deferred from the theory of continuous-time Markov processes, cf. [Res92,Lig10]. We now want to give a probabilistic description of the bosonic field. To that end, we define the Euclidean dispersion relation ω E : R d+1 → [0, ∞) as ω E (k, t) = ω 2 (k) + t 2 and the Hilbert space of the Euclidean field as Let φ E be the Gaussian random variable indexed by the real Hilbert space on the (up to isomorphisms unique) probability space (Q E , Σ E , µ E ) and denote expectation values w.r.t. µ E as E E . For the convencience of the reader, we have described a possible explicit construction in Appendix C.2. We note that the complexification R C is unitarily equivalent to E, by the map (f, g) → f + ig, and hence F (E) and L 2 (Q E ) are unitarily equivalent, by Lemma C.3. For t ∈ R, we define (2.13) Lemma 2.2.
(i) (2.13) defines an isometry j t : (iii) j * s j t = e −|t−s|ω for all s, t ∈ R. Proof. The statements follow by the direct calculation Remark 2.3. In the literature (2.13) is often defined via the Fourier transform | j t f = δ t ⊗ q f .
We set where Θ R denotes the Wiener-Itô-Segal isomorphism introduced in Lemma C.3 and Γ is the second quantization of the contraction operator j t , as defined in (2.3). Further, we define the isometry ι : where α i denotes the i-th entry of the vector α ∈ C 2 . We define the map I t := ι ⊗ I t , where To formulate the Feynman-Kac-Nelson (FKN) formula, it will be suitable to work with the following transformed Hamilton operator, which is unitary equivalent to H(λ, µ) up to a constant multiple of the identity. Explicitly, we apply the unitary U = e i π 4 σy = 1 √ 2 where we used Uσ z U * = −σ x and Uσ x U * = σ z . Our result holds under the following assumptions.
Hypothesis B. Assume Hypothesis A and the following: We are now ready to state the FKN formula for the spin boson model with external magnetic field.
Theorem 2.4 (FKN Formula). Assume Hypothesis B holds. Then, for all Φ, Ψ ∈ H and λ, µ ∈ R, We note that the integrability of the right hand side in above theorem follows from the identity which holds for any Gaussian random variable Z (see for example [Sim74,(I.17)]). We outline the argument in the remark below.
Remark 2.5. By (2.13), the map [0, T ] → E, t → j t v is strongly continuous. Hence, by (C.1), the x t is a piecewise continuous L 2 (Q E )-valued function on compact intervals of [0, ∞). Thus, the integral over t exists as an L 2 (Q E )-valued Riemann integral µ X -almost surely. Since Riemann integrals are given as limits of sums, the measurability with respect to the product measure µ X ⊗ µ E follows. In fact, again fixing x ∈ D f and using Fubini's theorem as well as Hölder's inequality, one can prove that the integral x t dt can also be calculated as Lebesgue-integral evaluated µ E -almost everywhere pointwise in Q E with the same result. This is outlined in Appendix A. Furthermore, x t dt is a Gaussian random variable, since L 2 -limits of linear combinations of Gaussians are Gaussian. We conclude that the right hand side of the FKN formula is finite, since exponentials of Gaussian random variables are integrable, cf. (2.18).
We now want to describe the expectation value of the semigroup associated with H(λ, µ) (cf. (2.7)) with respect to the ground state of the free operator H(0, 0), by integrating out the field contribution in the expectation value. To that end, let and define (2.20) Corollary 2.6. Assume Hypothesis B holds. Then, for all λ, µ ∈ R, we have Remark 2.7. For (x t ) t≥0 ∈ D f , the functions (s, t) → W (t − s)x t x s and t → x t are Riemannintegrable, since W is continuous. Further, the continuity also implies that the expression on the right hand side is uniformly bounded in the paths x and hence the expectation value exists and is finite by the dominated convergence theorem.
Remark 2.8. The expectation value on the right hand side can be interpreted as the partition function of a long-range continuous Ising model on R with coupling functions W . This model can be obtained as a limit of a discrete Ising model with long-range interactions, see [SD85,Spo89,HHS21a].

Ground State Energy
We are especially interested in studying the ground state energy of the spin boson model In this section, we want to use the FKN formula from the previous section to express derivatives of the ground state energy. Starting point of this investigation is the following well-known formula, sometimes referred to as Bloch's formula, expressing the ground state energy as expectation value of the semigroup, see for example [Sim79]. We verify it in Section 3.2 using a positivity argument.
Lemma 2.9. Assume Hypothesis A holds. Then, for all λ, µ ∈ R, The central statement of this section is that above equation carries over to the derivatives with respect to µ, provided that the ground state energy of H(λ, µ) is in the discrete spectrum, i.e., E(λ, µ) ∈ σ disc (H(λ, µ). We note that this spectral assumption has been shown in [AH95, Theorem 1.2] for µ = 0 if ess inf k∈R d ω(k) > 0 and we extend the result to arbitrary choices of µ in Appendix D.
Theorem 2.10. Assume Hypothesis A holds. Let λ, µ 0 ∈ R and suppose E(λ, µ 0 ) ∈ σ disc (H(λ, µ 0 )). Then, for all n ∈ N, the following derivatives exist and satisfy We now want to combine this observation with the FKN formula from Theorem 2.4. To that end, we define with W as defined in (2.20) and note that by Corollary 2.6. Thus, Lemma 2.9 gives We note that the stochastic integral in (2.24) was used in [Abd11] to show analyticity of λ → E(λ, 0) in a neighborhood of zero. The next two statements express the derivatives of the ground state energy in terms of a stochastic integral. To that end, for a random variable Y on (D, µ X ), we define the expectation (2.25) Further, we denote by P n the set of all partitions of the set {1, . . . , n} and by |M| the cardinality of a finite set M.
Then, for all n ∈ N, the following derivatives exist and satisfy In addition, we can express derivatives of the ground state energy in terms of the so-called Ursell functions [Per75] or cumulants. This allows us to use correlation inequalities to prove bounds on derivatives. In fact, we will use this in Corollary 2.14 below to estimate the second derivative with respect to the magnetic field at zero. Given random variables Y 1 , . . . , Y n on (D, µ X ), we define the Ursell function (2.26) Corollary 2.12. Assume Hypothesis B holds. Let λ, µ ∈ R and suppose E(λ, µ) ∈ σ disc (H(λ, µ)).
Then, for all n ∈ N, the following derivatives exist and satisfy ds n u n (X s 1 , . . . , X sn ).
Next, we show how the formulas in Theorem 2.11 and Corollary 2.12, respectively, can be used to obtain bounds on derivatives of the ground state energy. For this, we will use the following correlation bound of a continuous long-range Ising model, cf. Remark 2.7. Approximating this model by a discrete Ising model, we proved a bound on these correlation functions in [HHS21a]. HHS21a]). There exist ε > 0 and C > 0 such that for all h ∈ L 1 (R) which are even, continuous and satisfy h L 1 (R) ≤ ε, we have As an application of Theorems 2.11 and 2.13 we obtain the following result, giving us a bound on the second derivative of the ground state energy which is uniform in the size of the spectral gap.
Since the proof only demonstrates the application of these theorems, we state it here directly.
Then, for every λ ∈ R, the function µ → E(λ, µ) is twice differentiable in a neighborhood of zero and, choosing W as defined in (2.20), Further, there exists a λ c > 0 such that for all λ ∈ (−λ c , λ c ) the second derivative satisfies Proof. Due to the definition, we have ess inf k∈R d ω(k) ≥ m > 0 and hence E(λ, 0) ∈ σ disc (H(λ, 0)), by Theorem D.1. Thus, Theorem 2.11 is applicable. Due to the so-called spin-flip-symmetry of the model, i.e., X and −X being equivalent stochastic processes in the sense of their finite-dimensional distributions by the choice of the Bernoulli random variable in (2.9), we have Thus, by Theorem 2.11 By the definition in (2.20), the interaction function W ∈ L 1 (R) satisfies Setting λ c = ( 1 2 ε) 1/2 / ν −1/2 v 2 with ε given as in Theorem 2.13, we can apply Theorem 2.13 with This concludes the proof.

Proofs
In this section, we prove the results presented in Sections 2.2 and 2.3

The FKN Formula
We start with the proof of Theorem 2.4. To that end, we first derive a FKN formula for the spin part, which is described by the jump process. For the statement, we recall the definition of ι : Lemma 3.1. Let n ∈ N and t 1 , . . . , t n ≥ 0. We set s k = k i=1 t k for k = 1, . . . , n.
Observing that ιe 1 (x) = 1 and ιe 2 (x) = x for x = ±1 finishes the proof. We now move to proving the FKN formula for the field part. We recall the definition of the isometry j t : L 2 (R d ) → E in (2.13). For I ⊂ R, let e I denote the projection onto the closed subspace lin{f ∈ E : f ∈ Ran(j t ) for some t ∈ I}. Further, set e t = e {t} for any t ∈ R.
Proof. Lemma 2.2 (i) and the definition of e {t} directly imply (i). Further, (ii) follows from Lemma 2.2 (iii) by Furthermore, again by definition Ran(j t ) = Ran(e t ) for any t ∈ R. Hence, we can apply (ii) and obtain Since f and g were arbitrary, this proves the statement. Now, for t ∈ R and I ⊂ R, let and Then the next statement in large parts follows directly from Lemmas 2.2, 3.2 and C.1.
Proof. All statements except for (v)-(vii) follow trivially from Lemmas 3.2 and C.1 and the definitions. (v) follows from (iv), by the simple calculation Finally, (vi) and (vii) follow by combining Lemmas 2.2 (iii) and C.1. Repeated application of (vii) shows that (viii) holds for G a polynomial. That it holds for arbitrary bounded measurable G follows from the measurable functional calculus [RS72].
We can now prove the full FKN formula.
Proof of Theorem 2.4. Throughout this proof, we drop tensor products with the identity in our notation. Further, for the convenience of the reader, we explicitly state in which Hilbert space the inner product is taken.
) and H K (λ, µ) as in (2.17) with ϕ replaced by ϕ K . Since H K (λ, µ) is lowersemibounded and ϕ K is bounded, we can use the Trotter product formula (cf. [RS72, Theorem VIII.31]) and Lemma 3.3 Parts (vi) and (viii) (where the exponential is considered on the eigenspaces of σ x ) to obtain . Now we make iterated use of Lemma 3.3 (v). Explicitly, by Lemma 3.3 (viii), the vector to the left of any E k T N , i.e., is an element of Ran(E [0,k T N ] ). Equivalently, the vector to the right is an element of Ran(E [k T N ,T ] ). Hence, we can drop all the factors E k T N . Then, using Lemma C.3 and (2.14), we derive .

FKN Formula and Ground State Energy for the Spin Boson Model with External Magnetic Field
Hence, we can apply Lemma 3.1 to obtain is an L 2 (Q E )-valued continuous function. Thus, the sum in the exponential in (3.4) converges to an L 2 (Q E )-valued Riemann integral. By possibly going over to a subsequence the Riemann sum converges µ X ⊗ µ E -almost everywhere. Thus it follows by dominated convergence that (3.5) (Alternatively, the convergence could also be deduced by estimating the expectation.) Since ϕ(v) is bounded with respect to dΓ(ω) (cf. Lemma C.1), the spectral theorem implies that H K (λ, µ) converges to H(λ, µ) in the strong resolvent sense and hence the left hand side of above equation converges to Φ, e −T H(λ,µ) Ψ as K → ∞. On the other hand, using that for µ X ⊗ µ E -almost every x t dt almost everywhere. Hence, the right hand side of (3.5) converges to as K → ∞, by the dominated convergence theorem. For the majorant, we use that by Jensen's inequality where in the second line we used max{e x , 1} ≤ e x + e −x . Now the right hand side is integrable over Q E -space by (2.18). This proves the statement.
We end this section with the Proof of Corollary 2.6. First, observe that with U as in (2.16), we have (I t (U * ⊗ 1)Ω ↓ )(x) = 1 for x = ±1 and t ∈ R (cf. (2.3) and Lemma C.3). Hence, Theorem 2.4 implies Now, let x ∈ D f . Then, Fubini's theorem and the identity (2.18) yield Now, the definitions of the R-indexed Gaussian process (cf. (C.1)) and W (2.20) imply where we used j * s j t = e −|t−s|ω (cf. Lemma 2.2). This proves the statement.

Derivatives of the Ground State Energy
To prove our results on derivatives of the ground state energy, we start out with the proof of Lemma 2.9. Let us first state our version of Bloch's formula. For the convenience of the reader, we provide the simple proof. Similar arguments are, for example, used in [LMS02,AH12].
Since the notion of positivity is essential therein, we recall it here for the convenience of the reader. For an arbitrary measure space (M, µ), we call a function f ∈ L 2 (M, dµ) (strictly) positive if it satisfies f (x) ≥ 0 (f (x) > 0) for almost all x ∈ M. If A is a bounded operator on L 2 (M, dµ), we say A is positivity preserving (improving) if Af is (strictly) positive for all non-zero positive f ∈ L 2 (M, dµ). Proof. First, we note that by the spectral theorem where ν g denotes the spectral measure of H associated with g. This easily follows from the inequality it follows from e −T H being positivity preserving that Combined with (3.6), it follows that inf σ(H) ≤ E f = E h . Since f is strictly positive, the linear span of the set X of functions satisfying (3.7) is dense in L 2 (M, dµ). It follows that E f = σ(H), since otherwise E f > inf σ(H) and χ (−∞,E f ) (H)L 2 (M, dµ) would contain a nonzero vector which is orthogonal to X . Thus, the statement follows from (3.6).
We now prove that the transformed spin boson Hamiltonian from (2.17) is positivity improving in an appropriate L 2 -representation.
Lemma 3.5. Let ϑ be the natural isomorphism ) is the unitary from Lemma C.3. Then the operator ϑe −T H(λ,µ) ϑ * is positivity improving for all T > 0.
We now obtain Lemma 2.9 as an easy corollary of Lemma 3.5.
Proof of Lemma 2.9. Let ϑ be defined as in Lemma 3.5. By the definitions (2.7) and (2.17), we have .
Further, the following statement also is a direct consequence of Lemma 3.5. It will be a useful ingredient to our proof of Theorem 2.10.
Hence, we want to prove where (·) (n) as usually denotes the n-th derivative. We observe that the ground state energy e(µ 0 ) is a simple eigenvalue of h(µ 0 ), by Proposition 3.6. Further, by view of (2.7), it is obvious that the operator valued family µ → h(µ) = h(0) + µσ x ⊗ 1 is an analytic family of type (A), cf. [Kat80,RS78]. Then, by the Kato-Rellich theorem [RS78, Theorem XII.8], it follows that µ → e(µ) is analytic and e(µ) is an isolated simple eigenvalue of h(µ) in a neighborhood of µ 0 .
We introduce the distance of e(µ 0 ) to the rest of the spectrum by By the Kato-Rellich theorem [RS78, Theorem XII.8], we can choose an ε > 0 such that where the second inequality can be obtained using a Neumann series, cf. (3.12), or alternatively it can be obtained from the lower boundedness of Lemma 2.1 and a compactness argument involving that the set of (µ, z), for which h(µ)−z is invertible, is open, see [RS78,Theorem XII.7]. Henceforth, let µ ∈ (µ 0 − ε, µ 0 + ε). Then, by (3.10), we can write the ground state projection P (µ) of h(µ) as where Γ 0 is a curve encircling counterclockwise the point e(µ 0 ) at a distance δ/2 . Further, let and define the curve Γ 1 = −γ + + γ 0 + γ − surrounding the set σ(h(µ 0 )) \ {e(µ 0 )} (see Fig. 1). In view of (3.10), we can define where the integral is understood as a Riemann integral with respect to the operator topology. The spectral theorem for the self-adjoint operator h(µ) and Cauchy's integral formula yield e −T (h(µ)−e(µ)) = P (µ) + Q T (µ). (3.11) For z ∈ ρ(h(µ 0 )) and µ in a neighborhood of µ 0 we have (3.12) Using this expansion and the following bounds obtained from (3.10) we see that P (µ) and Q T (µ) are real analytic for µ in a neighborhood of µ 0 and, moreover, that the integrals and derivatives with respect to µ can be interchanged due to the uniform convergence of the integrand on the curves Γ 0 and Γ 1 . Hence, by virtue of (3.11), we see that the function µ → Ω ↓ , e −h(µ) Ω ↓ is real analytic on (µ 0 − ε, µ 0 + ε) for ε ∈ (0, ε) small enough. Let ψ µ be a normalized ground state of h(µ). Then, by Proposition 3.6, we find (3.14) Further, by the spectral theorem and (3.10) where ν Ω ↓ denotes the spectral measure of h(µ) associated with Ω ↓ , cf. [RS72, Section VII.2]. By (3.11) and the definition of e T (µ), we have Hence, we can calculate the n-th derivative of the expression on the left hand side at µ = µ 0 , by taking the n-th derivative on the right hand side. Using the Faà di Bruno formula (Lemma B.1) and recalling the notation from Theorem 2.11, we find By (3.14) and (3.15), the first factor is uniformly bounded in T . Hence, it remains to prove that Ω ↓ , Q (k) T (µ 0 )Ω ↓ is uniformly bounded in T for all k = 1, . . . , n. Therefore, we explicitly calculate the derivative of Q T (µ) at µ = µ 0 . This is done by interchanging the integral with the derivative, which we justified above. Note that, by the series expansion (3.12), we have Again using Faà di Bruno's formula (Lemma B.1) and the Leibniz rule, this yields Applying the bounds (3.13), we find Since P k,ℓ (T ) only grows polynomially in T , this implies Q We now combine Bloch's formula for derivatives of the ground state energy with the FKN formula.
Proof of Theorem 2.11. First, we recall the definition of Z T (λ, µ) in (2.22) and the notation ⟪·⟫ T,λ,µ from (2.25). By the dominated convergence theorem, one sees that Z T is infinitely often differentiable in µ and has the derivatives (3.16) Further, first using Theorem 2.10 and the Faà di Bruno formula Lemma B.1 to calculate the derivatives of the logarithm yields u n (X s 1 , . . . , X sn ) ds 1 · · · ds n .
Inserting this into Theorem 2.11 finishes the proof of Corollary 2.12.

Existence of Ground States
In this section, we use the bound on the second derivative of the ground state energy as function of the magnetic coupling from Corollary 2.14 to obtain the result that the spin boson Hamiltonian with massless bosons has a ground state for couplings which exhibit strong infrared singularities. This result is non-trivial, since the massless bosons imply that there is no spectral gap. Our main result needs the following assumptions.
Hypothesis C.
(iv) sup We can now state the main result of this section.
Theorem 4.1. Assume Hypothesis C holds. Then there exists λ c > 0, such that for all λ ∈ (−λ c , λ c ) the spin boson Hamiltonian acting on C 2 ⊗ F has a ground state, i.e., the infimum of the spectrum is an eigenvalue.
Remark 4.2. This improves the previously known results on ground state existence [HH11,BBKM17]. We also remark that in principal our method of proof not only gives existence of a small λ c , but could in fact be used to estimate the critical coupling constant, due to its non-perturbative nature. where χ : R d → R is the characteristic function of an arbitrary ball around k = 0. Obviously the assumptions on ω in Hypothesis C are satisfied. Further, Hypothesis C (iii) holds for any δ > −1 as is easily verified by integration in polar coordinates. The finiteness conditions (iv) of Hypothesis C also hold in this case by simple estimates. We remark that the previous results [HH11,BBKM17] covered the situation (4.2) with δ = − 1 2 . The method of proof relies on the approximation of the photon dispersion relation ν by the infrared-regularized versions ν m = √ ν 2 + m 2 with m > 0. We denote by H m and E m the definitions (2.7) and (2.21) with ω replaced by ν m . Since inf k∈R d ν m (k) ≥ m > 0, the operator H m (λ, 0) has a spectral gap for any m > 0 and hence also a ground state, cf. Theorem D.1. In the recent paper [HHS21b], we showed the following result, which together with Corollary 2.14 give a proof of Theorem 4.1.
We conclude with the proof for existence of ground states.
Proof of Theorem 4.1. Applying Corollary 2.14 to the function ν, we see that the assumptions of Theorem 4.4 are satisfied. This proves the theorem.

A L 2 -valued Riemann Integral and Pointwise Lebesgue Integrability
In Remark 2.5, we use the following lemma with where the integral on the right hand side is the L 2 (Q)-valued Riemann integral.
Proof. Using Fubini's theorem and Hölder's inequality, we find Hence, for µ-almost all q ∈ Q, the map t → f t (q) is Lebesgue-integrable. Let f n,t be an L 2 (Q)valued step function. Then using the triangle inequality, Fubini's theorem and Hölder's inequality, we find Now, by the piecewise L 2 (Q)-continuity of t → f t , the right hand side can be made arbitraritly small by making the mesh of the Riemann sum arbitrarily small. This implies (A.1).

B The Faà di Bruno Formula
The following formula is used in several places throughout the paper. A proof and historical discussion can be found in [Har06].
where P n denotes the set of partitions of the set {1, . . . , n}.

C.1 Standard Fock Space Properties
In this appendix, we collect well-known properties of the Fock space operators introduced in Section 2.1. In large parts, these can be found in standard textbooks such as [RS75,Par92,BR97,Ara18]. For the convenience of the reader, we give exemplary precise references to [Ara18] below.

C.2 Q-Space Construction
In this appendix, we define Gaussian processes indexed by a real Hilbert space r on a probability space (Q, Σ, µ). We then recall the isomorphism theorem connecting F (r ⊕ ir) and L 2 (Q). More details can be found in [Sim74,LHB11]. A random process indexed by r is a (R-)linear map φ from r to the random variables on (Q, Σ, µ). A Gaussian random process indexed by r is a random process indexed by r, such that φ(v) is normally distributed with mean zero for any v ∈ r, has covariance and Σ is the minimal σ-field generated by {φ(v) : v ∈ r}.
The following lemma states existence and uniqueness of Hilbert space valued Gaussian processes. Extensive proofs can, for example, be found in [Sim74, Theorems I.6 and I.9] or [LHB11, Prop. 5.6, Section 5.4]. For the convenience of the reader, we add a sketch of the proof below.
Lemma C.2. For any real Hilbert space r there exist a unique (up to isomorphism) probability space (Q r , Σ r , µ r ) and a unique (again up to isomorphism) Gaussian random process φ r indexed by r on (Q r , Σ r , µ r ).
Sketch of Proof. Existence: We present one possible construction here, further constructions can be found in [Sim74,LHB11]. Let {e i } i∈I be a (not necessarily countable) orthonormal basis of r. We set Q r = × i∈I (R∪ {∞}) and equip it with the infinite product measure of the probability measures π −1/2 exp(−x 2 i )dx i , i ∈ I , which obviously is a probability measure itself. The Gaussian random process is now defined by φ r (e i ) being the multiplication operator with the variable x i . Clearly, µ r • φ r (e i ) is normally distributed with mean zero and variance 1 2 . It also easily follows that Qr φ r (e i )φ r (e j )dµ r = 1 2 δ i,j , with δ denoting the usual Kronecker symbol. Finally, the Borel σ-algebra on Q r is generated by the set {φ r (e i ) : i ∈ I }. Hence, extending this definition to φ r (f ) for arbitrary f ∈ r by linearity finishes the construction. Uniqueness: The uniqueness can be deduced from the Kolmogorov extension theorem [Sim79, Theorem 2.1], which states that a probability space is uniquely determined by a consistent family of probability measures.
The Hilbert space isomorphism introduced in Lemma C.3, below, is often referred to as Wiener-Itô-Segal isomorphism. More details on its construction, which is sketched below, can be found in [Sim74,Theorem I.11] or [LHB11,Prop. 5.7]. Here, we denote the complexification of the real Hilbert space r as r C , which is the real Hilbert space r × r with the complex structure given by i(ψ, φ) = −(φ, ψ). Lemma C.3. There exists a unitary operator Θ r : F (r C ) → L 2 (Q r ) such that (i) Θ r Ω = 1, Sketch of Proof. We recursively define the Wick product of r-indexed Gaussian random variables by Then, the map Θ r : F (r C ) → L 2 (Q r ) given by (i) and f 1 ⊗ s · · · ⊗ s f n → √ 2 : φ r (f 1 ) · · · φ r (f n ) : for f 1 , . . . , f n ∈ r extends to a unitary. The property (ii) follows explicitly from the definitions (2.4), (2.5) and (C.1) and the fact that the pure symmetric tensors form a core for ϕ(f ) by construction.

D The Massive Spin Boson Model
In this appendix, we prove that the ground state energy of the spin boson model with external magnetic field H(λ, µ) is in the discrete spectrum for any choice of the coupling constants λ, µ ∈ R, if the dispersion relation ω is massive, i.e., The statement of the following theorem for the case µ = 0, except for some simple technical restrictions on Hypothesis A, can be found in [AH95].
We obtain the above theorem as a corollary of the following proposition.
Remark D.3. The statement can be seen as one half of an HVZ-type theorem for the spin boson model with external magnetic field. By a slight generalization of known techniques, one can prove see for example [AH95,DM20]. Here, we restrict our attention to the proof of the lower bound.
In the context of non-relativistic quantum field theory, HVZ-type theorems are often proven using spatial localization of quantum particles, cf. [DG99, GLL01, Møl05, LMS07, HS20]. To bound the error terms obtained by confining the system to a ball of radius L in position space, one needs to estimate the commutator of the multiplication operator ω and the Fourier multiplier η(−i∇/L), where η is a smooth and compactly supported function. Bounds on the commutator can be easily obtained, when ω is Lipschitz-continuous (cf. [HS20, Proof of Lemma 24]). However, for less regular choices of the dispersion relation, a generalization of the standard localization approach does not seem obvious.
Here, we use an approach used by Fröhlich [Frö74] and recently applied in [DM20], allowing us to work directly in momentum space and without any regularity assumptions on ω going beyond Hypothesis A. The proof needs several approximation steps, so we start out with a convergence lemma. In the statement, the norms · ∞ and · 2 are the usual norms in L ∞ (R d ) and L 2 (R d ).
We define a linear map U : C 2 ⊗ F → ∞ n=0 (C 2 ⊗ F (V)) ⊗ V ⊥ ⊗sn , where we set C := V ⊗s0 for any vector space V , by (f 1 ⊗ s · · · ⊗ s f n ) → p∈{0,1} n s j=1,...,n p j =1 P V f j ⊗ s j=1,...,n p j =0 (1 − P V )f j where P V denotes the orthogonal projection in L 2 (R d ) onto V. It is straightforward to verify that U is unitary. For n ∈ N 0 , we denote by Π n the projection onto the subspace C 2 ⊗ F (V) ⊗ V ⊥ ⊗sn in the range of U. From the definition (2.7) and the definitions of T and U above, it is easily verified that (D.6) Thus inf σ(T ) ≥ E(λ, µ) and hence Π k UH(λ, µ)U * Π k ≥ E(λ, µ) + m ω for k ≥ 1. Now, assume γ ∈ σ ess (H(λ, µ)). Then, by Weyl's criterion, there exists a normalized sequence (ψ n ) n∈N weakly converging to zero such that We now want to show that the last term converges to zero. To that end, we write Π 0 (Uψ n ), T Π 0 (Uψ n ) = SΠ 0 (Uψ n ), S −1 T Π 0 (Uψ n ) , (D.9) where S = 1 ⊗(1 + dΓ(ω)) as operator on C 2 ⊗F (V). By (D.7), SΠ 0 (U * ψ n ) is uniformly bounded in n. We write The assumption ω ≥ m ω > 0 implies that Since F (≤N ) (V) is finite-dimensional by construction S −1 T ↾ F (≤N) (V) has finite rank for any N ∈ N and it follows that S −1 T is compact, since it is the limit of compact operators. Hence, the last term on the right hand side of (D.9) (and hence that of (D.8)) converges to zero as n → ∞. This finishes the first step.
Step 2. We now relax the condition that ω and v must be simple functions: Assume M ⊂ R is a bounded measurable set, ω1 M is bounded and v = 0 almost everywhere on M c . By the simple function approximation theorem [Fol99, Theorem 2.10], we can pick a sequence (ω k ) k∈N of pointwise monotonically increasing simple functions on M uniformly converging to ω. Outside of M we set ω k equal to ω. Further, w.l.o.g., we can assume that m ω ≤ ω k . For given k ∈ N let M k,i , i = 1, . . . , N k , be a disjoint partition of M into measurable sets such that ω k ↾ M k,i is constant for all i = 1, . . . , N k . Further, w.l.o.g, we can assume that where diam denotes the usual diameter of a bounded set. Then, we define a projection P onto the vector space of simple functions with support in M by which can be easily verified to be well-defined for any f ∈ L 2 (R d ). If f is continuous and compactly supported on M, then it is straightforward to verify P k f k→∞ − −− → f in L 2 -sense. Since the continuous, compactly supported functions are dense in L 2 (M), this implies s-lim k→∞ P k = 1 L 2 (M ) in strong operator toplogy.
We now define v k = ω 1/2 k P k (ω −1/2 v) and observe this directly implies ω −1/2 k v k converges to ω −1/2 v in L 2 -sense. Further, by the triangle inequality and monotonicity of the integral we find By construction the right hand side goes to zero as k → ∞. Hence, we have shown that all assumptions of Lemma D.4 are satisfied and the operators H k (λ, µ), defined in (D.2), are uniformly bounded from below and converge to H(λ, µ) in norm resolvent sense. Further, ω k and v k satisfy by construction the assumptions of Step 1. The statement of the Theorem now follows under the simplifying assumptions of Step 2, since on the one hand the uniform convergence of ω k to ω implies m ω k converges to m ω and on the other the norm resolvent convergence and uniform lower boundedness imply convergence of the ground state energy, cf. [Tes14, Theorem 6.38], as well as the infimum of the essential spectrum, cf. [RS78, Theorem XIII.77].
Step 3. We now move to the general case.
Set v k = 1 M k v. Then, taking k → ∞, it is straightforward to verify that both v k and ω −1/2 v k converge to v and ω −1/2 v in L 2 -sense, respectively. Hence, we can once more apply Lemma D.4 to see that H k (λ, µ) with ω = ω k and v k in (D.2) is uniformly bounded below and converges to H(λ, µ) in the norm resolvent sense as k → ∞. Since, ω and v k also satisfy the assumptions of Step 2, the statement again follows by the spectral convergence as in Step 2.
We conclude this appendix with the Proof of Theorem D.1. Since the spectrum of H(λ, µ) is the disjoint union of discrete and essential spectrum, the statement follows from Proposition D.2.