Spin-Boson Model. Fumio Hiroshima ( ) Faculty of Mathematics, Kyushu University Fukuoka, , Japan

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1 Spin-Boson Model Fumio Hiroshima ( Faculty of Mathematics, Kyushu University Fukuoka, , Japan hiroshima@math.kyushu-u.ac.jp 1 Spin-boson model 1.1 Definition Spin-boson model describes an interaction between an ideal two-level atom and a quantum scalar field. Two eigenvalues of the atom are embedded in the continuous spectrum when no perturbation is added. See Figure 1. We are interested in investigating behaviors of embedded eigenvalues after adding a perturbation. In particular we consider properties of the bottom of the spectrum of spin-boson Hamiltonians by functional integrations. In this article we show an outline of HHL08. Figure 1: Embedded eigenvalues Let F = n=0 ( n sym L (R d be the boson Fock space over L (R d, where the subscript means symmetrized tensor product. We denote the boson annihilation and creation operators by a(f and a (f, f, g L (R d, respectively, satisfying the canonical commutation relations a(f, a (g = ( f, g, a(f, a(g = 0 = a (f, a (g. (1.1 We use the informal expression a (f = a (kf(kdk for notational convenience. Consider the Hilbert space H = C F. Denote by dγ(t be the second quantization of a self-adjoint 1

2 operator T in L (R d. The operator on Fock space defined by H f = dγ(ω is the free boson Hamiltonian with dispersion relation ω(k = k. The operator ϕ b (ĥ = 1 ( a (kĥ( k + a(kĥ(k dk, (1. acting on Fock space is the scalar field operator, where h L (R d is a suitable form factor and ĥ is the Fourier transform of h. Denote by σ x, σ y and σ z the Pauli matrices given by σ x = 0 1, σ 1 0 y = 0 i 1 0, σ i 0 z =. ( With these components, the spin-boson Hamiltonian is defined by the linear operator H SB = εσ z 1l + 1l H f + ασ x ϕ b (ĥ (1.4 on H, where α R is a coupling constant and ε 0 a parameter. 1. A Feynman-Kac-type formula In this section we give a functional integral representation of e th SB by making use of a Poisson point process and an infinite dimensional Ornstein-Uhlenbeck process. First we transform H SB in a convenient form to study its spectrum in terms of path measures. Recall that the rotation group in R 3 has an adjoint representation on SU(. In particular, for n = (0, 1, 0 and θ = π/, we have e (i/θn σ σ x e (i/θn σ = σ z and e (i/θn σ σ z e (i/θn σ = σ x. Let U = exp ( i πσ 4 y 1l = l be a unitary operator on H. Then H 1 1 SB transforms as H = UH SB U = εσ x 1l + 1l H f + ασ z ϕ b (ĥ. (1.5 If ĥ/ ω L (R d and h is real-valued, then ϕ b (ĥ is symmetric and infinitesimally small with respect to H f, hence by the Kato-Rellich theorem it follows that H is a self-adjoint operator on D(H f and bounded from below. To construct the functional integral representation of the semigroup e th, it is useful to introduce a spin variable σ Z, where Z = { 1, +1} is the additive group of order. For Ψ(+ (Hf + αϕ Ψ = H, we have HΨ = b (ĥψ(+ εψ(. Thus we can transform Ψ( (H f αϕ b (ĥψ( εψ(+ H on H to the operator H on L (Z ; F by ( HΨ(σ = (H f + ασϕ b (ĥ Ψ(σ + εψ( σ, σ Z. (1.6 In what follows, we identify the Hilbert space H with L (Z ; F, and instead of H we consider H, and use the notation H for H. Let (Ω, Σ, P be a probability space, and (N t t R be a two-sided Poisson process with unit intensity on this space. We denote by D = {t R N t+ N t } the set of jump points, and define the integral with respect to this Poisson process by f(r, N (s,t rdn r = r D f(r, N r r (s,t

3 3 for any predictable function f. In particular, we have for any continuous function g, g(r, N (s,t r dn r = g(r, N r D r. We write t+ dn s<r t s r for dn (s,t r. Note that t+ g(r, N s r dn r is right-continuous in t and the integrand g(r, N r is left-continuous in r and thus a predictable process. Define the random process σ t = σ( 1 N t, σ Z. In the Schrödinger representation the boson Fock space F can be realized as an L -space over a probability space (Q, µ, and the field operator ϕ b ( ˆf with real-valued function f L (R d as a multiplication operator, which we will denote by ϕ(f. The identity function 1l on Q corresponds to the Fock vacuum Ω b in F. Let (Q E, µ E be a probability space associated with the Euclidean quantum field. The Hilbert spaces L (Q E and L (Q are related through the family of isometries {j s } s R from L (R d to L (R d+1 defined by ĵsf(k, k 0 = e itk 0 ω(k ˆf(k. π k 0 Let Φ +ω(k E (j s f be a Gaussian random variable on (Q E, µ E indexed by j s f L (R d+1 with mean zero and covariance E µe Φ E (j s fφ E (j t g = 1 e s t ω(k ˆf(kĝ(kdk. R Also, let d {Js } s R be the family of isometries from L (Q to L (Q E defined by J s :ϕ(f 1 ϕ(f n : = :Φ E (j s f 1 Φ E (j s f n :, where :X: denotes Wick product of X. Then we derive that (J s Φ, J t Ψ L (Q E = (Φ, e t s H f ΨL (Q. We identify H as H = L (Z ; L (Q = L (Z Q. Proposition 1.1 Let Φ, Ψ H and h L (R d be real-valued. Then (ε 0 (Φ, e th Ψ H = e t E P E µe J 0 Φ(σ 0 e αφ E( t 0 σ sj s hds ε N t J t Ψ(σ t (1.7 σ Z (ε = 0 (Φ, e th Ψ H = e t E µe J 0 Φ(σe αφ E(σ t jshds 0 J t Ψ(σ. (1.8 σ Z Denote 1l H = 1l L (Z 1l L (Q. Using the above proposition we can compute the vacuum expectation of the semigroup e th. Corollary 1. Let h L (R d be a real-valued function. Then for every t > 0 it follows that (1l H, e th 1l H = e t E P ε N t e α t 0 dr t 0 W (N r N s,r sds, (1.9 σ Z where the pair interaction potential W is given by W (x, s = ( 1x e s ω(k ĥ(k dk. (1.10 R d Ground state of the spin-boson In the remainder of this paper we assume that h L (R d is real-valued. Let E = inf Spec(H. Assume that ε 0. Then e th, t > 0, is a positivity improving semigroup on L (Z Q, i.e., (Ψ, e th Φ > 0 for Ψ, Φ 0 such that Ψ 0 Φ. By this we can see that Ker(H E = 1 for ε 0. We consider the case of ε 0. Write Φ T = e T (H E 1l and γ(t = (1l H, Φ T = (1l H, e T H 1l H (1l H, e T H 1l H. (.1

4 4 A known criterion of existence of a ground state is LHB11, Proposition 6.8. Proposition.1 A ground state of H exists if and only if lim γ(t > 0. T By Corollary 1. we have = e T E σ Z (1l H, Φ T = e T E σ Z E P ε N T e α T T dt T T W (N t N s,t sds, (. E P ε N T e α 0 T dt 0 T W (N t N s,t sds. (.3 Note that 0 dt T W (N T 0 t N s, t sds 1 ĥ/ω uniformly in T and in the paths. Theorem. If ĥ/ω L (R d, then H has a ground state and it is unique. Proof: We can show that lim T γ(t > 0 by using (. and (.3. Then the theorem follows from Proposition.1. It is a known fact that H SB has a parity symmetry. Let P = σ z ( 1 N, where N = dγ(1l denotes the number operator in F. From Spec(σ z = { 1, 1} and Spec(N = {0, 1,,...} it follows that Spec(P = { 1, 1}. Then H can be decomposed as H = H + H and H can be reduced by H ±. Corollary.3 Let φ SB be the ground state of H SB. Then φ SB H. 3 Path measure associated with the ground state In this section we set ε = 1 for simplicity. Let X = D(R; Z be the space of càdlàg paths with values in Z, and G the σ-field generated by cylinder sets. Thus σ : (Ω, Σ, P (X, G is an X -valued random variable. We denote its image measure by W σ, i.e., W σ (A = σ 1 (A for A G, and the coordinate process by (X t t R, i.e., X t (ω = ω(t for ω X. Hence Proposition 1.1 can be reformulated in terms of (X t t R as (Φ, e th Ψ H = e t σ Z E µ E J 0 Φ(X 0 e αφ E( t Xsjshds 0 J t Ψ(X t. (3.1 Here E W σ = so that Eσ W X 0 = σ = 1. Let (Z, B be a measurable space with σ- field B = {, { 1}, {+1}, Z }. We see that the operator Q S,T = JS eφ E( α T S X sj s hds J T : L (Q L (Q is bounded.

5 5 Corollary 3.1 Let < t 0... t n < and A 0,..., A n B. Then where W (x, y, t = xy (1 (Φ, 1l A0 e (t 1 t 0 H 1l A1 e (t t 1 H e (tn tn 1H 1l An Ψ ( n = e tn t 0 E µ E 1l Aj (X tj Φ(X t0 Q t0,t n Ψ(X tn, σ Z ( (1l A0, e (t 1 t 0 H 1l A1 e (t t 1 H e (tn tn 1H 1l An = e tn t 0 e α t n t0 dt tn n t0 dsw (X s,x t,t s 1l Aj (X tj, σ Z R d e t ω(k ĥ(k dk. Now we make the assumption that ĥ/ω L (R d, so that there is a unique ground state φ g H. Let G T,T = σ(x t, t T, T be the family of sub-σ-fields of G and G = T 0 G T,T. Let G = σ(g. Define the probability measure µ T on (X, G by µ T (A = et Z T σ Z 1l A e α T T dt T T dsw (Xt,Xs,t s, A G, (3. where Z T is the normalizing constant such that µ T (X = 1. This probability measure is a Gibbs measure for the pair interaction potential W, indexed by the bounded intervals T, T. Let µ be a probability measure on (X, G. The sequence of probability measures (µ n n is said to converge to the probability measure µ in local weak topology whenever lim n µ n (A µ (A = 0 for all A G t,t and t 0. By the definition it is seen that whenever µ T µ in local weak sense, we have that lim T E µt f = E µ f for any bounded G t,t -measurable function f. We define below a probability measure ρ T on (X, G T,T and an additive set function µ on (X, G. The unique extension of µ to a probability measure on (X, Ḡ is denoted by µ. We shall prove that µ T (A = ρ T (A for all A G t,t with t T, and show that ρ T (A µ(a as T, which implies that µ T converges to µ in the sense of local weak. We define the finite dimensional distributions indexed by Λ = {t 0,..., t n } T, T with t 0... t n. Let ( µ Λ T (A 0 A n = et n 1l Aj (X tj e α T T dt T T dsw (Xt,Xs,t s (3.3 Z T σ Z be a probability measure on (Z Λ, B Λ, where Z Λ = n j=1 Zt j and B Λ = n j=1b t j for Λ = {t 1,..., t n }, and Z t j and B t j are copies of Z and B, respectively. Clearly, G is a finitely additive family of sets. Define an additive set function on (X, G by µ(a = e Et e t 1lA (φ g (X t, Q t,t φ g (X t H, A G t,t. (3.4 σ Z

6 6 Note that µ(x = (φ g, e t(h E φ g = 1. There exists a unique probability measure µ on (X, G such that µ G = µ. In particular, µ (A = µ(a, for every A G t,t and t R. In order to show that µ T (A µ (A for every A G t,t, we define the probability measure ρ T on (X, G T,T for A G t,t with t T by ρ T (A = e Et e t σ Z ( ΦT t (X t Φ T t (X t 1l A, Q t,t. (3.5 Remark 3. Both µ and ρ T are well defined. I.e., for A G s,s G t,t with s t T µ(a = e Es e s 1lA (φ g (X s, Q s,s φ g (X s σ Z = e Et e t σ Z ρ T (A = e Es e s σ Z = e Et e t σ Z 1lA (φ g (X t, Q t,t φ g (X t, ( ΦT s (X s Φ T s (X s 1l A, Q s,s ( ΦT t (X t Φ T t (X t 1l A, Q t,t The family of probability measures ρ Λ T on (ZΛ, B Λ indexed by Λ = {t 0,..., t n } T, T is defined by ρ Λ T (A 0 A n = e Et e ( n (ΦT t E σ t (X t Φ T t (X t W 1l Aj (X tj, Q t,t (3.6 Φ σ Z T for arbitrary t such that T t... t 0... t n t T. To show that µ T = ρ T, we prove that their finite dimensional distributions coincide. Lemma 3.3 Let Λ = {t 0, t 1,..., t n } and A 0 A n B Λ. Then µ Λ T (A 0 A n = ρ Λ T (A 0 A n, and µ T (A = ρ T (A follows for A G t,t and t T. Proof: The former statement follows from Corollary 3.1 and the later from Kolmogorov consistency theorm. Theorem 3.4 Suppose ĥ/ω L (R d. Then the probability measure µ T on (X, G converges in local weak sense to µ as T. Proof: Let A G T,T. Then µ T (A = ρ T (A. Since φ g as T, we can see that ρ T (A µ(a as T. Since µ(a = µ (A, the theorem follows. In the case when ε 1 a parallel discussion to the previous section can be made. We summarize this in the theorem below. Theorem 3.5 Suppose ĥ/ω L (R d. Then the probability measure µ ε T on (X, G converges in local weak sense to µ ε as T. We also write µ g for µ ε for notational convenience. Φ T.

7 7 4 Ground state properties In this section without proofs we show to be able to express ground state expectations of some observables in terms of the limit measure µ g discussed in the previous section. 4.1 Expectations of functions of the form ξ(σf (ϕ(f Theorem 4.1 Let f be a G εt,εt -measurable function on X. Then E µg f = e Eεt e ( εt φ g (X εt, Q (ε εt,εt φ g(x εt f. (4.1 σ Z An immediate consequence of Theorem 4.1 is the following. Corollary 4. Let f j : Z C, j = 0,..., n, be bounded functions. Then n E µg f j (X εtj = (φ g, f 0 e (t 1 t 0 (H E f 1 e (tn tn 1(H E f n φ g. (4. In particular, we have for all bounded functions ξ, f and g that E µg ξ(x 0 = (φ g, ξ(σφ g, (4.3 E µg f(x t g(x s = (f(σφ g, e t s (H E g(σφ g. (4.4 Theorem 4.3 Let ĥ/ω L (R d, f L (R d be real-valued, ξ : Z function, and β R. Then C be a bounded (φ g, ξ(σe iβϕ(f φ g = e β 4 f E µg ξ(x0 e iβk(f, (4.5 where K(f is a random variable on (X, G given by K(f = α (e r ω ĥ, ˆfX εr dr. By using Theorem 4.3 the functionals (φ g, ξ(σf (ϕ(fφ g can be represented in terms of averages with respect to the path measure µ g. Consider the case when F is a polynomial or a Schwartz test function. We will show in Corollary. below that φ g D(e +βn for all β > 0, thus φ g D(ϕ(f n for every n N. Corollary 4.4 Let ĥ/ω L (R d, f L (R d be real-valued, and ξ : Z C a bounded function. Also, let h n (x = ( 1 n e x / d n e x / be the Hermite polynomial of order n. Then dx n (φ g, ξ(σϕ(f n φ g = i n E µg ξ(x 0 h n ( ik(f f 1/ ( f 1/ n, n N. (4.6 In the next corollary we give the path integral representation of (φ g, ξ(σf (ϕ(fφ g for F S (R, where S (R denotes the space of rapidly decreasing, infinitely many times differentiable functions on R. Corollary 4.5 Let ĥ/ω L (R d, f L (R d be real-valued, F S (R, and ξ : Z C a bounded function. Then (φ g, ξ(σf (ϕ(fφ g = E µg ξ(x 0 G (K(f, where G = ˇF ǧ and g(β = e β f /4.

8 8 4. Exponential moments of the field operator In this section we show that (φ g, e βϕ(f φ g < for some β > 0. Theorem 4.6 Let ĥ/ω L (R d and f L (R d be a real-valued function. If < β < 1/ f, then φ g D(e (β/ϕ(f, e (β/ϕ(f φ g = and lim β 1/ f e (β/ϕ(f φ g =. 1 E µ 1 β f g e βk (f 1 β f, (4.7 Theorem 4.6 says that e (β/ϕ(f φ g <. Using this fact we can obtain explicit formulae of the exponential moments (φ g, e βϕ(f φ g of the field. Corollary 4.7 If ĥ/ω L (R d and f L (R d is a real-valued function, then φ g D(e βϕ(f and (φ g, e βϕ(f φ g = (φ g, cosh(βϕ(fφ g = e β 4 f E µg e βk(f, (4.8 (φ g, σe βϕ(f φ g = (φ g, σ sinh(βϕ(fφ g = e β 4 f E µg X0 e βk(f. ( Expectations of second quantized operators We consider expectations of the form (φ g, e βdγ(ρ( i φ g, where ρ is a real-valued multiplication operator given by the function ρ. An important example is ρ = 1l giving the boson number operator N = dγ(1l. We obtain the expression (Φ T, ξ(σe βdγ(ρ( i Φ T = E µ ε T ξ(x 0 e α 0 T dt T 0 W ρ,β (X εt,x εs,t sds, (4.10 where W ρ,β (x, y, T = xy R d ĥ(k e T ω(k (1 e βρ(k dk. Denote W ρ,β = 0 dt Notice that W ρ,β ĥ/ω / <, uniformly in the paths in X. 0 W ρ,β (X εt, X εs, t sds. (4.11 Theorem 4.8 Suppose that ĥ/ω L (R d and ξ : Z C is a bounded function. Then (φ g, ξ(σe βdγ(ρ( i φ g = E µg ξ(x 0 e α W ρ,β, β > 0. (4.1 Corollary 4.9 Suppose that ĥ/ω L (R d and ξ : Z C is a bounded function. Then (φ g, ξ(σe βn φ g = E µg ξ(x 0 e α (1 e β W, (4.13 where W = 0 dt W (X 0 εt, X εs, t sds. Furthermore φ g D(e βn for all β C and (φ g, e βn φ g = E µg e α (1 e β W (4.14 follows.

9 9 5 Van Hove representation The van Hove Hamiltonian is defined by the self-adjoint operator H vh (ĝ = H f + ϕ b (ĝ (5.1 in Fock space F. Suppose that ĝ/ω L (R d and define the conjugate momentum by π b (ĝ = i ( a (k ĝ(k ω(k a(kĝ( k dk. ω(k Then e iπb(ĝ H vh (ĝe iπ b(ĝ = H f 1 ĝ/ω and the ground state of H vh (ĝ is given by φ vh (ĝ = e iπb(ĝ Ω b. On the other hand, clearly the spin-boson Hamiltonian H with ε = 0 Hf + αϕ is the direct sum of van Hove Hamiltonians since H = b (ĥ 0 0 H f αϕ b (ĥ and H f ± αϕ b (ĥ are equivalent. Therefore the ground state of H with ε = 0 can be realized as φvh φ g = (αĥ φ vh ( αĥ. Thus in this case (φ g, e iβϕ(f φ g H = 1 (φ vh (σαĥ, eiβϕ b( ˆf φ vh (σαĥ F (5. σ=±1 and the right hand side above equals (Ω b, e iβ(ϕ b( ˆf+α(ĥ/ω, ˆf Ω b F = e β ˆf /4+iβα(ĥ/ω, ˆf. When ε 0 we can derive similar but non-trivial representations. Define the random boson field operator Ψ( ˆf = ϕ b ( ˆf + K(f on F. Let χ = α ω(kĥ(k e s ω(k X εs ds. Note that χ L (R d, K(f = (χ, ˆf, moreover, χ/ω L (R d, whenever ĥ/ω L (R d, and χ = σαĥ for ε = 0. We define the random van Hove Hamiltonian by H vh (χ. Theorem 5.1 If ĥ/ω L (R d, then (φ g, e iβϕ(f φ g = E µg (Ω b, e iβψ( ˆf Ω b = E µg (φ vh (χ, e iβϕ b( ˆf φ vh (χ. (5.3 Corollary 5. Suppose ĥ/ω L (R d and F S (R. Then we have (φ g, F (ϕ(fφ g = E µg (Ω b, F (Ψ( ˆfΩ b = E µg (φ vh (χ, F (ϕ( ˆfφ vh (χ, (5.4 e βϕ(f / φ g = E µg e βψ( ˆf / Ω b = E µg e βϕ b( ˆf / φ vh (χ. (5.5 Acknowledgments: This work was financially supported by Grant-in-Aid for Science Research (B from JSPS. References HHL08 M. Hirokawa, F. Hiroshima and J. Lőrinczi, Spin-boson model through a Poisson-driven stochastic process, preprint 01. LHB11 J. Lőrinczi, F. Hiroshima and V. Betz, Feynman-Kac-Type Theorems and Gibbs Measures on Path Space, de Gruyter Studies in Mathematics 34, 011.