On the existence and uniqueness of ground states of a generalized spin-boson model
The standard spin-boson Hamiltonian \[ H_{SB}(\alpha)= {\mu\over 2} \sigma_z\otimes I+ I\otimes H_b+ \sqrt 2\alpha\sigma_x\otimes \phi(\lambda) \] acting on the Hilbert space \(\mathbb{C}^2\otimes{\mathcal F}_b\) models a system of a spin and a quantized Bose field which are coupled to each other, with a coupling parameter \(\alpha\). Here \(\sigma_x\) and \(\sigma_z\) are the conventional Pauli matrices and \(\mu\) is a positive constant. \(H_b= \int_{\mathbb{R}^\nu} dk\omega(k)a(k)^*a(k)\) is the free Hamiltonian of bosons and, with some function \(\lambda\in L^2(\mathbb{R}^\nu)\), \(\phi(\lambda)= (1/\sqrt 2)\int_{\mathbb{R}^\nu} dk(\overline{\lambda(k)} a(k)+ \lambda(k)a(k)^*)\) the field operator, both on the boson Fock space \({\mathcal F}_b\) over \(L^2(\mathbb{R}^\nu)\), where \(a(k)\) and \(a(k)^*\) stand for the annihilation and creation operators with commutation relation \([a(k), a(k')^*]= \delta(k- k')\). The function \(\omega(k)\) is nonnegative and \(m:=\inf\omega(k)\) is regarded as the mass of the boson. The paper under review considers a generalization of this model with the Hamiltonian \[ H(\alpha)= A\otimes I+ I\otimes H_b+ \alpha \sum^J_{j= 1} B_j\otimes \phi(\lambda_j) \] acting on the tensor product \({\mathcal H}\otimes{\mathcal F}_b\) of a Hilbert space \({\mathcal H}\) and the boson Fock space \({\mathcal F}_b\), where \(A\) is a selfadjoint operator and \(B_j\), \(1\leq j\leq J\), for finite \(J\), are symmetric operators in \({\mathcal H}\). Assumed is that \(A\) is bounded from below by \(\mu_0\) and for each \(j\), \(B_j\) is \((A- \mu_0)^{1/2}\)-bounded with relative bound \(a_j\) such that \(|\alpha| \sum^J_{j= 1}a_j\|\lambda_j/ \sqrt\omega\|_{L^2}< 1\), and that \(\omega(k)\) is locally Hölder-continuous with \(\lim_{| k|\to \infty} \omega(k)= \infty\) and each \(\lambda_j(k)\) is continuous. Then the main result is: If \(A\) has a compact resolvent, there exists a ground state of \(H(\alpha)\), not only in the massive case \((m> 0)\) but also in the massless case \((m= 0)\), though this case needs to impose further the condition \(\lambda_j/\omega\in L^2(\mathbb{R}^\nu)\) with some auxiliary ones. Of course, the result includes the standard spin-boson model case [\textit{H. Spohn}, Commun. Math. Phys. 123, No. 2, 277-304 (1989; Zbl 0667.60108)]. The proof is operator-theoretical; in fact, it does not seem to be able to appeal to functional integral methods, because \(A\) is an abstract operator in a Hilbert space and so is \(H(\alpha)\). The massive case makes use of a finite volume approximation in constructive field theory. Then the ground state in the massless case turns out to be obtained as a nonzero weak limit of ground states in the massive case as \(m\to 0\). Further, in the massive case, the authors also treat some special models where \(A\) and the \(B_j\) are bounded so that \(\alpha\) may be arbitrary, and where the \(B_j\) commute mutually, to discuss uniqueness and degeneracy of the ground state with the min-max principle.
scientific article; zbMATH DE number 1117827
scientific article