Analyticity of the Gibbs state for a quantum anharmonic crystal: no order parameter
This work may be considered as a continuation of the research described in the article by \textit{R. A. Minlos, A. Verbeure} and \textit{V. A. Zagrebnov} [Rev. Math. Phys. 12, 981-1032 (2000; Zbl 0976.82010)]. The authors consider the following model of the physical system called by them quantum anharmonic crystal. To every point of the \(\nu\)-dimensional simple cubic lattice there is attached a quantum particle of mass \(m\) performing one-dimensional anharmonic oscillations. The particles interact with each other, the formal Hamiltonian describing this system is taken in the form \[ H = - {\hbar \over 2m} \sum_{x} {\partial^2 \over \partial q_x^2} + \varepsilon \sum_{\langle x, y\rangle}(q_x - q_y)^2 + \sum_x V(q_x) - h\sum_x q_x , \] where \(\varepsilon >0\) is the interaction intensity, \(q_x\) describes displacements of the particle attached to \(x\) and the second sum is taken over nearest neighbors. The one-particle potential is a polynomial \[ V(q) = a_0 q^{2s} + a_1 q^{2s-2} + \cdots + a_{2s-2} q^2 , \quad s>1 , \;a_0 >0. \] By means of a suitable modification of the cluster expansion method developed for this model in the paper mentioned above, the authors obtain the following result. There exists a positive constant \(m_0\), depending possibly on \(\varepsilon\), \(\nu\) and the parameters of \(V\) such that, for \(m\in (0, m_0)\), the thermodynamic limits obtained under the zero boundary conditions for the free energy density and expectation values of local observables are holomorphic functions of the external field \(h\) in the vicinity of the origin, the radius of which depends on \(m\). This effect holds for all temperatures including zero.
scientific article; zbMATH DE number 1854193