In this exercise, consider the Ising magnet that is isomorphic to the two-state quantal system. Suppose that the electric field, is a random fluctuationg field with a Gaussian probability...

In this exercise, consider the Ising magnet that is isomorphic to the two-state quantal system. Suppose that the electric field,
is a random fluctuationg field with a Gaussian probability distribution,

(a)   Show that on integrating over
 one obtains the partition function

Which is the partition function for a one-dimensional IsIsing magnet with long ranged interactions. Here, 2∆ is the energy spacing of the unperturbed two-state system, ε = ℬ/P, and k=

(b)   The long ranged interactions generated by integrating out fluctuations in the electric field can induce a transition in which tunneling is, the fluctuations in the environment cause a spatial localization of the quantal system. Demonstrate that this transition does occur by first evaluating
 and then showing that the Gaussian weighted integral over
that Q is a non-analytic function of σ when β → ∞. Identify the critical value of σ at which the transition occurs by considering where [(δm)²]  diverges. [Hint: The square fluctuation is a second derivative of In Q.] Note that at non-zero tempera-tures, β∆ is finite: the isomorphic Ising magnet is effectively a finite system in that case, and no localization transition occurs.