$y-z$ plane. The former is essentially a classical model while the latter corresponds to the quantum Ising model. In both cases, the Hamiltonian is analytically soluble. Of course, it is well known that the Ising model exhibits criticality when $B = J$, provided we find ourselves in the thermodynamic limit of an infinite spin chain. Aside from the two special cases outlined, however, the Hamiltonian will in general be non-integrable at any point along the circuit; this forces us to resort to numerical simulations of a limited number of spins, which moves the Ising model’s critical point outside our region of accessibility. On the other hand, when the magnetic field vector points in the $x$-direction, the Hamiltonian does possess a critical point, even for a *finite* number of spins, as outlined in the following.

The Critical Point {#section:CriticalPoint}
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A special case of the class of Hamiltonians (\[eq:generalham\]) occurs when the magnetic field vector points in the $x$-direction: $$\label{eq:specialham}
    H_x =
    J \sum_{k=1}^{N}\sigma^{x}_{k}\sigma^{x}_{k+1}
    + B \sum_{k=1}^{N}\sigma^{x}_{k}, \quad N \geqslant 2.$$ This Hamiltonian is purely classical: its two constituent terms commute and are diagonal in the {} eigenbasis. In the following we will demonstrate that $H_x$ possesses a critical point [@footnote1] 