Quantum phase transition

Remarks

The limit is the zero-temperature version of the thermodynamic limit, and $e(g)$ plays the role of the free energy density. At a continuous quantum phase transition the gap between the two lowest eigenvalues of $H_L(g_c)$ tends to $0$ as $L \to \infty$.

Example: the transverse-field Ising chain $H_L(g) = -J \sum_{i} \sigma^z_i \sigma^z_{i+1} - g \sum_i \sigma^x_i$ on a ring of $L$ sites, with $J > 0$ and $g \ge 0$. The Jordan–Wigner transformation maps it to free fermions with energies $\varepsilon_k = 2\sqrt{J^2 + g^2 - 2 J g \cos k}$. This gives $e(g) = -\frac{1}{2\pi} \int_{-\pi}^{\pi} \sqrt{J^2 + g^2 - 2Jg \cos k} \, dk$, which is not analytic at $g = J$. The gap $\min_k \varepsilon_k = 2\abs{J - g}$ closes exactly there. Check of the formula at $g = 0$: $e(0) = -J$, the energy per bond of the two aligned ground states.

Non-example: $H_L(g) = -g \sum_{i} \sigma^x_i$ for $g > 0$ has $e(g) = -g$, which is analytic, so there is no transition.