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In the context of Weyl semimetals, I often read the statement that a Lifshitz phase transition occurs when a Weyl cone is tilted so much that it tips over and crosses through the original Fermi level.
Hence my question: how is a Lifshitz phase transition defined? Is it just the statement that a Fermi surface goes from closed to open? Is there more to it? A reference to a minimal model of such a transition would be appreciated.
A Lifshitz transition is a change in the topology of a Fermi surface. They involve the appearance of zeros in the energy spectrum of a many-body fermion system, excluding the normal Fermi surface of course.
Click here for Lifshitz's original work. Also, this is another good reference for Lifshitz transitions (from Volovik).
A Lifshitz transition occurs when the gradient term in the Ginzburg-Landau (GL) free energy functional changes sign, meaning one must then include a higher-order gradient term to stabilize the free energy. Near the phase transition, one expands the free energy in powers of an order parameter. Consider the simplest case of a scalar order parameter $\phi$, such that the free energy takes the form
$$F \sim \int d^dx \left(\alpha\,(\partial_\mu\phi)^2 + \beta \,\phi^2 + \gamma \,\phi^4\right)$$
For simplicity, we assume that $\alpha > 0$, $\beta$ is a function of temperature, and $\gamma > 0$ is a constant. If $\beta$ is positive, then the free energy is minimized by $\phi = 0$ everywhere. Since $\alpha > 0$, the free energy is minimized by a homogeneous $\phi$. If at some temperature $T_c$ $\beta$ becomes negative, there will be a second-order phase transition, and the free energy will be minimized by a non-zero $\phi$. A Lifshitz transition occurs if $\alpha$ becomes negative, in which case we must include a $(\partial_\mu\phi)^4$ or $(\nabla^2 \phi)^2$ term to stabilize the system. The new minimum of the free energy is now obtained by a non-uniform $\phi$, such as spiral order in a magnet, or charge density waves in a metal. In reciprocal space, this would correspond to a minimum with a non-zero wavevector $\mathbf{q}$. Usually, the ordering wavevector is incommensurate with the underlying lattice. See, for example, chapter 2 section 4 of Daniel Khomskii's book Basic Aspects of the Quantum Theory of Solids.
