# What is a Lifshitz phase transition?

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 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.