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More concretely: When looking at the Ising model in the description of Bogoliubov fermions, we get an explicit expression for the energy gap, that vanishes for a particular value of the magnetic field. Many sources (including this one on page 36: http://edu.itp.phys.ethz.ch/fs13/cft/SM_Molignini.pdf) claim that this implies that this is the critical value at which the (quantum) phase transition occurs. What is the argument behind this? One can off course argue that the energy gap needs to disappear to make a CFT description possible, but this seems a bit backwards to me. Without invoking CFTs, what is the physical reason critical points are gapless?

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I assume your theory is described by a hamiltonian $H$ with a continuous parameter $x$. A phase transition means that the ground state will suddenly (and not smoothly) change for some value of this parameter. If the expression of the hamiltonian is smooth with respect with the parameter $x$, the ground state should be smooth too.

But there's an exception : if the two smallest eigenvalues cross, the ground state nature change and you can have a discontinuity. At the point where they cross the energy gap (the difference between the two lowest energy) disappears.

Really basic example not physical : $H = \begin{pmatrix} 1 & x \\ x & 1 \end{pmatrix}$ has eigenvalues $1-x$ and $1+x$, eigenvectors $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$. When $x = 0$, the energy gap disappear and the ground state change suddenly, otherwise it's smooth (constant even).

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The only relevant length scale close to a phase transition is the correlation length $\xi$. By dimensional analysis, the mass gap obeys $m_G \sim \xi^{-1}$, and at a second order phase transition we generically expect $\xi \to \infty$, implying a gapless system.

A more precise way of saying this is that at long distances, the two point correlation function behaves like (in all “reasonable” cases that we typically see) $$ \frac{e^{-r/\xi}}{r^{d-2+\eta}}$$ This actually defines the correlation length. By the argument above, the correlation length is the inverse mass gap, so gaplessness generically implies power-law correlations, which signal phase transitions.

The simplest example of this is the exactly solvable massless free scalar. When massive, the theory is described by the Lagrangian $(\partial \phi)^2 + m^2 \phi^2$ where I’ve neglected all constants. It’s easy to solve for the correlator exactly, and it decays like $e^{-mr}$ times a power as $r \to \infty$. When the mass gap vanishes, we are at criticality.

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