How to understand the negative specific heat of a massless Majorana fermion in one dimension? Let us consider a massless Majorana fermion on a ring of length $L$ with a periodic boundary condition at a temperature $T$.
Then the conformal field theory calculation tells us that the quantum partition function of this single Majorana is ($k_\text{B}\equiv1$)
\begin{eqnarray}
Z(T,L)=\chi_{1/16}(\tau)\chi^*_{1/16}(\tau)|_{\tau=i/(LT)}, 
\end{eqnarray}
where $\chi_{h}(\tau)$ is the Virasoro character of the primary field with conformal dimension $h$.
Then when the system size is large $LT\gg1$, the asymptotic behavior of the partition function is
\begin{eqnarray}
Z(T,L)&\approx&\exp\left\{\frac{\pi LT}{6}[c-12(h+\bar{h})]\right\},
\end{eqnarray}
where $h=\bar{h}=1/16$ namely the lowest value of the Virasoro generators $L_0$ and $\bar{L}_0$ of the massless Majorana with a central charge $c=1/2$.
Therefore, we can obtain the specific heat per unit length as
\begin{eqnarray}
c_v&=&\frac{1}{L}T\frac{\partial^2}{\partial T^2}[T\ln(Z(T,L))]\\
&=&\frac{\pi T}{3}(c-12h-12\bar{h})\\
&=&-\frac{\pi T}{3}\\
&<&0\text{ (when $T>0$)}.
\end{eqnarray}
My question is how to understand this negative specific heat and does it mean that Majorana fermion is thermally unstable at all at any finite temperature?
 A: The free energy per unit lenth of a chiral $c=1$ Dirac fermion, or a non-chiral  $c=1/2$ massless Majorana is
$$
\beta F/L = -\int_{-\infty}^{\infty} \frac{dk}{2\pi} \ln(1+ e^{-\beta v_f \hbar |k|})\\
=  - \frac{1}{\pi \beta v_f\hbar }\sum_{n=1}^\infty(-1)^{n+1}\frac 1{n^2}\\
= - \frac {\pi }{12 }\frac1 {\beta v_f\hbar } \nonumber
$$
For general central charge $c$ we have
$$
F/L= -\frac{\pi c}{6\beta^2 v_F \hbar}
$$
this is negative, but  the internal energy is
$$
\langle E\rangle/L=\frac{\partial}{\partial  \beta}(\beta F/L)= +\frac{\pi c}{6\beta^2 v_F \hbar}=\frac{\pi}{12\beta^2 v_F \hbar}= \frac{\pi k_B^2 T^2}{12 v_F \hbar}.
$$
The specific heat is then
$$
(1/L)\frac{\partial \langle E\rangle}{\partial T}=\frac{\pi c k_B^2 T}{3v_F \hbar}=\frac{\pi  k_B^2 T}{6v_F \hbar}
$$
This is positive and depends only on $c$ as it should. I don't know where you get the weight  $h$ bits from.  I think there are additional factors to be included in  expression  for the thermodynamic partition function in terms of the  the Virasoro characters, but it's too long since I worked on this stuff, and my copy of Di Francisco in in my inaccessible office.
