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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?

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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.

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    $\begingroup$ I obtained the $h$ and $\hbar$'s from Eq.(21.115) in Fradkin's lecture notes: eduardo.physics.illinois.edu/phys583/ch21.pdf Then, further by Eq.(21.120) there, I had an additional term. I am also trying to check my derivations. $\endgroup$ – Smart Yao Nov 29 '20 at 13:08
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    $\begingroup$ I am not sure, but I guess that your first equation might be calculating the free energy of the fermion in the Neveu-Schwarz (antiperiodic boundary condition) sector (in which the Majorana has partition function as $|\chi_0+\chi_{1/2}|^2$). There could be a correction $(2*1/16)$, due to normal ordering, on Casimir energy $(-2*c/24)$ between Neveu-Schwarz sector and the Ramond sector. $\endgroup$ – Smart Yao Nov 29 '20 at 14:01
  • $\begingroup$ You want Eduardo's 21.114 with $l\to \beta = 1/kT$ to get my expression $f=F/L$. The other equations don't address temperature effects as they are $T$ independent. The $L_0$'s in $H$ are not relevent either, nor is the Casimir energy $\epsilon_g$ in 21.20 as again it is temperature indepenedent. I had not read that chapter, although he had sent me a copy. It is a soon-to be published book. $\endgroup$ – mike stone Nov 29 '20 at 14:11
  • $\begingroup$ You are right and I indeed missed the subleading terms in the asymptotic relation of the Virasoro characters that I just checked from Di Francesco. After they are considered, the net remaining contribution is only the central charge part. $\endgroup$ – Smart Yao Nov 30 '20 at 1:55

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