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?


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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