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Recently I have read "Analyzing the two dimensional Ising model with conformal fi eld theory" by Paolo Molignini, but I don't understand clearly manipulations in the section about continuum limit of transverse field Ising model. Author gives the following lagrangian of model in terms of continuous Fermi fields $\Psi$: $$\mathcal{L}=\Psi^{\dagger}\frac{\partial\Psi}{\partial\tau}+\frac{v}{2}\left(\Psi^{\dagger}\frac{\partial\Psi}{\partial x}-\Psi\frac{\partial\Psi}{\partial x}\right)+\Delta\Psi^{\dagger}\Psi,$$ which gives (as I understand, may be it is wrong) motion equations: $$(\Delta - \partial_{\tau})\Psi=0;\quad (\Delta+\partial_{\tau})\Psi=0.$$ As was mentioned in these notes, this lagrangian describes Majoranna fermions. But I don't understand origin of the first term with time-derivative. From Zee, this term relates to Berry phase, but it is not useful for me. Can anyone please explain me origin of time-derivative term and connection of this lagrangian with Majoranna fermions?

My goal is to calculate fermionic path integral $\mathcal{Z}$ with this lagrangian and understand how it changes if I consider the limit $\Delta\rightarrow 0$. As one can see from the notes, case of $\Delta=0$ describes critical behaviour of 2D classical Ising model. Of course, I have obtained partition function for 1D quantum Ising model in transverse field but I would like to understand the connection to classical 2D model using the continuum limit.

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The first term gives you the right commutation relation when you canonically quantize the theory. Since the canonical conjugate variable of $\Psi$ is $$\Pi_\Psi=\frac{\partial \mathcal{L}}{\partial \Psi}=\Psi^\dagger,$$ it reproduce the desired commutation relation in Eq. (3.29), I think.

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(1) The first term of the following lagrangian, as I understand, can not be derived easily (I mean the Legendre transformation $\mathcal{H}\rightarrow\mathcal{L}$) because it comes form path integral for coherent states. In my view, this topic is perfectly disccused in Wen "Quantum field theory of many-body systems (p. 58).

(2) Then, for majorana fermion one can just use $\psi=\chi_1+i\chi_2$ or just use Giuseppe Mussardo "Statistical Field Theory" (chapter 9).

(3) For path integral calculation, it is important to keep in mind that $\psi$ is grasssmanian. Thus, for first term we can write $$\int d\tau\,\psi^{\dagger}\partial_{\tau}\psi=(\psi^{\dagger}\psi)|-\int d\tau\,\partial_{\tau}\psi^{\dagger}\psi=0+\int d\tau\,\psi\partial_{\tau}\psi^{\dagger},$$ which allows us to write the first term in the following form: $$\psi^{\dagger}\partial_{\tau}\psi=\frac{1}{2}(\psi^{\dagger}\partial_{\tau}\psi+\psi\partial_{\tau}\psi^{\dagger}).$$ Then, for the last term one can just write: $$\psi^{\dagger}\psi=\frac{1}{2}(\psi^{\dagger}\psi+\psi^{\dagger}\psi)=\frac{1}{2}(\psi^{\dagger}\psi-\psi\psi^{\dagger}).$$

These transformations permit to rewrite lagrangian in form $$\mathcal{L}=\frac{1}{2}\begin{pmatrix}\psi & \psi^{\dagger}\end{pmatrix}\mathcal{M}\begin{pmatrix}\psi \\ \psi^{\dagger}\end{pmatrix},$$ where $\mathcal{M}$ is 2x2 matrix. Finally, rewriting the lagrangian in matrix form, one can easily calculate path integral and obtain simple partition function for free fermions.

(4) However, there is still one open question. Performed all calculations of path integral, one can obtain the expression for free energy: $$\ln\mathcal{Z}\propto\int\frac{dk}{2\pi}\left(\frac{1}{2}\beta\omega_k+\log(1+e^{-\beta\omega_k})\right),$$ where $\omega_k^2=v^2k^2+\Delta^2$. It is interesting to define regular and singular part of this free energy. The first term is related to zero-point enegy and can be easily regularized by lattice spacing: all $k$ is smaller than $\Lambda\sim 1/a$. I am not sure about the second term, but at first glance there is no non-analiticity for $\Delta\rightarrow 0$. Secondly, it is also interesting to compare critical exponents of 2D classical Ising model and 1D Ising model in transverse field (I call it 1D quantum Ising model). But it is one incomprehensibility: in 1D model in transverse field transition happens due to magnetic field instead temperature.

May be it is not true but it seems reasonable to calculate (it is definition): $$C_{\Delta}\propto \frac{\partial^2(\ln\mathcal{Z})}{\partial\Delta^2}$$ and analyze it near $\Delta=0$. It is interesting to note that it diverges as $\ln\Delta$, which formally gives critical exponent $\alpha_q=0$ (compare with specifi heat exponent of 2D classsical model).

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