A

#### Aslan Monahov

##### Guest

Aslan Monahov Asks:

Let's assume that we have 2D square lattice with spins $\sigma=\pm 1$ and equal vertical and horisontal energy coefficients K: $$E_{interaction}=e^{K\sigma_i\cdot\sigma_j}$$ where $\sigma_i,\sigma_i$ are neighboring spins. Let's also use a disorder parameter $\mu_i$ (dual value to spin) that is located in the centers of squares of lattice:

So the fermion $\Psi$ will be given as a product of spin and neighboring disorder. Let's define for convenience of computing the shifts on lattice: $$\rightarrow =\vec{1};\quad \uparrow = \vec{2};\quad \leftarrow = \vec{3};\quad\downarrow = \vec{4}$$ and also semidiagonal shifts

Now, using the properties of disorder parameters: $$\mu_{j+e_1}=\mu_{j+e_2}\cdot e^{-2K\sigma_j\sigma_{j+\vec{2}}}$$ We can calculate how rotating of disorder affects on fermion: $$\Psi_j^{e_1}=\Psi_j^{e_2}e^{-2K\sigma_j\sigma_{j+\vec{2}}}=\sigma_j\mu_{j+e_2}[cosh(2K)-\sigma_j\sigma_{j+2}sinh(2K)]=cosh(2K)\Psi^{e_2}_j-sinh(2K)\Psi_{j+\vec{2}}^{e_3}$$ After simplifying $cosh(2K)=c$ and $sinh(2K)=s$ in general case we can write: $$\Psi^{a}_j=c\Psi_j^{a+1}+s\Psi^{a+2}_{j-a+1}$$ After desomposing the solution to monochromatic waves $$\Psi_j^a=\sum_pC_p^ae^{ipj\varepsilon};\quad \vec{j}\varepsilon=\left(\begin{array}{c} x_1 \\ x_2 \end{array}\right)$$ we can get a matrix equation

where nontrivial solutions require zero determinant, that gives us: $$cos(p_1\varepsilon)+cos(p_2\varepsilon)=\dfrac{c^2}{s}$$ where limiting the lattice step $\varepsilon$ to zero gives us spectrum: $$\dfrac{p^2}{2}=\dfrac{p_1^2}{2}+\dfrac{p_2^2}{2}=\dfrac{c^2-2s}{2s\varepsilon^2}$$ Where critical point $c^2-2s=0$ sets $sinh(2K_{crit})=1$.

I have two questions:

*How to get alternative critical point formula derivation for 2D Ising model on square lattice for fermions?*Let's assume that we have 2D square lattice with spins $\sigma=\pm 1$ and equal vertical and horisontal energy coefficients K: $$E_{interaction}=e^{K\sigma_i\cdot\sigma_j}$$ where $\sigma_i,\sigma_i$ are neighboring spins. Let's also use a disorder parameter $\mu_i$ (dual value to spin) that is located in the centers of squares of lattice:

So the fermion $\Psi$ will be given as a product of spin and neighboring disorder. Let's define for convenience of computing the shifts on lattice: $$\rightarrow =\vec{1};\quad \uparrow = \vec{2};\quad \leftarrow = \vec{3};\quad\downarrow = \vec{4}$$ and also semidiagonal shifts

Now, using the properties of disorder parameters: $$\mu_{j+e_1}=\mu_{j+e_2}\cdot e^{-2K\sigma_j\sigma_{j+\vec{2}}}$$ We can calculate how rotating of disorder affects on fermion: $$\Psi_j^{e_1}=\Psi_j^{e_2}e^{-2K\sigma_j\sigma_{j+\vec{2}}}=\sigma_j\mu_{j+e_2}[cosh(2K)-\sigma_j\sigma_{j+2}sinh(2K)]=cosh(2K)\Psi^{e_2}_j-sinh(2K)\Psi_{j+\vec{2}}^{e_3}$$ After simplifying $cosh(2K)=c$ and $sinh(2K)=s$ in general case we can write: $$\Psi^{a}_j=c\Psi_j^{a+1}+s\Psi^{a+2}_{j-a+1}$$ After desomposing the solution to monochromatic waves $$\Psi_j^a=\sum_pC_p^ae^{ipj\varepsilon};\quad \vec{j}\varepsilon=\left(\begin{array}{c} x_1 \\ x_2 \end{array}\right)$$ we can get a matrix equation

where nontrivial solutions require zero determinant, that gives us: $$cos(p_1\varepsilon)+cos(p_2\varepsilon)=\dfrac{c^2}{s}$$ where limiting the lattice step $\varepsilon$ to zero gives us spectrum: $$\dfrac{p^2}{2}=\dfrac{p_1^2}{2}+\dfrac{p_2^2}{2}=\dfrac{c^2-2s}{2s\varepsilon^2}$$ Where critical point $c^2-2s=0$ sets $sinh(2K_{crit})=1$.

I have two questions:

- How to get a mass from this? Some sources just say that $m^2=-p^2$ (where have they forgotten energy).
- How to get equation $sinh(2K_{crit})=1$ using j-independent solutions? The teachers notes just offer to use the fact, that system allows solution $$\Psi^{(a)}=const\cdot e^{\pm\dfrac{i\pi a}{4}}$$ that leads to critical point, but I don't understand how.

SolveForum.com may not be responsible for the answers or solutions given to any question asked by the users. All Answers or responses are user generated answers and we do not have proof of its validity or correctness. Please vote for the answer that helped you in order to help others find out which is the most helpful answer. Questions labeled as solved may be solved or may not be solved depending on the type of question and the date posted for some posts may be scheduled to be deleted periodically. Do not hesitate to share your thoughts here to help others.