Recent content by thatboi

Hi all, I'm trying to see if my question/logic makes sense. Suppose I have a classical Ising model on a 2-D Square lattice with ##N## sites and 0 external field. There is an exact formula for the average internal energy per site, and at criticality it turns out to be ##u = -\sqrt{2}## where I...

Haha no worries, I can't edit the original post now, so I'll proceed based off what you said, it's a bit of a mess so let me know if you see anything suspicious. $$\begin{equation} \begin{split} \xi_{1}\ket{\xi} &=...

Do you mean when I pass the ##\psi_{1}## through the ##\psi^{\dagger}_{2}## terms? I don't see why a minus sign would be picked up (##\psi^{\dagger}_{1}## should commute with both ##\xi_{1}\psi^{\dagger}_{2}## and ##\xi_{2}\psi^{\dagger}_{2}##).

I came across the following formula (2.68) in di Francesco's CFT book for a fermionic coherent state: $$\ket{\xi} = e^{\psi^{\dagger}T\xi}\ket{0}$$ where##\ket{\xi} = \ket{\xi_{1},...,\xi_{n}}##, ##\xi_{i}## is a Grassman number, ##T## is some invertible matrix, and ##\psi^{\dagger}## is the...

Ah that is true. I have edited the question.

Hi all, I came across the following stackexchange post and was wondering if anyone could give any elaboration for why the answer claims that evaluating the Taylor Series resulted in ##\mathcal{O}(\epsilon^{3})## errors? I have not encountered such an expansion before. EDIT: The equation at hand...

Sure, I already did some modifications and it seemed to match what I said above (for example if I put a negative sign on only the second element of the last column and second element of the last row, then the eigenvector corresponding to the largest magnitude eigenvalue only has a negative sign...

In case people are curious, see the following eqns (1.49) and (1.50) in these notes and references therein. It is a generalized real-space version of what is usually used in k-space.

Right, so the last row contributes to the eigenvalue in the sense that it gives the last entry of the resultant column vector when the matrix is multiplied by the eigenvector. So if the eigenvector also has entries that alternates signs, then the dot product between the eigenvector and the last...

Hey guys, I was wondering if anyone had any thoughts on the following symmetric matrix: $$\begin{pmatrix} 0.6 & 0.2 & -0.2 & -0.6 & -1\\ 0.2 & -0.2 & -0.2 & 0.2 & 1\\ -0.2 & -0.2 & 0.2 & 0.2 & -1\\ -0.6 & 0.2 & 0.2 & -0.6 & 1\\ -1 & 1 & -1 & 1 & -1 \end{pmatrix} $$ Notably, when one solves for...

I'm currently looking at the following set of notes and am confused at equation (1.15) where they discuss the Bogoliubov quasiparticle for the edge states. I understand up to equation (1.14), where they have solved for the edge state of the first-quantized Hamiltonian. What I don't understand is...

Consider a generic Hermitian 2x2 matrix ##H = aI+b\sigma_{x}+c\sigma_{y}+d\sigma_{z}## where ##a,b,c,d## are real numbers, ##I## is the identity matrix and ##\sigma_{i}## are the 2x2 Pauli Matrices. We know that the eigenvalues for ##H## is ##d\pm\sqrt{a^2+b^2+c^2}## but now suppose I have the...

Ok I took another crack at the problem and this is indeed the correct normalization condition.

Consider the state ##\ket{\Psi} = \sum_{1 \leq n_{1} \leq n_{2} \leq N} a(n_{1},n_{2})\ket{n_{1},n_{2}}## and suppose $$|a(n_{1},n_{2})| \propto \cosh[(x-1/2)N\ln N]$$ where ##0<x=(n_{1}-n_{2})/N<1##. The claim is that all ##a(n_{1},n_{2})## with ##n_{2}-n_{1} > 1## go to ##0## as...

Hi all, I am starting with the following equation: ##2\cot\left(\frac{\theta}{2}\right) = \cot\left(\frac{k_{1}}{2}\right) - \cot\left(\frac{k_{2}}{2}\right)## with the following definitions: ##k_{1} = \frac{K}{2} + ik, k_{2} = \frac{K}{2}-ik, \theta = \pi(I_{2}-I_{1}) + iNk##, where...