Construction of Wannier function for optical lattice potential
Parameters of the Bose-Hubbard model require the knowledge of the Wannier functions from the lowets band of the optical lattice potential $V(x) = V_{0}sin^{2}(kx)$ according to equations: $$J = int dx...
View ArticleDensity of states in a system of interacting electrons
When we are introduced to the density of states in typical band-theory problems we neglect interaction between electrons, and thus we define the density of states of a sigle particle as:...
View ArticleNonequilibrium Green's functions weakly interacting two-component Bose gas
I am planing to describe time evolution of two-component BEC. I was thinking about non-equilibrium Green’s functions, but I don’t if the method can be applied to the problem describe below. I know...
View Articlesecond quantization – time dependent basis
In the second quantization time-independent field operator can be expanded in the orthonormal basis: $$hat{Psi}(mathbf{x}) = sumlimits_{i}hat{a}_{i} phi_{i}(mathbf{x})$$ Time evolution of the field...
View ArticlePerturbation method for Hubbard hamiltonian
I’ve got a hamiltonian to solve from perturbation methods: $$H=underbrace{-tsum_sigmasum_{i=1}^N(a^dagger_{isigma}a_{i+1sigma}+a^dagger_{i+1sigma}a_{isigma})}_{hat...
View Articlehow to construct self-energie diagrams
I am working on self-energie and feynmann diagrams. they are not very easy to get but i think i am starting to understand how it works but of course i am not realy sure.and before going any further i...
View ArticleWhat does a bucked honeycomb lattice mean?
I was going through some literature where they have mention about bucked honeycomb lattice, but I was unable to understand about the bucked honeycomb term.
View ArticleProblem using spin-restricted form of the second-quantized nonrelativistic...
I have a problem that confuses me a lot. The two-electron part of the electronic nonrelativistic Hamiltonian can be written begin{equation} frac{1}{2}sum_{pqrs} (pq|rs)...
View ArticleNumerical problem with Hartree-Fock equations for dilute Bose gas
I have to solve the following set of equations self-consistently: $$begin{align} n_c(mathbf{r}) & = frac{1}{g}left[mu - V_{rm ext}(mathbf{r}) - 2 g n_{T}(mathbf{r}) right] \[3mm]...
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Consider a particle of reduced mass $mu$ orbiting in a central force with...
Consider a particle of reduced mass $mu$ orbiting in a central force with $U=kr^n$ where $kn>0$. (a) Explain what the condition $kn>0$ tells us about the force. Sketch the effective potential...
View ArticleGaussian integral on a Riemannian manifold
How do I estimate the Gaussian integral $int d^nx sqrt{g(x)}~e^{-x^2} $ on a Riemannian manifold $(M,g=det~g_{munu})$? I’ve tried to consider $sqrt{g(x)}$ as an analytic function and expanded it. My...
View ArticleDifferentiating between Tensor Networks
I am trying to study tensor networks and their application to quantum phase transitions. However, I had a question concerning the connection between the projected entangled-pair states (PEPS) and the...
View ArticleGrand potential $leftrightarrow$ ground state energy of interacting electrons...
I want to calculate the ground state energy $E_0$ of interacting electrons in a solid at $T=0$ via pertubation theory and Feynman diagrams, i.e. I want to understand the connection between the...
View ArticleUnitarity of Symmetrisation operators
How can i proove that the symmetrisation operators $S_{pm}$ are unitary? Defining $S_{pm}$ on the $N$ Particle Fock Space by it’s effect on $|Psi rangle=| Psi_1 rangle otimes… otimes| Psi_Nrangle $...
View ArticleBCS-BEC crossover
It would be really helpful if somebody could describe what does one mean by a BEC-BCS Crossover. I was going through articles available on the topic, but I was unable to grasp the gist of the topic.
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Tidal tails of galaxies after collision
When there is a collision of 2 disc shaped galaxies, there is a tail formation created from both the galaxies. I read here that this was due to tidal forces, but I couldn’t figure out how this happens....
View ArticleHartree Fock equations
I don’t understand how the Hartree Fock equations define an iterative method! For this discussion, I am referring to the HF equations as described here: click me! Basically if you guess a bunch of...
View ArticleHamiltonian for electron hole
I found in lectures notes that the Hamiltonian containing the energy of a electron hole without any interaction is given by $$H = sum_k d_k^{dagger} d_k left( frac{hbar k^2}{2m_V} – E_{0,V} right)$$...
View ArticleWave function of N electrons in a superconductor
Assuming that the wave function consisting of $N$ electrons is $Psi_{N}(bf{r_1,r_2,cdots r_N)}$ then in the presence of a magnetic field ($bf{B}=nabla times A$), how do I show that the current can be...
View ArticleIndistinguishable particles and probability density
I am given the following (probably simple) exercise, but I think I misunderstand something: Let $psi_{a,b}(r_1,r_2)$ be a two-particle state, calculate the probability density for distinguishable and...
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