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The short answer is that BCS theory is derived bottom-up from quantum mechanics (you assume that there is some local attractive interaction between electrons, and perform a mean field approximation), while the older Ginzburg-Landau theory is derived top-down from thermodynamics (you assume that superconductivity can be described by some order parameter, and ...


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Introduction: Superconductivity is phenomenon when certain materials electrical resistance drops sharply to zero when their temperature is lowered below it's critical temperature ($T_c$), There are two types of superconductors $\mathrm{Type\text{ }I}$ and $\mathrm{Type\text{ }II}$. Type I Superconductors: In $\mathrm{Type\text{ }I}$ superconductors ...


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any eigenfunction to this Schrödinger operator is automatically periodic with the potential's period, is this true? No!! The eigenfunctions are Bloch waves $\psi(x) = u(x)e^{ikx}$, where $u$ is periodic (with the period of the lattice). But the product $\psi$ is not periodic (with the period of the lattice) unless $k=0$. I put up an example on Wikipedia ...


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What are phonons? Phonons aren't particles like electrons or protons are, phonons are quasi particles, these type of particles are just used to describe excitations of a field: in phonons case, phonons are used to describe elementary lattice vibrations which have certain frequency. Electron-Phonon Interaction: Basically Cooper pairs are just pairs of ...


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Then, there is the case that such an operator is defined on the full interval I assume that by "full interval" you mean the whole real line. First question: Do we then need any boundary conditions? Yes, as noted by Sam Bader, boundary conditions are part of the Hamiltonian. In my physics lecture we used so-called Born von Karmann boundary ...


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Both! The fundamental physics comes from the fact that these were the first systems (in the pre-graphene era) to really exhibit two-dimensionality! In two dimensions, physics can be fundamentally different which is exhibited by the integer and fractional quantum hall effects, the quantum spin hall effect and Kosterlitz-Thouless-type phase transitions just to ...


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This is certainly unexpected, for as Mark Mitchison commented the $J=0$ Heisenberg model is equivalent to free fermions in one dimension. Moreover I suspect that something is amiss even before that, for the $J=-1$ is the ferromagnetic model and $J=+1$ the antifferomagnetic one, and certainly they have different ground state energies. In fact, for $J\leq -1$ ...


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Consider the partition function $$Z = \int D\phi ~ e^{-S_0 - S_I},$$ where $S_0$ is the Gaussian/free part and $S_I$ is the interaction part of the action. Within a perturbative framework we may aim to systematically include the contributions of fast modes to the (effective) action for slow modes. For this we expand in the interaction strength as $$Z = \int ...


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You don't have to discretize your problem (XY model). For each step, just take some value as the new $\theta$, and calculate the transition rate accordingly. Of course, when choosing the new value of $\theta$, better don't do it in a completely random way, otherwise your transition rate might be usually too small and you are just wasting time. Having said ...


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Yes 2DEG offers some fundamental physics by eliminating one spatial degree of freedom. Of most important physics the quantum Hall measurement comes to mind. From a technological perspective, the bandgap engineering of heterostructure devices allows the formation of 2DEG, i.e. a high concentration of charge without recourse to mechanisms such as doping, ...


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It's good that you're considering questions like this; I find that this type of questions really forces a student to a deeper understanding of the math involved. Do we then need any boundary conditions? Yes, boundary conditions should be considered as part of the definition of the Hamiltonian and its domain. Different boundary conditions can result in ...


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Equation 12 is $H' = -\frac{e}{i \omega m} \mathcal{E}_- P_+ e^{i\omega t}$, where $\mathcal{E}_-$ is the strength of the AC electric field $\omega$ is its frequency, $m$ is the electron mass, $e$ is its charge, and the other quantities are defined in the text. Plugging in the definitions, we find \begin{equation} \begin{aligned} H' &= \frac{ie}{\omega ...


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The answer lies in the fact that, in graphene, there is an effective long range interaction mediated by the inverse biharmonic operator (which in 2D goes as $x^2\ln(x)$ and is extremely long-ranged) coupling the gaussian curvature at any two points on the sheet. Due to this, any static ripples or thermally produced dynamic ripples interact at arbitrary ...


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In quantum chemistry two-center integrals refer to exchange or coulomb integrals involving 1-electron atomic wave functions (orbitals) centered on two different atoms in a molecule. Say the coulomb repulsion between an electron described by an orbital belonging to atom A and another electron described by orbital belonging to atom B is given in a simplified ...



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