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2

To grasp the relevant physics at a sloppy level, perhaps you simply need a few examples. You know a concept is commonly constructed by the manner you refer to it together with other concepts. Symmetry breaking usually results in ground state degeneracy and long range order. Order parameter field aids you in identifying degenerate sectors with the symmetries ...


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In spin liquids, the ordered state is broken by zero-point fluctuations even at $T=0$. Even though it is common for spin liquids to be frustrated, it is not necessarily so. The $S=1$ Heisenberg spin chain (AFM), for example, is a spin liquid without being frustrated. The name spin liquid comes (I believe), from the exponentially decaying correlation (like ...


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You would only be able to separate variables when your potential is of the form $$U_\text{separable}=U_a(x)+U_a(y)+U_c(z).$$ But your potential is of another form: $$U_\text{your}=U_a(x)U_b(y)U_c(z),$$ which doesn't make it possible to separate variables to compute the wavefunction. Then spectrum would then appear more complex than just sum of energies of ...


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There are a few ways to extract transport properties from your single-particle temperature Green's functions. By analytically continuing it to real time $t$, one gets information about how a particle propagates in the medium. More exactly, you get the probability of the particle traveling a distance $x = |x_1-x_2|$ in the interval of $t$. From this, you ...


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By inhomogeneous I assume you mean disordered, i.e., a system with a noisy/random potential landscape. I'm not sure which Mahan book you are referring to, however I found Akkermans and Montambaux' Mesoscopic Physics of Electrons and Photons to give a good discussion of the problem of wave propagation in disordered media. Essentially the problem is "solved" ...


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A sharp knife is still several molecules thick on the edge; dull blades are even wider. So when you attempt to cut material, it needs to be ripped apart. As explained in other answers, the material either fractures along faults in the lattice, or you separate molecules (as when you cut bread). The only materials where you might split chemical bonds are ...


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From the wikipedia article linked in your question, we see that Wannier functions can also be defined as $$ \underbrace{\phi_{\textbf{R}}(\textbf{x})}_{\textbf{R} \text{ Bravais lattice vector}} \equiv \frac{1}{\sqrt{N}} \underbrace{\sum_{\textbf{k}}}_{\substack{\text{summed over first}\\\text{Brillouin zone}}} e^{-i\textbf{k}\cdot\textbf{R}} ...


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The problem is: $(AB)^\dagger=B^\dagger A^\dagger$. Look how you treat $c^\dagger c$.


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Hopping and tunneling are often used as synonyms, but they are really very different terms with a fundamentally different basis. Tunneling is an inherently quantum-mechanical feature which means that a particle wave-function tends to overlap into it's energetically disallowed area which leads to a non-zero probability of finding it "where it should not be". ...


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When "hopping", the particle has enough energy to surmount the potential barrier. Its like water molecules passing from liquid state to gas: only those who happen to have enough kinetic energy KE to escape the average bounding of the other water molecules. This can happen even in room temperature, since their KE follows a Boltzman distribution and it can be ...


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You can use either potential for either purpose; it's just that some potentials are better for the different purposes. The reason is that these are empirical potentials; their constants are tweaked to work for a certain purpose. For example, if you're looking at the phonon band structure, you want $\omega\left(\vec{q}\right)$ to be as accurate as possible. ...


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When you Fourier transform the tight-binding Hamiltonian, $$c_{j,s}=\sum_{k,s} a_{k,s} e^{i R_j k},$$ with periodic boundary conditions, you will be left with a diagonalized Hamiltonian in the desired form. For details see section 2.3 of these lectures notes, starting on p.18: http://manybody.skku.edu/Lecture%20notes/Solid%20State%20Physics.pdf


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Phonons are collective modes in solids and a general derivation is needed independent of particular lattice constants to first order. Lattice constants define individual solids. Are you aware of the harmonic oscillator approximation? All symmetric potentials have as a first term in their expansion the quadratic, thus the harmonic oscillator ...


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For a metal, the permittivity can is typically described by the Drude model with a permittivity given by, \begin{equation} \epsilon = \epsilon' - i\epsilon'' = \epsilon_\infty - \frac{\omega_p^2}{\omega(\omega - i\gamma)} = \epsilon_\infty - \frac{\omega_p^2}{\omega^2 + \gamma^2} + i\gamma\omega\frac{\omega_p^2}{\omega^2 + \gamma^2} \end{equation} where ...


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It is not thermodynamics that controls crystal formation at the atomic level, but quantum mechanics. Large crystals, from diamonds to clear ice crystals are a macroscopic manifestation of the underlying quantum dynamical level. The molecules that build up the crystal have such field properties, dipoles and quadrupole and even higher moments that have ...


5

First let me make two comments before answering the question. The difference between metal and insulator rest in the existence of the itinerant electron Fermi surface or not. Ising (or Heisenberg) model is just an effective theory of local moments (localized electrons in the atoms), which contains no information of the itinerant electron, so there is no ...


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The main justification for considering the Ising model is that it's exactly solvable in one & two dimensions (and that it shows critical behavior which is universal in some sense). It is not particularly meaningful as an approximation to a real physical system. The Heisenberg model does a much better job, but it is also a lattice model. If you really ...


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Well, when we talk of stability of systems, at least for equilibrium systems, we require the free energy to be bounded below and be convex. As the free energy is obtained by a Legendre transformation (which preserves convexity), the energy functional is required to be convex. This essentially allows us to minimize energies to find ground states. Within the ...


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The thing you are looking for is called the Sommerfeld Expansion. The integral you specify can be approximated quite well to calculate the chemical potential (different to $E_F$ when the electrons are not completely degenerate) and expressions for the number density and energy density of the electrons when the chemical potential (or $E_F$) is larger than ...


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There's a nitinol wire that stiffens when warm and softens when cool. It's been used in various patented heat engine applications. see this reference http://www.imagesco.com/articles/nitinol/09.html



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