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Usually, when talking of the "band structure" of such a system one either refers to the non-interacting band structure (which relates to the free Green functions occuring in many methods to handle the interactions, like perturbation expansions or DMFT), or to the sharp features usually visible in the spectral function (which is more or less experimentally ...


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The most truthful answer, to my mind, to this is simply "because it often works in practice." It is not obvious, a priori, that band structure should apply to any realistic solid. The Coulomb interaction is typically of the order of the Fermi energy. Nonetheless, thanks to the magic of Fermi liquid theory, this strong interaction somehow only results in ...


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In the long run I don't think it matters much which of the two you study now. If you truly understand calculus in 2-3 dimensions, you won't have too much trouble generalizing your understanding to $N$ dimensions. On the other hand, if you want to do research in condensed matter, you will need linear algebra anyway, so there's no harm in picking up that topic ...


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In condensed matter "bulk" does not refer to the dimensionality of the problem but the location in the material. It refers to the volume of the crystal, as opposed to, e.g., surface effects. Many organic conductors behave as 1D systems, yet you can talk about bulk properties. Copper oxide superconductors have a 2D physics. However, often you will find ...


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Fundamentally it is that the $1/N!$ for the classical system only correctly compensates for overcounting of indistinguishable states if the particles are always in different states. For a system of Bosons at low temperature, where it is quite likely that many particles are in the same state, this breaks down. For a very understandable introduction to this ...


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You've confused two things. The business with adding vectors whose direction is determined by the stopwatch tells you what kind of interference pattern you'd see on a screen after the light diffracts through an aperture. In this case you're assuming you have an opaque screen with which to view the diffraction pattern. This is entirely independent of the ...


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Quantum electrodynamics is part of quantum mechanics and is the mathematical method used to calculate quantities rather than hand wave explanations. What you call "quantum mechanics " is a hand waved explanation, not wrong, but no numbers can come out of it because light is composed of zillions of photons impinging on other zillions of electrons making up ...


2

First of all, the expression for the magnetic length that you give is wrong: there is a square root missing: $l_B=\sqrt{\frac{\hbar c}{eB}}$. Secondly, to understand the meaning, you don't really need to think about lattices or phases of the electron wavefunction as the previous answers would have it. Instead, begin by thinking about a motion of a classical ...


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Your equation $$ v_F (\sigma \cdot k)\psi(k) = E \psi(k) $$ seems to be the "momentum representation" via Fourier transform of the envelope function equation for graphene. In this case $k$ is just the wavevector introduced by the Fourier transform, not a "crystal wave vector" as in the Bloch ansatz, and the $\psi_k$-s are the Fourier coefficients for the ...


2

The relevant part of the sum as $\sum_{k^*,s_1,s_2}\delta_{k^*,s_1s_2}d_{s_1}d_{s_2}B^k_P$ Let me assume that the fusion category has no multiplicities, so $N_{ab}^c=0,1$, which I think Levin and Wen also assumed. We can write the sum as $\sum_{k^*,s_1}d_{s_1}B^k_P\sum_{s_2\in s_1\times \bar{k}}d_{s_2}$ This is because if $\delta_{k^*,s_1s_2}=1$, it ...


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They are all the identical electrons. Its just that for the purpose of the paper or whatever they are behaving according to the rules of Dirac's equation, etc, because of the circumstances the electron is in. 'Free electron' means an electron flying about on its own, while a bound electron is in an atom and a Dirac electron is one that needs to be modelled ...


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It works just like every other kind of thermal energy. If a resistor can give out energy to the environment, it can also receive it. For example, if it gives it out by radiating, it can also absorb radiation; if it gives it out by having its fast-moving atoms smash into air molecules, then fast-moving air molecules can also smash into it. When it's in ...


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You can find an excellent description of what a topological insulator is in this brief presentation from the Yazdani Group at Princeton: Topological Insulators. To answer your question on the meaning of negative effective mass: The effective mass is actually determined by the behavior of the energy levels $E({\bf k})$ as functions of the crystal wave ...


1

In the continuum limit the lattice spacing $a$ goes to zero, therefore the Brillouin zone grows to infinity. If the Fermi velocity shall remain constant, the hopping parameter has to be rescaled as $t \propto 1/a$ (remember that the bandwidth is on the scale of $t$ and $v_F = \nabla_k E(\vec k)$), therefore only the features close to the Dirac points remain ...


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Is the existence of deconfined gauge charges a sufficient condition to ensure gaplessness? I think the answer is NO, such as the $Z_2$ gauge theory in 2+1D and 3+1D. I believe that the existence of deconfined gauge charges of a continuous gauge group is a sufficient condition to ensure gaplessness? Hastings and I have a paper ...


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The fractional dimensional space approach (FDSA) can be adopted to introduce flexibility in examining optoelectronic properties in anisotropic systems (quantum dot, wells etc). Here the material are fitted to models that utilize a variable dimension, (alpha) which has provided good agreement with experimental results in many works. This is because the ...


1

My answer will be brief unfortunately. Here's how I think about it. Electrons in the band that is near the Fermi energy interact through some attractive force. The ground state of this system is the superconducting state, with all those electrons paired up and you have a superconducting condensate which is that "Cooper sea" Now we consider low-energy ...


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The lower part is not filled with quasi-particles. At zero Kelvin, in zero magnetic field and with zero disorder all free electrons condense and form the superconducting condensate. The semiconductor model now describes the breaking of Cooper pairs not as resulting in two electron-, but in one electron- and hole-like excitation. As you are potentially ...


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$\rho_{xy}$ is related to the ratio of the $x$ component of current density to the voltage in the $y$ direction, when the $y$ component of the current is $0$. Decreasing the scattering time will increase the $x$ component of the drift velocity and so will increase mean force on a charge carrier, leading to a larger Hall voltage. However the current is also ...


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Does this mean that there are two different fields, one static field and one induced by a laser? Yes, that is exactly right. There is a static (meaning not time dependent) electric field $\vec{E}_\text{static}$ $(*)$ and there is also a time-varying electric field $\vec{E}_\text{laser}$ from the laser. Since we are told the laser field is linearly ...


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There are many resources on many-body Green's functions (propagators) both on-line and in print. You may want to search "quantum field methods in many-particle systems" or "quantum field methods for condensed matter systems" or variations thereof. In any case, I personally recommend the oldie-but-goodie book by Fetter and Walecka, Quantum Theory of ...


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I highly recommend Richard D. Mattuck A Guide to Feynman Diagrams in the Many-Body Problem. You can read some pages here. It's a very surface level introduction, but the first 3 or so chapters are presented at what he calls a "kindergarten" level so you shouldn't have any problems understanding it. However, the last part is most definitely not ...


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Every phase transition has an order parameter: something that vanishes above the transition temperature and is finite below. In superconductors, the order parameter is a complex quantity related to the superconducting gap: $\Delta = |\Delta| e^{i \phi}$. In BCS theory, there is a self-consistent equation for the gap: $\Delta_k = -\sum_q V_{kq} ...



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