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Your statement itself is not quite right. What is not conserved is the chiral current, namely the current of fermions at one of the Weyl nodes. The physics can be understood essentially in one-dimensional version of the Weyl metal: consider a 1D electron gas. There are two Fermi points, and the low-energy theory is given by two "Weyl fermions" in 1D with ...


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The conduction and valence band electron are only differed by their Wannier orbital (wave function within each unit cell), so exciting a Bloch electron from one band to another changes the Wannier orbital of the electron, for example, from a bounding orbital to an anti-bounding orbital. In the simplest 1D example of a periodic potential, exciting the ...


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The statement is not true. Counter example: quantum spin ice or U(1) spin liquid. In gapless spin liquid phase, the boson (spin excitations) are emergent U(1) photons in the deconfined phase, which are gapless but not condensed.


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When you are exciting an electron from VB to CB, you normally talk in terms of transition from a state $|k\rangle$ to $|k'\rangle$ and $E(k')>E(k)$. Specifying $|k\rangle$ automatically rules out an precise determination of $|x\rangle$ as the state $|k\rangle$ is spread over the entire real space with specific weight at each point.


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In simplest terms, the presence of sub-gap zero energy localized modes (Majorana modes) makes a superconductor topological. A superconducting ground state is just a bunch of Cooper pairs and the BdG Hamiltonian describes excitations above the ground state. If the excitation spectrum has these localized modes then it is a topological superconductor otherwise ...


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A very interesting question that has undeservedly waited too long for its answer! I will try to address the different parts of the question one-by-one. However, excellent references are available on the subject. I can recommend Basic superfluids by T. Guenault as a simple intro to superfluidity in general, while a more difficult read An introduction to the ...


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Yes, the statement can be explicitly verified from the matrix representation of the spin operators acting on different spins. Acting on the spin-1/2 object, the spin operators read $$S^x=\left( \begin{array}{cc} 0 & \frac{1}{2} \\ \frac{1}{2} & 0 \\ \end{array} \right), S^y=\left( \begin{array}{cc} 0 & \frac{i}{2} \\ -\frac{i}{2} & 0 \\ ...


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I met the author of that paper today. He said it's more of an empirical observation rather than a statement based on solid arguments. He asked for a counterexample, which I don't have. Please post it here once you got one.


2

The Hamiltonian is time-reversal invariant: $c_{k\uparrow}\rightarrow c_{-k,\downarrow}, c_{k\downarrow}\rightarrow -c_{-k,\uparrow}$. You can check that explicitly. The ground state is also invariant, because Cooper pairs are all spin singlet. One of the significant implications of time-reversal symmetry for s-wave superconductors is the Anderson's ...


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DFT is exact concerning ground state properties. However, the bandgap is not a ground state property. Not sure, if this simple explanation is correct, but I find it somehpw intuitive: in order to speak about a bandgap, you either need a (at least fictitious) electron in the conduction band, which therefore is in an excited state, or you need a ...


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Your method is correct. I just followed your approach and successfully observed the localized zero modes at the end of the chain. Following is my Mathematica code. First generate a list of alternating hopping amplitudes with random fluctuation: L = 20; ts = 0.5 + Boole@EvenQ@Range[L-1] + RandomVariate[NormalDistribution[0, 0.1], L-1]; ListLinePlot@ts ...


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In a system with non-interacting particles you may know that the right quantity to look at is $\rho \Lambda^3 \sim \left(\Lambda/l \right)^3$ where $\Lambda$ is the (thermal) de Broglie wavelength and $l = \rho^{-1/3}$ the typical inter-particle distance in the system. Basically if $\Lambda/l \ll 1$, then your system behaves classically i.e. you do not ...


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Strictly speaking, a localized state is well defined only on an infinite system. Therefore, the natural idea is to change the size of the system. Localized states respond differently to the size-changing than the extended states. For example, the center-of-mass of the wave function is located at a constant distance to the edge as long as the system size is ...


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Time reversal essentially means a system looks the same if you reverse the flow of time. The only difference beeing that things like velocity go in the opposite direction. In condensed matter systems it is represented as a unitary matrix times complex conjugation $\mathcal{T} = U\mathcal{K}$. A simple system that follows T-symmetry would be a system ...


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I can provide an example for bosonic models. \begin{eqnarray} \mathcal{H} & = & \mathcal{K} + \mathcal{T}_\text{soc} +\frac{U}{2}\sum_{i\tau} \hat n_{i\tau}( \hat n_{i\tau}-1) \nonumber \\ & & + U^{\prime} \sum_i \hat n_{i\uparrow} \hat n_{i\downarrow} + V\sum_{i\tau} \hat{n}_{i\tau}\hat{n}_{i+1\tau} \nonumber ...


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A short answer to questions 2 and 3: In Mermin-Wagner's paper the short-range condition is stated as $\sum_{\bf R} {\bf R}^2 |J_{\bf R}|<+\infty$. For interactions with (or more precisely majorized by a) power law decay $|J_{\bf R}| \sim R^{-\alpha}$, this requires $\alpha > D+2$, where $D$ is the space dimensionality (i.e., $\alpha >4$ for $D=2$ ...


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A further investigation led me to the desired reference, which discusses this precise problem: Hard Superconductivity: Theory of the Motion of Abrikosov Flux Lines Work of Anderson and Kim, at Bell Labs, around 1964


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This is not a full answer to the question, but just to point out existing related studies. This kind of question was considered a lot in the context of Josephson junction, which is basically a superconducting ring but with a weak link (i.e. the junction), where intuitively vortices tunnel through the junction. The simplest model of such a system is just the ...


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In essence, you are taking the Fourier transform of the Heaviside theta function, $\theta(t)$, to try and get $$ \tilde\theta(\omega)=\int_{-\infty}^\infty e^{i\omega t}\theta(t)\text dt=\int_{0}^\infty e^{i\omega t}\text dt. $$ For nonzero $\omega$, this is perfectly fine and easily evaluates to ${1}/{i\omega}$ once you discard the term at infinity (which ...


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The $\mathrm{i}0$ is short for $$\lim_{\epsilon\rightarrow 0^+}\mathrm{i}\epsilon$$ and is used in the so-called $\mathrm{i}\epsilon$-prescription. It is necessary for some integrals so that the contour does not hit a pole and render the integral ill-defined. In this case, you must insert $\lim_{x\rightarrow 0}\mathrm{e}^{-x\epsilon}$ into your Fourier ...


3

$i0$ is meant as $$\frac{1}{\epsilon - \omega_0 + i 0} = \lim_{x\to 0^+} \frac{1}{\epsilon - \omega_0 + i x}\, . $$ The idea here is that the Fourier transform of something containing a theta function is not well defined. This comes from the fact that the integral $$ G(\epsilon) = -i \int_{-\infty}^{\infty} \theta(\tau) \text{e}^{-i \tau ...


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This is why people start to abandon calling the zero modes in vortices "Majorana fermions" because they are NOT fermions. $\gamma$ is a Majorana zero mode, which means it always has to pair up with another Majorana zero mode to form a 2-dimensional Hilbert space. Exchanging vortices generates a nontrivial unitary transformation on the degenerate space.


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Band theory does not properly describe systems with strong electron-electron interactions. Specifically, the transition metal oxides you have mentioned are both examples of strongly correlated electron systems, which show all kinds of novel behavior which escapes description of band theory. In general strong interactions can induce phases such as Mott ...


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The Green function you gave above is not the Green function for an insulator, which should generically has no pole if there is no Fermi surface. The Volovik argument is a bit circular, since you know from the beginning that $p=p_{F}$ is the Fermi momentum, defining the Fermi surface. When $p\approx p_{F}$ you can infer the form of the Green function as you ...


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Exact statements The Hohenberg-Kohn theorems, which are the theoretical foundation of DFT, essentially say that the ground state properties of a many-electron system are only a function of the electron density. Any quantity you want to calculate can be re-expressed in terms of the electron density $n(r)$, including the many-body ground state wavefunction, ...


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In principle it's an exact theory, except you don't know the expression of the exchange-correlation functional. Also exact theory is not opposite to mean field theory, in terms of which DFT can be understood.


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I assume this is the formula for Chern number in a Chern insulator. The physical reason that such a formula exists is that this is exactly what Kubo formula gives you for the Hall conductance, which holds for interacting systems too.


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Ferromagnetism is a collective behavior so the answer to your 1st question is no. In a ferromagnetic material the moments of the magnetic atoms not only align themselves with an external magnetic field but they also align spontaneously in the same direction even without any external field. The regions in the material where the magnetic moments have the same ...


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Starting with some background information from Wikipedia, we have that under time reversal the position is unchanged while the momentum changes sign. In quantum mechanics we can express the action of time reversal on these operators as $\Theta\,\mathbf{x}\,\Theta^\dagger = \mathbf{x}$ and $\Theta\,\mathbf{p}\,\Theta^\dagger = -\mathbf{p}$. It is worth ...


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It might be useful to consider the physical meaning of the term $aA_1A_2$ in a gauge theory. Compactify the theory on a "thin" torus, say the length of the $y$ direction $l_y$ is much smaller than $l_x$. The two ground states are distinguished by the value of the Wilson loop along $y$. Heuristically, we can just substitute $a=0,\pi$ (I'm sloppy about the ...


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In general, Symmetry protected topological (SPT) phases are formed by gapped short-range entangled quantum states that do not break any symmetry (Phys. Rev. B 80, 155131 (2009)). Contrary to trivial symmetric states, a nontrivial SPT state cannot be transformed into direct product state (or Slater determinant state for fermions) of local atomic basis via ...


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The dielectric constant, or more appropriately, the dielectric function, can be thought of as a measure of screening. A simple relation for which to picture this is: $V_{eff} = V_{ext}/\epsilon$ Therefore, in TMDs, since the electrons are more mobile in the planes, they tend to screen potentials with a greater efficiency. This gives a higher dielectric ...


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Hook and Hall is probably my personal favourite as it is very clear and concise without a lot of fuss. For a totally different style to the classics maybe try "The Oxford Solid State Basics". The lecture notes on which this book was based are available (in part) online (google steve simon solid state lecture notes and you should get there without much ...



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