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The most immediate answer would seem to be that a great variety of different crystal phases can exist because their long-range order makes it possible to classify them based on the different symmetries of their lattice structure. Since the liquid (or amorphous solid) phase only has short-range order and the gaseous phase doesn't even have that, it seems ...

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Yes. Bose-Einstein condensation was experimentally achieved in systems of: Rubidium atoms (first experimental realization, 2001 Nobel prize) Potassium atoms Cesium atoms Lithium atoms Sodium atoms Exciton polaritons Photons Phonons (?) If you are using atoms, they must behave like bosons (see also here).

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Nothing in the laws of thermodynamics forbids multiple liquid phases for a single substance. The only limit is the simultaneous coexistence of at most three phases (at triple points). Water has a solid-liquid-gas triple point and several soid-solid-liquid and solid-solid-solid triple points; see the phase diagram of water and ice. In addition, although not ...

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A eigenstate of a crystal hamiltonian can be written as a Bloch function in space representation $$\psi(\mathbf{r}) = e^{i\mathbf{k}\mathbf{r}} u_\mathbf{k}(\mathbf{r})$$ $u$ is periodic with respect to the unit cell. The momentum is now given by $$\langle \psi|\hat{\mathbf{p}} |\psi\rangle = -i\hbar \int e^{-i\mathbf{k}\mathbf{r}}u_\mathbf{k}^*(\mathbf{... 3 First of all, I have not directly worked on BEC or laser cooling per se, but what I am writing is my understanding after discussing this subject with a person who is directly involved in this activity. Hence if anything is wrong or inconsistent please let me know. As rightly said by @Lagrangian that it is not required that the particles should be at same co-... 3 The general misconception is that the bosons in BEC and superfluids are in the same quantum state including the same spatial coordinates. This would result in stacking of each particles' wave function and unlimited reduction in volume of such a substance. The truth is that the particles in BEC or superfluids do not necessarily crowd, but have the capability ... 2 Charge density waves and Wigner crystals are two ways of understanding the same broken symmetry. In the Wigner crystal picture we imagine the electrons sitting on lattice sites; the electronic charge density thus has broken translation and rotation symmetry. In the CDW picture we imagine an essentially uniform distribution of electrons that develops a ... 2 Solids are rigid and resist a change in volume. There are three general types of solid: As you can see from the picture, a crystal is a solid for which the atoms/molecules are highly ordered to form a periodic lattice. Examples of (poly)crystalline solids include diamond and table salt. Examples of non-crystalline solids include glass and amorphous ... 2 I have asked a professor about this and he gave me the answer. After replacing \mathbf{k} by -i\nabla in H(h\mathbf{k}), we are actually getting a new Hamiltonian that acts on envelop of wave functions. To make this answer relatively complete, I will briefly introduce the main steps focusing on only one band. Suppose the band we are interested in ... 2 When you take the limit q \rightarrow 0, your formula becomes$$\chi( 0,0) \sim - \sum\limits_{\bf{k}} \frac{d f(\varepsilon_k)}{d \varepsilon_k}$$where f is the occupancy number of the electronic states. This susceptibility represents the response of the system to some external perturbation. Usually, this perturbation tends to displace electrons to ... 2 The canonical example for MPS (in fact, the first MPS ever) is the AKLT model (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.799, https://projecteuclid.org/euclid.cmp/1104161001). The 2nd reference also discusses the 2D (=PEPS) version of the state. Another example of an exact MPS/PEPS model are (nearest-neighbor) RVB states (https://arxiv.org/... 2 As you pointed out, the phonon-mediated BCS-type superconductors exhibit a gap \Delta_0 which is isotropic in k-space, we call it an s-wave gap. As @leongz pointed out, it comes from the fact that the electron-phonon interaction used in the BCS model does not depend a momentum ; inserting it into the gap equation gives a s-wave gap. The precise ... 2 Any free fermion Hamiltonian where the Fermi energy is chosen such that it is exactly at the bottom (or top) of the band (in the case of a single band) is of this form: It is obviously gapless, and its ground state is the vacuum, i.e., short-range correlated (or rather uncorrelated). One such example would be the 1D XX model,$$ H=-\tfrac12\sum (\sigma_x^i\...

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I can see some mistakes : "I realized that we must have $|F\rangle=\sum_\alpha\int\;\mathrm{d}\textbf{k}\,\Theta(k_F-k)\,\hat{a}^\dagger_{\textbf{k},\alpha}|0\rangle$" This is wrong : the ground state of an non-interacting fermion gas (Fermi sea) is not some sort of superpositions on the individual $\textbf{k}$ states as you seem to suggest. It has to ...

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As pointed out by FraSchelle, your first question (why we can replace by the Brillouin zone by a sphere when calculating winding numbers) has been asked (and answered) a few times. The same goes for your tag-on question of why we get a $\mathbb Z_2$ invariant in the case of extra symmetries. So I will focus on your middle question, which I find the most ...

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I'm afraid you heard wrong: there is no exact solution for the Bose-Hubbard model (BHM) in 1D. There are approximate solutions in the superfluid ($J\gg U$) and Mott insulator ($J\ll U$) limits, but neither works in the intermediate regime $J\sim U$. For $D>1$, mean-field theory is often used to interpolate between these limits, but is obviously an ...

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About the limit: $\frac{\sin[\frac{\pi}{d}(1/N+1)x]}{\sin[\frac{\pi x}{dN}]}= \sin[\frac{\pi}{d}(1/N+1)x]\times \frac{\frac{\pi x}{dN}}{\sin[\frac{\pi x}{dN}]}\times \frac{dN}{\pi x}\approx \sin[\frac{\pi}{d}x] \times 1 \times \frac{Nd}{\pi x}$ in the last step I used $\lim_{x\to 0}\frac{\sin x}{x}=1$ and $1/N+1\approx 1$. after rearranging the terms it ...

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Someone might give a more precise answer in terms of group theory but I'll give it a go anyway ; feel free to edit my post. Instead of considering the case of an odd number of fermions, one can consider just a single spin $1/2$ - fermion to discuss $2 \pi$ rotations. Spin $1/2$ is a representation of dimension 2 of the rotation group, which is called the ...

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So spin 0 particles can pass freely through fermions and other particles if there is lack of EM repulsion between them? There is never lack of EM repulsion because baryonic matter is made of charged particles. The superfluid will stay into the container.

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Technically speaking all solids have crystalline structure. Anything that is truly amorphous is known as supercooled liquid (such as glass). Depending on how well crystals are oriented they can be divided into two categories, as pointed out by @lemon, single crystals and polycrystals. Single crystals have nearly perfect orientation of sizes few mm long. ...

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The susceptibility quantifies the response of a material to an external electric field due to the redistribution of electronic charge within the material. The formula you give for the charge in a particular electronic band (in practice the total susceptibility comes from the contributions from all bands). It is known as the Lindhard model and is derived from ...

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A topological invariants is a continuous map $n$: $$\mathfrak{H}\ni H\mapsto n\left(H\right)\in S$$ where $H$ is the Hamiltonian of your system and $S$ is some topological space. $\mathfrak{H}$ is the space of all admissible Hamiltonians. Unfortunately the exact definition of $\mathfrak{H}$ is still a matter of current research. To keep things short, we ...

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There is actually only one disordered phase - from a physicist's perspective, the liquid and the gas are actually the same phase because one can continuously vary the external parameters (temperature and pressure, in this case) to get from the liquid to the gas without passing through any phase transition, because the phase transition line terminates within ...

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