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Probably the best place to start classically is with integrable systems. A crude physicist definition is that these are systems that have, in the words of Nandkishore et al, "an infinite set of extensive conserved quantities that are sums of local operators" (1). Roughly speaking, such systems will never approach an equilibrium because none of these ...

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It is indeed possible to change between these phases adiabatically. Since, as you noted, the ground state changes between being a superfluid and a Mott insulator, starting in the ground state and making an adiabatic change means that you track that change in state by definition. Note that this diagram is only formally true for the Grand Canonical ensemble, ...

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You need to be careful about how you go from the full system to the subsystem $A$. You define $\rho^\text{eq}(T) = Z^{-1} \exp(-H/T)$ as the thermal state of the whole system, but then you use $\rho_A^\text{eq}(T)$ without defining how you are reducing the density matrix of the whole system onto just the subsystem. There are two reasonable ways to do so: ...

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In the thermodynamic limit (linear size of the system $L$ to infinity), boundary conditions don't really matter, and most physical observables will be the same for all boundary conditions. The use of periodic boundary conditions is mostly for practical reasons, in particular, translation symmetry is conserved, which really helps. One could in principle do ...

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To me, the Hubbard interaction per site is defined as $H_{int}=U n_{\uparrow} n_{\downarrow}$. (I suppressed the $i$, and also there is a factor of two because of your spin sum.) Then, the mean-field approximation is defined as $n_{\uparrow} n_{\downarrow}\to n_{\uparrow} \langle n_{\downarrow}\rangle+n_{\downarrow} \langle n_{\uparrow}\rangle-\langle ... 1 Index of refraction and the extinction coefficient are related to real and imaginary parts of dielectric function via a simple functions. Empirical real and imaginary parts of a dielectric function (or Lorentz oscillator fits to them) are usually used when simulating macroscopic Maxwell equations. So, what is missing (assuming you do not mean the trivial, ... 1 For a gas: Density Nuclear half-life of the components Viscosity Temperature of liquefaction Specific heat Speed of sound Rayleigh scattering coefficient 0 Take the fluorescence properties of a material for example. You cannot predict the time decay properties of the material subject to illumination just by knowing the refractive indexes. 5 I assume you're talking about the numerical instabilities that arise from having an infinite potential at$r=0$. Here are three common solutions: Use a soft-core potential that behaves like$1/r$except very close to$r=0$where it levels off to a finite value. For example,$1/\sqrt{\epsilon+r^2}$instead of$1/r\$ is common. Add hard sphere collision ...

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Momentum does not have a direction associated with it. A wave vector does. So the electron and hole can have the same +/- sign for their respective momenta.

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