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2

Hints: OP's identity follows from standard manipulations in the interaction picture, cf. e.g. Ref. 1. Start with the evolution operator $$\tag{1} U(t_f,t_i)~:=~\exp\left(-\frac{i}{\hbar}H (t_f-t_i) \right), \qquad H~=~H_0+V,$$ which satisfies the Schrödinger equation $$\tag{2} i\hbar\frac{\partial}{\partial t_f}U(t_f,t_i)~=~HU(t_f,t_i).$$ Define the ...

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This question is subtler than one might think at a first glance. Three points may be worth noting. It will make a difference whether the manifold $M$ is compact or not. The nice answer by Michael Seifert implicitly assumes the non-compact case. Otherwise the infinite Gaussian integrals (over $\mathbb{R}^n$) will not make sense. In the compact situation ...

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You can get a nice expression for the leading-order correction to the flat-manifold result via the use of Riemann normal coordinates. Basically, imagine expanding the metric in a power series at the point $x_0$: $$g_{\mu \nu}(x) = g_{\mu \nu} (x_0) + \partial_\rho g_{\mu \nu} (x^\rho - x_0^\rho) + \frac{1}{2} \partial_\rho \partial_\sigma g_{\mu \nu} ... 2 Another way of showing this is to consider the definition of the ket$$|ab\rangle = a^{\dagger}_ab^{\dagger}_b |0\rangle$$and the bra$$\langle ab| = \langle 0 | a_b a_a$$and to look at the matrix element of H as it is defined.$$ \langle ab | \hat{H} | cd \rangle = \langle ab | V | cd \rangle \times \langle 0 | a_b a_a ...

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Now I know why the Logarithmic discretization are take place in Anderson Model for low temperatures. We want to discretize the energy band-width $[-D,D]$ such that we can perform a numerical calculation. But we want to answering questions of low temperature, and we need to have very careful to apply the thermodynamic limit $N\rightarrow \infty$ before the ...

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Because these are actually Fourier transform of the usual Green functions. Considere the Schrödinger equation : $$\hat{\mathcal{H}}|\Psi(t)\rangle=\mathrm{i}\partial_t|\Psi(t)\rangle$$ The general solution $|\Psi(t)\rangle$ of such equation for a time-independant hamiltonian $\hat{\mathcal{H}}$ can be expressed in terms of Green function $G(x',x,t)$ :  ...

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In one dimension, MERA naturally capture critical systems (i.e., systems with power-law decaying correlations and a log-divergence in the entanglement entropy). MPS (i.e., one-dimensional PEPS), one the other hand, have exponentially decaying correlations and a constant entanglement entropy. (Note: This is for a constant bond dimension and does not preclude ...

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The Hartree-Fock method treats the interaction between particles in a mean-field approximation. So the potential felt by particle i is given by the average over the wave functions of all the other particles. However - speaking semi-classically - you could imagine that particle j has a position as a function of time, and when it's on the "left" side of the ...

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