# Tagged Questions

Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.

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### Hedin's equations and the ground state energy

Hedin's equations are an iterative scheme to calculate the Green's function $G$, the self-energy $\Sigma$, the vertex $\Gamma$, the polarizability $\chi$, and the screened interaction $W$. However,...
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### Existence of effective Hamiltonian for variational ansatz

I recently saw on Wikipedia that in perturbation theory one can define in a systematic way effective Hamiltonian $\hat{H}_{\rm eff}$ that produces the same kind of states and energies as if one ...
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### Exact expression for the ground state energy in the 1D Bose-Hubbard model

I heard that Bose-Hubbard model is exactly solvable in 1D. What is then expression for the ground state energy $E_{0}(N, M, ...)$ as a function of total number of particles $N$, number of lattice ...
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### Proper definition of the Wannier function

I am studying the construction of the Wannier functions for 1D system with periodic boundary conditions and my reasoning seems to be a little bit different from what I can find in textbooks. First ...
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### many body wavefunction and exchange correlation

Everywhere I ready about HF or DFT the term exchange correlation functional comes up. I have a couple of fundamental questions about these: 1) Books say that the correlation energy is the difference ...
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### Bose-Einstein Condensation at higher critical temperature

The critical temperature $T_{c}$ of a Bose-Einstein Condensate is directly proportional to $n^\frac{2}{3}$, where $n$ is the density of the system which is to be condensed. The current $T_{c}$ for ...
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### Showing phase change for fermions

When discussing identical particles books often use that the states are eigenstates of the permutation operator: $P_{ij}|\psi\rangle = \lambda |\psi\rangle$ for bosons this is easy to see if I use ...
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### Capturing superfluid condensation with exact diagonalization

Doing exact diagonalization on bosonic systems is tricky, because the possibility of multiple occupancy means that even the single-site Hilbert space is infinite-dimensional. It's my understanding ...
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### What is the recursive relation for three-particle Green's functions?

In condensed matter physics, one often choose to study the many-body Green's functions (GF) with the diagram (perturbation) expansion technique. In what follows only two-body interaction is concerned. ...
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### Derivation of open boundary conditions for Non-Equilibrium Green's Function (NEGF)

It is widely claimed that the Non-Equilibrium Green's Function (NEGF) equations for the study of quantum transport have been derived from the many-body perturbation theory (MBPT). Yet the bridge ...
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### Hamiltonian lattice gauge theory with physically observable local degrees of freedom

In my answer at What, in simplest terms, is gauge invariance?, I mentioned that in certain contexts there can be a "gauge theory" with a local symmetry that leave the Lagrangian/Hamiltonian invariant ...
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In http://journals.aps.org/prb/pdf/10.1103/PhysRevB.64.195109 Eq.(4), why does the Fourier transform of the quadrupolar interaction function takes the form F(\mathbf{q})=\frac{F_2}{1+...
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### Definition of a gapless spin liquid

I understand the definition of a gapped spin liquid: it's a gapped, topologically ordered spin state - i.e. there does not exist a local unitary transformation that takes it to a product state in ...
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### Is that possible to derive Landau-Fermi liquid theory from microscopic equation?

This question arises from reading Wen's book "Quantum Field Theory of Many-body Systems (Oxford 2004)" p204 To appreciate the brilliance of Landau-Fermi liquid theory, let us look at the many-...
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### Is 3D optical lattice just a stack of 2D lattices?

I am confused about the idea of 3D optical lattice. Many papers use 3D optical lattice to study bosons behavior, but is it really a 3D system where atoms interact in all three directions or is it just ...
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### Are symmetries of a degenerate ground-state manifold always broken?

If a Hamiltonian has a global symmetry and a degenerate ground state, then in the thermodynamic limit, the ground states $| \psi \rangle$ that are eigenstates of the symmetry operator typically become ...
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### Representation of U(1) on fock space

I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making ...
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### Is there anything comparable to many-body localization in classical physics?

I've only just started looking into many-body localization, so this question may come off as a little vague. But my understanding is that it relates to how some quantum systems do not thermalize, as ...
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### reconstruct the wave-function from one body reduced density matrix?

Given a many body wave-function for a Fermion system, we can calculate the one-body reduced density matrix straightforwardly. Now suppose we know the one-body reduced density matrix, is there a way to ...
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### Adiabatic transition from superfluid to Mott insulator?

I have a question about the dynamical passage from superfluid to Mott insulator state in the Bose-Hubbard model. Is it possible to go from superfluid region to the Mott insulator by changing the ...
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### A seemingly paradox for Eigenstate Thermalization Hypothesis (ETH)

ETH states that for a system, all of its eigenstates thermalize. To be more specific, consider an energy eigenstate of the full system $H|n\rangle=E_n|n\rangle$. If the full system is in this ...
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### Is a phason a Goldstone mode?

Suppose we have a lattice system whose ground state is an incommensurate charge-density wave. Strictly speaking, this ground state does not have Goldstone modes because the only symmetry that is ...
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### Why are periodic boundary conditions used for the derivation of phonons? [duplicate]

I am currently reading "Quantum Field Theory for the Gifted Amateur". In chapter 2 Phonons are introduced as solutions (in k-space) of a coupled harmonic oscillator. In real space the oscillator is ...
In second quantization, the Hamiltonian describing the hopping process between two neighboring sites is given ($N$ - number of particles and $M$ - number of sites) by: $$\hat{\mathcal H} = J\sum\... 1answer 65 views ### Hubbard model within mean-field: three different approaches While reading doi:10.1016/j.carbon.2012.03.009 , the authors mention three types of Hubbard models within mean-field approximation. The first one describes the electron-electron interaction, and to my ... 0answers 18 views ### Creation operators that differ by a reciprocal lattice vector In very general terms, if you have an infinite lattice of atoms you can describe the physics in terms of creation (and annihilation) operators \hat{a}_{\mathbf{R}}, that create (and annihilate) an ... 3answers 57 views ### What physical properties can't be predicted based on index of refraction? [closed] If I tell you the real and imaginary parts of the index of refraction for all frequencies, name a property that can't be predicted based on that information. If you're assuming this is a gas, specify ... 1answer 68 views ### Methods for handling close approaches in N-body simulations In direct gravitational N-body simulations, what are the preferred methods for handling close approaches between bodies in order to preserve the accuracy of the evolution of the system? 2answers 364 views ### The momentum of a hole I'm currently working through "A Guide to Feynman Diagrams in the Many-Body Problem" by R.D. Mattuck (self study, not a homework problem) and am stumped by the following problem: "In a system of free ... 0answers 113 views ### How to write the second quantization form of spin-orbit coupling(Dzyaloshinskii-Moriya interaction)? Spin orbit coupling is the single particle term, so the second quantization form can be written like:\langle \alpha\sigma|s\cdot(\nabla V\times P)|\beta\sigma'\rangle c^{+}_{\alpha\sigma}c_{\beta\... 0answers 18 views ### Magnetic susceptibility in the spin triplet channel In the literature and articles I sometimes see the phrase magnetic/electric susceptibility(or other kinds of correlation functions) in the triplet channel. I don't know what does it exactly mean. ... 1answer 41 views ### What is invalidated when turning on many body interactions in a crystal? I have just started to think about strongly interacting particles and Fermi liquid theory, and I have two questions. For non interacting particles moving in an potential field, we know that the ... 1answer 57 views ### Why do we use the anticommutation relation for particle-hole and chiral symmetries? In physics we say that a quantity is conserved if its operator commutes with Hamiltonian. For example, in condensed matter systems, when the momentum k commutes with the Hamiltonian H as [H,k]=0... 0answers 124 views ### time-dependent Hartree-Fock for two-component bosons How does the ansatz for the time-dependent Hartree-Fock wavefunction look like in the second quantization if we have two-component boson system and in one case the Hamiltonian commutes with number of ... 0answers 60 views ### Alternative derivation of Gross-Pitaevskii equation I wanted to derive time-dependent Gross-Pitaevskii equation in an alternative way, but I don't know if something presented below is allowed. Hamiltonian is the following (I do not assume translational ... 1answer 59 views ### What is special about many-body localization as opposed to Mott insulator What is the difference between localization in a many-body localization system and localization in a Mott insulator? Is the difference that many-body localization is driven by disorder while Mott ... 1answer 86 views ### Density of states in a system of interacting electrons When we are introduced to the density of states in typical band-theory problems we neglect interaction between electrons, and thus we define the density of states of a sigle particle as: D(E)=2\int_{... 0answers 25 views ### Nonequilibrium Green's functions weakly interacting two-component Bose gas I am planing to describe time evolution of two-component BEC. I was thinking about non-equilibrium Green's functions, but I don't if the method can be applied to the problem describe below. I know ... 0answers 40 views ### Perturbation method for Hubbard hamiltonian I've got a hamiltonian to solve from perturbation methods:$$H=\underbrace{-t\sum_\sigma\sum_{i=1}^N(a^\dagger_{i\sigma}a_{i+1\sigma}+a^\dagger_{i+1\sigma}a_{i\sigma})}_{\hat H_\text{kin}}+\...
In the second quantization time-independent field operator can be expanded in the orthonormal basis: $$\hat{\Psi}(\mathbf{x}) = \sum\limits_{i}\hat{a}_{i}\ \phi_{i}(\mathbf{x})$$ Time evolution of ...