Questions tagged [many-body]
Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.
888 questions
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Quasi-particle decay in Fermi Liquids Involving Matsubara Sums [closed]
My course notes had a part to calculate the decay rate of fermi particle using the second order terms in the self-energy operator which contain an imaginary part and hence can be used to calculate the ...
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How to select a bond dimension for Matrix Product States (MPS) calculations?
I am trying to get a better understanding of tensor networks and Matrix Product States (MPS) before implementing any code. I will preface this by saying I am very new to these concepts.
My ...
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Why is a sharp discontinuity necessary to define the Fermi surface in interacting fermion systems?
In the study of interacting fermion systems, the Fermi surface is often defined by the presence of a sharp discontinuity in the momentum distribution function n(k) at zero temperature. For example, in ...
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Form of wide-band limit for the broadening
When considering the Green's function (GF) of a system coupled to a reservoir, the wide-band limit (WBL) is often assumed to simplify the discussion.
For instance, the reservoir retarded GF reads $g_k(...
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Does the local density approximation exchange functional obey the spin-scaling relation?
In density functional theory, the exchange energy functional has an analytic form under the local density approximation (LDA):
$$ E^\text{LDA}_\text{X}[\rho] = C_\text{X} \int \mathrm{d}^3r\, \rho^{4/...
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Greens Function formalism for the independent boson model
This may be a very specific question, but since almost everyone seems to be quoting this book, I want to understand the derivation(s) of the solution for the independent boson model (IBM) from Mahan's ...
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Auxiliary Particles Green's Function Factoring - Just Tensor Algebra Calculation and Consequences?
I have a question regarding the single-particles Green's Function in Imaginary time $\tau$ for a physical fermion $c^\dagger$ expressed by the quasi-particles auxiliary fields
$$ c^\dagger_{i, \sigma} ...
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Quadratic fermionic Hamiltonian of two modes [closed]
I was reading the thesis "A Quantum Information Perspective of Fermionic Quantum Many-Body Systems" by Christina V. Kraus (https://mediatum.ub.tum.de/doc/811670/811670.pdf) where it is shown ...
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Time evolution and anti-unitary operators
Let $\hat{H}$ be a system of lattice fermions with two internal states per site, so they can be described using operators $\hat{c}_{m,\alpha}$, with $m$ denoting the lattice site and $\alpha$ the ...
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Transformations on correlation functions
Let $\hat{U}$ be a unitary transformation that acts on fermionic fields (defined on a lattice) as
$$\hat\psi_j \rightarrow \hat{U}\hat\psi_j\hat{U}^{-1},$$
$$\hat\psi_j^\dagger \rightarrow \hat{U}\hat\...
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Bloch Hamiltonian and chiral symmetry
Let $h_k$ be a "first-quantized" (single-particle) Hamiltonian for free fermions with two internal states e.g. spin states, $c_{k,\uparrow}$ and $c_{k,\downarrow}$ respectively. $h_k$ is ...
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Energy shift in generalized BCS Theory
I'd like to understand the constant energy shift in the (generalized) mean-field theory of superconductivity.
Consider a system with the Hamiltonian $$H= \sum_k \epsilon_{k\sigma} c_{k\sigma}^\dagger ...
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Why is parity not used to classify topological insulators?
The ten-fold classification of topological insulators employs two distinct symmetries: time reversal and charge conjugation, as well as their combination, known as the sublattice or chiral symmetry. ...
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Decomposition of Fock space
Consider the regular (bosonic) Fock space in 1D (for simplicity):
$$\mathcal{F}_{+} = \bigoplus\limits_{n=0}^{\infty} S_{+}H^{\otimes n}(\mathbb{R}),$$
and say $H(\mathbb{R}) = L^2(\mathbb{R})$. Since
...
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Two completely different-looking adiabatic connection formula in DFT
In my density functional theory (DFT) course, we learned that what is called the adiabatic connection formula in DFT is some formula of the form
$$ E_{\lambda = 1} - E_{\lambda = 0} = \int ^1_0 \...
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Does there exist a version of the Slater-Condon rules for a sum of Slater determinants?
The Slater-Condon rules are useful in working out the non-zero single and double electron matrix elements.
For example if $\vert \Psi \rangle = \frac{1}{\sqrt{2}} \vert \phi_{1s}^{\alpha}(1) \phi_{2s}^...
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Number of pairs of indistinguisable particles
On pg. 18 of Quantum Many-Particle Systems, by Negele and Orland, the authors introduce the following operator which "counts the number of pairs of particles in the states $\vert\alpha\rangle$ ...
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Does Anderson localization hold in low dimensional systems with correlated disorder?
My understanding is that Anderson localization is highly generic for non-interacting Fermi systems in low dimension. More precisely, let us consider the Anderson model for fermions, with Hamiltonian
$$...
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Other than DMRG, what's the advantage of representing operator in MPO form?
I have been working with density matrix renormalization group (DMRG) (which a numerical method based on tensor network to find ground state energy), but I don't really get that, other than doing DMRG, ...
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Existence of gap when the first energy level has an infinite degeneracy
Consider a Hamiltonian $H_N$ on a finite system with the following properties:
It has a unique ground state
The first excited energy level has a degeneracy $N$ (there are $N$ eigenstates for the ...
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Non-zero resitivity in perfect crystal
It is very commonly stated that a perfect crystal has zero resistivity at zero temperature due to its translation invariance. However, in Critical drag as a mechanism for resistivity by Else and ...
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Drag forces and Lorentz covariance
I am looking for a simple example where a drag force on a macroscopic body moving in a fluid is derived from a microscopic model.
I am not so much interested in a realistic model but in a Lorentz-...
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Meaning of residual resistivity in high-$T_c$ materials
I recently came across this paper titled "Universal Tc depression by irradiation defects in underdoped and overdoped cuprates" studying the effect of disorder on high-$T_c$ materials. The ...
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Inverse spacial fourier transform of Retarded Green's Function
I'm currently taking a course on many-body QFT and came across this integral when computing the full-Born approximation for a random external potential. The book is "Many-Body Quantum Theory in ...
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If a non-interacting particle behaves like an undamped wave, can an interacting particle behave like a damped wave?
As a standard textbook introduction to quantum mechanics, there are often examples, such as single particle in a box, described as waves. I'd like to better understand problems involving more ...
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Physical realization of Bose-Hubbard model in solid state physics
Are there any supposed physical realizations of the Bose-Hubbard model in solid state physics? Obviously the Fermi version is more relevant to this purpose, but I wonder if there are any materials ...
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Robinson Approximation for the exchange integral
I am trying to deduce Robinson approximation for the exchange integral in Hartree-Fock equation (https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.9.215), but without success. I obtained the ...
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Is the quasi-particle band gap of a quantum system always non-negative?
The quasi-particle band gap of a system of electrons is defined as
\begin{equation}
\tag{1}
E_{\text{gap}} = \varepsilon_{0+}^{QP} - \varepsilon_{0-}^{QP} = E_{0}^{N+1}+E_{0}^{N-1} - 2E_{0}^N,
\end{...
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When can you neglect terms like $a_n a_{n+1}$ or $a_n^\dagger a_{n+1}^\dagger$? [duplicate]
I am looking for a detailed explanation of when and why we can neglect terms like $a_n a_{n+1}$ or $a_n^\dagger a_{n+1}^\dagger$ in Hamiltonians like
$$
H = \omega \sum_n a^\dagger_n a_n + \lambda\...
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How to obtain the Hamiltonian of the Magnon through the Hamiltonian of the electron
Suppose I already have the Hamiltonian with the atomic basis for the electron in a certain crystal structure. As usual, the atomic basis is not orthogonal, which means this Hamiltonian consists of two ...
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Ground state energy: Spin vs Fermionic language
Background
Consider the Hamiltonian of 1D periodic Ising model given by
$$ \hat{H}_{\mathrm{PBC}} = -J\sum_{i=1}^L X_i X_{i+1} - h\sum_{i=1}^L Z_i, $$
where $X_i$ and $Z_i$ are Pauli operators.
Now, ...
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Photon-induced excitations of electrons in interacting systems
We consider a two-spin system of the type
$$
H = E_{\uparrow} \sum_{i = 1, 2} c_{i, \uparrow}^{\dagger} c_{i, \uparrow}
+ E_{\downarrow} \sum_{i = 1, 2} c_{i, \downarrow}^{\dagger} c_{i, \downarrow}
$$...
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Is electron correlation in Quantum Chemistry a consequence of many-body entanglement?
Electron correlation is largely defined in Quantum Chemistry as the set of properties that the celebrated Hartree Fock Approximation cannot model accurately. One popular example is the phenomenon ...
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Unitary transformations in quantum mechanics
I have a question regarding the unitary transformation in quantum mechanics. Suppose the original Hamiltonian $H_1(t)$ satisfies the Schrodinger equation
\begin{equation}
i\frac{\partial}{\partial t} |...
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Confusion on evaluating $S$-Matrix with Interaction turned on in P.Coleman’s Many-Body Physics
I’ve been teaching myself many-body physics with Introduction to Many-Body Physics by Piers Coleman,which is a wonderful book, but I am confused with some details on page 198. All the following ...
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Lattice gauge theories and commuting-operator Hamiltonians
Since the Hamiltonian of the toric code is made of mutually commuting terms that are therefore local conserved quantities, it must be naturally linked to lattice gauge theories.
What is in general the ...
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What is the second quantized 1D non-interacting spinless Fermi gas Hamiltonian in position space?
So I understand that the Hamiltonian in momentum space for a 1D spinless fermion chain with $N$ sites $j = 1\cdots N$ is written:
$$ H = \sum_k c^\dagger_k c_k$$
with $c_k$ be the annihilation ...
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Calculate MPS tensor for a given translationally invariant state (numerically)
Assume you are given a many-body state that lives on a 1D chain with local Hilbertspace dimension $d$ on each site and length $N$ as a vector $|\psi \rangle\in \mathbb{C}^{d^N}$. Let the state be ...
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Thermal propagator for free phonons evaluation
I'm trying to evaluate the thermal propagator for free phonons $$-D(\bar{x}\tau, \bar{x}'\tau') = \left\langle \mathcal{T} \varphi(\bar{x}\tau)\varphi(\bar{x}'\tau') \right\rangle $$
where its ...
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Why does the Schriefer-Wolff transformation works for phonons?
One way to derive a Hamiltonian with attractive electron interactions is to start from the Hamiltonian with a part quadratic in electrons, quadratic in phonons, and a standard electron phono coupling ...
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What is the proper ansatz for describing an electron-photon many-particle System?
I am somewhat used to simplified non-relativistic quantum mechanics (both canonically and grand canonically), describing a system by a Hamiltonian containing a kinetic part, an external potential as ...
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Slater determinant with complex coefficient
Suppose we have a system of three particles with states $\alpha,\beta,\gamma$. We can write a state of the form (up to a normalization factor ):
\begin{eqnarray}
\Psi=|\alpha,\beta,\gamma\rangle + e^{...
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Is the Luttinger liquid a limit of the Kitaev chain model?
From what I understand, they are both models of electrons on a nanowire, but the Hamiltonian is different, just in the pairing term. When I look up the connection between them, there's surprisingly ...
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What does it mean to break a Cooper pair?
I am studying BCS Theory for the first time, and I did encounter the Bogoliubov-Vitalin transformation for the BCS hamiltonian, that gives
$$
\hat{\mathcal{H}} = - \sum_\mathbf{k} \sqrt{\xi_\mathbf{k}^...
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Fermi poles expansion
I want to prove the following formula:
$$\frac{e^{-tE}}{1+e^{-\beta E}} = \frac{1}{\beta}\sum_{k \in \frac{2\pi}{\beta}\mathbb{Z}}\frac{e^{-ikt}}{-ik+E},$$
for $\beta > t > 0$. I know the trick ...
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Hirsch's discrete Hubbard-Stratonovich transformation
Suppose
$$\hat{H}_1 = U\left(\hat{n}_{d,\uparrow} - \frac{1}{2}\right)\left(\hat{n}_{d,\downarrow} - \frac{1}{2}\right)\tag{1}$$
To decouple the many-body operator, many articles suggest using the ...
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How do I compute the ground state starting from a Nambu-like representation of the system?
I'm aware that the ground state
$$|{\Omega}>= \Pi_{|k|<k_f} c^\dagger_{k,\sigma} |{0}>$$
is a product up to the Fermi momentum of construction operators acting on the vacuum. In this case, in ...
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SYK Normalization
In the usual SYK model described via
$$
H = -\frac{1}{N^{3/2}}\sum_{ijkl}^{N}J_{ij,kl}\chi_{i}\chi_{j}\chi_{k}\chi_{l}.
$$
The normalization factor out front ($N^{-3/2}$) is chosen such that the ...
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Ergodic hierarchy and the two-point correlation function
I'm currently looking at a paper about dual unitary circuits (https://arxiv.org/pdf/1904.02140) where the authors derive an expression for the correlation function looking like
$$C_{\alpha\beta} = \...
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How to tell if a state written in second quantization is a Slater determinant?
How do I tell if a state written in second quantization is a Slater determinant? I was solving some basic quantum-many body systems, and, for numerical purposes, I would like to determine if the ...
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