Questions tagged [many-body]
Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.
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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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Can I use this trick to write the one-body Laplacian as a two-body Laplacian? [migrated]
Suppose we have a Hilbert space $\mathscr{H} = L^{2}(\Omega)$, where omega is some finite torus $\Omega = [-L/2,L/2]^{3}$, with $L \ge 1$. Wave functions of $\varphi_{p}(x) = L^{-3/2}e^{i\langle p, x \...
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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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Ground state multiplicity of highly charged atomic ions
I tried to estimate what the electron configuration of charged atomic ions would have. I assumed they would be configured like their isoelectronic neutral element, i.e. Kr$^{8+}$ should be configured ...
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Perturbation theory of spectral gap at specific point in spectrum
Suppose I have a family of finite system-size $N$ Hamiltonians $H^{(N)}$ (i.e. hermitian matrices of dimension $2^N\times2^N$). I know that for a certain scalar $E^{(N)}$ $(H^{(N)} - E^{(N)})^2$ has ...
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Jellium Hamiltonian in the thermodynamic limit
In Fundamentals of Many-body Physics by W. Nolting, 1e, the author arrives at the following formula for the electron-electron contribution to the Hamiltonian of Jellium:
$$
\hat{\mathcal{H}}_{ee}=\...
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When is the temperature relevant for a quantum many body system?
Let me illustrate my question with an example. Suppose I want to consider a system of $N$ identical bosons. The Hamiltonian of the system is typically of the form:
$$H_{N} = \sum_{j=1}^{N}(-\Delta_{x_{...
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Phase Coherence in the BCS wavefunction and the Cooper Pair Wavefunction
I have a couple question regarding the following BCS wavefunction ($|0\rangle$ is the vacuum state):
$$|\psi\rangle = \Pi_k \big(|u_k|+|v_k|e^{i\varphi}c^\dagger_{k\uparrow} c_{-k\downarrow}\big)|0\...
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Example of an injective matrix product state (MPS)
I am struggling to understand what is an injective matrix product state (MPS). From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ ...
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X-ray absorption edge, accurate theoretical predition possible?
It is well known that in X-ray absorption spectroscopy, the absorption spectrum shows discontinuities at some critical frequencies.
Is it possible to predict locations of these jumps theoretically? I ...
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Do different bases of Fock space commute?
$\newcommand\dag\dagger$
Suppose we have a Fock space $\mathcal{F}$ with two different bases of creation and annihilation operators $\{a_\lambda, a^\dag_\lambda\}$ and $\{a_{\tilde \lambda}, a^\dag_{\...
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Determining if Self-Energy is Complex from Action Alone?
My understanding of quantum field theory is that an interacting two-point function of spin-0 bosons will have the form: $ \frac{i}{p^2-m_0^2-\Sigma(p)}$, where $\Sigma(p)$ is the self-energy, the sum ...
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Difference between GW gap and BSE gap. How does net charge play a role?
The energy gap from BSE (GW-BSE) is just the lowest optical excitation energy.
The energy gap from GW, from what I read, is the electron affinity (energy of adding an electron).
I suppose this means ...
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Perturbation theory of Anderson impurity model
I’ve been learning about DMFT(Dynamical mean field theories) these days and have encountered rather simple questions in many-body perturbation theory.
It is about IPT Impurity solver (applying ...
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Hubbard-Stratonovich (HS) transform (or similar) for higher order-interactions
I have a question about a generalization of the Hubbard-Stratonovich (HS) transformation to decouple two-body interactions. When dealing with Hamiltonians of the kind
\begin{equation}
H = -\sum_{a}...
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1
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Total ground state energy calculation of an electron-ion system using plane wave basis sets
Recently I have begun working on a project which involves constructing a "simple" density functional theory (DFT) code using a plane wave basis set in Python. My first step has been to try ...
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Fourier transform of spin Matsubara Green function
Is the spin Matsubara Green function of a generic spin operator (or product of spin operators) bosonic? How can one obtain its frequency decomposition?
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Photo-excited Radiation/Electric Current with Green Function Method
Suppose there is an incident photo with a specific frequency hitting the material; the incident photo is absorbed by this material. Meanwhile, the material is excited by this incident photo and ...
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Permanent operation's result
N-body fermionic systems are constructed by Slater determinant, and it is equal to Vandermonde polynomial. Are there any special polynomial for the permanent which is used to construct N-body ...
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What's the difference between the one-particle perturbation theory and many-body perturbation theory?
When I reading Hewson's book (The Kondo Problem to Heavy Fermions) about Kondo's explanation to resistance minimum, I encounter a sentence which saying
'The s-d Hamiltonian cannot be reduced to the ...
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Rigorous definition of the average value $\langle a^{*}(f)a(g)\rangle$ on Fock spaces for arbitrary states [duplicate]
One of the axioms of quantum mechanics states that a quantum state $\rho$ is a positive (hence self-adjoint) trace class operator with trace one. Given an observable $A$, the expected value of $A$ in ...
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Self Consistency of Wave Function Given By Green Function ($\psi(r,t) = \psi^0(r,t) + \int dr'\int dt G_0(r,r',t,t') V(r')\psi(r',t') $)
In "Introduction to Many-body Quantum theory in condensed matter physics" by Bruus and Flensberg there is an exercise regarding Green functions.
We want to solve the time dependent ...
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Why is the "decision" version of the local Hamiltonian problem promised to have a positive gap?
The Wikipedia article on the local Hamiltonian problem is ungrammatical and unclearly written. I think that this is what it is supposed to say:
The decision version of the $k$-local Hamiltonian ...
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Why are PEPS more frequently used when simulatind 2D systems rather than branching MERA?
From what I've read it seems that PEPS is a go-to method while simulating 2D quantum systems. Why is it the preferred method rather than branching MERA?
The contraction of PEPS is a #P-complete ...
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Current for Anderson-type model in linear response
I would like to raise an issue regarding calculation of current for Anderson-type model as discussed in the corrected version (14 January 2016) of the book "Many-body Quantum Theory
in Condensed ...
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Quantum Dimer Model with Multiple Chemical Potentials
Regarding the commonly known quantum dimer model on the honeycomb lattice
See References: Phys. Rev. B 64, 144416 and Phys. Rev. B 69, 224415
only a chemical potential $v$ for the one possible type ...
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Does BKT transition exist in 2-dimensional non-interacting (ideal) Boson gas and why?
In 2D continuum system, the transition from the BKT phase to the normal phase seems to represent the superfluid is broken by the excitation of the single vortex. But why does superfluid exist in 2D ...
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Two-particle Green's function, possible typo in the book referred?
I'm trying to follow a computation in some QFT book, p64. The goal is to derive the equation of motion for the lesser Green's function $G^<$ defined as
$$
G^< = \mp i {\rm Tr}\left(\rho \Psi^\...
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In an $n$-body system, do all bodies accelerate towards center of mass?
I've been building an $n$-body simulator as a way to learn a new game engine, and I've come across something that isn't necessarily a problem, but has intrigued me. Most other simulators I'm seeing ...
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Force-simulation for graph layout: How to avoid particle collapsing into a single point?
In a force-based graph-layout simulation using Barnes-Hut, what are the conditions for collapse? With collapse I mean multiple (or even all) nodes "collapsing" into a single point.
Is there ...
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The relation between spectral function, Green function and particle number in many-body systems
The spectral function is defined as the imaginary part of Green function multiplied by $-2$ (Ref. Mahan. Many-Particle Physics 3ed. Kluwer Academic, 1990.),
$$
A(\mathbf{k},\mathrm{i} \omega_n) = -2 \...
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Two-body operators in second-quantized form
I'm currently reading through Altland and Simons' book "Condensed Matter Field Theory" (2nd ed.), where in chapter 2, page 49 they derive the form of two-body second-quantized operators. ...
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Two species of bosons [closed]
Consider the Bose-Hubbard model for a single species of boson in a square lattice,
\begin{equation}
H_{a}=-t\sum_{<ij>} a^{\dagger}_{i}a_{j}+U \sum_{i}a^{\dagger}_{i}a^{\dagger}_{i}a_{i}a_{i}-\...
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Scattering by point potentials
Suppose we consider the quantum scattering of two particles. The interaction Hamiltonian is given by:
$$H = -\Delta_{\vec{x}}-\Delta_{\vec{y}} +V(x-y)$$
where $V$ is the interaction potential between ...
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Majorana Boson Coherent States
Consider $a$ be a bosonic operator, and we define $\Phi = a+a^{\dagger}$ and it is clear that $\Phi^{\dagger}=\Phi$ that implies "Majorana Boson". Now, i want to find the coherent states for ...
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Correlation functions zero for repeated creation or annihilation operators [duplicate]
In the derivation of the damped harmonic oscillator, at some point, see for instance here one has to compute certain correlation functions for the bath of harmonic oscillators in thermal equilibrium. ...
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General expression for fermionic creation operator subspace
This is perhaps a particular case of the question discussed here.
Given a fermionic Fock space $H$ of dimension $2^N$, that is, with $N$ fermionic modes, let $H_n$ be the subspace of states with $n$ ...
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Constant in mean-field Hamiltonian
When one obtains the mean-field Hamiltonian of a (classical or quantum) spin system and then needs to find the mean-field parameters by minimizing the expectation value of the Hamiltonian, does one ...
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