Questions tagged [lattice-model]

Lattice is a way of discretizing a quantum field theory for numerical simulations.

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Is it possible for the Kagome lattice to have edges of different lengths?

Is it possible for the edges of the Kagome lattice to be of different lengths? In other words, in the attached picture, is it possible to have $x\neq y$? If so, how will be the shape of that network? ...
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Is there any relation between Lieb-Robinson velocity bounds and micro-causality?

Background So I recently asked a question about relativistic quantum mechanics and the answerer invoked micro-causality (from QFT) to show me that the assumption the information would propagate ...
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Mixing Fourier Transform and Real Space for lattice with only one perodic direction

I have a 2D lattice consting of two different regions A and B and in real space a closed set of equations for a quantity $f_{\mu\nu}$ at every lattice site $\mu,\nu$. To solve this, I normally would ...
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Jordan-Wigner transformation for lattice models without $U(1)$ symmetry

The Jordan-Wigner transformation is a powerful approach to studying one-dimensional spin models. The following dictionary between spin operators and creation/annihilation operators for fermions allows ...
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How is the unit cell of a Trihexagonal lattice? [closed]

Considering a Trihexagonal lattice as a periodic system, how is its unit cell? and how many vertices it will have?
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Rotor representation of creation and annihilation operators

I read the paper "Edge–Entanglement correspondence for gapped topological phases with symmetry" Where a lattice $U(1)$ model is introduced with bosons both on the sites and links of the ...
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Worldsheet SCFT on a lattice

My question is clear from the title. I'm curious whether it is possible to put the string world sheet SCFT on a lattice. I expect when the world sheet theory is chiral, then it's not possible. But I ...
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Are there any quantum theories where particles have discrete probability distributions?

I have done a bit of searching for an answer to this question, but I am an amateur and suspect I lack the proper "language" to describe what I'm actually looking for, so a pointer in the ...
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Question about discretizing Laplace equation of potential flow

I've learned from fluid mechanic class that the laplace equation of stream function $\Phi$ can be discretized by $\Phi_{0}=\frac{\Phi_{A}+\phi_{B}+\Phi_{C}+\Phi_{D}}{4}$, if the flow is irrotational ...
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How to understand Peierls substitution?

The Peierls substitution is given by the following formula: $$t_{12}\rightarrow t'_{12}=\exp\left[\int_{{\bf R}_{1}}^{{\bf R}_{2}}{\bf A}({\bf r})\cdot d{\bf r}\right]t_{12}.$$ What are the meanings ...
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Why is the ground state degeneracy of the toric code 4?

Hi I'm kind of confused about the ground state degeneracy in the toric code model. The generic ground state of the TCM is a state $|\Omega\rangle$: $A_v |\Omega \rangle = B_p | \Omega \rangle = | \...
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How to identify a triangular reciprocal lattice as sum index?

I have the following Fourier expansion for a wavefunction in the Bloch basis $$\psi_{0}\left(\boldsymbol{r}\right)=\sum_{\boldsymbol{Q}}C_{\boldsymbol{Q}}e^{i\boldsymbol{Q}\cdot\boldsymbol{r}}, (1)$$ ...
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Relating the topological behaviour in the toric code to cohomology?

I've been working on the Toric Code Model (by Kitaev in his 2003 paper on quantum computation), and the model is a lattice realisation of a topological phase. The local operators in the model are ...
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Basis vectors for square kagome (squagome) lattice

I was wondering if someone might be able to explain how you could determine what the basis vectors would be for a decorated square lattice otherwise known as a square kagome (squagome lattice)? My ...
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Discretization of QFT via Finite Volume method

When working with discretized QFT one moslty works with the Finite Difference Method (FDM) on the hypercubic lattice, and the fields live on sites, links, faces ... of the lattice. Also QFT can be ...
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How is it possible to differentiate or integrate with respect to discrete time or space?

As far as I have understood, the case is that there is nothing that argues that time or space is continuous, but at the same time we must assume this in order to be able to calculate derivatives or ...
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Proof of commutation relation for lattice QFT

How do you prove the following commutation relation for the lattice QFT \begin{equation} [\phi(X),\Pi(y)]=\text{i}a^{-d}\delta_{x,y}\mathbb{I}? \end{equation}
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Number possible polygons in a body centred lattice

I wanted to ask how many possible polygons of side 6 and side 8 can be constructed in a body centred lattice(with a^3 sites), considering that the sides of the polygon are of length of the nearest ...
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Symmetries of the square lattice

According to the literature, the square lattice has $C_{4v}$ symmetry. This point group does not contain inversion. However, the square lattice is obviously inversion-symmetric. Is this because ...
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Equivalence between rotation and magnetic flux in lattice models

I am trying to understand the presence of complex hopping amplitudes in Hubbard-like lattice models. The hopping term features the so called "Peierls phase": $$ - t\sum_{j=1}^L \left( c_{...
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What is the connection between vertex/spin models and gauge theory?

In the usual formulation of lattice gauge theories, one considers gauge variables on the links of a lattice (often hypercubic) taking value in some representation of a gauge group, $U_{ij} \in G$. The ...
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How to write velocity operator of a given Hamiltonian in quantum mechanics?

Let us have a 2D lattice model with three sites in one unit-cell (basically 2D Kagome lattice). In Fourier space, the Hamiltonian is written as $$ H = \sum_{k_x,k_y} \begin{bmatrix} a^\dagger & b^\...
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Fermions from lattice model

In TASI Lectures on Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems from lattice Hamiltonian, which describe fermions on honeycomb lattice: $$ H_f = -t_f \sum_{\langle ...
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Time reversal symmetry implies that fermions are massless?

In TASI Lectures on Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems some continuous limit of lattice model with fermions considered. And on page 6 there is a statement: ...
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Completely Integrable Frustrated Lattice Systems

The Toda lattice is a prime example of a lattice system that is completely integrable, in the sense that it admits a Lax pair, https://doi.org/10.1143/PTP.51.703, making it easy to find soliton ...
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Recommend books for learning lattice QCD

I want to learn lattice QCD by myself, but I don't know how to start. Can you recommend some books for lattice QCD?
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Fourier transform of a 2D stripe lattice model

Let us have a 2D square lattice with the distance between two sites equal to $a=1$. The hopping Hamiltonian can be written as: $$H = -J\sum_{\langle\textbf{n,n}'\rangle} c_{\textbf{n}}^+c_{\textbf{n}'}...
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Is symmetric band structure mandatory for particle-hole symmetry?

The particle-hole transformation acts on fermionic annihilation operator so that $c_j \rightarrow e^{i\pi j}c^\dagger_j$ Using this we can redefine the Fourier transform of an operator, $c_k = \sum_j ...
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A doubt regarding discrete spacetimes

Shorter version of a deleted post which hopefully is focused enough to not be closed. Basically, I'd like to ask: How is it that a discrete spacetime (Taking our universe as a basis, of course) can be ...
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From spins to fields

In statistical field theory, one usually considers the so-called Landau Hamiltonian: $$\beta H = \int d^{d}x\bigg{[}\frac{t}{2}m^{2}(x) + \alpha m^{4}(x)+\frac{\beta}{2}(\nabla m)^{2}+\cdots+ \vec{h}\...
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Clarifications about definition of Bravais lattice

I have a doubt about the definition of Bravais lattice for periodic materials. Precisely, here it is defined as: a discrete set of vectors closed under vector addition and subtraction If I look at ...
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Inhomogeneous Classical Gravitational wave equation discretization and boundary conditions?

On a previous posting I asked this community for information on forms of gravitation (classical or relativistic) that take into account the wave like nature and finite speed of a propagation for ...
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Are gravitons equivalent to extra localised lattice points?

So imagine space is a regular square mesh or lattice. In a theory like QCD, the photon lines are placed along the edges of this graph to form paths. The space is supposed to represent simple ...
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How does the reciprocal lattice takes into account the basis of a crystal structure?

I am reading about solid state physics, and I think I got right the concept of crystal lattice. We first define a Bravais lattice as the set of vectors spanned by $\{\vec{a}_1,\vec{a}_2,\vec{a}_3 \}$ ...
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Physical meaning of the Yang-Baxter equation

I'm a graduate student in mathematics, and I have lately been interested in the relation between knot theory and statistical mechanics. As I understood, the Yang-Baxter equation (shown below) is the ...
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1D Ising Model with magnetic field on even sites: Transfer Matrices

I have been trying to work out a practice problem after reading about Transfer Matrices method for solving 1D Ising Model. Please, if you are able to, tell me whether the way I introduced transfer ...
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Is the definition of gap of a Hamiltonian, i.e. difference between two distinct eigenvalues, restrictive?

The spectral gap of a quantum model or a Hamiltonian, in the context of whether it is a gapped or gapless model, is often defined as the difference between the two lowest distinct eigenvalues of the ...
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Ising universality class

Ising model is defined as lattice model with interactions only between nearest sites if lattice. If we deform Ising model, include non-nearest interactions or interactions between more than two ...
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Velocity driven Poiseuille flow using the incompressible lattice Boltzmann method

My goal here is to develop a Poiseuille flow in a channel, using constant pressure and zero velocity as initial conditions. From what I have read, people seem to be mostly using pressure-driven, or ...
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What is the lattice temperature?

I am wondering how the lattice temperature is defined in this context. Is it the temperature on which the energy is big enough to split the crystal in it's components (atoms)?
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Find tools to better understand crystal structure

I am currently studying solid state physics. But I have trouble understanding the crystal structure (or lattice structure), such as the structure in this picture (just to give an example). When I look ...
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Can someone tell what are primitive translation vectors for BCC lattice?

I referred internet and two very renowned books ( Puri-Babbar and SO PILLAI ). COULDN'T find any primitive translation vector explanation for BCC or FCC. Eventually when I moved on to Reciprocal ...
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What do the points in the reciprocal lattice stand for?

I'm wondering for what the points in the reciprocal lattice physically stand for, I know that they are the k-vectors, that are the fourier-transformed vectors of the lattice in the real space. Do ...
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Ising model operators

Ising model formulated as lattice theory with local degrees of freedom described by $s_i$ $i\in 1, \dots, N$ and energy: $$ E[\sigma_i] = -J\sum_{<ij>} s_i s_j $$ From $s_i$ I can construct a ...
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Phase diagram of ${\rm O}(3)$ lattice model and mean field theory

In David Tong: Lectures on Statistical Field Theory Problem Sheet 1 exist task: I fully understand this problem and I know solution: section 17.2. I am interested in application of mean field theory ...
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Symmetries in the Hubbard model

I would like to understand both in an intuive and in a mathematical way the meaning of the sentence "The Hubbard Hamiltonian has an SU(2) symmetry". What are the symmetry transformations that leave ...
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Partition function of 2D Ising model on a squared lattice in the canonical ensemble in the low temperature limit

I'm currently working through David Tong's script on statistical mechanics (http://www.damtp.cam.ac.uk/user/tong/statphys/sp.pdf) and came across something that I don't quite understand (page 166). ...
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Initial conditions in incompressible lattice Boltzmann method

I have very recently come to LB methods, and, after some reading, have implemented a little 2D code of my own. It uses some of the most basic assumptions : D2Q9 lattice BGK collision operator Zou-He ...
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Spin-orbit coupling Hamiltonian in tight-binding models

Consider spin-orbit coupling (of strength $\lambda_1$) on lattice, with the below Hamiltonian $$H = i \lambda_1 \sum_{<ij>} ~\frac{E_{ij} \times R_{ij}}{|E_{ij} \times R_{ij}|} \cdot \sigma ~...
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Can anyone help me describe the physics of my problem mathematically? [closed]

I was unsure if I should post this here or on the Mathematics page, so I decided to do both. Here's the problem I want to describe: Suppose I have a sphere, discretely represented by a large set of ...

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