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Questions tagged [tight-binding]

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Strain field and periodic boundary conditions

Let's say I have a lattice, and I impose periodic boundary conditions. I want to construct a tight-binding model on a strained lattice, and I can determine the change in the hopping parameter based on ...
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Lattice hopping at boundary in graphene lattice with magnetic field

Let's say I have a tight binding model for graphene, where I have a two-atom basis and three nearest neighbor vectors. I've applied a homogenous magnetic field $B$ in the z-axis, and can take the ...
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1answer
69 views

Momentum Space Representation of the Tight Binding Hamiltonian

I am trying to represent the tight-binding Hamiltonian \begin{equation} \hat{H}_{TB} = \sum_{\sigma} \sum_{\alpha,\beta} \sum_{\mathbf{R}_1,\mathbf{R}_2} t^{\alpha,\beta}_{\mathbf{R}_1,\mathbf{R}_2} \...
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Is the tight binding model an effective free fermion model?

The tight-binding Hamiltionian has the form $$H=-t\sum_i\left(c_i^\dagger c_{i+1} + c_{i}c_{i+1}^\dagger\right)$$ But does this mean that it can be represented in the form of free fermion modes?
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Identity when Diagonalising Single-Particle Hamiltonian

Sorry the title is not precise; wasn't sure how to make it so (this is perhaps a straightforward question). The following is an identity I see quite often when reading lecture notes about ...
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Is the SSH model a tight binding model?

Sorry if this is an obvious question. I have trouble understanding where the Hamiltonian of the Su-Schrieffer-Heeger model comes from? May I confirm if it is from the Tight Binding Model? The creation ...
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Gutzwiller renormalization factors

I am computing the expectation value of the kinetic term of a tight-binding model, respect to the Gutzwiller wavefunction, in the limit of infinite lattice-coordination, i.e using these constraints (...
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2D BHZ tight binding model for Quantum spin Hall insulator

I am currently reading this article : https://arxiv.org/abs/cond-mat/0611341 and want to derive the k-space tight binding model of 2D BHZ. The tight binding model is written as \begin{equation} H = \...
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Physical Meaning of the Gutzwiller Constraints

I have a doubt on the constraints for the expecation values obtained by Bünemann et all. First i want to introduce my notation To analytically solve a tight-binding model, \begin{equation} \hat{H}= ...
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1answer
58 views

Is numerical lattice wavefunction smooth? — graphene tight binding case

I tried to follow exactly Sec. II.K [page 112-113, Hamiltonian after Eq. (113)] of the standard Review of Modern Physics paper on graphene, which is a tight-binding model of a graphene stripe under ...
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Loschmidt Amplitudes In Tight Binding Models

I am trying to understand the calculation in this paper which experimentally looks at dynamical phase transitions in a driven (shaken) haldene-like model. The definition of the Loschmidt Amplitude is ...
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1answer
35 views

Tight-binding extended attractive Hubbard model for unconventional superconductivity

BCS theory suggest that the effective attraction between two electrons, due to electron-phonon coupling, is in momentum ($k-$)space. However, in literature, (real space) tight-binding Hubbard model ...
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1answer
65 views

Why does the 2D hexagonal lattice have a different tight binding band structure than Graphene?

Here you can find band structures for various tight binding models. I was wondering, why the 2-D hexagonal lattice has a different band structure than Graphene, even though they have the same lattice.
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1 Dimensional quantum chain

I had a question about the transformation of the hamiltonian of a 1 dimensional quantum chain, from real space to reciprocal space. Using the given discrete fourier transforms. $\hat{H}$ = $\sum_j$ $\...
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1answer
57 views

What is nesting/ what is a nesting vector in energy contour plots?

I am making different plots for a 2-d non-interacting tight binding Hamiltonian $$ H = - t \sum_{<ij>, \sigma} c_{i \sigma}^{\dagger} c_{j \sigma} + h.c$$ I get the dispersion relation $$\...
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1answer
58 views

Can someone give a simple mathematical explanation of the tight binding method?

I'm trying to understand the tight binding method but I'm struggling with a lot of the mathematical formalism. A lot of the mathematical formalism I read jumps into explaining it a few too many steps ...
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60 views

Derivation of Slater and Koster Table

Im currently struggling with the description of the Slater-Koster (SK) method. I should calculate the SK table. But I do not know how they calculated the formulas. There are some other references ...
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217 views

One-band 1D tight-binding model: how to find the two-particle eigenstates?

Consider the simple hopping model in second quantization, $\hat{H} = -J \sum_{i,j=1}^\infty \left(\hat{c}_i \hat{c}^\dagger_j + h.c.\right)$ where $J$ is real and $\hat{c}_i$ are annihilation ...
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Writing the Tight Binding Hamiltonian as a Sum Over Unit Cells

Typically, the Tight Binding Hamiltonian of a lattice is written as a sum over nearest neighbours. Is there a straight forward way in which one can rewrite this as a sum over unit cells instead?
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71 views

Dispersion relation in tight binding model with even indices only

Given a tight binding model with Hamiltonian $H= \sum_{i(even)}t[c_{i+1}^\dagger c_i+h.c]$ containing even indices only, how can we find out the dispersion relation? Attempt: My guess is that the ...
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1answer
533 views

Number of bands in 1D tight-binding model

I was reading about the one-dimensional tight-binding Hamiltonian (TBH) with one quantum state per atom $$H=E_0\sum\limits_{n}|n\rangle\langle n|-t\sum\limits_{n}\Big(|n\rangle\langle n+1|+|n+1\rangle\...
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591 views

Bloch Hamiltonian of a Hexagonal Lattice

As explained here (link in the first comment. couldn't post more than 2 links here!), the Bloch Hamiltonian of a lattice is obtained as $$h_k = \sum_je^{i\mathbf{k.R_j}}h_j$$ where $h_j$ is the ...
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Can we write energy band of square lattice with vertical magnetic field?

I am interested in a square lattice with the vertical magnetic field. Without a magnetic field, we can know the energy dispersion of the square lattice easily. But, how about in case of with magnetic ...
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1answer
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Does a core already need to be magnetized in order to establish flux in the core?

I was going through working of a real transformer and saw that there are two different branches of currents in the primary, one to magnetize the core I(m) and another whose flux links with the ...
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How to show transfer integral is a real number?

I want to know how to show transfer integral is a real number. I am learning tight binding approach. Now let's consider the two-dimensional system whose unit cell has two atoms (for example, ...
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1answer
501 views

Tight binding Hamiltonian for a slab

I want to study the surface states in a material using the tight binding model (the goal is to find surface states inside the bulk band gap). The material in question has rock salt crystal structure ...
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1answer
252 views

Graphene Tight binding 2D Hamiltonian

I got stuck on Homework problem, where I need to construct Hamiltonian of 2D Graphene layer and obtain Dispersion graph from it. I already went trough a lot of materials but all I find is $2\times 2$ ...
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1answer
192 views

How can I compute the eigenstates of a tight-binding Hamiltonian describing a system?

I have the following set-up in a 3-site tight-binding system: \begin{align} i\hbar\frac{dc_1}{dt}&=-Ac_2,\\ i\hbar\frac{dc_2}{dt}&=-Ac_1-Ac_3,\\ i\hbar\frac{dc_3}{dt}&=-Ac_2, \end{align} ...
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1answer
288 views

Green's function approach in Tight-Binding

I am studying single-particle Green's functions using Economou's textbook, and am interested in using them to calculate surface states in tight-binding models. What I don't really understand is what ...
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1answer
200 views

Symmetry of spectrum of tight binding model with quasiperiodic potential

In the Aubry-André model, a tight binding model with nearest neighor hopping and a cosine-like potential $\lambda_n = \lambda \cos(2\pi \beta n)$ (where $n$ is the lattice site, $\lambda$ is the ...
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1answer
364 views

explicit representation of creation/annihilation operators & its fourier transform (matrix form) (tight-binding hamiltonian, graphene)

While I'm a mathematician/computer-scientist myself, I've some problems trying to understand some paper about the electronic states of graphene nanoribbons modelled by the tight-binding Hamiltonian (...
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372 views

Position and type of carrier in 1st Brillouin zone

Provided the following dispersion relation for a band in a 2D crystal using the tight binding model: E(k)=$E_0-2t_1cos(k_xa_1)-2t_2cos(k_ya_2)$ How can I identify the carrier types and their ...
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1answer
471 views

Getting tight binding density of states more accurately

I calculated numerically the density of states (DoS) for the 3-D tightbinding dispersion $\epsilon(k_x,k_y,k_z)=-2t\,(\cos k_x + \cos k_y + \cos k_z)$ and obtained the following plot [$t=1$ has been ...
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1answer
184 views

Terms $\hat{c}^{\dagger}_{i\uparrow} \hat{c}^\phantom{\dagger}_{i+1\downarrow}+\text{h.c.}$ in tight-binding hamiltonians

The basic tight-binding hamiltonian consists of terms of the form $$\hat{c}^{\dagger}_{i\uparrow} \hat{c}^\phantom{\dagger}_{i+1\uparrow}+\text{h.c.}$$ (where $\text{h.c.}$ denotes the adjoint of the ...
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Tight-binding model of diatomic solid

I am working on a problem based on Problem 1.6 in Gerald D. Mahan - Many Particle Physics, 3rd edition. The problem: Consider a tight-binding solid which has alternating atoms of type A and B. The ...
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1answer
416 views

What holds metal atoms together? And what accounts for the strength of metallic bonds?

From the wikipedia page for metallic bonding, I've noticed that there seem to be a few things at play: (1) the delocalization of electrons, and (2) the fact that there are a far larger number of ...
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2answers
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How can localization of electrons in a solid be defined in a basis-independent way?

In the tight binding model, it's said that in a certain limit, we can regard electrons in the solid as localized to individual atoms. This statement shows up in most introductory condensed matter ...
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341 views

Which book do you recommend for extensive review of tight binding model in second quantization formalism?

I need a book that explains the tight binding model in second quantization formalism. for example by telling the meaning of the eigenfunctions of kernel of Hamiltonian in $k$ space. also explains $2$ ...
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1answer
322 views

Have edge modes of the SSH model (or Kitaev Chain) been observed?

I am putting together a presentation on topological phase transitions in 1D tight binding models for a course in Solid State Physics, and while I have found many sources for theoretical descriptions I ...
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1answer
189 views

Diagonalizable matrix tight-binding model

Given the following model for a particle on a lattice: $$ \hat{H} = -t \sum_j[\left|\ j-2\right\rangle \left\langle j \right|+ \left|j+2\right\rangle \left\langle j \right|]$$ We introduce vectors $\...
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Finding the lowest energies of electrons in a box with a delta separator with a tight-binding model

I am asked to find the two lowest energies of a 1D situation with two electrons in two adjacent $0$-potential wells of width $L$ with infinitely high barriers and a 'coupling potential' of $V(z)=\...
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1answer
804 views

Slater-Koster parameter

When we use Slater-Koster parameter for tight binding model, we represent hopping integral using direction cosine (l,m,n) and bond intergral $V$. For example, the energy between s and px orbital is ...
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1answer
730 views

Tight binding model

The origin of the phrase "nearly free" in the nearly free electron model comes from the fact that we introduce a small periodic potential in a metal lattice as a perturbation to free electrons, so ...
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203 views

Project a Bloch Hamiltonian to subspace

I have a Bloch Hamiltonian H(k) which describe a triangular lattice with six atoms inside one unite cell. This lattice is nothing but group six atoms instead of two in a honeycomb lattice. So as you ...
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356 views

Hamiltonian with periodic potential in second quantization

I'm working with the following Hamiltonian $$\hat{H}=\int\mathrm{d}\mathbf{x}\sum_{\sigma\in\left\lbrace\uparrow,\downarrow\right\rbrace}\hat{\psi}_\sigma^\dagger(\mathbf{x})\left[-\frac{\hbar^2\...
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Total energy in tight binding methods

How to calculate total energy in tight binding methods? The total energy is calculated as the sum of the eigenvalues of the occupied state?
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1answer
326 views

How to convert a Hamiltonian for the tight-binding model in silicene into $k$-space?

I am trying to convert the Hamiltonian from the paper "A topological insulator and helical zero mode in silicene under an inhomogeneous electric field" (also on arXiv) into $k$-space. $$ \begin{array}{...
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415 views

Tight binding Hamiltonian in the k-space

I want to find the band structure of this 2 dimensional lattice which isn't completely flat: Using a tight binding model.And take unit cells as they are shown in the figure. And assuming that each ...
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Derivation of effective mass equation in carbon nanotubes

I am trying to reproduce the calculations in the paper here by Ando and Nakanishi and am already stuck on equation 1. It is stated that in the vicinity of $\epsilon=0$ the amplitude of the ...
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1answer
347 views

Paradox in topological phase of SSH model

Consider the SSH model, i.e. the dimerized tight-binding model with Hamiltonian $$H = \sum_i (t+\delta t) c^\dagger_{Ai} c_{Bi} + (t-\delta t) c_{A(i+1)}^\dagger c_{Bi} + \text{h.c.}.$$ This describes ...