Questions tagged [tight-binding]

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How does the Hubbard hamiltonian change when considering a Peierls distortion (bipartite lattice)?

The following is the Hubbard contribution to the hamiltonian in the Hubbard-Tight Binding model. $$H_{hubbard}=U \sum_i n_{i \uparrow}n_{i\downarrow}$$ where $n_{i \sigma}=c_{i\sigma}^\dagger c_{i\...
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How to calculate optical conductivity from numerical eigenstates of tight-binding model?

Let's say we have a 1D spatially inhomogeneous tight-binding model that does not have momentum as a good quantum number. We can numerically diagonalize it to get the spectrum and eigenstates. But how ...
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How many bands does one expect to get in the band diagram for a 4 atoms, 12 total orbitals and 15 valence electrons in the Tigh Binding Aproximation?

The question, as the title say, refers to understand how many bands do we expect to get in the band diagram for a solid with 4 atoms, 12 total orbitals and 15 valence electrons in the Tigh Binding ...
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44 views

Why is there a need to add the complex conjugate in the tight binding hamiltonian?

So we start with the following hamiltonian describing non-interacting free fermions: $$ \hat{H}_{\text{free}} = \sum_{i,j,\sigma}\tilde{t}_{ij} \hat{c}_{i\sigma}^\dagger\hat{c}_{j\sigma}.$$ Then ...
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Harper-Hofstadter model in symmetric gauge

If I have l a square lattice, with the total flux = $\pi$, I can work in the symmetric gauge, which will have my vector potential be $A = \frac{\pi}{2}(-y,x)$. In a tight-binding model with Peirels ...
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1answer
23 views

Why hopping amplitude with no negative sign?

I'm learning SSH model now. I notice people use tight-binding model of this form, $$H=t\sum_{<i,j>} c_i^†c_j+\mathrm{H.c}$$ where $t>0$ in Lecture 1 : 1-d SSH model, or A Short Course on ...
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48 views

Tight Binding Hamiltonian for graphene

The TB Hamiltonian for the tetragonal lattice is $ \hat H_0 = -J\sum_{m,n} (\hat a_{m+1,n}^\dagger \hat a_{m,n}+\hat a_{m,n}^\dagger \hat a_{m,n+1}+h.c.) $ How can this be derived for the hexagonal ...
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42 views

Calculating topological invariants under different conventions of tight-binding models

There are two widely used conventions to construct the Bloch-like basis in a tight-binding model [1]. Convention I: $$ \psi_\mathbf{k}=\frac{1}{\sqrt{N}}\sum_{\mathbf{R},j}c_j(\mathbf{k})e^{i\mathbf{...
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27 views

Wave Function of the Tight Binding Model

The Gutzwiller wavefunction, i talked in brief in this other question, is introduced to compute the expectation value of the Hubbard Hamiltonian. It is composed by a uncorrelated Slater determinant (...
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What is the tight binding Hamiltonian for Graphene in terms of the Pauli Matrices?

I have been unable to find an expression for the tight binding Hamiltonian of Graphene in terms of the Pauli Matrices. Please share any reference available. Thank You
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Slater-Koster parameter for SC

I want to use SK method to calculate the interatomic matrix element for $P_{y}$ and $P_{z}$ for the nearest, second and third neighbors in simple cubic crystal. based on paper written by Slater-Koster ...
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42 views

Tight binding hopping

The definition of the tight-binding model is that the electrons are sited on certain points(the Wannier centers) and that they can hop in linear paths, when we refer to sites that belong to the ...
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37 views

Peierls phase in graphene

In the introduction of the paper presented here, a derivation of the Peierls phase is presented, using a Wannier base of eigenfunctions and the Kohn-Sham Hamiltonian. After it symbolises the hopping ...
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26 views

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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44 views

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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91 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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90 views

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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106 views

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
67 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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21 views

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
61 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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148 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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86 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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74 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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92 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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237 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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102 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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608 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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1answer
808 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
28 views

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
563 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
367 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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238 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
346 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
244 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
454 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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1answer
547 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
215 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
566 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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201 views

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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390 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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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 ...