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Using particle-hole symmetry of the Hubbard model to study the model at different densities

In Condensed Matter Field Theory by Altland and Simons, they state that the Hubbard Hamiltonian $$ H = \sum_{\text{nearest neighbors } ij \text{ and spin } \sigma} a^\dagger_{i\sigma} a_{j\sigma} + U \...
zeroknowledgeprover's user avatar
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How do you see that the Wannier functions are localized on ion sites?

In Condensed Matter Field Theory by Altland and Simons, when discussing the tight-binding approximation for a lattice system with a periodic potential, they define the Wannier states as follows: $$ |\...
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Equivalence of Wannier Functions and Atomic Orbital Wave Functions in the Tight Binding Approximation

In tight binding approximation, what I've learned is that we can write the wave function of an electron which satisfies Bloch theorem in lattice as $$ \psi(\mathbf{r})=\sum_{\mathbf{R}_s} e^{i\mathbf{...
Gao Minghao's user avatar
2 votes
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Variables dependency after unitary transformation

Currently working on tight-binding model with external field that induced an extra phase factor, e.g. Peierls phase $e^{i\frac{q}{\hbar}{\int_{\vec{r}_j}^{\vec{r}_i}}\vec{A}\cdot d\vec{r}}$ which will ...
まこと's user avatar
1 vote
1 answer
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Question about electrons in tight binding model?

I was wondering, in the 1D tight-binding derivation, is the expansion of the ket: $$|\Psi\rangle=\sum_n\phi_n|n\rangle,$$ a superposition of all the electrons or only one thus $|n\rangle$ only ...
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Why are zero-modes preserved on turning on coupling in twisted bilayer graphene model?

In the paper https://arxiv.org/abs/1808.05250 on page 3, they talk about how when the parameter $\alpha$ in the Hamiltonian in Eq. (5) equals $0$, we get zero modes at the Dirac points K and K’. This ...
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Why is the first Brillouin zone size of a body center cubic is 4π/a?

According to many online sources that the first Brillouin zone of a body center cubic (bcc) has the shape illustrated below. Along the $k_x$ (or $k_y$, $k_z$, $(100)$...) direction, basically the $\...
physstudent11's user avatar
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Tight binding of monoatomic chain

For the the 1D atomic chain tight binding calculation, if we choose 1 atom per unit cell, the band dispersion is simply: $\epsilon-2t\cos{(ka)}$. However, if we redefine our unit cell as 2 atoms, 3 ...
physstudent11's user avatar
1 vote
1 answer
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Understanding electric conduction in tight binding model

Let's consider a system of free electrons moving in a one dimensional lattice with dispersion $\varepsilon(k) = -2t\cos{ka}$, ($a$ is the lattice spacing and $t$ the hopping amplitude). Let's now ...
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Matrix representation of Slater-Koster parameters

I'm coding a program to calculate electronic bandstructures using the Slater-Koster formalism (I am aware that such programs exist already- this is a pedagogical exercise). I notice that the $p-p$ ...
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How to find the energy bands of a kagome lattice?

I've been trying to solve the tight binding for a kagome lattice. The thing is that I find a cubic equation and have no idea on how to solve it. There's this article though (https://arxiv.org/abs/2310....
Gean Araujo's user avatar
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Must Tight binding model use real space Basis?

To construct a tight binding model, the basis must be real space?
雾岛董香's user avatar
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Renormalization of two-sublattice tight-binding Hamiltonian

I have a generic tight-binding Hamiltonian of the form $$H=\sum_n A c_n^\dagger d_n+Bd_n^\dagger c_n +C c_{n+1}^\dagger d_n + D d_n^\dagger c_{n+1}$$ where $A,B,C,D$ are parameters (hopping amplitudes)...
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What is the difference bewteen atomic orbitals and wannier functions?

They both use the "s, px , py, pz, dxy...." formalism Both of them are in real space. Wannier is orthogonal but atomic orbitals are not... But what's the fundamental difference between them?
雾岛董香's user avatar
3 votes
2 answers
70 views

How does symmetry act on general case fermionic operator?

I am trying to understand symmetries and how they work in condensed matter physics to understand some concepts from topology. In general second-qunatized Hamiltionian can be written in the following ...
IhateDonuts's user avatar
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What's the difference in chemical potentials between nearly free electron model and tight-binding model?

In nearly free electron model, we know that when temperature is zero, the chemical potential $\mu$ is same as the Fermi energy $E_F$, $\mu=E_F$. For a good metal, $E_F$ is roughly $10\mathrm{eV}$, so ...
Houmin Du's user avatar
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How to build up the creation/anihilation operators in Numerical Renormalization group?

Let's define a simple 1D linear tight-binding model (no matter that NRG is trivial here or even fails, this is just an simple example to illustrate my point) $$H = \sum t_j \hat{c}^\dagger_{j}\hat{c}_{...
Qwertuy's user avatar
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Tight binding on a square lattice with three orbitals symmetries

I came across a tricky problem while studying tight binding within the second quantization frame: Consider a square lattice with one atom per unit cell, where each atom has three active hydrogen atom ...
Chris Ze Third's user avatar
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Triangular Tight Binding Model

I have the following Hamiltonian for a triangular tight binding model: $$ H = e^{i\phi} \Big(|2\rangle \langle1| + |3\rangle \langle2| + |1\rangle \langle3| \Big) + e^{-i\phi} \Big(|1\rangle \langle2| ...
Sid's user avatar
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How can I identify momentum eigenstates in a tight binding model with degenerate energy eigenstates?

Summary: I numerically diagonalize a tight binding Hamiltonian to get energy eigenvectors, some of which are degenerate. However, the numerically diagonalized degenerate eigenvectors are not ...
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Orthonormality of Wannier functions [duplicate]

In "Modern Condensed Matter Physics" by Girvin and Yang, a question invites the reader to prove the orthonormality of the Wannier functions. My proof will be complete if $$ \langle n',0\vert ...
CW279's user avatar
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319 views

Dispersion Relation for one dimensional monoatomic lattice in Kronig-Penny model and in Tight-Binding Approximation model

In the tight binding approximation model, we have the dispersion relation for a one-dimensional atomic lattice given as: $$E(k) = E_0 - \alpha - 2\beta \cos(ka)$$ Here, $\alpha$ and $\beta$ are ...
Dinesh Katoch's user avatar
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Confusion about linear combination of atomic orbitals (LCAO)

In an YouTube video, that I used some information from it to work out my homework, Link: https://youtu.be/5vOvjeTRcqM?si=NzIcx_9QTNQ9Nhff the linear combination of atomic orbitals (LCAO) was discussed ...
Anky Physics's user avatar
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1 answer
118 views

Orthogonality in the Tight-binding model

In the linear combination of atomic orbitals and the Tight-binding model we write the wave functions for some k as a linear combination of Bloch states like this $$ \Psi_n(\textbf{r}, \textbf{k}) = \...
Anky Physics's user avatar
1 vote
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22 views

Impact of labeling in a bloch-crystal with orbital basis

If I'm diagonalizing a hamiltonian of electrons in a crystal that is written in the orbital basis, does it matter whether I calculate the matrix element between one atom and another atom (or the image ...
Mikkel Rev's user avatar
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Current Operators on Lattice

Peierls substitution method by taking the functional derivative of Hamiltonian can be used to determine the form of current-operator in continuum model (See Bruus-Flensberg) as well as lattice model. ...
Snpr_Physics's user avatar
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65 views

Signal in hopping term in diatomic chain using the tight binding method

I am currently studying the tight binding method, and while solving solving a problem I came across something I don't understand. There are two atoms A and B, A has only type s orbitals and B has only ...
Ana Branco's user avatar
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1 answer
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How the $-2t\cos(k)$ term appears in the dispersion of the $1D$ tight binding model?

I am trying to derive the tight binding dispersion relation with periodic boundary condition with $N$ lattice sites of the simplest Hamiltonian: \begin{equation} \label{ham} \tag{1} H = -t\sum\...
kal92's user avatar
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Complex values for the dispersion relation obtained through an $s$-band only tight binding model for diamond cubic crystal

Any given atom in a diamond cubic lattice (Like Si or Ge) has four nearest neighbours at at a distance $\sqrt{3}a/4$, being $a$ the lattice constant. The translation vectors to these neighbours can be ...
NeonGabu's user avatar
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1 answer
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How to calculate a momentum space of a semi-finite lattice?

If we have a 2D square lattice of lattice constant a whose $x$ axis has only $N_x$ cells each with one atom and no with spin degeneracy, and periodic boundary conditions on $y$ with $N_y$ cells along ...
chen jiiong's user avatar
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1 answer
122 views

Position space representation of tight-binding model with composite lattice

Consider a composite lattice such as silicon or graphene (with sublattice A and B). We may call the sublattice tight-binding Bloch sums $\left| k, \alpha \right>$ with $\alpha$ being the sublattice ...
Frank Wang's user avatar
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2 answers
75 views

Can we use hybridised orbitals in Tight-Binding Method for calculation energy dispersion?

I am working on a model of a Fe square planar complex with nitrogen and oxygen given below is a monomer of that polymer. While constructing the hamiltonian matrix of this monomer I'm confused as to if ...
Harshdeep Chhabra's user avatar
1 vote
0 answers
15 views

Term simplification in single atomic orbital (LCAO) model

I was following the text of one of our lectures where we derived for a single-orbial $\phi(\vec{r})$ LCAO model for the band-structure of a crystal: $$E_{\vec{k}} = E_a + \frac{\sum_{\vec{R}}e^{i\vec{...
Simon Fromme's user avatar
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29 views

Why is the effective mass of top band just $-1$ times that of the bottom band? [duplicate]

In band theory, the formula for effective mass is $$m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}}$$ and $m_{top}^* = -1 \times m_{bottom}^*$ the effective mass of top band is just -1 times the bottom band. ...
Harshdeep Chhabra's user avatar
1 vote
0 answers
34 views

Why perturbational approach fails to describe the bands?

I'm trying to find the band structure of Graphene using tight-binding for a unit cell with 6 carbon atoms(this is a toy model for my own research). The hamiltonian is as below: $$H= \left[\begin{...
Ali Rayat's user avatar
2 votes
3 answers
106 views

Time evolution of non-interacting field operators [closed]

I learned that for a non-interacting tight-binding system $H=\sum_{n}\varepsilon_na^\dagger_n a_n$, the time evolution of $a_n$ is simply $a_n(t)=e^{-i\varepsilon_nt}a_n$. I tried to prove this: \...
user835469's user avatar
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1 answer
178 views

Can I calculate partial density of states (PDOS) using tight binding approximation? [closed]

I am using tight binding approximation for a 2D material by 2*2 Hamiltonian, and I have ploted the density of states correctly. Can I also calculate the partial density of states using tight binding? ...
Mohammad Mortezaei Nobahari's user avatar
0 votes
1 answer
176 views

Berry connection in SSH model

In the SSH model for the 1D case, we get the eigenvectors as $$|(\pm)k>= \begin{pmatrix} \pm e^{-i\phi} \\ 1 \end{pmatrix}$$ where $\phi = tan^{-1}(\frac{wsin(k)}{v+wcos(k)})$. We can calculate the ...
AntMan's user avatar
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1 answer
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Dispersion Relation and Eigenvectors of SSH Model in Tight Binding

Consider a one-dimensional chain of atoms as shown in the figure. Let the spacing between the atoms be $a$. Assume that the onsite energy is the same at each point and is equal to $0$ (without any ...
AntMan's user avatar
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Empirical tight-binding sp$^3$s* band structure of semiconductors

I'm simulating on code the tight-binding sp3s* bandstructure of certain semiconductors, such as GaAs, AlP, InP, ZnSe, etc. with spin-orbit coupling at a temperature of $T = 0$ K but I'm having trouble ...
user9867's user avatar
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0 answers
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Is there any method for folding a Hamiltonian matrix to lower dimension?

I want to solve a tight-binding Hamiltonian which is $6\times6$. I'm only interested in two of the six bands which lie near zero energy at $\vec{k}=(0,\frac{4\pi}{3\sqrt{3}a})$. Is there any way to ...
Ali Rayat's user avatar
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0 answers
37 views

How should I plot a tight-binding dispersion when I use multiple gridpoints in real space?

Consider a simple tightbinding calculation with spacing $a$. I can write down a dispersion relation $$E(k) = 2t - te^{ika} - te^{-ika}$$ Say I am solving a simple tightbinding problem numerically, on ...
DJames's user avatar
  • 411
2 votes
1 answer
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Variational method in the tight binding approach

I'm trying to read Professor David Tong's notes to understand the principles behind the tight-binding model - section 2.3.5 'Deriving the Tight-Binding Model'. He first considers the Hamiltonian of ...
Angry Physicist's user avatar
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0 answers
92 views

Tight binding model, get dispersion relation in crystal

I know the theory of tight binding model but don't know how to apply them in the real lattice
Anchal Kumar Sharma's user avatar
1 vote
0 answers
88 views

What does band inversion mean for occupation?

I am thinking about Topological Insulators at the moment and am not clear about how to understand the occupation of the inversed band. I understand that due to energetic and symmetry reasons the ...
sir_khorneflakes's user avatar
1 vote
0 answers
308 views

Chemical potential $\mu$ controls filling?

If total magnetization of a spin 1/2 system is zero, does it mean that system is at half filling? or chemical potential $\mu=0$? I was trying to show that chemical potential controls filling by taking ...
Barry's user avatar
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1 answer
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Green's functions of disordered tight binding models

A research problem has led me to calculate a Green's function of a tight binding model with both onsite disorder and hopping amplitudes which vary in space. Since so much is known about tight binding ...
miggle's user avatar
  • 759
2 votes
1 answer
294 views

Is one-body tight-binding Hamiltonian always diagonal in Fourier space?

A generic tight-binding Hamiltonian for one-body operators is (assume one state per site) $$ H=\sum_{ij}C_i^\dagger t_{ij} C_j \quad; \quad t_{ij}=\int d\mathbf{r} \phi_i^*(\mathbf{r}) t(\mathbf{r})\...
Sana Ullah's user avatar
3 votes
1 answer
321 views

Second quantization: Hamiltonian in field operators vs Tight binding form

Hamiltonian written in terms of field operators: The kinetic energy (KE) part of Hamiltonian is $$ H=-\frac{\hbar^2}{2m}\int d\mathbf{r} \Psi^\dagger(\mathbf{r}) \nabla^2\Psi(\mathbf{r}) \tag{1} $$ ...
Sana Ullah's user avatar
2 votes
1 answer
206 views

Confusion tight binding model graphene

I think I have a conceptual doubt on the calculation of the matrix elements in momentum space for the tight binding Hamiltonian of graphene. I will break down my question into sections to make it as ...
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