Questions tagged [tight-binding]

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Does particle-hole symmetry always imply half-filling and real correlations $\langle c^\dagger_n c_{n+1} \rangle$?

Suppose we had a lattice Hamiltonian $H$ which was symmetric under the particle-hole transformation $$ c_n \mapsto U^\dagger c_nU=(-1)^nc^\dagger _n$$ such that $[H,U] = 0$, where $c_n$ are Fermionic ...
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Un-equal time correlation via non-interacting tight-binding Hamiltonian

Let's assume we have a model, which is initially defined by the tight-binding Hamiltonian with a random on-site energy $f_n$, as follows: $$H^i=-J\sum_n^{L-1}\left(a_n^\dagger a_{n+1}+h.c\right)+\...
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8 votes
1 answer
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Alternating Tight Binding Hamiltonian

The alternating Hamiltonian may be written as: $$H = t \sum_{n} (-1)^{n} \left[c^{\dagger}_{n+1}c_{n} + c^{\dagger}_{n}c_{n+1} \right] \; \; .$$ I wanted to know the energy dispersion for this system, ...
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Tight-Binding method and orthogonality of Bloch functions

Tight binding summary When computing the electronic bands of a crystal in the tight-Binding approximation, the standard way to do it is to construct Bloch solution as $$ \Psi_n(\vec{r}, \vec{k}) = \...
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-2 votes
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How to understand the tight binding chain?

I am wondering how to extend the mono-/di-atomic chain models here, say $Q_{diatomic}$ to describe behaviors of electrons in solids. (References: https://www.physics.rutgers.edu/grad/601/CM601/...
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1 vote
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Number density equation in BCS Theory

In this paper, The author has given two-equation $$\frac{1}{U}=\sum_k \frac{1}{E(k)}\tanh\left(\frac{\beta E(k)}{2}\right)$$ and $$2\rho-1=-\sum_k \frac{\epsilon(k)}{E(k)}\tanh\left(\frac{\beta E(k)}{...
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1 vote
1 answer
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Questionable Taylor expansion for Peierls substitution

In this paper, on page 3, the authors go from the tight binding model w the Peierls substitution $$ H = \sum_{i,j} \sum_{a,b} t_{a,b} \exp\left(i \int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr'_{\mu} ...
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3 votes
1 answer
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What's a charge density wave?

I'm reading chapter 2 of Condensed Matter Field Theory by Alexander Altland, Ben D. Simons, Section on Interacting fermions in one dimension. From what I understood, they considered the system of ...
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Matrix components of energy of simple cubic crystals

Im working through the derivation of the paper by Slater and Koster (1954) http://users.wfu.edu/natalie/s15phy752/lecturenote/SlaterKoster.PhysRev.94.1498.pdf Im having trouble working through table 2....
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2 votes
1 answer
84 views

What is the meaning of this wave function?

In these notes here the tight binding model for graphene is worked out. The tight Binding Hamiltonian is the usual: $$H=-t\sum_{\langle i,j\rangle}(a_{i}^{\dagger}b_{j}+h.c.)$$ where two different ...
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How to calculate surface states in Weyl semimetals?

I'm reading an article https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235127. Fig. 2 in this article shows band structures calculated from Eq. (9), (13), (14), (15), and (16). For example, ...
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How to fit Tight-Binding parameters to DFT results in a general situation?

I've been searching for an answer to this for several days and couldn't find what I need. I'm explaining myself: I got the Bandstructure of a MoS2 monolayer using DFT (PBE functionals) and I wanted to ...
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Evaluating a commutation relation - (Mahan's book)

I am trying to replicate one of the equations from Mahan's Many-body theory book. First the Hamiltonian $H$ is defined as: $$ H = \sum_i h_i \quad;\quad \text{where} \quad h_i=\frac{t}{2}\sum_{a}[c_{...
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Reference for empirical Tight-binding Hamiltonian of spds* vs sps* (ZB)

Is there a clear reference article/note for the 20X20 Hamiltonian matrix of the spds* Zinc Blende system similar to the sps* reference in [1] Table (A) of Vogl P, Hjalmarson HP, Dow JD. A Semi-...
2 votes
1 answer
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Calculating thermal average of an observable in quantum mechanics [closed]

Single-band Hamiltonian: If we have a Hamiltonian given as $$ \hat H_s = \sum_k \mathcal{E}_k c_k^\dagger c_k $$ then the thermal average of operator $c_k^\dagger c_k$ is $$ \langle c_k^\dagger c_k\...
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2 votes
0 answers
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Sign in Peierls substitution

A method often used to couple a lattice tight binding model to a magnetic field is the Peierls substitution, whereby one changes all hopping elements (schematically) as $t_{ij}\mapsto t_{ij}\exp(\...
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Graphene energy bands at Dirac points

I know that the energy bands in Graphene (possible energies that an electron can take in the material) are given by : $E_{\pm}(k)=\pm t\sqrt{3+f(k)}$ where $f(k)=2\cos(\sqrt{3}k_{y}a)+4\cos(\frac{3}{...
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4 votes
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Is the continuum limit equivalent to the low-energy limit?

It is frequently stated that the continuum limit of a lattice model is equivalent to the low-energy limit, e.g. here, here and section IIB of this. I do not know how to show this for myself. Take for ...
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2 votes
1 answer
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How to unfold electronic bands? How do the reduced/extended schemes work? Confusion regarding the quasi-momentum

One of the simplest models for electronic band structures is a tight binding model on a one dimensional chain with spacing $a$, one atom per cell and interactions only for nearest neighbours. This ...
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1 answer
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Tight binding hamiltonian with spin

I'm looking at the tight binding Hamiltonian which describes non interacting fermions on a bravis lattice, and has the form (in real space): $H=-t\sum_{<ij>,\sigma}(c^{\dagger}_{i\sigma}c_{j\...
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Assumptions of tight binding model

I am referring to chapter 10 of Ashcroft & Mermin's Solid State Physics text (from the 2d edition): In developing the tight-binding approximation. we assume that in the vicinity of each lattice ...
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2 votes
1 answer
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What is the difference between lattice models and tight-binding simulations?

In condensed-matter physics, people use different methods to solve the many-particle Schrödinger equation. I was wondering about two of those methods, the lattice model and tight-binding simulation. ...
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0 votes
0 answers
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Nuclear binding and internal energy [duplicate]

A section of some article: When you cool a body at rest its internal energy decreases. Since energy is related to mass, here $E=m_0c^2$, where $m_0$ is the rest mass, the rest mass of the body ...
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3 votes
1 answer
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Why aren't the eigenvectors of a tight-binding Hamiltonian periodic?

I try to calculate the Berry connection for a simple graphene model and stumbled across the following question. Suppose I have a tight binding Hamiltonian (further details here or here): $$H = \begin{...
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2 votes
1 answer
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Tight-binding model for decorated square lattice

How do I go about determining the tight-binding Hamiltonian for the crystal structure below? I have identified the primitive lattice vectors $\mathbf{a}_1=(a,0)$ and $\mathbf{a}_2=(0,a)$ for lattice ...
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0 votes
0 answers
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Pair correlation function in a 1D tight binding chain

I want to calculate the pair correlation function in a non-interacting Fermi system which is defined in "Quantum Theory of the Electron Liquid" by Gabriele Giuliani and Giovanni Vignale as: $...
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2 votes
1 answer
99 views

Complex energies of non-reciprocally coupled chain with hoppings of equal absolute value

In the Hatano-Nelson chain (i.e. the simplest 1D tight-binding model with nonreciprocal hopping) for positive hoppings $t_{1,2}>0$ you get a purely real spectrum. However, as soon as you change the ...
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1 vote
1 answer
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Fermionic occupation for an inhomogeneous tight-binding model

The model Consider the simple one-dimensional fermionic tight-binding chain of $N$ sites with inhomogenous hopping couplings $t_n$: $$ H = - \sum_n t_n c^\dagger_{n+1} c_n + \text{h.c.} \equiv \sum_{n,...
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1 vote
1 answer
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Peierls substitution, periodic boundary condition, gauge invariance

I am considering a tight-binding model in a magnetic field, and studying the Peierls substitution $$t_{jk} \to t_{jk} e^{i\frac{q}{\hbar}∫_j^k \vec{A} \cdot d \vec{r}}$$ In some papers, such as this ...
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Wave-function (in real space) of electron in Graphene nano-ribbons

I was trying to solve the tight binding model of Graphene nano-strip in the zig-zag configuration. It looks something like this: It has a very beautiful band structure. In order to calculate the band ...
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6 votes
2 answers
395 views

Numerically transforming Hamiltonian into $k$-space

A rather computational question: Suppose you have a very simple tight-binding Hamiltonian in matrix form, i.e. for example something like this for a 1D chain with open ends: $$H =\begin{bmatrix} 0 ...
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Greens function of tight-binding chain (literature request)

The retarted one-particle Greens function $G = \frac{1}{E-H+i\epsilon}$ of a tight-binding chain: $$H = -J \sum_n c_{n}^\dagger c_{n+1} + c_{n+1}^\dagger c_{n}$$ can easily be evaluated using contour ...
3 votes
2 answers
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Why can $\hbar q_x$ and $\hbar q_y$ be replaced by $\hat{p}_x=-i\hbar\frac{\partial}{\partial x}$ and $\hat{p}_y=-i\hbar\frac{\partial}{\partial y}$?

It is written in the book The Physics of Graphene (Page 10 and 17) that when the intervalley scattering is neglected, we can make the following substitution in the Hamiltonian of the graphene when ...
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Pseudo-Hamiltonian of atom

Consider the pseudo-Hamiltonian of atom i, It is said that "the more the states are localized on the atom and the less they overlap with the neighbouring orbitals, the more this formalism is ...
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2 votes
1 answer
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Group velocity (open vs periodic boundary conditions) [duplicate]

I'm trying to understand the meaning of the group velocity for Bloch electrons given by $$ \mathbf{v}=\frac{1}{\hbar}\frac{\partial E(\mathbf{k})}{\partial \mathbf{k}} $$ where $E(\mathbf{k})$ is the ...
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0 answers
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Particle current operator in tight binding

For general non-interacting Hamiltonian $H = \frac{-\hbar^2}{2m}\int dr \Psi_r^\dagger\nabla_r^2\Psi_r$, the particle current operator $J$ can be derived using continuety equation $\nabla_r\cdot J = -\...
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Position of the Wannier center in tight-binding model

With the Fourier transform of the annihilation and creation operators, $$c_{m,n} = \frac{1}{\sqrt{N}} \sum_{k_x} \sum_{k_y} c_{kx,ky} e^{-i\mathbf{k}\cdot \mathbf{r}_{m,n}} \quad\text{and}\quad c^{\...
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2 votes
1 answer
66 views

Hamiltonian density of a spin Hamiltonian

A many-body Hamiltonian in second quantization is written as $$ H = \int d\vec r \Psi^\dagger_{\vec r} H_1 (r) \Psi_{\vec r} $$ where $H_1(r)$ is one-body Hamiltonian. For example, for a non-...
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Constructing an operator from k.p hamiltonian

I have a question regarding to how to construct an operator from k.p hamiltonian. May be there are some problems in my understanding, I hope you can point me out and correct my description if I made ...
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2 votes
0 answers
65 views

Line integral in Peierls substitution

I'm trying to understand the reasoning behind Peierls substitution. The final result seems to be simply replacing the hopping elements $$t_{ij} \to t_{ij} e^{i \frac{q}{\hbar} \int_i^j \vec{A} \cdot d ...
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2 votes
1 answer
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Expanding the Graphene Hamiltonian near Dirac points upto second order term

I was trying to solve the Graphene Hamiltonian near the Dirac points upto the second order term for the nearest neighborhood points. So expanding the function near the Dirac Point, we get$$g(K+q)=\...
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How to calculate spin texture in $k$-space?

I have a triangular lattice model. In $k$-space, it is written as: $$ H = \sum_k\sum_\sigma \Psi_{k\sigma}^+ h_k\Psi_{k\sigma} $$ where $\Psi_{k\sigma} = [a_{k\uparrow}, b_{k\uparrow}, c_{k\uparrow},...
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1 vote
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Given a crystal with mirror symmetry along a lattice plane, how to find the correspond plane in first Brillouin zone

Given a crystal with mirror symmetry along a lattice plane, how to find the correspond invariant plane in first Brillouin zone?
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1 answer
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Hermiticity of Slater-Koster hamiltonian

From the paper written by Slater and Koster, some of the tight binding Hamiltonian seemed to change sign under a sign flip. For example $$E_{s,x}=l(sp\sigma)$$ Suppose we have two atoms (A and B), ...
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1 vote
1 answer
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Quantum description of the tight-bonding model

In an introduction to the tight-bonding model in Condensed Matter Physics, I came across this statement: When two atoms go towards one another, the electrons in the atoms start to build existence of ...
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  • 495
1 vote
1 answer
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Bands versus reciprocal lattice vectors in the Bloch basis

In Bloch theory, using the reduced zone scheme, we can index the states of a crystal by either $k,G$ where $k$ is in the first Brillouin zone and $G$ is a reciprocal lattice vector, or $n,k$ where $k$ ...
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4 votes
1 answer
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Thermal averages with opposite sign in the exponential

I've recently seen an example where a thermal average was carried using a plus sign instead of the usual minus sign inside the exponential. $$\langle \mu \rangle = \frac{1}{Z} tr(e^{\beta H }\mu) \...
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  • 155
3 votes
1 answer
283 views

Continuum limit of tight-binding models

Suppose I had a simple $1D$ tight-binding Hamiltonian $$ H = -t \sum_i a^\dagger_n a_{n +1} + \text{h.c.}$$ with $N$ sites and lattice spacing $a$. This Hamiltonian can be diagonalised with a discrete ...
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Derivation of Lyapunov exponent in 1D disordered system

What I am considering is a tight-binding model of 1D disordered system. According to the literature (page 1500, equation (60)), Lyapunov exponent $\gamma$ is calculated as follows which I am not ...
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Meaning of the discrete Fourier transform in condensed matter

I'm in a CMP course now and I think I'm taking for granted what it means for a creation/annihilation operator to be Fourier transformed. I understand what a Fourier transform is for some function, but ...
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