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Questions tagged [lattice-model]

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

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Einstein solid: one or three dimensional quantum harmonic oscillator?

The Einstein model for solids assumes all atoms vibrate with the same frequency $\omega$, each atom being modeled as a quantum harmonic oscillator. The thing is: solids are three-dimensional objects, ...
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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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In Hamiltonian lattice QFT, does low energy imply low momentum?

In relativistic quantum field theory (QFT), the spectrum of the total energy and total momentum operators is supposed to be restricted to the forward light-cone (the relativistic spectrum condition), ...
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What is the meaning of the notation $\langle a_1, \ldots, a_n \mid X_i(u) \mid a_1', \ldots, a_n' \rangle$? [closed]

I am from the math department and reading Belavin & Gebner's On the Algebraic Approach to Solvable Lattice Models. I am trying to understand the left-hand side of Equation (2.2) on page 4. What ...
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Formula for Boltzmann weight

I am reading the paper. I am trying to understand the formula (2.1): \begin{align} \lim_{u \to i \infty} g(u) \omega \left( \begin{matrix} a & b \\ c & d \end{matrix} \mid u \right) = C_{c,d} ...
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Wigner Seitz cell

While searching the difference between primitive cell and unit cell I have seen that "Primitive unit cells contain only one lattice point, which is made up from the lattice points at each of the ...
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56 views

Discretization of Hamiltonian with first derivative

In a particular 1D system, the Hamiltonian can be writen as $$H=\mathrm{i}\left(f(r)\frac{\partial}{\partial r}+\frac{1}{2}f'(r)\right)\; ,$$ wher $\mathrm{i}$ is the imaginary unit, and $f(r)$ is a ...
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Why can the spin operator be written as a product of fermions?

I was studying the Hubbard model, where we define the spin operator $\vec{S} = \frac{1}{2} c^\dagger \vec{\sigma} c$, where the creation and annihilation operators are both vectors of the form $c^\...
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A computation problem on reciprocal lattice

I am reading David Tong’s lecture notes on Application of Quantum Mechanics. My confusion is about the following paragraph: Consider a function $f(\vec{x}) $, suppose $f(\vec{x})=f(\vec{x}+\vec{r})$...
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Drawing reciprocal lattice structures

I am currently trying to understand why and how reciprocal lattice relates to the diffraction plane, so I did some research on Wiki about reciprocal lattice, but I seem to be stuck as I would like to ...
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How does one construct the $n$th Brillouin Zone in a rectangular lattice geometrically?

What are the general rules of constructing the Brillouin Zone in a rectangular lattice geometrically? While the construction of the 1st and 2nd Brillouin zone is rather simple, starting to construct ...
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How to find groundstate energy of a simple Hamiltonian at $N/L$-filling using Jordan-Wigner (JW) transformation?

$\underline{\textbf{Model:}}$ Let we have the $t-V$ model for spinless fermions on a 1D lattice, which is defined in second quantization operators as follows: $$H_1 = -t\sum_i \big(c_i^\dagger c_{i+...
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Collimated Beam through Palladium Powder to identify the lattice structure

I came across the following Problem: A collimated beam with a wavelength of $0.162 \ [nm]$ is incident upon a powdered sample of simple cubic metal palladium. Now, five angles are given, which ...
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1answer
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Different ways to write t-V model Hamiltonian

The $t-V$ model is usually written as: $$H = -t\sum_{i}^{L}c_i^\dagger c_{i+1} + c_{i+1}^\dagger c_i + V\sum_i n_i n_{i+1}$$ where $c_i^\dagger(c_i)$ are creation (annahilation) operators and $n_i$ is ...
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What's wrong with lattice quantum gravity?

Assume one can write the metric field on a lattice, so on each lattice point one has a value of $g^{\mu\nu}$. Similar to the way lattice QCD is formulated. Then later taking the distance between ...
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Lattice model realization of $SU(2)$ WZW model at level $k$?

Is there any lattice model realization of the following model: $c=1$ boson at the self-dual radius, or the $SU(2)$ WZW model at $k=1$. This is a question inspired by: Orbifolds of the $c =1$ ...
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Temperature-dependence of quark potential in Abelian lattice gauge theory

I am working with Kapusta's "Finite-Temperature Field Theory" textbook, and am working through the first part of chapter 10. When building the correlator of the two quarks a distance $R$ apart in the ...
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Questions on how Wilson loops relate to field & charge conservation, and lattice QFT

The path-ordered exponential from which the Wilson loop is traced is, crudely, $$ \prod (I+ A_\alpha dx^\alpha) = \mathcal{P}\,\mathrm{exp}(i \oint A_\alpha dx^\alpha )$$ which returns a matrix $\...
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Why does Bell's field theory model need to be stochastic on a space-time lattice?

In "Beables for quantum field theory", John Bell has presented a realistic interpretation of any fermionic quantum field theory, along the pilot-wave ideas. This model is formulated on a spatial lattice ...
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Negative Miller indices and parallel planes

The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. https://en.m.wikipedia.org/wiki/Miller_index Does this mean, that parallel planes are generally ...
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Possible ground state wavefunctions of anti-ferromagnetic Heisenberg spin chains

What are the ground state wavefunctions of the anti-ferromagnetic (AFM) Heisenberg spin chains? Is that which of the following? $$| ↑↓↑↓ · · · ↑↓>$$ or $$| ↓↑↓↑ . . . ↓↑>$$ or $$| ↑↓↑↓ · · · ↑↓...
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Fractal discrete networks approximations and Poincaré invariance?

In the link below one may find interesting paper by Sabine Hossenfelder, about finite networks symmetry due to Poincaré group. According to this thesis locally finite networks cannot be Poincaré ...
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What is the meaning of propagator in the context of lattice theory?

Say in $1+1D$ free fermion theory, it is easy to calculate the propagator in the (effective) field theory to be $$\langle \psi^\dagger(z)\psi(z')\rangle = \frac{1}{2\pi}\frac{1}{z-z'}$$ (in the ...
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Perturbative vs Wilsonian renormalization in 2D

In 2D a scalar field is dimensionless so terms in the Lagrangian $\phi^n$ of arbitrary power are renormalizable (indeed we can even have two derivatives). This has two consequences that seem to be in ...
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One dimensional paramagnet

I have a lattice model consisting of $N$ spins $s_{j}$ which can take the values $s_{j}=\pm1$. The spins are considered to be non interacting. The probability for a spin spin to be 1 is p and the ...
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Non-vanishing matrix elements in QFT after discretization

In QFT, introducing the lattice size $L$ implies that the momentum of plane wave solutions is discretized as $$\vec{p}_\vec{n} = \dfrac{2\pi}{L}\vec{n}\quad,\qquad \vec{n}\in\mathbb{Z}^D$$ I would ...
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Constructing PEPS representation of an arbitrary quantum state

Given a quantum state we can construct its MPS (Matrix Product State) representation by doing a series of singular value decompositions. Given the freedom to choose arbitrary bond dimensions the MPS ...
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Quantum phase transitions in a finite lattice

Sachdev begins his book on Quantum Phase Transitions by asserting that, for a system on a finite lattice, the ground state energy of a Hamiltonian H(g) (where g is some coupling) is a smooth, analytic ...
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Lattice QFT of the Jones Polynomial

Start with a gauge theory with Chern-Simons action $$ S[A] = \frac{k}{4 \pi} \text{Tr} \intop_{M} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right) $$ and a Wilson loop observable in the ...
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What is an intuitive explanation of wave ordering vector $Q$? (Pierls Instability)

How does the wave ordering vector $Q$ order a CDW? I saw this vector while studying the following system. We have a system with $N$ sites and $N/2$ spinless fermions and system is in the fully ...
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Comparing momentum cutoff and lattice regularization in Quantum Field Theory

Usually, it is heuristic to say that we can understand a QFT with a momentum cutoff $|k|<\Lambda$ by imagining that the system is living on a lattice. I would like to ask: (1) Is there any ...
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What is the effect of including additional representations in the action of a lattice gauge theory?

I'm reading Introduction to Quantum Fields on a Lattice by Jan Smit. When introducing the lattice gauge-field action as a sum over plaquettes, Smit says that in general the action should include a sum ...
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How does the Dirac equation follow from the algebra of order and disorder operators in the Ising model?

In his book "Statistical Field Theory", Mussardo derives the Dirac equation for the Majorana fermions in the scaling limit of the Ising model. He does this by defining order operators (the Pauli ...
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Defining the renormalized coupling constant in a lattice phi fourth simulation

I'm currently studying a scalar quantum field theory with a $\lambda\phi^4$ interaction (commonly referred to as $\phi^4$ theory. To study this theory non perturbatively I've written a program in ...
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Determining bound state masses from a lattice $\phi^4$ simulation

I've recently written a program in python that simulates the $\phi$ to the fourth scalar quantum field theory in a 4 dimensional euclidean spacetime. The lagrangian for this theory is that of a free ...
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Is simulating quantum field theoretical equations in the continuous limit the same as simulating really quantum continuous processes?

At the end of this article: https://hal.archives-ouvertes.fr/hal-00542373/document it is said that: "For example, several works simulate quantum field theoretical equations in the continuous limit of ...
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Peierls substition in square lattice

I want to calculate the current operator of a model which is a 2d square lattice model. I first start with a tigth binding model of this system and make a Peierls substition to it and if I expand the ...
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Computation of String Tension in Lattice QCD

There is a quantity called String Tension in lattice QCD calculation. How is this quantity (String Tension) defined and computed in Lattice QCD? Are there some useful formulas to define it both in ...
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Why is $x(t) = \frac {x_n + x_{n+1}}{2}$ when proving the Euler-Lagrange Equation?

When proving the Euler-Lagrange Equation for a single degree of freedom, we replace continuous time with stroboscopic time. We replace $$\dot x=\frac{x_{n+1}-x_n}{\Delta t}$$ but why do we replace $$...
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Zero interaction limit of one-dimensional lattice of spinless fermions

I am working on reflection of energy at a boundary that has a one-dimensional Wigner crystal on one side and non-interacting electron gas on the other side. I naively assumed that a single Hamiltonian ...
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What is the physical meaning of the Fourier transform of the creation/annhilation operators in the nearest neighbour model?

It is possible to take the Fourier transform of the creation operator as $$a_k=\frac{1}{\sqrt N}\sum_n e^{-ik\cdot n}a_n$$ with $k=2\pi l/N$ and $l=\{-N/2+1, -N/2 +2 ... N/2\}$ but I am really ...
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1answer
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How to describe charge density waves in k-space?

I am reading this article about variational ground state wavefunction. In equation 10 they said that charge density wave (CDW) for spinless fermionic system at half-filling can be written as following:...
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How to calculate the lattice constant of NaCl using chemical information? [closed]

More specifically, how would I show that: $a_0=\sqrt[3]{\frac{4M}{N_A\rho}}$ where $a_0$ is the lattice constant, M is the molecular mass of NaCl, $\rho$ is the density of NaCl, and $N_A$ is ...
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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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What is effect of choice of unit cell on 1st Brillouin zone?

Let's say we have a 1D lattice with $a$ as lattice constant: and hopping strength between two nearest neighbors (NN) is $t$. We can choose unit cell as one lattice site per unit cell. in k-space $k$...
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114 views

Basis transformation of eigenvectors of Hamiltonian written in different basis

There is a very famous topological model, Su-Schrieffer Heeger (SSH) model, according to which the hopping strength between even and odd sites is different. i.e. Now, there are two ways one can write ...
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Peskin's duality in XY model (Mandelstam-'t Hooft duality in abelian lattice models)

I am studying the old paper by Peskin (1978): Mandelstam-'t Hooft duality in abelian lattice models (https://doi.org/10.1016/0003-4916(78)90252-X). However, I am confused about some details of ...
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Schrödinger equation on a lattice

Consider the following problem: The given Hamiltonian is: $$ H=-J\sum_{m,n}\left( e^{-i\phi n}a^{\dagger}_{m+1,n}a_{m,n}+a^{\dagger}_{m,n+1}a_{m,n} +h.c.\right)$$ Inserting this Hamiltonian in the ...
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Action of Operator in lattice model

i'm dealing with a paper about square lattice in a magnetic field. There are defined these operators: $ T_{x}=\sum_{m,n}a^{\dagger}_{m+1,n}a_{m,n}e^{i\theta ^{x}_{m,n}} $ and $T_{x}=\sum_{m,n}a^{\...
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Mean field theory Vs Gaussian Approximation?

I am getting confused about the distinction between Mean-field theory (MFT) and the Gaussian approximation (GA). I have being told on a number of occasions (in the context of the Ising model) that the ...