Questions tagged [lattice-model]

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

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Finding eigenvalue and eigenvector of non-Hermitian matrix product operator

Suppose we have a matrix product operator (MPO) $X$ with a periodic boundary, which is not necessarily Hermitian. That is, $$X^{s_1\cdots s_n}_{s^{\prime}_1\cdots s^{\prime}_n}:=\mathrm{Tr}(G_1[s_1,...
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Getting the Bose-Hubbard Hamiltonian from cold atoms

In the famous paper by Dieter Jaksch, it is shown that the usual Hamiltonian for cold bosonic atoms interacting by s-wave scattering (Equation (1) in the paper): $$ \hat{H}=\int d^3 x\hat{\psi}^\...
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Is is cheating to use alpha strong (instead of V_QCD) to argue that there is asymptotic freedom of QCD?

For asymptotic freedom of QCD, people give the argument that alpha_strong decreases at high energy (thus there is freedom at low distance). But if we make the reasoning on the QCD potential, for ...
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Uniform discrete physical theories [closed]

I'm looking for a list of physical theories which are discrete and uniform (in some sense). For example, if space-time has a regular lattice like structure this would be uniform. If possible, are ...
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102 views

Is the QCD potential really monotonic ? How does it prevent two quarks from meson to annihilate?

The QCD potential is made of two terms -(4/3) * alpha_s / r that describes the short distance and the term +k*r that describes the long distance Of course, alpha is a function of energy, so it is ...
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Quantum Monte Carlo Loop Algorithm for quantum spin: why is the freezing graph present in ferromagnetic Ising model?

I study the loop algorithm (Evertz et al). I cannot understand, why the freezing graph type where we have to flip all 4 spins together is not present for the quantum-XY model and the anti-/...
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227 views

Correlation length anisotropy in the 2D Ising model

In the Ising model, the two-spin correlation function is $$ C(\vec{r}) = \langle \sigma_{\vec{r}_0+\vec{r}}\sigma_{\vec{r}_0}\rangle - \langle \sigma_{\vec{r}_0+\vec{r}}\rangle \langle \sigma_{\vec{r}...
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44 views

Spectrum of the momentum operator for free Dirac fermions on a lattice

I am studying lattice field theory and would like to understand the momentum operator for free Dirac fermions on a square lattice. In this case one needs to discretize the momentum operator (which ...
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45 views

Emergence of rotational symmetry on 2D square lattice

On page 74 of David Tong's Statistical Field Theory lecture notes, it is said that $(\partial_1\phi)^2 + (\partial_2\phi)^2 $ respects both $D_8$ (that includes discrete four-dimensional rotation ...
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For what choice of the basis atoms origin the Structure Factor is real?

I have a Cuprite Structure: Now, given that I described this structure as a Simple Cubic lattice with a 6 atoms basis, I have to choose the origin for these basis atoms such that the structure ...
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Symmetry properties of Goldstone mode dispersions

I've recently worked on a lattice model relevant to condensed matter physics. The model features an ordered phase with a broken $U(1)$ symmetry, so I anticipated the presence of a Goldstone mode. One ...
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Question about notation of a Jimbo's paper

I am reading Jimbo's Introduction to Yang-Baxter Equations. And I am confused by the notation he used in the definition: Here he uses $u\in C$ without previously mentioning what is $C$. I guess ...
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56 views

How many atoms are in the primitive unit cell for diamond?

The primitive unit cell for diamond is pictured (the parallelepiped inside the cube). How many atoms are in the unit cell? My first guess is 3.5 and i know this must be wrong as the number of atoms in ...
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What is the lattice vector description of diamonds structure for a conventional fcc unit cell?

I understand how diamond has a conventional fcc unit cell with 4 extra atoms in it's basis. I also know that we can construct a primitive lattice cell using the lattice vectors: $\pmatrix{\frac{1}{2} ...
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Lattice construction of 2D topological field theory and Frobenius algebras vs. associative algebras

I have a basic confusion about 2D topological field theories (TFTs). In the lattice construction of 2D TFTs introduced by Fukuma et al (https://arxiv.org/abs/hep-th/9212154) only associative algebras ...
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1answer
20 views

Am I double counting an interaction in this Heisenberg-like model with next nearest-neighbor interactions?

I'm attempting to model a 1D system that consists of 4 lattice sites. When only considering the nearest-neighbor interactions, the Hamiltonian is simply $$ H_1 = \sum\limits_{i=1}^4 E(i,i+1) $$ where ...
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High temperature phonon heat capacity and equipartition theorem

I have seen the derivation of the high temperature heat capatcity sing integration of the density of states. The heat capacity is $C=3Nk_B$ with $N$ the number of atoms in the lattice. According to my ...
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Using Brillouin zones to write Hamiltonian and find momenta values

This is a homework problem that I have spent a large amount of time on..I will try my best to simplify the questions down to conceptual ones, but I will also write more or less that whole question ...
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1answer
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What is the physical significance of links between sub lattices in the staggered fermion formulation of lattice gauge theories?

I am currently learning about Lattice Gauge theories and have come across the Lattice formulation in terms of staggered fermions (see e.g. Susskind 1977, Kogut & Susskind 1975). As I understand ...
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How to calculate the chemical potential for a 2D lattice gas with excluded volume?

The chemical potential of a 2D lattice gas is given by $\mu = k_bTln(\frac{c}{(1-c)q}) $ , where c is the concentration of particles on the lattice, T is the temperature, and $k_b $ is the ...
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“Data structure” for a fermion field

I am understanding the path integral formalism of fermion fields. Most textbooks told me that grassmannian integration is only algebaric notation. It shouldn't be understood in a Lebesgue Integral ...
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How to find propagator for domain wall fermions

I am working on domain wall fermions right now and I am trying to understand how Luescher finds the propagator for the domain wall fermions in this review https://arxiv.org/abs/hep-th/0102028 on pages ...
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Does the polarized Kagome antiferromagnet contain Dirac or Weyl points?

I've been reading about frustrated quantum magnets lately and a prominent topic is the study of antiferromagnets on the Kagome lattice. A calculation of the spectrum for the sort of model I have in ...
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53 views

Two Dimensional Self-Reciprocal BravaisLattice

I've been reading Quantum States of Atoms Molecules and Solids by Morrison et al. for a condensed matter course. They make the claim that all 2D Bravais lattices are self-reciprocal, but I'm having ...
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70 views

Can we numerically find ground-state of a 1D tight binding Hamiltonain with odd sites at half filling?

We can numerically find ground state energy and wavefunction of a 1D Hamiltonian at half-filling ($L = \#$ of sites and $N = \# $ of particles) using exact diagonalization. i.e at $L = 10$ and $N = 5$,...
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Why is this Pilot-wave model on a discrete spacetime is stochastic? [duplicate]

In Gluza & Kosek (2015) (DOI 10.1007/s10701-016-0026-7; paper available at Springer (NB: PDF)) It introduces a pilot-wave model on a discrete spacetime lattice. However, the pilot-wave model is ...
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How do irrational numbers give incommensurate potential (in lattice models)?

I am trying to understand Aubry-Andre model. It has the following form $$H = \sum_n c_n^\dagger c_{n+1}+H.C.+V\sum_n \cos{(2\pi\beta n)}c_n^\dagger c_n$$ This reference (at 3rd page) says that if $\...
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116 views

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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60 views

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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68 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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72 views

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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178 views

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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106 views

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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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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110 views

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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137 views

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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61 views

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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141 views

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 ...