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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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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 $c =1$ ...
5
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1answer
119 views

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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0answers
80 views

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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+50

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 ...
2
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1answer
29 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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0answers
126 views

Computational content of Feynman integral

I am looking for references with rigorous (not just numerical) studies deriving convergence rates of discretized path integrals to their "true" values in some interesting special cases (since the ...
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33 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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2answers
126 views

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 ...
3
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1answer
110 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 ...
0
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1answer
34 views

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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40 views

Derivation of improved gauge action, starting from Wilson plaquette action

I'm looking for an understandable, detailed derivation of the improved gauge action up to a form like (taken from arXiv:hep-lat/0404007, p.3), starting from the Wilson plaquette action: I mean an ...
2
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0answers
51 views

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 ...
1
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1answer
31 views

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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79 views

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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93 views

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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24 views

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 ...
4
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2answers
196 views

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 ...
3
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1answer
135 views

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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40 views

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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78 views

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 ...
8
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1answer
137 views

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 ...
0
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0answers
80 views

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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41 views

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 ...
0
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1answer
71 views

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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0answers
50 views

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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0answers
11 views

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 ...
0
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1answer
29 views

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 ...
1
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1answer
51 views

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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0answers
60 views

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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52 views

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 $k$...
0
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1answer
101 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 ...
2
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1answer
77 views

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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28 views

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 ...
0
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1answer
32 views

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^{\...
14
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5answers
489 views

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 ...
-1
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1answer
72 views

Hubbard Model Hamiltonian

Right now I try to understand the Hubbard Model. I do have a question to the Hamiltonian written in this form: $H=\sum_{m,n}(\hat{c}^{\dagger}_{m+1,n}\hat{c}_{m,n}+\hat{c}^{\dagger}_{m,n+1}\hat{c}_{m,...
8
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1answer
393 views

Continuum Field Theory for the Ising Model

My problem is to take the $d$-dimensional Ising Hamiltonian, $$H = -\sum_{i,j}\sigma_i J_{i,j} \sigma_j - \sum_{i} \tilde{h}_i \sigma_i$$ where $J_{ij}$ is a matrix describing the couplings between ...
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0answers
26 views

Lattice to continuum transformation in one dimension (arbitrary range interaction)

Please help me convert the ionic displacement $u_l$ of the $l^{th}$ ion to a continuous field $u'(y)$ in the following problem. I am trying to derive the Hamiltonian of a 1-D lattice (spacing $=n_0^{-...
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0answers
57 views

Correlation between spins using delta function in Potts model

In reading about the Potts model, I found this correlation: $$\langle s_{i}s_{j} \rangle = \frac{q}{q-1}\frac{1}{N_{p}} \sum_{s_{i},s_{j}} (\delta(s_{i} - s_{j})-\frac{1}{q})$$ with the following text:...
0
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1answer
127 views

Can the minimum energy state of an Ising-Hamiltonian be negative?

I am simulating 1D many body (N>2) quantum chains, governed by Ising-Hamiltonians. More on the Ising-Hamiltonian. My simulation generates the Hamiltonian based on the following formula: $$\hat H = J\...
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0answers
37 views

Local Density of the Medium for Classical 1D Harmonic Chain

In the first chapter of Altland and Simons (2nd Edition), pg. 34, there is the following exercise: Consider the one-dimensional elastic chain discussed in Section 1.1. Convince yourself that the ...
3
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1answer
146 views

Literature on lattice quantum field theory

Could you please list some books on lattice quantum field theory? Hardcopy books or electronic books. I'm interested in quantum field theory with only discrete impulses, with impulse cut-offs, both ...
1
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2answers
290 views

Fourier Transform and Lattices

I've been always confused Fourier transforms on lattices because some times a continuous version is used and others a discrete version is used. I don't understand well when should use one or the other....
2
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0answers
75 views

Toric Code model with an extra projector? (Levin-Gu)

In the seminal work by Levin and Gu in 2012 ( Braiding statistics approach to symmetry-protected topological phases ) they give a concrete prescription for how to gauge a global symmetry to a local ...
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0answers
22 views

Electron positioning question

Random question, but this is related to probability and electron positioning on atoms. For this example, I will use water and ice. As I'm sure we all already know, water forms ice under the right ...
3
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1answer
83 views

Lorentz Symmetry Group as continuous symmetry for limit of discrete spacetime

There is a variety of models of quantum field theory, where discrete spacetime is used as technical support, or even suggested as physical reality. As far as I know, all of such models faced serious ...
1
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1answer
47 views

When does the structure function have parity symmetry?

I'm working on a magnetic system and in the calculations this function appears \begin{equation} \gamma(\vec{k})=\frac{1}{Z}\sum_{\vec{\mu}}e^{i \vec{\mu} . \vec{k}} \end{equation} where $\vec{\mu}$ ...
1
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1answer
127 views

What is average plaquette?

I'm reading papers with numerical lattice results that have this quantity called 'average plaquette' but I can't find a good definition anywhere. I know what a plaquette is, the smallest loop in a ...