Questions tagged [lattice-gauge-theory]
The study of (particle physics) gauge theories on a spacetime that has been discretized into a lattice.
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Non-perturbative approach to high-energy physics
I know that main numerical approach to modeling high-energy physics events are Monte-Carlo event generators. But they are using perturbative description of collision and decay processes of particles.
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Wilson loops as representations of the Lorentz group
Wilson loops in lattice $4d$ Yang-Mills theory are used to build various glueball states of different spins when they are applied to the vacuum. The spin dependence of such states is related with the ...
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Continuum Limit of Lattice QCD
I was trying to verify that the continuum limit of lattice QCD is indeed, regular old QCD, but I ran into an issue where when I tried to take the limit $a \rightarrow 0$ ($a$ is the lattice spacing), ...
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Lattice QCD Link Variables Meaning of $\mu$ and $\nu$
I'm currently coding a lattice QCD project and ran into an issue with my understanding.
A link variable connecting two points could be in the $\mu$ or $\nu$ direction, for example, $U_\mu(x)$ goes ...
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Are the definitions of chirality in continuum QFT and the Nielsen-Ninomiya theorem equivalent?
I have seen two definitions of chirality in quantum field theory:
According to the Wikipedia article, chirality is defined as whether a particle transforms under a left- or right-handed ...
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Lattice QCD $SU(3)$ Pseudo Heat Bath Algorithm in Practice
I'm doing a Lattice QCD project and would like to use the pseudo heat bath algorithm for updating links. I've been following Gattringer and Lang's "Quantum Chromodynamics on the Lattice". ...
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Gauss law with staggered fermions
I was wondering if someone could explain how to derive the discrete version of Gauss law in 1+1 QED using staggered Fermions.
The result I am trying to reproduce is found in multiple references [see ...
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Derivative of Function of Unitary matrices
I need some help in understanding derivative of function of matrices, Unitary matrices in my case.
I am studying lattice-qcd, there i need to take derivative of Wilson gauge action, $S[U]$ w.r.t link $...
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Holomorphicity of Functions of unitary matrices
I am studying lattice QCD and there I encounter functions of unitary matrices. For ex. The action, $S = \sum$Tr( plaquettes), where each plaquette, $P$ is written as,
$$ P = U_{\mu}(x)U_{\nu}(x+\mu){U}...
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Phase transition in scalar $\phi^4$-theory
In the case of scalar $\phi^4$-theory, there is spontaneous symmetry breaking of $Z_2$ symmetry. And in lattice formulation of $\phi^4$ theory, we observe divergence of magnetization and avg squared ...
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Different adjoints in particle physics
I am currently reading Quantum Chromodynamics on the Lattice by C. Gattringer C.B. Lang and I am confused about an expression in the book.
The expression is
$$\langle \text{tr}[S(\textbf{m}, \textbf{n}...
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Transitions in Ising lattice gauge theories in 3+1 dimensions
What is known about the character of the transition (apart from the self-duality of the model and its self-dual point marking the transition point) in the Z2 lattice gauge theory in 3+1 dimensions?
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Action of $\phi^4$ theory in lattice field theory
Euclidian action of $\phi^4$ thory is given by:
\begin{equation}
\int d^Dx L_E=\int d^Dx\left ( \frac{1}{2}(\partial_\mu \phi)^2 +\frac{m_0 ^2}{2}\phi^2+\frac{g_0}{4!} \phi^4\right),
\end{equation}
...
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Boundary conditions and CPT theorem on lattice [duplicate]
Take scalar fields for simplicity. How can one guarantee that the fields obeying the boundary conditions have the correct symmetries from the perspective of CPT theorem? A little bit of details will ...
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How do the boundary conditions on lattice fields ensure that the particle has the correct symmetries in the perspective of CPT theorem?
Imagine for simplicity that we have scalar fields. Why don't we just impose on it the symmetries dictated by CPT theorem instead of using the boundary conditions as given in literature? A detailed ...
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How to derive anyons fusion rules by given spin liquid Hamiltonian and PSG?
In many textbooks, the gapped Z2 spin liquid (equivalent to toric code) with parton method is well illustrated. It has 1, e, m, f four basic choson anyons and their fusion rules are easy to obtain and ...
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Can lattice field theory be used with $1$-loop effective actions?
It's all in the title. If I have, say, a $1$-loop effective action, is it possible to perform reliable lattice computations? I am asking because I heard lattice is not that accurate if the coupling is ...
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What are clover fermions?
I've seen the term been used quite a lot when reading about lattice gauge theory calculations. So far what I've gathered is the following, from this source [1].
Lorentz invariance of the action is ...
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Why is the gauge group of pseudo-fermion mapping referred to as $\mathrm{SU}(2)$ and not $\mathrm U(2)$?
The representation of spin $\frac{1}{2}$ operators $\hat{S}^{a}$ by pseudo-fermions (also called Abrikosov fermions) is defined by the mapping
$$
\hat{S}^{a} = \frac{1}{2} \text{Tr}\big[ \hat{\psi}^{\...
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Confinement in 3+1 dimensions from confinement in 2+1 dimensions
It is well known that Yang-Mills theories in $2+1$ dimensions exhibit the color confinement property. This property is characterized by the average of a Wilson loop that is the exponential of a term ...
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Book for lattice field theory for somebody with basic understanding of Quantum Field Theory [duplicate]
I have finished a first course in Quantum Field theory and I'm looking to get into lattice field theory (mostly QCD on the lattice). What are some resources for somebody wanting to learn how things ...
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Does the existence of the monopole in a 2+1D $U(1)$ gauge theory require the gauge field compact?
I find the monopole is allowed in a 2+1D compact $U(1)$ gauge theory in lattice (Hermele, PRB 69,064404 (2004)), there the gauge field $A$ is also compact and takes value in $[0,2\pi)$, so there is a ...
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Question with computing the chiral scalar condensate in lattice QCD about the dagger of the discretized fermionic propagator
My professor defined, given $a$ the lattice spacing the massive propagator according to the Wilson prescription as:
$$D_m \equiv D+m \left( 1-\frac{aD}{2} \right)$$
Then he computes the chrial ...
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The heat bath algorithm for $SU(3)$ lattice gauge field
I'm studying lattice gauge theory and succeeded in simulating $U(1)$, $SU(2)$ with a heat bath algorithm. However, I have difficulty in applying the algorithm to $SU(3)$. I refer to Gattringer and ...
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Physical meaning of quartic observable in abelian Higgs model
Consider the $\mathrm{U}(1)$ gauge-Higgs model defined by the lagrangian
\begin{equation}
\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+D^\mu\phi^\dagger D_\mu\phi-V(\phi^\dagger\phi),
\end{equation}
...
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Why is magnetic ’t Hooft loop operator independent of its path in$Z_2$ gauge theory?
One important concept for $Z_2$ gauge theory is magnetic ’t Hooft loop operator $\tilde{W}_{\tilde{\Gamma}}$ along a non-contractible loop $\tilde{\Gamma}$ on the dual lattice of the torus is:
$$\...
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Periodic boundary condition and hadronic correlator
Recently I have been learning about lattice QCD in a self-taught way. I have a question about the 18th page of the following link:
https://www.jlab.org/hugs/Slides/Sufian_HUGS2018.pdf
It seems to me ...
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$\mathbb{Z}_2$ gauge theory and disorder
I am confused about basics of $\mathbb{Z}_2$ (and likely other) gauge theories and plain disorder. Let
$$H=H_F + h\,H_{EM}$$
$$H_F = -t\sum_l (c^\dagger_l \sigma^z_{l,l+1} c_{l+1} + h.c.)$$
be (the '...
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Lattice spacing in lattice QCD
It is known that the lattice spacing in lattice QCD is not an external parameter and needs to be calculated, also the lattice beta parameter scales the lattice spacing ($a$) and goes as a function of ...
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How to calculate the chiral condensate from wilson fermions in lattice qcd?
In lattice qcd, respective more specifically in (1+1)-dimensional massless schwinger model on a lattice, iam trying to reproduce the chiral condensate by using correct constructed wilson overlap ...
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Wilson action equations of motion
Let $S_W$ be a Wilson action of $1\times 1$ plaquettes for a gauge group $G$:
\begin{equation*}
S_W = \beta a^4 \sum_P \left( 1-\frac{1}{N_G} \text{Re Tr}(U_P) \right),
\end{equation*}
where $\beta$ ...
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What intuition led to J. Wang and X.G. Wen's lattice formulation of the 3450 chiral gauge theory?
In the paper cited below, Juven Wang and Xiao-Gang Wen give an example of a lattice model that reduces to a chiral $U(1)$ gauge theory at low energy. The low energy theory is called the $3450$ model. ...
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Question about the measure in the partition function of a lattice Yang-Mills theory
This can seem like a dumb question but the partition function of a lattice pure gauge field theory in euclidean space is:
\begin{equation}
Z=\int \prod_{x,\mu} dU_\mu(x)\,e^{-S_W[U_\mu(x)]}\,\,\,,\,\,\...
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How to determine the ground state configuration for a Hamiltonian as a function of expansion in terms of some parameter in the Hamiltonian?
I have been learning about lattice gauge theories, in particular about the Ising gauge theory on the 2D square lattice. The Hamiltonian for a system with no matter fields is given by (for eg. from ...
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How to visualize the $U(1)$ instanton event in (2+1)D compact lattice gauge field?
In the continuum limit of (2+1)-dimensional compact $U(1)$ gauge field, the instantons are input by hand in terms of nonconservation of magnetic flux $\int b$:
\begin{eqnarray}
\int dxdy [b(x,y,t+\...
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How does a lattice regulator work if we don't want observables to be invariant under "large" gauge transformations?
In quantum field theory (QFT), observables must be invariant under gauge transformations that are continuously connected to the identity, but invariance under "large" gauge transformations ...
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Proof of commutation relation for lattice QFT
How do you prove the following commutation relation for the lattice QFT
\begin{equation}
[\phi(X),\Pi(y)]=\text{i}a^{-d}\delta_{x,y}\mathbb{I}?
\end{equation}
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What is the relationship between lattice field theory and grid-based numerical methods?
I recently learned of lattice field theory wherein quantum fields are defined on a discrete spacetime. Naively this sounds to me a lot like certain numerical methods for solving differential equations ...
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Wilson loop shapes and glueball operators
In the AdS/QCD correspondence, glueballs operators are given, for example, by $\text{Tr}[F_{\mu \nu}F^{\mu \nu}]$ for $0^{++}$ or $\text{Tr}[F_{\mu \nu}\widetilde{F}^{\mu \nu}]$ for $0^{-+}$. However, ...
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"Axial" gauge in the $Z_2$ lattice gauge theory
I am reading the paper by Fradkin and Susskin on the lattice gauge theory (Order and disorder in gauge systems and magnets). In section III. C, where they were trying to introduce the duality ...
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How fermion doublers cause practical issues?
These days I learn about the lattice gauge theory, and in particular learned when one naively discretizes the fermion action, doublers, superfluous poles for a propagator, emerge. I wonder what issue ...
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Different sectors of the Ising gauge theory
The Hamiltonian of Quantum 2D Ising gauge theory is given by:
$$ H=-\sum_p \prod_{i\in \square}\sigma^z_i -g \sum_{i\in \text{links}} \sigma^x_i$$
This $H$ is invariant under the local symmetries:
$$ ...
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Quantization of Flux in Polyakov's 3D Compact QED
In his his book "Gauge Fields and Strings" Polyakov introduces the compact QED on a cubic lattice in 3D Euclidean space as:
$$ S\left[ \left\{ A_{\mathbf{r},\mathbf{\alpha}}\right\} \right]=\...
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On-site symmetries can be gauged, but is a gaugeable symmetry necessarily on-site?
I've always liked lattice QFT because it's mathematically unambiguous and non-perturbative, but it does have two drawbacks: (1) the lattice is artificial, and (2) some features are messy. One of those ...
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Equivalence between rotation and magnetic flux in lattice models
I am trying to understand the presence of complex hopping amplitudes in Hubbard-like lattice models. The hopping term features the so called "Peierls phase":
$$
- t\sum_{j=1}^L \left( c_{...
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What is the connection between vertex/spin models and gauge theory?
In the usual formulation of lattice gauge theories, one considers gauge variables on the links of a lattice (often hypercubic) taking value in some representation of a gauge group, $U_{ij} \in G$. The ...
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Recommend books for learning lattice QCD
I want to learn lattice QCD by myself, but I don't know how to start. Can you recommend some books for lattice QCD?
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Is it possible to have a compact abelian $U(1)$ lattice gauge theory on a non-compact manifold?
We have a compact lattice gauge theory if we let $A_{i}(n)\in[-\pi,\pi]$, and if we identify $A_{i}(n)\sim A_{i}+2\pi$. A simple lattice gauge theory in 2+1D then has an action
$$S=\sum_{x}1-\cos(F_{\...
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Lattice differentiation and Locality
Assume we define the locality of a theory in the following way:
Assume we have a theory of real scalars, so this theory is non local if the action has terms like
$$\int d^dx\,\phi(x)V(x-y)\phi(y).$$
...
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Are gravitons equivalent to extra localised lattice points?
So imagine space is a regular square mesh or lattice. In a theory like QCD, the photon lines are placed along the edges of this graph to form paths.
The space is supposed to represent simple ...
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