Questions tagged [lattice-gauge-theory]
The study of (particle physics) gauge theories on a spacetime that has been discretized into a lattice.
62
questions with no upvoted or accepted answers
8
votes
0
answers
208
views
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:
$$ ...
- 1,417
7
votes
0
answers
164
views
Existence of Schwinger Functions for QCD?
It seems to me the 'naive' approach to proving the existence of Yang-Mills in a rigorous context (via Osterwalder-Schrader $\to$ Wightman axioms), would be:
Study gauge invariant lattice QCD ...
- 6,896
6
votes
0
answers
265
views
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. ...
- 54.5k
5
votes
0
answers
112
views
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+\...
- 813
5
votes
0
answers
347
views
How Is Are Parton Distribution Functions (PDFs) Determined In Practice And In Theory?
I have put my actual questions in bold normal font, and the rest of this question clarifies what I mean to be asking, especially in terms of the depth and kinds of information I am looking for. ...
- 3,957
4
votes
0
answers
487
views
Can I do anything instructive by simulating QED on a lattice?
For learning something about the degrees of freedom and underlying path integral math, is it possible to do some kind of scalar QED or normal QED simulation on a lattice in the same way Lattice QCD is ...
- 7,219
4
votes
0
answers
213
views
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 ...
- 2,263
4
votes
0
answers
88
views
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 ...
- 3,710
3
votes
0
answers
101
views
Why doesn't Peierls substitution capture the Lorentz force?
It's well known that the classical Hamiltonian governing the dynamics of a charged particle in a static magnetic field is
$$ S_{cl}[x,\dot{x}] = \int_0^t dt' \frac{1}{2}m\dot{\vec{x}}^2 + e\vec{A}(x)\...
- 2,829
3
votes
0
answers
118
views
Resource recommendation for Kogut-Susskind Hamiltonian formalism for lattice gauge theory
Recently, quantum simulations for quantum field theories have been a hot topic of research. In these calculations, the lattice calculations are done using the Hamiltonian formalism in contrast to the ...
3
votes
0
answers
64
views
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
$$\int d^dx\,\phi(x)V(x-y)\phi(y).$$
...
- 1,369
2
votes
0
answers
102
views
Instantons in the Global $O(2)$ Model (Compact scalar field) - Polyakov textbook
This question is related with Polyakov, "Gauge Fields and Strings" section 4.2
In section 4.2, partition function is
\begin{equation}
Z=\sum_{n_{x,\delta}}\int_{-\pi}^{\pi}\prod_x\frac{d\...
- 21
2
votes
0
answers
34
views
Restoring Poincaré symmetries in Hamiltonian lattice field theories
I can imagine how the continuum limit of a non-relativistic quantum field theory discretized on a spatial lattice restores the Galilean symmetries of the original theory. But how does this work for ...
- 2,921
2
votes
0
answers
58
views
Extracting a gauge-invariant variable from a given Wilson line? (NOT Wilson loop)
Let $W[x_i,x_f]$ be the Wilson line as defined here.
Under a local gauge transform $g(x)$, it transforms as
\begin{equation}
W[x_i,x_f] \to g(x_f)W[x_i,x_f] g^{-1}(x_i)
\end{equation}
which is shown ...
- 1,669
2
votes
0
answers
52
views
Topological classification of (classical, Abelian) vortices on a lattice
Consider the XY model on the square lattice. A field configuration $\theta$ is specified by an element of the Abelian group $\mathbb{R}/2\pi \simeq U(1)$ at each vertex of the lattice. The gradient of ...
- 3,710
2
votes
0
answers
73
views
How to update $SU(2)$ Higgs fields with Heat Bath algorithm?
I'm trying to update the Higgs field coupled with a pure gauge $SU(2)$ theory through Heat Bath algorithm. Pure gauge and Higgs configurations should be updated separately. For the pure gauge part the ...
- 31
2
votes
0
answers
38
views
The "overlap function" of a $Z_2$ gauge theory
Consider a $Z_2$ gauge theory on a square lattice (Ising spins on edges) with classical degrees of freedom, i.e.
\begin{equation}
E = -\sum_{\square} \sigma_i\sigma_j\sigma_k\sigma_l
\end{equation}
...
- 321
2
votes
0
answers
54
views
Lattice gauge theory with $A_\mu$ instead of $E_\mu$ or $B_\mu$
In most formulations of the lattice gauge theory one uses the Hilbert space basis defined by the eigenstates of the electric or magnetic field. For example, in the "electric basis" on one ...
- 2,921
2
votes
0
answers
71
views
Path integral in Lattice gauge theory with fixed gauge really the same as without fixing the gauge?
In 1 the question why in lattice gauge theories with gauge group $G$, there was no need for gauge fixing to obtain finite path integrals was answered. Thus observables could be calculated as
\begin{...
- 586
2
votes
0
answers
238
views
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 ...
- 868
2
votes
0
answers
74
views
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 $\beta$ ...
- 2,613
2
votes
0
answers
32
views
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)]}\,\,\,,\,\,\...
- 2,613
2
votes
0
answers
162
views
"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 ...
- 1,181
2
votes
0
answers
50
views
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":
$$
- t\sum_{j=1}^L \left( c_{...
- 1,232
2
votes
1
answer
93
views
Degenerate link variable configuration in $Z_2$ lattice gauge theory (Wen's QFT book)
I'm reading through Xiao-Gang Wen's Quantum Field Theory of Many-body Systems, and I just begin reading $Z_2$ lattice gauge theory.
In page 255, the author constructed a four-fold denegerate (in the ...
- 71
2
votes
0
answers
156
views
2D toric code intuition
I’m trying to self-learn QFT in condensed matter and I’ve hit a stumbling block with gauge theory and toric code. I’m trying to solve problem 2 from this webpage http://www.its.caltech.edu/~motrunch/...
- 301
2
votes
0
answers
841
views
What's wrong with lattice quantum gravity?
Assume one can write the metric field on a lattice, so on each lattice point one has a value of $g^{\mu\nu}$. Similar to the way lattice QCD is formulated. Then later taking the distance between ...
user84158
1
vote
0
answers
37
views
Parity of a 1d Ising model, and with higher order terms
I don't know if this should be asked here or in a math stack exchange, but I'll try here first.
Consider the classical 1d Ising model with periodic boundary condition:
\begin{equation}
H_2 (\vec{\...
- 368
1
vote
1
answer
76
views
What is the reverse operation of gauging a global symmetry?
As far as I understand, gauging a global symmetry means taking a model with a global symmetry and transforming it into a model such that the previous symmetry group is now the gauge symmetry of your ...
1
vote
0
answers
45
views
What means charge-$N$ scalar field $\varphi$?
Let $G = \oplus_{i=1}^N (\mathbb{Z}/N_i)$ be an Abelian group, for sake of simplicity eg a cyclic group $\mathbb{Z}/N$ . We consider abstract $G$- gauge theories.
What is in this context the precise ...
- 1,395
1
vote
0
answers
22
views
How do I numerically compute the interquark potential from the correlation function of Polyakov Loops?
I know that the potential can be calculated in the following way:
$$
aV(r) =-\ln(<\sum_{\textbf{x}} (P(\textbf{x}+R)P^{\dagger}(\textbf{x}))>)/N_T.
$$
Now, suppone I have some procudure to ...
- 185
1
vote
0
answers
51
views
How to couple the Higgs field with $SU(3)$ (and/or $SU(2)$) Yang-Mills theory in numerical simulations?
I'm trying to couple the Higgs field to numerical simulations of pure gauge theory with heatbath and overrelaxation update of link variables. I don't know how to insert the Higgs field into the ...
- 31
1
vote
0
answers
124
views
Gauge degrees of freedom in Schwinger model
Schwinger model is the (1+1)-D QED. The number of gauge degrees of freedom (DOF) after the gauge fixing of the Schwinger model depends on the boundary condition of the model. For example, one can ...
- 964
1
vote
0
answers
38
views
Can anyone suggest me some papers to understand the mathematics behind the form factor
I am trying to study semileptonic decays of $B$ mesons and different models are also there but I am not understanding how specific form factors are assigned to specific mesons. For example, right now ...
1
vote
0
answers
43
views
Status of Approach of constructing Hamiltonians from Transfer Matrix
I am studying this old paper from J.B.Kogut on lattice gauge theories and spin systems [Rev. Mod. Phys. 51, 659(1979)].
This paper discusses about the way of constructing a quantum Hamiltonian using ...
1
vote
0
answers
88
views
Meaning of lattice gauge theories
I would like to ask about the physical interpretation of lattice gauge theories. Coming from a condensed matter background, I know only that lattice gauge theories are constructed by adding additional ...
- 325
1
vote
0
answers
90
views
Understanding fermion doubling in lattice QFT
I'm studying Rothe's book on lattice gauge theory. For the case of a scalar field, we can use lattice discretization to find (using equations 3.18 and 3.19 on page 41)
$$\langle 0|T\phi(x)\phi(y)|0\...
- 789
1
vote
0
answers
115
views
How well does Lattice QCD handle relativity?
In Lattice QCD space-time is approximated by a grid.
To me this doesn't seem to handle relativity well. Due to
(1) A Lorentz transformation of the grid will distort the hyper-cube volumes into ...
user84158
1
vote
0
answers
139
views
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), ...
- 968
1
vote
1
answer
140
views
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 ...
- 153
1
vote
0
answers
149
views
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 $...
1
vote
1
answer
153
views
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?
- 111
1
vote
1
answer
123
views
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
$$
\hat{S}^{a} = \frac{1}{2} \text{Tr}\big[ \hat{\psi}^{\...
- 11
1
vote
0
answers
164
views
Why is magnetic ’t Hooft loop operator independent of its path in$Z_2$ gauge theory?
One important concept for $Z_2$ gauge theory is magnetic ’t Hooft loop operator $\tilde{W}_{\tilde{\Gamma}}$ along a non-contractible loop $\tilde{\Gamma}$ on the dual lattice of the torus is:
$$\...
- 1,602
1
vote
0
answers
110
views
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 ...
- 21
1
vote
0
answers
148
views
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}
- 67
1
vote
0
answers
85
views
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 ...
- 21
1
vote
0
answers
109
views
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_{\...
- 1,369
1
vote
0
answers
237
views
Polyakov loops and Wilson loops as order parameters
At zero temperature, the confinement/deconfinement criterion is the area/length law of the following non-local parameter called the Wilson loop:
\begin{eqnarray}
W=\text{Tr}\exp\left(\oint_CA_idx^i\...
- 813
1
vote
0
answers
142
views
How do we derive the Hamiltonian of the Wilson Loop Action?
I'm reading Fradkin and Susskind's 1978 paper "Order and disorder in gauge systems and magnets" to try and understand how they derive the Hamiltonian for the U(1) compact lattice gauge theory. The ...
