# Is the wilson loop always positive in the lattice simulation?

• sufive
In summary, the sign of the Wilson loop can vary in different lattice simulations due to its dependence on lattice spacing and size. However, it is always positive and represents the strength of the gauge field in a given region of the lattice. The Wilson loop is calculated by taking the product of link variables and averaging over all possible loops, giving the expectation value. It can also be used to determine the confinement or deconfinement phase transition in lattice simulations, with a decrease and negative sign indicating deconfinement and an increase and positive sign indicating confinement.
sufive
More precisely, my question is, Is the wilson loop used to calculate the heavy quark potential always positive in the lattice simulation?

(i) as we are usually told, wilson loops of the following form are related with
the heavy quark potentials
.........
_________ <WC_______...
|......|../\ ...
|WD...../\...|...|...
|......|...r...
| \/......WB...|...|...
|_____WA>__________|..\/...
<---------- t ----------->...
.........

<W>=<W_{r,t}> ~ e^{-V( r )*t}
so we expect <W> is always positive(or negative,
any way, it should not change signs we we change t and r)

(ii) but by definition

<W>= <tr[e^{iAa ta} * e^{i Ab tb} * e^{-iAc tc} * e^{-i Ad td}]>

= <tr[WA * WB * WC * WD]>

since in numerical simulations, the link variables WA, WB, WC, WD et al
are random unitary matrix, we cannot expect the trace of their products
is positive.

I encounter this problem because in one of my little exercises to write
lattice QCD codes, I got results just as I explained in (ii), so I cannot extract
the heavy quark potentials, because some times I get W(r,t)>0, some times
W(r,t)<0. Especially worse is, I encounter the case
W(r,t)>0
W(r,t+a)<0
So the proportion W(r,t+a)/W(r,t) ~ e^{-V( r )*a} < 0, this is really a disappointing
outcome.

What key points did I ignored ? Who can tell me or give me some
thread or write me email, dfzeng2000@hotmail.com

Last edited:

Thank you for your question regarding the use of Wilson loops in lattice simulations to calculate the heavy quark potential. As a scientist familiar with this topic, I would like to provide some clarification and address the issues you have encountered.

Firstly, it is important to note that the Wilson loop is not used directly to calculate the heavy quark potential, but rather it is used as a tool to extract the potential from lattice simulations. The relation you have mentioned, <W>=<W_{r,t}> ~ e^{-V( r )*t}, is an approximation that is valid in the large t limit, where t is the temporal extent of the lattice. In this limit, the Wilson loop average is dominated by the ground state of the system, which corresponds to the heavy quark potential.

However, in numerical simulations, we are limited by the size of the lattice and cannot take t to be arbitrarily large. This means that the approximation <W>=<W_{r,t}> ~ e^{-V( r )*t} is not exact and the Wilson loop average can be affected by excited states of the system. This can result in fluctuations in the sign of the Wilson loop, as you have observed.

To improve the accuracy of the extraction of the heavy quark potential, one can use a technique called smearing, where the link variables are modified in a way that enhances the ground state contribution to the Wilson loop. This can reduce the effects of excited states and improve the stability of the Wilson loop average.

Additionally, it is important to consider the choice of lattice spacing and the size of the lattice, as these can also affect the accuracy of the calculation. A finer lattice spacing and a larger lattice size can improve the accuracy of the extraction of the potential.

In summary, the key points to keep in mind when using Wilson loops to extract the heavy quark potential in lattice simulations are the limitations of the approximation <W>=<W_{r,t}> ~ e^{-V( r )*t}, the use of smearing techniques, and the choice of lattice parameters. I hope this helps to address your concerns and improve the accuracy of your results. If you have any further questions, please do not hesitate to reach out.

## 1. Does the sign of the Wilson loop change in different lattice simulations?

The sign of the Wilson loop can vary in different lattice simulations. This is because the Wilson loop is a gauge-invariant quantity, meaning that it is dependent on the chosen lattice spacing and lattice size.

## 2. Can the Wilson loop ever be negative in lattice simulations?

No, the Wilson loop is always positive in lattice simulations. This is because it is related to the square of the trace of the gauge field around a closed loop, which is always positive.

## 3. How is the Wilson loop calculated in lattice simulations?

The Wilson loop is calculated by taking the product of the link variables around a closed loop on the lattice and then averaging over all possible loops. This is known as the trace of the gauge field and gives the expectation value of the Wilson loop.

## 4. What does the sign of the Wilson loop represent in lattice simulations?

The sign of the Wilson loop represents the strength of the gauge field in a given region of the lattice. A positive sign indicates a dominant gauge field, while a negative sign indicates a subdominant gauge field.

## 5. Can the Wilson loop be used to determine the confinement or deconfinement phase transition in lattice simulations?

Yes, the Wilson loop can be used to determine the confinement or deconfinement phase transition in lattice simulations. At high temperatures, the Wilson loop decreases and becomes negative, indicating deconfinement, while at low temperatures it increases and remains positive, indicating confinement.

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