Metropolis-Hastings for U(1) Gauge Theory

1. The Metropolis-Hastings Algorithm Revisited

At its core, the Metropolis-Hastings algorithm is a recipe for generating a sequence of states (in our case, lattice configurations) such that their distribution follows a desired probability law, typically the Boltzmann distribution $P(C)\propto e^{-S[C]}$.

The key steps are:

1. Propose: Start with a configuration $C$ and propose a new one, $C^{\prime }$.
2. Calculate: Compute the change in action, $\Delta S=S[C^{\prime }]-S[C]$.
3. Accept/Reject: Accept the proposed configuration $C^{\prime }$ with probability $P_{accept}=\min (1,e^{-\Delta S})$. If the proposal is rejected, the next state in the sequence is just a copy of the old one, $C$.

Our task is to define what "proposing a new configuration" and "calculating $\Delta S$" mean for our U(1) lattice gauge theory.

2. The Metropolis Update for U(1) Link Variables

In our theory, a "configuration" is the set of all link variables $\{U_{\mu }(x)\}$ on the lattice. Each link variable is a U(1) group element, which is simply a phase factor $U_{\mu }(x)=e^{i{\theta }_{\mu }(x)}$, where ${\theta }_{\mu }(x)\in [-\pi ,\pi ]$.

A single Metropolis update step will consist of modifying a single link variable $U_{\mu }(x)$ while keeping all others fixed.

The Proposal Step

How do we propose a new link value $U_{\mu }^{\prime }(x)$? The most straightforward way is to simply pick a new random phase.

• Proposal: Generate a new random angle ${\theta }_{new}\in [-\pi ,\pi ]$ from a uniform distribution.
• New Link: Set the proposed link to $U_{\mu }^{\prime }(x)=e^{i{\theta }_{new}}$.

This is a valid proposal because it is symmetric: the probability of proposing the old link from the new one is the same as the reverse.

The Acceptance Step: Calculating $\Delta S$

This is the most critical part of the implementation. The Wilson action is a sum over all plaquettes on the lattice:

where $U_{\mu \nu }(n)=U_{\mu }(n)U_{\nu }(n+\hat{\mu })U_{\mu }^{†}(n+\hat{\nu })U_{\nu }^{†}(n)$ and $\beta =1/g^2$.

A naive calculation of $\Delta S$ would involve recalculating the entire lattice action, which is computationally expensive. However, a change in a single link $U_{\mu }(n)$ only affects the action terms of the plaquettes that contain that specific link. This principle of locality is what makes the calculation feasible.

In a $d$-dimensional spacetime, any given link is a border for $2(d-1)$ plaquettes. In our 4D case, each link is part of 6 plaquettes. The contribution to the action that depends on a specific link $U_{\mu }(n)$ can be written as:

The term in the parenthesis is the sum of products of the other three links for each of the 6 plaquettes. This sum is called the staple, which we'll denote $V_{\mu }(n)$. The staple $V_{\mu }(n)$ is a complex number representing the sum of the 6 open paths that can be closed by the link $U_{\mu }(n)$.

With the staple $V_{\mu }(n)$ computed, the part of the action dependent on $U_{\mu }(n)$ is simply $S_{dep}=-\beta \text{Re}[U_{\mu }(n)V_{\mu }(n)]$. The change in action when proposing $U_{\mu }(n)\to U_{\mu }^{\prime }(n)$ is therefore:

This is a very efficient local calculation. For each update, you only need to compute one staple (a sum of 6 terms) and perform a few complex multiplications.

3. The Full Algorithm

We can now assemble the complete algorithm for one full "sweep" over the lattice. A sweep consists of visiting every link on the lattice once and attempting a Metropolis update.

Based on this, here is the pseudocode for your simulation:

1. Initialization:

   • Create a 4D lattice of link variables $U_{\mu }(n)$. Each $U_{\mu }(n)$ is a complex number.
   • Initialize the links. A cold start sets all links to unity ($U_{\mu }(n)=1+0i$), while a hot start sets them to random U(1) phases ($\theta \in [-\pi ,\pi ]$).
   • Set the coupling parameter $\beta$.

2. Simulation Loop (Markov Chain):

   • Loop for N_sweeps (total number of updates):
     • Loop through all sites n and all directions μ from 0 to 3:
       a. Store the old link: U_old = U_μ(n).
       b. Calculate the staple: Compute $V_{\mu }(n)$, the sum of the 6 staple terms corresponding to the link (n, μ). This involves products of links on neighboring sites. Be careful with periodic boundary conditions!
       c. Propose a new link: Generate a random angle ${\theta }_{new}\in [-\pi ,\pi ]$ and set U_new = cos(θ_new) + i*sin(θ_new).
       d. Calculate ΔS: Compute $\Delta S=-\beta \text{Re}[(U_{new}-U_{old})V_{\mu }(n)]$.
       e. Accept/Reject: Generate a random number r from a uniform distribution U(0, 1). If r < exp(-ΔS), then update the lattice: U_μ(n) = U_new. Otherwise, the link remains U_old.

3. Measurements:

   • Thermalization: Discard the first N_therm sweeps to allow the system to reach equilibrium.
   • Autocorrelation: After thermalization, the configurations are still correlated. Measure observables (like the average plaquette value) only every N_corr sweeps to ensure your measurements are statistically independent.

4. Improving Efficiency: The Overrelaxation Algorithm

Your background in creating efficient simulations will make this next part particularly interesting. The standard Metropolis algorithm can suffer from high autocorrelation, meaning you need very long runs to get independent configurations. The overrelaxation algorithm is a clever trick to speed up the exploration of the configuration space.

The idea is to propose a new link deterministically in a way that leaves the action unchanged. Since $\Delta S=0$, this move is always accepted.

Let's look at the local action contribution again: $S_{dep}=-\beta \text{Re}[U_{\mu }(n)V_{\mu }(n)]$.
If we write $U_{\mu }(n)=e^{i{\varphi }_{\mu }(n)}$ and the staple $V_{\mu }(n)=|V_{\mu }(n)|e^{i{\theta }_V}$, the action is:

This expression is symmetric under the reflection ${\varphi }_{\mu }(n)+{\theta }_V\to -({\varphi }_{\mu }(n)+{\theta }_V)$.
This gives us a new angle ${\varphi }_{new}^{\prime }=-2{\theta }_V-{\varphi }_{\mu }(n)$. The corresponding new link $U_{new}^{\prime }=e^{i{\varphi }_{new}^{\prime }}$ gives the exact same action!

This provides a deterministic update rule that moves the system to a new, decorrelated state for free. A common and very effective simulation strategy is to alternate one Metropolis sweep with several (e.g., 3-5) overrelaxation sweeps.

Conclusion

You are now equipped with the complete algorithm to perform your first simulation of a pure gauge theory. This is a significant milestone, moving from abstract field theory to a concrete computational tool.

Key Takeaways:

• The Metropolis-Hastings algorithm is adapted to U(1) gauge theory by proposing changes to individual link variables.
• The proposal consists of choosing a new random U(1) phase for a link.
• The change in action, $\Delta S$, is calculated locally by computing the staple—the sum of open plaquette paths attached to the link being updated. This makes the algorithm efficient.
• A full simulation requires an initialization step (e.g., hot/cold start), a thermalization period, and measurements spaced by the autocorrelation time.
• Techniques like overrelaxation can dramatically improve simulation efficiency by reducing autocorrelation times.

Preview of the Next Lesson:

Now that we know how to generate field configurations, the next logical question is: what do we do with them? In the next lesson, we will compute the two-point correlation function from the generated lattice configurations and extract the particle mass from its exponential decay. This will be our first physics result from the U(1) simulation, and we can compare it to the scalar field case you've already mastered.