Villain Worm?

[evanberkowitz]

The Villain formulation with W>1 can be read as having a horrible sign problem, since the action contains $2\pi i v dn /W$. But we could think another way: execute the sum over v and get a constraint that dn≡0 (mod W). If we start with dn=0 and find an ergodic scheme to update n while maintaining the constraint, we could do small-κ (Villain-formulation) W≠1 simulations.

[alcherman]

This is a great idea! The trouble is that dn = 0 mod W (or just dn = 0) is not a 'closed loop' constraint, in contrast to \delta n = 0. We need some `geometric' interpretation of dn = 0, beyond the simple statement that this says the discrete gauge field n is flat...

[evanberkowitz]

Hm, yes. I assumed there would be a worm simply because there is a worm in the dual formulation.

[evanberkowitz]

I think there may still be a worm algorithm. It goes like this:

• Start on any plaquette.
• Pick a direction, go to that plaquette, change the link you crossed by +1, violating the dn=0 (mod W) constraint.
• Repeat until you get back to the starting plaquette (if you turn you might have to adjust the sign to maintain the constraint on the plaquette you're leaving).
• When you get back to the starting plaquette, the link you just crossed should have now restored the constraint on the starting plaquette.

It's sort of like fig 2 of 2310.17539.

The main thing is that this would leave dn fixed. We'd still need an algorithm that could change the multiple of W, like changing any given n by ±W in the local algorithm of #113.

[SethBuesing]

I agree, this combination should be ergodic and preserve dn.

[alcherman]

To allow dn = 0 mod W, how about modifying the algorithm as follows:

• Start on any plaquette.
• Pick a direction, go to that plaquette, change the link you crossed by +1 \pm W, with a metropolis test. Whether you do the \pm W should be random.
• Repeat until you get back to the starting plaquette (if you turn you might have to adjust the sign to maintain the constraint on the plaquette you're leaving).
• When you get back to the starting plaquette, the link you just crossed should have now restored the constraint on the starting plaquette, except that if the number of places where you did \pm W is random, the constraint you'll satisfy will be dn = 0 mod W, not dn = 0.

On Thu, Jan 25, 2024 at 11:46 AM SethBuesing @.***> wrote:

I agree, this combination should be ergodic and preserve dn.

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[SethBuesing]

The simplest answers are often the best. I think we've been in the muck too long and we're overthinking it this, seems like the way to go instead

[evanberkowitz]

This wouldn't be instead of the worldline worm, it would be for small κ.