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I'm currently studying a scalar quantum field theory with a $\lambda\phi^4$ interaction (commonly referred to as $\phi^4$ theory. To study this theory non perturbatively I've written a program in python that simulates the theory in the euclidean (imaginary time) formalism. I'm trying to numerically look at the potential triviality of the theory in 3 + 1 dimensions. The problem I'm having is that I can't find a good definition of the renormalized coupling constant that I can compute with relative ease on the lattice.

On Scholarpedia they have a formula for the renormalized coupling constant in the symmetric phase is given by $$\lambda_r=(Lm_r)^4(3\left\langle M^2\right\rangle ^2 - \left\langle M^4 \right\rangle)/\left\langle M^2 \right\rangle ^2$$ where $$ M = a^4\sum_{i}\phi(x_i),$$ and where $\left\langle X \right\rangle$ deontes the expectation value or mean of said quantity. Also note, $m_r$ is the renormalized mass of the scalar particle.

I'm having a few problems with this coupling constant as it's defined. The first is that it is unreasonably large even for highly coarse lattices with a lattice spacing $a = 1/2$ and with a lattice size $L = 8$. The bigger problem however is that if we're in the the symmetric phase where $\langle\phi\rangle = 0$, it follows that $ M = a^4\sum_{i}\phi(x_i)$ should be zero as well.

So my question is the following; have I messed up somewhere with my definition of the renormalized coupling constant? Has Scholarpedia messed up with their definition? Does anyone have an equally easy to calculate definition of the coupling constant that doesn't have these (apparent) problems?

Thanks

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  • $\begingroup$ Off-topic question: why labeled as lattice model? I thought lattice is not a model but first principle. $\endgroup$
    – Turgon
    Commented Jul 20, 2018 at 2:31
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    $\begingroup$ $\langle\phi\rangle=0$ does not imply that $\phi=0$. Therefore, $M\neq 0$. If anything, $\langle M\rangle=0$, but this does not imply that $\langle M^2\rangle=0$. I'm not sure I really understand your question though. Good luck! $\endgroup$
    – AccidentalFourierTransform
    Commented Jul 20, 2018 at 3:09
  • $\begingroup$ A lattice model in quantum field theory is simply quantum field theory defined on a discretized space-time lattice with imaginary times. You then use algorithms like monte-carlo to generate relevant field configurations. You compute quantities based on those configurations, and use them to extract things like the renormalized mass of the particle, the masses of bound states, etc. $\endgroup$
    – chuckstables
    Commented Jul 20, 2018 at 3:15
  • $\begingroup$ @AccidentalFourierTransform Here's the algorithm I use to compute M. I do a bunch of iterations and generate a new field configuration. I compute $M^2$ or $M^4$ based on said field configuration. I then average $M^2$ and $M^4$ over a bunch of minimally correlated configurations to get their expected values. At the end of this all I calculate $\lambda_r$ $\endgroup$
    – chuckstables
    Commented Jul 20, 2018 at 3:24
  • $\begingroup$ How large do you consider "unreasonably large"? I notice that Fig. 2 in arXiv:1502.03714 shows a coupling ~30 for L=8 (and a=1). That proceedings also advocates using the two-point correlation function (rather than $M$) to define a renormalized coupling. Reproducing that figure might be a useful exercise. $\endgroup$
    – David Schaich
    Commented Aug 29, 2018 at 16:02

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