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Let's say that you are doing some Monte-Carlo simulations of a statistical system on a lattice and you observe scale invariance, meaning that you are at a conformal point. Can you get a numerical appreciation of the central charge?

I know how the central charge is related on the free energy (on a cylinder for example) or to the stress-energy tensor but these are not direct observable in a Monte-Carlo.

Is there a systematic method for that? Has it already been done?

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    $\begingroup$ Are you by any chance working in (1+1) dimensions? $\endgroup$ – fqq Sep 4 '14 at 16:43
  • $\begingroup$ Yes, in 2d. If that matters I'm looking at two dimensional spin systems. $\endgroup$ – Learning is a mess Sep 5 '14 at 10:56
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    $\begingroup$ I found this, and this. Googling "Monte Carlo central charge" comes up with lots more besides. $\endgroup$ – Holographer Sep 5 '14 at 15:21
  • $\begingroup$ @Holographer: I had found these, just haven't been able to implement them in my current system, so I'm seeking alternative methods. $\endgroup$ – Learning is a mess Sep 5 '14 at 15:33
  • $\begingroup$ @Learningisamess the method I have in mind works for one-(space)-dimensional quantum systems, which has some links to the 2d classical model described by the same CFT, but it's not exactly the same. I will however try to write an (at least partial) answer. $\endgroup$ – fqq Sep 10 '14 at 17:37
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Since you found the critical point via numerical simulations, you probably have little analytical insight into its properties. This makes it hard to extract the central charge, because it often appears in expressions combined with speed of sound or other quantities (e.g. in the free energy for a 2d CFT). So you need to find a universal quantity, easily measurable in numerical simulations, which gives you the central charge explicitly.

For one-dimensional quantum critical systems, and their corresponding 2d CFT, the entropy of entanglement is such a quantity. A complete reference is 0905.4013, and the rest of this special issue of JPhysA (note that it is five year old, so it doesn't include some recent developments). The entanglement entropy is a measure for the entanglement between two complementary parts A and B of a system in a pure state. Given the reduced density matrix of one of the parts, $$\rho_A=\mathrm{tr}_B\rho,$$ it is defined as the Von Neumann entropy $$S_A=-\mathrm{tr}(\rho_A\log\rho_A),$$ $S_B=S_A$ for pure states.

As an entanglement measure, this was considered in quantum information theory. However, it has some properties that made it interesting for the statistical physics community. It diverges at critical points, in interesting ways.

In particular, consider an infinitely long chain, and its bipartition in an interval of length $L$ and its complement. At a critical point described by a CFT with central charge $c$, it has a logarithmic divergence given by the Cardy-Calabrese formula $$S=\frac{c}{3}\ln L+c_1,$$ where $L$ is measured in units of some short-distance regularisation (e.g. the lattice spacing) and $c_1$ is not universal.

This provides a universal asymptotic result, which is often easy to compare to numerical simulations. Knowing not much about both simulations in general and your specific case, I can't tell much more; the only common difficulty with this approach is that you may need quite large values of $L$, since you need to extract the coefficient of the logarithm and separate it from the non-universal constant (and other corrections vanishing for large $L$), however you are probably able to handle this with MC simulations. You also need to account for the finite size of the system, which is nicely discussed by Cardy and Calabrese. You can also find much more in the literature, such as universal and non-universal corrections to increase your precision, more complicated setups etc.

All of this strictly holds for 1-d systems. You may be interested in higher dimensional models, of which I know little about. If you work with a 2-d classical model, you might be able to exploit the correspondence between 1-d quantum and 2-d classical models, and use the same results. Many similar results are available in the literature, but they are usually not as nice and concise. Cardy discusses some in this paper.

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  • $\begingroup$ I'm sorry about the sketchiness of the last part of the answer, my knowledge is mostly limited to analytic results in 1+1d. I hope you can find something more helpful in the literature. $\endgroup$ – fqq Sep 10 '14 at 18:17
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Well, theoretical suggestions are as many;  but lab techniques suggest for scaled electroscopes to do as such. There also are technical issues

Measuring the q charge - contacting method 1) Sure one has to contact the "point" charge with the top head of the electroscope. 2) The scale has to agree with the physical systems of measurements, SI or else.

Measuring the V potential capabilities - induction method 1) Sure one has to keep the "point" charge at several distances with the top head of the electroscope. 2) The scale has to agree with the physical systems of measurements, SI or else.

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    $\begingroup$ Is you answer about measuring an electric charge? I'm interested in 'measuring' numerically the central charge which is a number characterizing a conformal field theory. Thank you anyway, I was confused when I started to read your answer =) $\endgroup$ – Learning is a mess Sep 8 '14 at 10:32
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    $\begingroup$ How do you measure numerically central charge of the system ? Of course, I mean measuring the electrostatic charge, which sure be at the center of any net field or field net, whatever! ( any means theoretical or experimental ) $\endgroup$ – Murtuza Vadharia Sep 11 '14 at 9:44

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