How do you measure numerically central charge of the system ?

[Murtuza Tipu]

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?

 

[alionalizoti]

Measuring electrostatic charge

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.

 

[Murtuza Tipu]

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.

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.

 

[alionalizoti]

Measuring the charge

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 )

 

[sardar]

To measure the central charge numerically in a statistical system, you can use the finite size scaling method. This involves studying the behavior of the system as you vary the size of the lattice. At the conformal point, the correlation length of the system diverges, but the correlation functions remain finite. This leads to a scaling relation between the correlation length and the size of the system, which can be expressed as:

ξ ∝ L^(1/ν)

where ξ is the correlation length, L is the size of the system, and ν is the correlation length exponent. The central charge can then be calculated using the following formula:

c = 6(1 - 1/ν)

By performing Monte-Carlo simulations on lattices of different sizes, you can obtain a numerical value for the correlation length exponent and use it to calculate the central charge.

Additionally, the central charge can also be calculated using the entanglement entropy of the system. The entanglement entropy measures the amount of entanglement between two regions of a system. At the conformal point, the entanglement entropy follows a logarithmic scaling relation, which can be expressed as:

S ∝ (c/6)log(L)

where S is the entanglement entropy and L is the size of the system. By measuring the entanglement entropy for different sizes of the system, you can obtain a numerical value for the central charge.

In summary, there are systematic methods for numerically measuring the central charge in a statistical system at the conformal point. These methods involve studying the scaling behavior of the system and using the scaling relations to calculate the central charge. These techniques have been successfully applied in various Monte-Carlo simulations to obtain the central charge of different systems.