Extracting the central charge of a 1d critical quantum system

DMRG and Numerical Methods
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1

Hi,

I basically want to extract the central charges at the different critical regions of a 1 D quantum system. I know of three methods to do this-

  1. Using PBC in a finite size chain.
  2. Using IDMRG or VUMPS.
  3. Using MERA.

As far as I know the three methods listed above have different levels of accuracy in representing a the ground state of the system, MERA being the most accurate for critical systems.

Now to extract the central charge I know that I can use the form given by Cardy et al.

S= \frac{c}{6} log(l) where l is the subsystem size. So i was thinking in the PBC case I would get the ground state MPS , then see how S scales with l and find the value of c by fitting the known form.

The other method is via IDMRG or VUMPS, where I would use the form S = \frac{c}{6}log(\Xi), where \Xi is the correlation length which i obtain from the spectrum of the transfer matrix.

I can guess the first method from above should be applicable to MERA as well.

Could you point out the pros and cons of all the three ways mentioned above in studying critical systems in particular extracting the central charges. Also are there any other ways one could use to study critical system properties.

Thank you for your response.

Sincerely,
Sunny.

2

Both 1 and 2 are good choices. The nice thing about 1 is that it can work well in practice and the scaling law is clear. For 2, extracting c may be more subtle since you will need to scale with various bond dimensions and relate these to the effective finite correlation length induced by choosing a finite bond dimension.

Using MERA is conceptually the best option for critical systems, but I would not actually recommend it because algorithms for optimizing MERA are not nearly as good as algorithms for optimizing MPS, and MPS can also represent critical systems very well in practice.

There are many other ways to study critical systems, such as studying correlation functions, critical exponents, etc. Please consult the physics literature and books for more information about this.

3

Thanks for your detailed answer.