Symmetry of hamiltonian under renormalization

[simoncks]

Hi everyone,

Currently, I am self-learning Renormalization and its application to PDEs, nonequilibrium statistical mechanics and also condensed matter. One particularly problem I face is on the conservation of symmetry of hamiltonian during renormalization.

Normally renormalization of hamiltonian, say Ising model in 2D square lattice, must preserve its form, i.e.
H = S_i S_j + h S_i
a) H' = S'_i S'_j + h' S'_i (S' is block spin)
b) H' = S'_i S'_j + h' S'_i + α S'_i S'_j S'_k
Only a) is permissible while b) is not allowed for non-zero α.

1. I want to understand why, for this transformation of S to S', the symmetry is preserved and which symmetry is preserved under this transformation.
2. Why the hamiltonian in this form is guaranteed to work under transformation? i.e., why such transformation guarantees no extra term coming into the hamiltonian? Is there any mathematical theorem behind it?
3. Is there any other example?
4. In reality, the renormalization operation is discrete rather than continuous which discretizes the RG flow into steps of mapping in parameter space. So, we cannot evaluate the transformation of parameters "smoothly". However, this may only be true in Ising model. Is there any possible system which permits a continuous rescaling for continuous renormalization?

Thank you very much.

 

[radium]

The purpose of the renormalization group is to see how a system behaves at different length and energy scales. It consists of rescaling transformations where the couplings scale according to their mass dimension. This will give you differential equations for the couplings, which will be modified if you consider loop corrections (called anomalous dimensions). So this transformation basically just changes the resolution of the system.

When you reach a fixed point, the system is scale invariant for those values of the couplings which in the Ising model corresponds to the critical temperature and the divergence of the correlation length. The critical quantum Ising model corresponds to a free fermion conformal field theory.

The example you are giving is real space renormalization in the Ising model, which is a discretized lattice model. You can rescale continuously if you were in the continuum limit of some field theory, but here you usually want to think of energy scales in momentum space instead of length scales.

 

[simoncks]

May I ask why the rescaled hamiltonian must preserve its form under renormalization? Also, is there any example which demonstrates the rescaling of momentum space continuously in practice, so I can understand how to use a continuous rescaling to change into ODEs of parameter flow?

 

[DrDu]

Whether the form changes or not also depends on the definition of "form". E.g. with a block spin transformation starting from a model with only nearest neighbour interactions, you get a hamiltonian with interactions between all spins. Clearly you also loose symmetry, as the original translations by a (the lattice constant) are no longer symmetries of the renormalized hamiltonian.

 

[radium]

In general, you can think of the rescaling transformation in the continuum (good starting examples are the Gaussian fixed point and phi^4 theory) as generated by an series of infinitesimal rescalings the same way angular momentum generates infinitesimal rotations. This will give you differential equations for the couplings. At tree level you can just write these down from scaling dimensions.