Probably I am going to receive many down-votes for this post but I really need to ask this question here.

I am new to statistical mechanics.

I wanted to learn Density Matrix Renormalization Group (DMRG) to simulate a 1D many-body system. Before going into DMRG, I decided to learn RG. But when I searched RG on Google I come to know that there are many other concepts which are related with RG, for example, scaling, infinite correlation length $\zeta$ and universality.

After reading some introductory level material about RG, I feel like I have some idea of all these terms. If I am to give one-liner of all my (naive) understanding about RG, I would say "near a phase transition point, the correlation length is the most important length scale (which diverges to infinity), as we can not have infinite system, so we rescale our finite system to bigger scales and then renormalize the physical quantities of interest."

Now from here, I can not see in which direction DMRG is.

My question

How to get to DMRG from RG? Is there any relation between these two?


The name DMRG is somewhat of a historical accident, and its modern day incarnations are not directly linked to the renormalization group or phase transitions. Instead, it is better understood as a variational technique based on matrix-product states (MPS) ansatzes. Still, there is a historical link between the two tools, and it can be useful to know about it.

Before continuing, your description of RG is a bit restrictive. What you write is RG as it's applied to critical phenomena and phase transitions, which, as you've found already is a very rich field. However, more generally RG is the systematic study of a system as a scale is changed.

Some history

After introducing RG for phase transitions, Ken Wilson introduced a numerical RG technique with the aim to find the groundstate of a 1D problem. It basically boiled down to repeatedly diagonalizing the Hamiltonian of ever larger finite size blocks, keeping the lowest set of states at each iteration. (The block size playing the role of the scale for RG purposes.) This method worked wonders for the Kondo problem, where the coupling to an impurity spin falls off with distance, but fails for ordinary spin models - and famously also for the problem of a particle in a box.

The reason it fails is that keeping only the lowest energy states is not the right method of truncating the Hilbert space basis. Still, we obviously want some way of truncation, or else any hope of large scale computer calculations goes out the window. This is where DMRG originally developed from during the 90s. Steve White invented it with the idea of throwing away states with low probability, which is naturally framed in the language of density matrices. Hence, the origin of DMRG is real-space block RG.

But when you have a changing system size, it's reasonable to think of all this in terms of bipartite entanglement. That is, we can divide the lattice into a system part and an environment part, and calculate the entanglement between the two subsystems. Then DMRG can be reformulated as a way of throwing away high entanglement states, and keeping low entanglement ones. It turns out that most ground states have low entanglement, and are efficiently expressed in MPS states. When this was realized in the 2000s, the RG roots of DMRG were largely shoved under the rug, and DMRG was reformulated as a variational method using a basis of MPS states.


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