# Density Matrix Renormalization Group (DMRG) Simulation of a String-Net Model

+ 2 like - 0 dislike
395 views

In the following paper, Dr. Xiao Gang-Wen et. al. introduce the idea that string-net condensed states can be represented in terms of tensor product states:

http://arxiv.org/pdf/0809.2821.pdf

The authors mention the usefulness of DMRG for the study of 1D systems described by TPS, but I would like to know more of the details. How, exactly, would the algorithm proceed for, say, an N=1 string-net model of a spin-1/2 honeycomb lattice? Would it be exceedingly difficult to implement?

This post imported from StackExchange Physics at 2015-05-04 13:49 (UTC), posted by SE-user Bronzeclocksofbenin

edited May 4, 2015

+ 1 like - 0 dislike

A full introduction to the DMRG algorithm definitely does not fit here, and you can find many well-written introductory materials online.

DMRG has been applied to simulate perturbed toric code model, which is the simplest example of string-net model, see http://arxiv.org/pdf/1205.4289.pdf. Generally speaking, DMRG for 2D spin models is indeed much more difficult to implement, however there have been remarkable progress in recent years along this direction. Usually in this type of simulations spin models are placed on a long cylinder. The complexity grows exponentially with the circumference of the cylinder, while the length can be very long, so in a sense this is a quasi-one-dimensional geometry.

I suppose what you looked for is some kind of algorithms that take advantage of the tensor network structure in 2D. A lot of effort has been put into the development of such algorithms, but to date it is fair to say that nothing as useful as DMRG in 1D has come out yet.

This post imported from StackExchange Physics at 2015-05-04 13:49 (UTC), posted by SE-user Meng Cheng
answered May 3, 2015 by (550 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.