• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,075 questions , 2,226 unanswered
5,348 answers , 22,744 comments
1,470 users with positive rep
818 active unimported users
More ...

  How do physicists use solutions to the Yang-Baxter Equation?

+ 7 like - 0 dislike

As a mathematician working the area of representation of Quantum groups, I am constantly thinking about solutions of the Yang-Baxter equation. In particular, trigonometric solutions.

Often research grants in this area cite this as an "application" of their research. This being said, many mathematicians(definitely including myself) don't know why these solutions are important. So, I wonder;

What exactly do physicists do with solutions to the Yang-Baxter Equation once they have them?


This post imported from StackExchange Physics at 2014-04-01 16:25 (UCT), posted by SE-user BBischof
asked Nov 2, 2010 in Theoretical Physics by BBischof (35 points) [ no revision ]
I do not really work in this part of the field, but my very vague impression is that they are primarily useful in finding exact solutions for lattice models for statistical systems in 2D. These "integrable models" may or may not have much direct relevance to the real world, but are theoretically interesting, as most realistic statistical mechanical models admit no analytical solutions. Hopefully someone else will come along and correct me or expand on this further.

This post imported from StackExchange Physics at 2014-04-01 16:25 (UCT), posted by SE-user j.c.
@j.c. Well thanks for these initial thoughts. :)

This post imported from StackExchange Physics at 2014-04-01 16:25 (UCT), posted by SE-user BBischof

1 Answer

+ 4 like - 0 dislike

Ah. Finally a topic I know something about !

There are many places in physics where the YB equation pops up. I can think of two at the moment.

a. Exactly solvable lattice models

b. Quantum Computation (QC)

It is the second application I find most exciting, so I'll focus on it.

The canonical reference (IMHO) on the link between the YB equation and QC is the wonderful paper by Lomonaco and Kauffmann (LK04) http://arxiv.org/abs/quant-ph/0401090

In topological quantum computation, the hope is to be able to perform unitary operations on qubits by moving them around each other. A typical arena is a 2D electron gas, where our qubits are the quasiparticles of the system. In 2D when we exchange two objects we get a richer symmetry group than in 3D, where we get the permutation group whose eigenvalues $ \pm 1$ correspond to the case of bosons and fermions respectively. However, in 2D this symmetry group is enlarged to the braid group - one can exchange two objects by moving them around such that their worldlines "braid" around each other. This braiding cannot be eliminated by deforming the trajectories, because we don't have the third dimension to utilize.

Anyhow to cut a long story short, the YBE can be shown diagramatically as a relationship between three particles under exchange (see fig. 1 on pg. 8 of above ref.). What LK04 then show is that solutions of the YBE are unitary matrices which are universal for quantum computation. In much the same way that any classical binary circuit can be built out of NAND gates along, any quantum circuit can be built out of a set of universal quantum gates.

This post imported from StackExchange Physics at 2014-04-01 16:25 (UCT), posted by SE-user user346
answered Nov 11, 2010 by Deepak Vaid (1,985 points) [ no revision ]
Awesome, thanks.

This post imported from StackExchange Physics at 2014-04-01 16:25 (UCT), posted by SE-user BBischof

Your answer

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):
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:
Then 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).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights