Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Visualization of 1D spin chain wave fuction

+ 11 like - 0 dislike
1279 views

What are the known methods for visualizing quantum states of one-dimensional spin chains? They can be based either on their wave functions or density matrices.

My particular interest is in plotting eigenstates of 1D spin chains (e.g. Heisenberg model) in a way in which correlations between particles are related to the self-similarity of the image.

As a side note, I found a Wikipedia Commons image with its description on the topic (a thumbnail below). However, it is not supported neither by a peer reviewed publication nor by a decent preprint. The only text related is extremely drafty.

However, I'm the most interested in published results, and even more when they relate properties of the image to the properties of the state.

enter image description here

This post has been migrated from (A51.SE)
asked Sep 20, 2011 in Theoretical Physics by Piotr Migdal (1,260 points) [ no revision ]
Eventually, I co-wrote a paper on it: [Qubism: self-similar visualization of many-body wavefunctions](http://arxiv.org/abs/1112.3560).

This post has been migrated from (A51.SE)

3 Answers

+ 8 like - 0 dislike

You seem to be answering your own question. Obviously there is far too much information to represent in a reasonably sized 2d or 3d plot, so no matter what you do, you will be losing something. If you want a plot that shows correlations between particles, you can do just that: For a spinchain of $N$ sites, create a $N\times N$ image, where the value of the pixel at each point $(i,j)$ is given by the expectation value of the operator $\sigma^A_i \sigma^B_j$, where $\sigma^{A/B}_{i/j} \in \{\sigma_X, \sigma_Y, \sigma_Z\}$. This gives you 9 possible plots.

It seems likely that you may only care about the case where $\sigma^A = \sigma^B$, in which case there are only 3 plots, and you could take the expectation values for $XX$, $YY$ and $ZZ$ as the R, G and B values for a colour plot.

This kind of thing produces things very similar to the type of plot you have in your question.

This post has been migrated from (A51.SE)
answered Sep 20, 2011 by Joe Fitzsimons (3,575 points) [ no revision ]
I edited my question not to seem that I am answering myself. I know quite a few methods how to make a plot out of a spin chain wave function. However, I'm searching for references, especially with detailed analysis of the graphical representations.

This post has been migrated from (A51.SE)
@PiotrMigdal: It seems to me that you want references to papers that specifically discuss how to draw such plots, but I doubt these really exist. We have excellent measures of correlation (i.e. concurrence, etc) between sites, but how you plot them is up to you. For two-site correlations, a 2D image or surface plot in 3D of the value of the correlation (for whatever metric you choose) is the natural choice.

This post has been migrated from (A51.SE)
+ 6 like - 0 dislike

The image you have posted looks like a recurrence plot to me. It is primarily used to visualize correlation information in time-series data, but it can be used with any discrete set. A particularly good source of information is this website.

This post has been migrated from (A51.SE)
answered Sep 20, 2011 by rcollyer (240 points) [ no revision ]
@rollyer: Actually, for translation-invariant states recurrence plot is a function only of the difference of the position, i.e. $r(x,x') = f(x-x')$ (tough for other applications it is a nice tool).

This post has been migrated from (A51.SE)
@PiotrMigdal, point. But, I think you could use it for the cross-correlation between sites.

This post has been migrated from (A51.SE)
+ 5 like - 0 dislike

This isn't really an answer on its own, more an addendum to Joe's...

You are not, I think, going to find a "detailed analysis" of how correlation function have been plotted in most papers. Your best bet is to look at various examples - Nature and Science papers will be most helpful for this, as they are heavy on the graphical representation and have space for lots of supplementary material.

A good example of the sorts of plots Joe talked about, in sets that make it easy to see the transitions, is http://www.nature.com/nphys/journal/v7/n5/fig_tab/nphys1919_F6.html

This post has been migrated from (A51.SE)
answered Oct 5, 2011 by Clare (180 points) [ no revision ]

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:
p$\hbar$ysicsOver$\varnothing$low
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
...