Visualization of 1D spin chain wave fuction

Question

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.

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Eventually, I co-wrote a paper on it: [Qubism: self-similar visualization of many-body wavefunctions](http://arxiv.org/abs/1112.3560).

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Answer 1

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.

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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.

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@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.

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Answer 2

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.

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@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).

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@PiotrMigdal, point. But, I think you could use it for the cross-correlation between sites.

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Answer 3

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

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