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 ...

  Information geometry of 1D Ising model in complex magnetic field regime

+ 5 like - 0 dislike
1250 views

Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by

$$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = 1}^N\sigma_i$$

where $\sigma = \{\sigma_i\}_{i = 1,\dots, N}\in\Omega := \{\pm 1\}^N$, $\{J_i\}_{i = 1,\dots, N}$ are nearest neighbor interaction strength couplings, and $h \in \mathbb{R}$ is the magnetic field. Let's consider the ferromagnetic case, that is, $J_i \geq 0$ for $i = 1, \dots, N$, and for the sake of simplicity (though this doesn't matter in the thermodynamic limit), take periodic boundary conditions. Neither in the finite volume, nor in the thermodynamic limit does this model exhibit critical behavior for finite temperatures.

On the other hand, as soon as we allow $h$ to be complex (and fix the temperature), even in the finite volume $N$, the partition function has zeros as a function of $h$. In the thermodynamic limit these zeros accumulate on some set on the unit circle in the complex plane (Lee-Yang circle theorem).

Now the question: let's consider information geometry of the Ising model, as described above, when $h$ is real. In this case the induced metric is defined and the curvature does not develop singularities (obviously, since there are no phase transitions). Now, what about information geometry of the Ising model when $h$ is complex? This is a bit puzzling to me, since then the partition function attains zeros in the complex plane, so that the logarithm of the partition function is not defined everywhere on the complex plane, and the definition of metric doesn't extend directly to this case (the metric involves the log of the partition function), let alone curvature.

Is anyone aware of any literature in this direction? I thought it would be a good idea to ask before I try to develop suitable methods from scratch.

One could of course try to define the metric and curvature first for real $h$ and only then extend the final formulas to complex $h$. This seems a bit unnatural to me, and even dangerous.

EDIT: Allow me to elaborate on what I mean by "information geometry." Let us consider for simplicity the finite volume model (i.e., $N < \infty$ above). The Gibbs state (i.e., the probability distribution on $\Omega$ of maximal entropy), given by

$$P(\sigma) = \frac{e^{\beta H(\sigma)}}{\sum_{\sigma\in\Omega} e^{\beta H(\sigma)}}$$

where $\beta$ is inverse temperature, obviously depends on the temperature and the magnetic field. So it is convenient to write, for example, $P := P_{(\beta, h)}$ to make this dependence explicit.

Now, the Gibbs states can be identified with points in the parameter space

$$M := \{(\beta, h): \beta, h\in (0,\infty)\}$$

On this space one can define a metric, the so-called Fisher information metric, that naturally measures the distance between two Gibbs states $P_{(\beta, h)}$ and $P_{(\beta', h')}$. The definition of this metric (as you may have guessed!) involves the partition function. This metric then induces a geometry on the parameter space $M$, the so-called statistical manifold (see, for example, http://en.wikipedia.org/wiki/Information_geometry for more details). The curvature (induced by the metric) is an interesting quantity to study. As it turns out, the curvature develops singularities at phase transitions (also something you may have guessed, since the metric involves the partition function, and hence so does the curvature tensor).

Actually, the definition of the metric involves the logarithm of the partition function. All is fine so far, since $h$ is real (assume nonzero), and everything is well-defined. However, as soon as we pass to complex $h$, the partition function admits zeros, and it is no longer clear (at least to me) how the above constructions should generalize. See, for example, Section 4 in http://eprints.nuim.ie/268/1/0207180.pdf).

This post has been migrated from (A51.SE)
asked Jan 6, 2012 in Theoretical Physics by WNY (45 points) [ no revision ]
cross-listed from http://physics.stackexchange.com/q/19132/2451

This post has been migrated from (A51.SE)
While I have a reasonable amount of experience with the Ising model, I'm not quite clear how exactly it makes sense to make $h$ complex. The dynamics of the system would then not only not be unitary, but would not even conserve total probability. Could you clarify?

This post has been migrated from (A51.SE)
Lee and Yang proposed, in their classical papers (C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952); T. D. Lee and C. N. Yang, ibid. 87, 410 (1952).) to study phase transitions in terms of these zeros. Later, Fisher extended their ideas to complex-temperature plane. Please see Lee-Yang theory in references above. (I should mention that I'm a mathematician, and am a bit lacking on the physical interpretations. Could you point me to a place where I can learn about the dynamics you mentioned above? Thanks!)

This post has been migrated from (A51.SE)
@JoeFitzsimons to consider $h$ in the complex plane is just a trick that allow us to use the powerful results of complex analysis in order to prove real analyticity of pressure and other thermodynamical functions which some models implies the absence of first order phase transition, for example.

This post has been migrated from (A51.SE)
@WNY could you elaborate more in this sentence : " Now the question: let's consider information geometry of the Ising model, as described above, when h is real." How and where the metric is defined ?

This post has been migrated from (A51.SE)
@Leonardo: Your comment above to Joe is excellent. Don't mean to be immodest, but that is exactly how I proved analyticity in https://webfiles.uci.edu/wyessen/papers/piq.pdf. Now, for your question, see the edit in the original post (editing it now...)

This post has been migrated from (A51.SE)
@WNY: I meant that if you plug such a Hamiltonian into the time dependent Schroedinger equation you get completely unphysical results. I understand from the above comments that you are simply using complex analysis on the conventional Hamiltonian, so I think my confusion has been cleared up. Sorry.

This post has been migrated from (A51.SE)
@WNY, probably the Fisher metric diverges at these zeros, so what? Why do you expect it to be defined there?

This post has been migrated from (A51.SE)
@Squark: I know that the Fisher metric diverges at those points. The reason I want it to be defined, is so that we get meaningful geometry. Otherwise, the statistical manifold is just a set, and it isn't clear what sort of topology we should assign on it, to be able to read statistical information from this topology. Your question seems to me a bit naive; it is like saying: why do we need the Fisher metric at all? It is exactly those points that are the interesting ones; to investigate them geometrically, we need a geometry (i.e. a metric). Can you suggest another good candidate for a metric?

This post has been migrated from (A51.SE)
@WNY, probably you should study the geometry of the parameter space with these points removed. The points then become sort-of asymptotic boundaries. It might be though that some structure can be continued to these points e.g. the conformal structure.

This post has been migrated from (A51.SE)
@Squark: Good point, and I am suggesting that in the original post. But I wanted to see whether any literature in this direction exists.

This post has been migrated from (A51.SE)

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$\hbar$ysicsOve$\varnothing$flow
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
...