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

+ 5 like - 0 dislike

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 ]
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@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)
Most recent comments show all comments
@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)

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