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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.

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Excellent question :-) It's out of my area of expertise so I don't think I can provide an answer, but hopefully someone else can. (If you don't get anything you could ask on [theoreticalphysics.SE], but give it a couple days to see if anyone answers here first.) – David Z Jan 5 '12 at 5:32
Indeed, it is natural for two reasons. 1) because in one dimension complexification is the only way to get critical behavior, so it is natural to ask how this is reflected in information geometry; 2) consider the one dimensional quantum Ising model with transverse magnetic field. This model is equivalent to a tight binding Hamiltonian. Question: is there a relation between the spectrum of this Hamiltonian and partition function zeros in classical model with complex field? Idea: consider information geometry of both, and see if there is some relation (a metric preserving diffeomorphism, say). – user12918723509187 Jan 5 '12 at 5:42
Another quick comment: one might try calculating the metric and the curvature for real $h$ first, and only then continue the results to complex $h$. But in my opinion this is very dangerous, and seems rather artificial. – user12918723509187 Jan 5 '12 at 5:58
cross-listed on – Qmechanic Jan 6 '12 at 7:27
up vote 2 down vote accepted

Should have read the cross-list first - you are already aware of the reference below :)

This might be of some use:

B P Dolan, D A Johnston and R Kenna The information geometry of the one-dimensional Potts model J. Phys. A: Math. Gen. 35 (2002) 9025–9035 [arXiv:cond-mat/0207180]

An information geometry metric is calculated for real h for 1D Potts/Ising models and then naively continued to complex h "to see what happens", the resulting curvature diverges along the Lee-Yang line.

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