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).

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    $\begingroup$ 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? $\endgroup$ – Joe Fitzsimons Jan 6 '12 at 7:48
  • $\begingroup$ 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!) $\endgroup$ – user3657 Jan 6 '12 at 11:52
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    $\begingroup$ @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. $\endgroup$ – Leandro Jan 6 '12 at 16:55
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    $\begingroup$ @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 ? $\endgroup$ – Leandro Jan 6 '12 at 16:57
  • $\begingroup$ @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. $\endgroup$ – Joe Fitzsimons Jan 6 '12 at 19:34

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