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- About the continuum limit of Ising lattice

I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including

As far as I understood it, the state of the lattice is described with the spin variables ##\sigma_\mathbf{x}##, and additionally the disorder variables ##\mu_{\mathbf{x}}##, where the position vectors ##\mathbf{x}## can form either a discrete lattice or a continuum.

The fermion variables ##\psi_{a,\mathbf{x}}## are defined as

##\psi_{a,\mathbf{x}} = \sigma_{\mathbf{x}}\mu_{\mathbf{x}+\mathbf{e}_a}##

where ##a## seems to be the spinor index (taking values from 1 to 4) and the ##\mathbf{e}_a## are the four lattice unit vectors.

Some questions:

1. How can the ##\psi_{a,\mathbf{x}}## be a complex field obeying the Dirac equation in the continuum limit if ##\sigma_{\mathbf{x}}## and ##\mu_{\mathbf{x}}## are binary variables taking only values ##\pm 1## ? I would expect the components of a Dirac spinor to be able to have a plane-wave dependence on the position ##\mathbf{x}##, so at least the complex phase of ##\psi## should be able to be something other than ##0## or ##\pi##.

2. Are there other ways to define the ##\psi_{a,\mathbf{x}}## without changing anything physically relevant in the model? In other words, is it possible to reconstruct the ##\sigma_{\mathbf{x}}## for all ##\mathbf{x}## if I know ##\psi_{a,\mathbf{x}}## for all ##\mathbf{x}##, or do several sets of ##\sigma_{\mathbf{x}}## produce the same set of ##\psi_{a,\mathbf{x}}## (or the other way around)?

As far as I understood it, the state of the lattice is described with the spin variables ##\sigma_\mathbf{x}##, and additionally the disorder variables ##\mu_{\mathbf{x}}##, where the position vectors ##\mathbf{x}## can form either a discrete lattice or a continuum.

The fermion variables ##\psi_{a,\mathbf{x}}## are defined as

##\psi_{a,\mathbf{x}} = \sigma_{\mathbf{x}}\mu_{\mathbf{x}+\mathbf{e}_a}##

where ##a## seems to be the spinor index (taking values from 1 to 4) and the ##\mathbf{e}_a## are the four lattice unit vectors.

Some questions:

1. How can the ##\psi_{a,\mathbf{x}}## be a complex field obeying the Dirac equation in the continuum limit if ##\sigma_{\mathbf{x}}## and ##\mu_{\mathbf{x}}## are binary variables taking only values ##\pm 1## ? I would expect the components of a Dirac spinor to be able to have a plane-wave dependence on the position ##\mathbf{x}##, so at least the complex phase of ##\psi## should be able to be something other than ##0## or ##\pi##.

2. Are there other ways to define the ##\psi_{a,\mathbf{x}}## without changing anything physically relevant in the model? In other words, is it possible to reconstruct the ##\sigma_{\mathbf{x}}## for all ##\mathbf{x}## if I know ##\psi_{a,\mathbf{x}}## for all ##\mathbf{x}##, or do several sets of ##\sigma_{\mathbf{x}}## produce the same set of ##\psi_{a,\mathbf{x}}## (or the other way around)?

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