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With Fourier convention $c_n=N^{-1/2}{\sum }_k{e}^{-ikn}{c}_k$, the Transverse-field Ising model has $H=J{\sum }_k{\Psi }_k^{†}[(g-\cos k){\sigma }_z-\sin k{\sigma }_x]{\Psi }_k$. Expanding the spinor retains the constant $-JgN$ and pairs $k$ with $-k$. The physical quasiparticle energy is twice the positive matrix eigenvalue times $J$, because both momentum partners occur in the Nambu sum.

At the critical field,The quasiparticle gap closes and the low-energy branch is linear, unlike a massive excitation. This is the Ising-chain quantum critical point of the Transverse-field Ising model; the cusp in the nonnegative dispersion represents the two opposite propagation directions.

The Transverse-field Ising modelcommutes with translations, reflections, and the global spin flipFor $\lambda \gg 1$, the unique product ground state has $Z_i=-1$ at every site; for $\lambda \ll -1$, it has $Z_i=+1$. Both are symmetric paramagnets. For $|\lambda |\ll 1$, the two thermodynamic ground states are ferromagnets $|+{⟩}_X^{\otimes L}$ and $|-{⟩}_X^{\otimes L}$, exchanged by $G$; on a finite ring their even and odd cat combinations have an exponentially small splitting.

The large-field phase has a unique symmetry-preserving ground state, whereas the small-field phase spontaneously breaks the ${\mathbb{Z}}_2$ symmetry. These distinct symmetry realizations cannot be connected while retaining both a nonzero thermodynamic gap and the symmetry, so a phase transition must intervene. Kramers--Wannier duality locates the self-dual transitions at $|\lambda |=1$.