Spin operator matrix elements in the quantum Ising chain: fermion approach

Using some modification of the standard fermion technique we derive factorized formula for spin operator matrix elements (form-factors) between general eigenstates of the Hamiltonian of quantum Ising chain in a transverse field of finite length. The derivation is based on the approach recently used to derive factorized formula for Z_N-spin operator matrix elements between ground eigenstates of the Hamiltonian of the Z_N-symmetric superintegrable chiral Potts quantum chain. The obtained factorized formulas for the matrix elements of Ising chain coincide with the corresponding expressions obtained by the Separation of Variables Method.

💡 Research Summary

The paper presents a rigorous derivation of factorized expressions for spin‑operator matrix elements (form‑factors) in the finite‑size quantum Ising chain (QIC) subjected to a transverse magnetic field. The authors start from the standard Hamiltonian
(H=-\sum_{j=1}^{L}\sigma_{j}^{x}\sigma_{j+1}^{x}-\lambda\sum_{j=1}^{L}\sigma_{j}^{z})
and recall the conventional Jordan‑Wigner transformation that maps spin operators to Majorana fermions. While the Jordan‑Wigner map is sufficient for diagonalizing the Hamiltonian, it does not directly yield compact formulas for off‑diagonal spin matrix elements. To overcome this limitation, the authors introduce a modified fermionic representation: after the Jordan‑Wigner step they perform a Bogoliubov rotation that defines new quasiparticle operators (often called Bogoliubov fermions). This two‑stage transformation respects the boundary conditions (periodic or antiperiodic) and the value of the transverse field (\lambda), and it reorganizes the Hilbert space into sectors labelled by the occupation numbers of the Bogoliubov modes.

With the new basis, any eigenstate (|\Phi_{{k}}\rangle) of the QIC can be written as a product of creation operators acting on the Bogoliubov vacuum. The spin operator (\sigma_{j}^{x}) (or (\sigma_{j}^{z})) is expressed as a bilinear combination of the Bogoliubov fermions, which in turn can be interpreted as a sum over mode‑changing operators. Consequently, the matrix element between two arbitrary eigenstates (|\Phi_{{k}}\rangle) and (|\Phi_{{p}}\rangle) reduces to a sum of determinants of Cauchy‑type matrices whose entries are simple rational functions of the mode momenta (k_{a}) and (p_{b}).

Exploiting the classic Cauchy determinant identity, the authors factorize each determinant into a product of simple sine ratios. The final result for the form‑factor takes the compact, fully factorized form
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