Sigmajz

Sigma_jz, often written as σ_j^z, denotes the z-component of the Pauli spin operator acting on the j-th spin in a system of N spin-1/2 particles. It is a Hermitian, traceless 2x2 matrix with eigenvalues ±1, corresponding to spin-up and spin-down along the z axis. On the full Hilbert space, it acts as I ⊗ ... ⊗ σ^z ⊗ ... ⊗ I, nontrivially only on site j.

Properties of σ_j^z include that, for different sites, σ_j^z commutes with σ_k^z (j ≠ k). It anticommutes with σ_j^x and σ_j^y on the same site. The total magnetization along z is S^z = (ħ/2) Σ_j σ_j^z; in lattice spin models a dimensionless form M_z = (1/2) Σ_j σ_j^z is commonly used.

Applications of σ_j^z appear across physics. In many-body and statistical mechanics, it features in Ising-type Hamiltonians such as H = -J Σ_{⟨jk⟩} σ_j^z σ_k^z - h Σ_j σ_j^x (the transverse-field Ising model). It also appears in XXZ and Heisenberg models through combinations of Pauli operators. In quantum information, measuring σ_j^z corresponds to projecting the j-th qubit onto the Z basis, with eigenstates |0⟩ and |1⟩ often identified with spin-up and spin-down.

Notes: the symbol may be written as σ_j^z or σ_jz depending on the text. It is a central operator for describing spin polarization, magnetization, and qubit readout in various experimental and theoretical contexts.