The quantum Heisenberg model, developed by Werner Heisenberg, is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mechanically. It is related to the prototypical Ising model, where at each site of a lattice, a spin σ i ∈ { ± 1 } {\displaystyle \sigma _{i}\in \{\pm 1\}} represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction.
Overview For quantum mechanical reasons (see exchange interaction or Magnetism § Quantum-mechanical origin of magnetism), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are aligned. Under this assumption (so that magnetic interactions only occur between adjacent dipoles) and on a 1-dimensional periodic lattice, the Hamiltonian can be written in the form
H ^ = − J ∑ j = 1 N σ j σ j + 1 − h ∑ j = 1 N σ j {\displaystyle {\hat {H}}=-J\sum _{j=1}^{N}\sigma _{j}\sigma _{j+1}-h\sum _{j=1}^{N}\sigma _{j}} , where J {\displaystyle J} is the coupling constant and dipoles are represented by classical vectors (or "spins") σj, subject to the periodic boundary condition σ N + 1 = σ 1 {\displaystyle \sigma _{N+1}=\sigma _{1}} . The Heisenberg model is a more realistic model in that it treats the spins quantum-mechanically, by replacing the spin by a quantum operator acting upon the tensor product ( C 2 ) ⊗ N {\displaystyle (\mathbb {C} ^{2})^{\otimes N}} , of dimension 2 N {\displaystyle 2^{N}} . To define it, recall the Pauli spin-1/2 matrices
σ x = ( 0 1 1 0 ) {\displaystyle \sigma ^{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}} ,
σ y = ( 0 − i i 0 ) {\displaystyle \sigma ^{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}} ,
σ z = ( 1 0 0 − 1 ) {\displaystyle \sigma ^{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}} , and for 1 ≤ j ≤ N {\displaystyle 1\leq j\leq N} and a ∈ { x , y , z } {\displaystyle a\in \{x,y,z\}} denote σ j a = I ⊗ j − 1 ⊗ σ a ⊗ I ⊗ N − j {\displaystyle \sigma _{j}^{a}=I^{\otimes j-1}\otimes \sigma ^{a}\otimes I^{\otimes N-j}} , where I {\displaystyle I} is the 2 × 2 {\displaystyle 2\times 2} identity matrix. Given a choice of real-valued coupling constants J x , J y , {\displaystyle J_{x},J_{y},} and J z {\displaystyle J_{z}} , the Hamiltonian is given by
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