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The AKLT parent Hamiltonian is $H={\sum }_j{P}_{j,j+1}^{(2)}$, where $P^{(2)}$ projects two neighboring spin-one particles onto total spin two. It is frustration free, and the periodic AKLT MPS is its unique ground state on a sufficiently long chain.

For $H={\sum }_{m<n}{c}_{mn}(X_m{X}_n+Y_m{Y}_n)+{\gamma }_{mn}{Z}_m{Z}_n$, Pauli commutators give cancellation between the $XX$ and $YY$ terms in $[H,Z_m+Z_n]$, while each $ZZ$ term commutes individually. Hence total spin magnetization is conserved. Local $Z_k$ controls preserve the same law, so dynamics split into fixed-excitation sectors even when a particular sector is controllable.

A connected isotropic spin-$S$ ferromagnet with positive exchanges has a fully polarized ground state and a maximal-total-spin multiplet. For $N$ sites the total spin is $NS$, with $2NS+1$ magnetic states generated by total-spin lowering. Global rotations span the same degenerate space. Slowly varying reorientations produce gapless ferromagnetic magnons; finite-volume degeneracy must be distinguished from a thermodynamic statement about spontaneous order.

The Heisenberg model couples neighboring or more general pairs of spins through ${\mathbf{\sigma }}_i\cdot {\mathbf{\sigma }}_j$. For all-to-all equal coupling, the Hamiltonian can be written in terms of the square of the total spin.

In a Cartesian basis of the physical spin-one space, the AKLT MPS matrices are proportional to ${\sigma }_x,{\sigma }_y,{\sigma }_z$. Products of two such matrices span the full virtual matrix algebra, and the two-site parent term projects onto total spin two.

A channel's electrical conductivity weights its density of states by squared velocity and the transport momentum relaxation time. In an elementary spin-conserving $s$–$d$ scattering model, ${\tau }_s^{-1}\propto |V_{sd,s}{|}^2{D}_{d,s}(E_F)$, so available heavy final states can reduce the current carried by light states. Band filling, defects and interfaces affect the asymmetry; total spin population alone does not determine its sign. A density-of-states sketch without scattering matrix elements is not a universal transport prediction.

A wavefunction intended for a specified total-spin state contains components of other total spins. For an unrestricted determinant with one alpha and one beta orbital, $⟨S^2⟩=1-|⟨{\varphi }_{\alpha }|{\varphi }_{\beta }⟩{|}^2$ in units of ${\hslash }^2$.

For two spin-one-half particles, the singlet isEvery component of the total spin annihilates it, and ${\mathbf{S}}^2$ has eigenvalue zero.

Coupling three distinguishable spin-one systems giveswhere the labels denote total spin and coefficients denote multiplicities. The dimensions are respectively $1$, $3\cdot 3$, $2\cdot 5$, and $7$, summing to $27$.

The total spin is the vector sum of subsystem spin operators and generates their simultaneous rotation. For a rotationally invariant Heisenberg model, both its squared magnitude and any chosen component commute with the Hamiltonian. An anisotropic exchange or field can preserve only a component.

Because ${\sigma }_i{\sigma }_j={\delta }_{ij}I+i{\epsilon }_{ijk}{\sigma }_k$, the two-site image of the Pauli-matrix AKLT tensor is the direct sum of the scalar and antisymmetric subspaces, which have total spin zero and one. Its orthogonal complement is the symmetric traceless total-spin-two subspace.