Heisenberg model

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.

With $S={\sum }_{n=1}^N{Z}_n$, a computational basis vector containing $k$ excitations has eigenvalue $N-2k$. There are $N+1$ distinct eigenspaces, of dimensions $\left(\genfrac{}{}{0ex}{}{N}{k}\right)$. Any Hamiltonian commuting with $S$ preserves each sector and their population weights. In particular, a state supported in one sector cannot be transferred to a different sector by such controls.

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.

For a network of $N$ spin-one-half sites with $Z|0⟩=|0⟩$ and $Z|1⟩=-|1⟩$, the single-excitation subspace has exactly one spin in state $|1⟩$ and dimension $N$. Excitation-conserving exchange interactions preserve it. Its restricted Hamiltonian is a hopping matrix whose entries connect the corresponding sites. Controllability within this subspace need not imply controllability on the full $2^N$-dimensional space.

The isotropic XY exchange interaction between two qubits is $E=(X\otimes X+Y\otimes Y)/2$. Its action isIndeed, $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Y|0⟩=i|1⟩$, and $Y|1⟩=-i|0⟩$. The two terms cancel on equal bits and add on unequal bits. Thus a network Hamiltonian operator $H={\sum }_{a,b}{K}_{ab}{E}_{ab}$ preserves excitation number; in the single-excitation subspace, it is the weighted adjacency matrix, moving an excitation from $a$ to $b$ with hopping coefficient $K_{ab}$.

Consider $q$ identical length-$n$ arms joined to one hub, with XY exchange interactions of strength $J_0/\sqrt{q}$ from the hub to each first site and strength $J_k$ between sites $k,k+1$ on each arm. Let $|s_0⟩$ be the hub excitation and $|s_k⟩=q^{-1/2}{\sum }_{a=1}^q|a,k⟩$. This orthonormal symmetric single-excitation subspace is invariant, and its restricted Hamiltonian operator iswith endpoint terms omitted. At the first link the $q$ hub contributions add to $J_0$; all other links act identically on the arms. Thus the linear isometry $F|k⟩=|s_k⟩$ intertwines $H$ with the path Hamiltonian operator $H_T={\sum }_{k=0}^{n-1}{J}_k(|k⟩⟨k+1|+|k+1⟩⟨k|)$: $HF=F{H}_T$. Expanding the matrix exponential gives $e^{-iHt}F=F{e}^{-i{H}_{T}t}$. Perfect quantum state transfer from zero to $n$ on the path therefore produces an equal excitation superposition at the $q$ arm tips. Since all other qubits are zero, the tip state is a pure W state, up to the transfer's common phase.

In the single-excitation subspace, take a connected real nearest-neighbor hopping matrix $A$ with every coupling nonzero and endpoint control $B=\mathrm{diag}(-1,1,\dots ,1)$. Endpoint commutators isolate the first transition, then propagate along the chain to generate both off-diagonal skew-Hermitian directions and adjacent diagonal differences. These generate $\mathfrak{su}(N)$. For $N\ge 3$, $\mathrm{Tr}B=N-2\ne 0$ supplies the central direction, giving $\mathfrak{u}(N)$. For $N=2$ the algebra is $\mathfrak{su}(2)$, which still gives full state controllability but omits an arbitrary global phase. A broken coupling separates an unreachable chain component.

A Heisenberg antiferromagnet has positive exchange coupling in $H=J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, so neighboring spins energetically prefer antiparallel alignment.

A bipartite spin rotation rotates the spin on one sublattice by $\pi$, converting an alternating classical Néel state to an all-up reference state. Rotation about $x$ sends $S^z\mapsto -S^z$ and $S^{\pm }\mapsto S^{\mp }$ on that sublattice. For an isotropic nearest-neighbor Heisenberg antiferromagnet, a bond becomes $-S_i^z{S}_j^z+(S_i^{+}{S}_j^{+}+S_i^{-}{S}_j^{-})/2$. The transformation is unitary and preserves the spin commutation relations; it changes the reference frame, not the model's spectrum.

A Heisenberg ferromagnet has negative exchange coupling in $H=J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, so neighboring spins energetically prefer parallel alignment.

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.

For two spin-one-half particles,has two eigenvalues $\alpha {\hslash }^2/4$ and two further eigenvaluesFor positive $\alpha$, the zero-field limit has a unique spin-one-half singlet state ground state and a triply degenerate spin-one-half triplet state excited level.

For $N$ spin-one-half systems and one term per unordered pair,Its minimum lies in the smallest available total-spin sector.

For a bipartite spin-one-half Heisenberg antiferromagnet with coordination $z\to \infty$, the two-sublattice mean-field state becomes exact. Antiparallel pure Bloch vectors give bond energy $-J/4$ for the convention $J{\mathbf{S}}_i\cdot {\mathbf{S}}_j$.