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.

For two spin one-half systems, the dimensionless isotropic Heisenberg model coupling is $C=X\otimes X+Y\otimes Y+Z\otimes Z$, where $X,Y,Z$ are Pauli matrices. In the ordered computational basis $(|00⟩,|01⟩,|10⟩,|11⟩)$,where $S$ is the SWAP gate. This follows by acting on each basis state: the $XX+YY$ terms exchange the two opposite-spin states with coefficient two and annihilate the parallel-spin states. The symmetric three-dimensional sector has eigenvalue $1$ and the antisymmetric sector has eigenvalue $-3$. HenceThis identity explains how isotropic exchange interaction generates a SWAP gate at $a=\pi /2$ up to a global phase.

For $H=(\Delta /4){\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2-({\mu }_{0}B/2)(g_1{\sigma }_{1z}+g_2{\sigma }_{2z})$, the parallel-spin energies are $\Delta /4\mp {\mu }_{0}B(g_1+g_2)/2$. The zero-projection block has energies $-\Delta /4\pm \sqrt{{\Delta }^2+[{\mu }_{0}B(g_1-g_2){]}^2}/2$. Signed moment conventions must be fixed before assigning a physical opposite-charge case.

In the two-spin exchange spectrum with unequal signed magnetic moments, the axial Hamiltonian conserves $M=S_{1z}+S_{2z}$. Parallel-spin states have $M=\pm \hslash$, while the two mixed states lie in the $M=0$ block. Levels in different $M$ sectors may cross as the axial magnetic field changes because the conserved projection forces their coupling matrix element to vanish.

This is a symmetry qualification of the non-crossing rule. A nonzero isotropic exchange interaction does not by itself turn every field-dependent crossing into an avoided crossing. A transverse perturbation that breaks conservation of $M$ can mix the sectors and open a gap, provided its relevant matrix element is nonzero. Plotting the energy eigenvalues by their conserved projection therefore distinguishes protected crossings from the avoided splitting already present within the $M=0$ block.

If $g_2=-g_1$ in a shared magneton convention, the Zeeman energy of both parallel-spin states cancels. Their energy stays $\Delta /4$, with exact degeneracy two for every field. Opposite charges alone do not enforce this equality because the masses and magnetic factors may differ.

If the differential Zeeman energy $|w|=|{\mu }_{0}B(g_1-g_2)|$ greatly exceeds $|\Delta |$, the two mixed levels are $-\Delta /4\pm |w|/2+O({\Delta }^2/|w|)$ and approach opposite-spin product states. Four energies are generically distinct. Equal signed moments have $w=0$, so that block remains singlet/triplet rather than being unmixed by a stronger field.

The XXZ spin chain has equal transverse exchange interactions and a possibly different longitudinal coupling. For $J>0$ the transverse coupling is antiferromagnetic, while the sign and size of $\Delta$ determine the longitudinal preference. A periodic even chain is bipartite. Its continuous symmetry is rotation about $z$ for generic $\Delta$, enhanced to full spin rotation at $\Delta =1$.

For an even periodic classical chain with spin length $S$ and $J>0$, each bond attains the negative largest singular value of $\mathrm{diag}(1,1,\Delta )$. For $\Delta >1$ the two minimizing configurations are alternating $z$ spins. For $|\Delta |<1$ they are planar Néel states with arbitrary in-plane angle. For $\Delta <-1$ the two configurations have uniform $z$ spins. At $\Delta =1$ the Néel direction is arbitrary on a sphere; at $\Delta =-1$ a staggered transverse component and uniform longitudinal component parameterize another spherical degeneracy. These are classical configuration manifolds, not a claim about finite quantum-chain degeneracy.

The classical Heisenberg model assigns a three-component unit vector to each site and couples pairs by their dot product. Positive $J$ favors parallel alignment. With no applied magnetic field or anisotropy, simultaneous rotations of all vectors preserve the energy, so an ordered state chooses a direction and exhibits spontaneous symmetry breaking. The vector order parameter is the spatially averaged orientation. A quartic Landau free energy describes its mean-field onset near a continuous transition; fluctuations can change the critical exponents.^[ref]

For unit three-component classical spins and energy $-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ on a lattice of coordination $z_c$, mean-field decoupling gives $E_{\mathrm{MF}}=-z_{c}JS{\sum }_i{S}_{iz}+N{z}_{c}J{S}^2/2$. Integrating each spin over solid angle yieldsThe constant term corrects double counting. A normalized angular measure removes $(4\pi {)}^N$ but does not change the self-consistency equation. The lattice dimension and spin-space dimension are different data.

The single-site average in molecular field $z_{c}JS$ is $S=\coth x-1/x$, where $x=\beta z_{c}JS$. Thus $\tau x=\coth x-1/x$ with $\tau =k_{B}T/(z_{c}J)$. The small-$x$ slope $1/3$ gives $k_B{T}_c^{\mathrm{MF}}=z_{c}J/3$. On a square lattice $z_c=4$. This finite mean-field transition does not imply actual positive-temperature magnetization for the ideal two-dimensional short-range isotropic model, where the Mermin-Wagner theorem applies.

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.

With $H={\sum }_{ij\mathrm{bonds}}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ counting bonds once, type-I and type-II classical collinear energies per spin are $S^2(-4J_1+3J_2)$ and $-3J_2{S}^2$. Their boundary is $J_2/J_1=2/3$. The corresponding ordering reflections have integer indices with odd sum, and three half-integer indices, respectively, in units of $2\pi /a$. This compares classical order, not an exact quantum product-state ground state.

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 $J{\mathbf{S}}_i\cdot {\mathbf{S}}_j$.