Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.

Molecular PhysicsJun 10, 200416 citations

Application of the coherent state representation in the theory of magnetism: I. The heisenberg model

Annals of PhysicsDec 1, 19700 citations

Projective symmetry group classification of chiral Z2 spin liquids on the pyrochlore lattice: Application to the spin- 12 XXZ Heisenberg model

Benedikt Schneider, Jad C. Halimeh, Matthias Punk

Simplified models for quantum spin-ice materials, such as the XXZ model on the pyrochlore lattice, may host interesting new spin-liquid ground states. In this work, the authors provide a complete classification of nearest-neighbor chiral ${\mathbb{Z}}_{2}$ spin-liquid states on the pyrochlore lattice using a projective symmetry group analysis. Moreover, the newly classified states are applied to the XXZ model within a self-consistent mean-field scheme. Observable signatures in neutron scattering experiments are discussed.Simplified models for quantum spin-ice materials, such as the XXZ model on the pyrochlore lattice, may host interesting new spin-liquid ground states. In this work, the authors provide a complete classification of nearest-neighbor chiral ${\mathbb{Z}}_{2}$ spin-liquid states on the pyrochlore lattice using a projective symmetry group analysis. Moreover, the newly classified states are applied to the XXZ model within a self-consistent mean-field scheme. Observable signatures in neutron scattering experiments are discussed.

arXiv:2107.13574Physical review. B./Physical review. BMar 16, 202220 citations

Bound states in Heisenberg antiferromagnets

Stephen William Lovesey, Ewald Balcar

Physics Letters ASep 1, 19697 citations

Statics and dynamics of quantum XY and Heisenberg systems on graphs

Tobias J. Osborne

We consider the statics and dynamics of distinguishable spin- $\frac{1}{2}$ systems on an arbitrary graph $G$ with $N$ vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the $XY$ Hamiltonian ${H}_{XY}$ , which contains an $XY$ interaction for every pair of spins connected by an edge in $G$ ; and (ii) the Heisenberg Hamiltonian ${H}_{\mathrm{Heis}}$ , which contains a Heisenberg interaction term for every pair of spins connected by an edge in $G$ . We find that the action of the $XY$ (respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence ${G}_{k}$ , $k=0,\dots{},N$ of graphs derived from $G$ (with ${G}_{1}=G$ ). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs ${G}_{k}$ . Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the $XY$ model in a magnetic field on the path consisting of $N$ vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the $XY$ model and Heisenberg model in a magnetic field on the complete graph.We consider the statics and dynamics of distinguishable spin-$\frac{1}{2}$ systems on an arbitrary graph $G$ with $N$ vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the $XY$ Hamiltonian ${H}_{XY}$, which contains an $XY$ interaction for every pair of spins connected by an edge in $G$; and (ii) the Heisenberg Hamiltonian ${H}_{\mathrm{Heis}}$, which contains a Heisenberg interaction term for every pair of spins connected by an edge in $G$. We find that the action of the $XY$ (respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence ${G}_{k}$, $k=0,\dots{},N$ of graphs derived from $G$ (with ${G}_{1}=G$). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs ${G}_{k}$. Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the $XY$ model in a magnetic field on the path consisting of $N$ vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the $XY$ model and Heisenberg model in a magnetic field on the complete graph.

arXiv:quant-ph/0312126Physical Review BSep 12, 200624 citations

A hybrid variational quantum circuit approach for stabilizer states classifiers

Hamna Aslam, Frédéric Holweck

Entanglement classification of pure multipartite quantum states is a challenging problem in quantum information theory that can be mathematically stated as orbit classification for some given group action on the ambient Hilbert space. The group action depends on the grained classification one expects, the finer-grained one being the classification up to local unitary transformation (LU). In this article, we show how a variational quantum circuit approach can be used to learn entanglement orbits, and we apply our findings to build a classifier for four-qubit states.Entanglement classification of pure multipartite quantum states is a challenging problem in quantum information theory that can be mathematically stated as orbit classification for some given group action on the ambient Hilbert space. The group action depends on the grained classification one expects, the finer-grained one being the classification up to local unitary transformation (LU). In this article, we show how a variational quantum circuit approach can be used to learn entanglement orbits, and we apply our findings to build a classifier for four-qubit states.

quant-pharXiv:2511.09430Nov 13, 20250 citations

{\mathbb{Z}}_{2}

spin-liquid states on the pyrochlore lattice using a projective symmetry group analysis. Moreover, the newly classified states are applied to the XXZ model within a self-consistent mean-field scheme. Observable signatures in neutron scattering experiments are discussed.

We consider the statics and dynamics of distinguishable spin-

\frac{1}{2}

systems on an arbitrary graph

G

with

N

vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the

XY

Hamiltonian

{H}_{XY}

, which contains an

XY

interaction for every pair of spins connected by an edge in

G

; and (ii) the Heisenberg Hamiltonian

{H}_{\mathrm{Heis}}

, which contains a Heisenberg interaction term for every pair of spins connected by an edge in

G

. We find that the action of the

XY

(respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence

{G}_{k}

k=0,\dots{},N

of graphs derived from

G

(with

{G}_{1}=G

). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs

{G}_{k}

. Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the

XY

model in a magnetic field on the path consisting of

N

vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the

XY

model and Heisenberg model in a magnetic field on the complete graph.