Ground states of the infinite q-deformed Heisenberg ferromagnet

Abstract

We set up a general structure for the analysis of ``frustration-free ground states'', or ``zero-energy states'', i.e., states minimizing each term in a lattice interaction individually. The nesting of the finite volume ground state spaces is described by a generalized inductive limit of observable algebras. The limit space of this inductive system has a state space which is canonically isomorphic (as a compact convex set) to the set of zero-energy states. We show that for Heisenberg ferromagnets, and for generalized valence bond solid states, the limit space is an abelian C*-algebra, and all zero-energy states are translationally invariant or periodic. For the $q$-deformed spin-$1/2$ Heisenberg ferromagnet in one dimension (i.e., the XXZ-chain with S${}_q$U(2)-invariant boundary conditions) the limit space is an extension of the non-commutative algebra of compact operators by two points, corresponding to the ``all spins up'' and the ``all spins down'' states, respectively. These are the only translationally invariant zero-energy states. The remaining ones are parametrized by the density matrices on a Hilbert space, and converge weakly to the ``all up'' (resp.\ ``all down'') state for shifts to $-\infty$ (resp.\ $+\infty$).

42 pages, plain TeX, no figures

References in corpus (7)

• Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
• Correlation Functions of the XXZ model for $\Delta <-1$
• Matrix-product-groundstates for one-dimensional spin-1 quantum antiferromagnets
• q-deformations of quantum spin chains with exact valence-bond ground states
• Quantum spin chains with quantum group symmetry
• The classical limit of quantum theory
• On the Two-Point Correlation Function for the $U_q[SU(2)]$ Invariant Spin One-Half Heisenberg Chain at Roots of Unity

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• The classical limit of quantum theory
• The complete set of infinite volume ground states for Kitaev's abelian quantum double models
• The spectral gap of the ferromagnetic XXZ chain
• Convex effective potential of $O(N)$-symmetric $ph{i}^4$ theory for large $N$
• Some properties for the low-lying spectrum of the ferromagnetic, quantum XXZ spin system
• Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model
• Interfaces and droplets in quantum lattice models
• Rigidity of interfaces in the Falicov-Kimball model
• Entanglement subvolume law for 2D frustration-free spin systems
• Expansions for Droplet States in the Ferromagnetic XXZ Heisenberg Chain
• A continuum approximation for the excitations of the (1,1,...,1) interface in the quantum Heisenberg model
• Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Transport of interface states in the Heisenberg chain
• Interface states of quantum spin systems
• Isolated Eigenvalues of the Ferromagnetic Spin-J XXZ Chain with Kink Boundary Conditions
• Path integral representations for the spin-pinned quantum XXZ chain
• Low Temperature Results for the Heisenberg XXZ and XY Models
• Interrelations among frustration-free models via Witten's conjugation
• Topologically ordered states in infinite quantum spin systems
• Gapless Excitation above a Domain Wall Ground State in a Flat Band Hubbard Model