Exact dimer ground states for a continuous family of quantum spin chains

Abstract

Using the matrix product formalism, we define a multi-parameter family of spin models on one dimensional chains, with nearest and next-nearest neighbor anti-ferromagnetic interaction for which exact analytical expressions can be found for its doubly degenerate ground states. The family of Hamiltonians which we define, depend on 5 continuous parameters and the Majumdar-Ghosh model is a particular point in this parameter space. Like the Majumdar-Ghosh model, the doubly degenerate ground states of our models have a very simple structure, they are the product of entangled states on adjacent sites. In each of these states there is a non-zero staggered magnetization, which vanishes when we take their translation-invariant combination as the new ground states. At the Majumdar-Ghosh point, these entangled states become the spin-singlets pertaining to this model. We will also calculate in closed form the two point correlation functions, both for finite size of the chain and in the thermodynamic limit.

11 pages

References in corpus (23)

• Surprises on the Way from 1D to 2D Quantum Magnets: the Novel Ladder Materials
• Matrix product states represent ground states faithfully
• Dimerization and Incommensurate Spiral Spin Correlations in the Zigzag Spin Chain: Analogies to the Kondo Lattice
• Experiments on Ladders Reveal a Complex Interplay between a Spin-Gapped Normal State and Superconductivity
• Diverging Entanglement Length in Gapped Quantum Spin Systems
• Matrix-product-groundstates for one-dimensional spin-1 quantum antiferromagnets
• Variational study of hard-core bosons in a 2-D optical lattice using Projected Entangled Pair States (PEPS)
• Renormalization group transformations on quantum states
• Quantum phase transitions in matrix product systems
• Elementary excitations in the $\Delta$-chain
• Non-Haldane Spin Liquid Models with Exact Ground States
• On Quantum Solitons in the Sawtooth Lattice
• Finitely Correlated Generalized Spin Ladders
• Ground state approximation for strongly interacting systems in arbitrary dimension
• Second order quantum renormalisation group of XXZ chain with next nearest neighbour interactions
• Quantum spin models with exact dimer ground states
• Phase diagram of an asymmetric spin ladder
• Semi-classical description of the frustrated antiferromagnetic chain
• Excitations of anisotropic spin-1 chains with matrix product ground state
• Quantum Phase Transitions and Matrix Product States in Spin Ladders
• Exact ground states of spin-2 chains
• Exact ground states of quantum spin-2 models on the hexagonal lattice
• Optimum ground states for spin-3/2 ladders with two legs

Cited by in corpus (10)

• The matrix product representations for all valence bond states
• Quantifying entanglement with scattering experiments
• A complete characterization of the spectrum of the Kitaev model on spin ladders
• Control of entanglement transitions in quantum spin clusters
• Exact models for trimerization and tetramerization in spin chains
• Fourfold degenerate columnar-dimer ground state in square lattice antiferromagnets
• Exact symmetry breaking ground states for quantum spin chains
• Exact ground states for two new spin-1 quantum chains, new features of matrix product states
• DMRG studies on linear-exchange quantum spin models in one dimension
• Classification of matrix product ground states corresponding to one dimensional chains of two state sites of nearest neighbor interactions