Spectral gaps of AKLT Hamiltonians using Tensor Network methods

Abstract

Using exact diagonalization and tensor network techniques we compute the gap for the AKLT Hamiltonian in 1D and 2D spatial dimensions. Tensor Network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies e_0<0, which is a quantum critical point. With the Tensor Network Renormalization Group methods we provide evidence of a finite gap in the thermodynamic limit for the AKLT models in the 1D chain and 2D hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.

10 pages, 12 figures. References added, minor changes

References in corpus (12)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Tensor renormalization group approach to 2D classical lattice models
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Scaling of entanglement support for Matrix Product States
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Novel schemes for measurement-based quantum computation
• Measurement-based quantum computer in the gapped ground state of a two-body Hamiltonian
• Quantum computational capability of a 2D valence bond solid phase
• Quantum computation by local measurement
• Symmetry breaking and criticality in tensor-product states
• Field-induced superfluids and Bose liquids in Projected Entangled Pair States

Cited by in corpus (31)

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• Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice
• Universal measurement-based quantum computation with spin-2 Affleck-Kennedy-Lieb-Tasaki states
• Theory of triangulene two-dimensional crystals
• Hybrid valence-bond states for universal quantum computation
• Quantum spin systems for measurement-based quantum computation
• Detection of symmetry-protected topological order in AKLT states by exact evaluation of the strange correlator
• Preparing Valence-Bond-Solid states on noisy intermediate-scale quantum computers
• Topological polarons, quasiparticle invariants and their detection in 1D symmetry-protected phases
• Symmetry-protected topological order and negative-sign problem for SO(N) bilinear-biquadratic chains
• AKLT models on decorated square lattices are gapped
• Quantum state preparation via engineered ancilla resetting
• Heisenberg antiferromagnet on the Husimi lattice
• Graph states as ground states of two-body frustration-free Hamiltonians
• Efficient verification of Affleck-Kennedy-Lieb-Tasaki states
• Efficient Verification of Ground States of Frustration-Free Hamiltonians
• Some aspects of Affleck-Kennedy-Lieb-Tasaki models: tensor network, physical properties, spectral gap, deformation, and quantum computation
• Phase transitions of a 2D deformed-AKLT model
• The AKLT model on a hexagonal chain is gapped
• Spectral properties for a family of two-dimensional quantum antiferromagnets
• Spectral gaps of two- and three-dimensional many-body quantum systems in the thermodynamic limit
• The AKLT models on the singly decorated diamond lattice and two degree-4 planar lattices are gapped
• Haldane phases and phase diagrams of the S = 3/2, 1 bilinear-biquadratic Heisenberg model on the orthogonal dimer chain
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• The Complexity of Approximating Critical Points of Quantum Phase Transitions
• Deformations of the Boundary Theory of the Square Lattice AKLT Model
• Low-energy dynamics of the Affleck-Kennedy-Lieb-Tasaki model in the one- and two-triplon basis
• Generalization of the Affleck-Kennedy-Lieb-Tasaki Model for Quantum Ferromagnetism