Valence bond ground states in isotropic quantum antiferromagnets

Abstract

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds. Table of Contents 1. Introduction 478 2. The One-Dimensional Model (1.1) 2.1. The Ground State 481 2.2. The Ground State Two-Point Correlation Function 485 2.3. The Energy Gap 487 2.4. The Infinite Chain 492 3. The Spin 3/2 Model on the Hexagonal Lattice 3.1. The Ground State 496 3.2. Properties of the Ground State 499 3.3. Proof of Exponential Decay 500 4. VBS States on an Arbitrary Lattice 4.1. The Ground States 504 4.2. The VBS State on the Cayley Tree 506 4.3. SU{n) VBS State and Random Loop Representation 510 4.4. SU(n) Quantum "Spin" System 514

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

“…However, nontrivial SPT states cannot be transformed to product states via the LU transformations that preserve the symmetry, and different SPT states cannot be transformed to each other via the LU transformations that preserve the symmetry. The onedimensional (1D) Haldane phase for spin-1 chain [7,8,33,34] and topological insulators [35][36][37][38][39][40] are nontrivial examples of SPT phases. Some examples of two-dimensional (2D) SPT phases protected by translation and some other symmetries were discussed in Refs.…”

Quantum computational capability of a 2D valence bond solid phase

“…The construction can be visualized like in the Figure 3.2 of Ref. [4] for instance. It is straightforward, for our convenience, to describe it as a tensor network state via the celebrated Schwinger boson method [20],…”

“…This is because conditioned on the specification of the tensor of Eqs. (4,5) to be, for instance, |0 x 1 x | ⊗ |0 x by the outcome 3 2…”

“…Moreover, if such a resource is available as a stable ground state of a naturally-occurring system, that gives us several possibilities to prepare, such as dissipative coolings or adiabatic evolutions, in a similar way that nature presumably does. Our target of the naturallyoccurring two-dimensional (2D) system is the valence bond solid (VBS) phase of spin 3 2 's on the 2D hexagonal lattice, modeled by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [3,4], which is widely recognized as a cornerstone in condensed matter physics. Their VBS construction of the ground state in terms of the distributed spin singlets (or the valence bonds) has become one of most ubiquitous insights in quantum magnetism as well as in high-T c superconductivity, and leads to modern trends of spin liquids and topological orders.…”

Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

“…On the other hand, in the last two decades several different ansatz were introduced in the literature under the general name of matrix product ansatz (MPA). The first formulation was done for the description of the ground-state eigenfunction of some special non-integrable quantum chains, the so called valence-bond solid models [11][12][13][14]. The MPA becomes also a successful tool for the exact calculation of the stationary probability distribution of some stochastic one dimensional systems [15][16][17].…”