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arXiv:0904.0550 · Physical Review B

Topologically distinct classes of valence-bond solid states with their parent Hamiltonians

Hong-Hao Tu, Guang-Ming Zhang, Tao Xiang, Zheng-Xin Liu, Tai Kai Ng

Published Jul 1, 2009• 36 citations
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Abstract

We present a general method to construct one-dimensional translationally invariant valence-bond solid states with a built-in Lie group $G$ and derive their matrix product representations. The general strategies to find their parent Hamiltonians are provided so that the valence-bond solid states are their unique ground states. For quantum integer-spin-$S$ chains, we discuss two topologically distinct classes of valence-bond solid states: one consists of two virtual $SU(2)$ spin-$J$ variables in each site and another is formed by using two $SO(2S+1)$ spinors. Among them, a spin-1 fermionic valence-bond solid state, its parent Hamiltonian, and its properties are discussed in detail. Moreover, two types of valence-bond solid states with $SO(5)$ symmetries are further generalized and their respective properties are analyzed as well.

DOI
10.1103/physrevb.80.014401

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We present a general method to construct one-dimensional translationally invariant valence-bond solid states with a built-in Lie group GGG and derive their matrix product representations. The general strategies to find their parent Hamiltonians are provided so that the valence-bond solid states are their unique ground states. For quantum integer-spin-SSS chains, we discuss two topologically distinct classes of valence-bond solid states: one consists of two virtual SU(2)SU(2)SU(2) spin-JJJ variables in each site and another is formed by using two SO(2S+1)SO(2S+1)SO(2S+1) spinors. Among them, a spin-1 fermionic valence-bond solid state, its parent Hamiltonian, and its properties are discussed in detail. Moreover, two types of valence-bond solid states with SO(5)SO(5)SO(5) symmetries are further generalized and their respective properties are analyzed as well.
We introduce several exact models for SU(3) spin chains: (1) a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3‾\overline{\mathbf{3}}3 of SU(3) if the original spins of the model transform under representation 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement, and a Haldane gap in the excitation spectrum.
To begin with, we introduce several exact models for SU(3) spin chains: First is a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3‾\overline{\mathbf{3}}3 of SU(3) if the original spins of the model transform under representation 3. Second is a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and, hence, a Haldane gap in the excitation spectrum. We generalize some of our models to SU(n)\mathrm{SU}(n)SU(n). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of SU(n)\mathrm{SU}(n)SU(n) spins, in general, possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes $\ensuremath{\lambda}$ which is divisible by nnn. If $\ensuremath{\lambda}$ and nnn have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If $\ensuremath{\lambda}$ and nnn have a common divisor different from nnn, it will depend on the specifics of the model including the range of the interaction.
We introduce a one-dimensional valence bond solid (VBS) state with symplectic symmetry SP(n)\text{SP}(n)SP(n) and construct the corresponding parent Hamiltonian. We argue that there is a gap in the spectrum. We calculate exactly the static correlation functions, which fall off exponentially. Hence the model introduced here shares all properties of the Haldane scenario for integer-spin quantum antiferromagnets. We further show that the VBS state possesses string order and discuss its generalization to higher dimensions.
We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence bond states proposed in the Affleck--Kennedy--Lieb--Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest-neighbor interactions, and the partially dimerized or AKLT/valence bond solid states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions.
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking, but most typically manifest a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate, regardless of the parity of bulk (super)spins. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.
We present a detailed analysis of topological properties of the valence-bond solid (VBS) states doped with fermionic holes. As concrete examples, we consider the supersymmetric extension of the SU(2) and the SO(5) VBS states, dubbed UOSp(1$|2)andUOSp(12) and UOSp(12)andUOSp(1|$4) supersymmetric VBS states, respectively. Specifically, we investigate the string-order parameters and the entanglement spectra of these states to find that, even when the parent states (bosonic VBS states) do not support the string order, they recover it when holes are doped and the fermionic sector appears in the entanglement spectrum. These peculiar properties are discussed in light of the symmetry-protected topological order. To this end, we characterize a few typical classes of symmetry-protected topological orders in terms of supermatrix product states (SMPS). From this, we see that the topological order in the bulk manifests itself in the transformation properties of the SMPS in question and thereby affects the structure of the entanglement spectrum. Then, we explicitly relate the existence of the string order and the structure of the entanglement spectrum to explain the recovery and the stabilization of the string order in the supersymmetric systems.