Supersymmetric valence-bond solid models are extensions of the VBS model, a paradigmatic model of ``solvable'' gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix-product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin or the charge excitation spectrum, the superconducting order parameter. A generalized hidden order is proposed, and the corresponding string nonlocal order parameter is also calculated. The behavior of the string-order parameter is discussed in the light of the entanglement spectrum.
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.
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when
2S is not an integer multiple of the coordination number
z.
S=2 spins on the triangular lattice with
z=6 is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.
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 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). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of
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
n. If $\ensuremath{\lambda}$ and
n have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If $\ensuremath{\lambda}$ and
n have a common divisor different from
n, it will depend on the specifics of the model including the range of the interaction.
Most of the present understanding of the S=1 quantum spin chains displaying the Haldane gap is coming from the so-called valence-bond-solid (VBS) Hamiltonian which has an exactly known ground state. We show that this point is characterized by the onset of short-range incommensurate spin correlations in the one-parameter family of Hamiltonians ${\mathit{H}}_{\mathrm{\ensuremath{\theta}}}$= cos\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$
Si\ensuremath{\cdot}
Si+1+sin\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$ (
Si\ensuremath{\cdot}
Si+1)2. This gives a physical meaning to this special point. We establish precise values for the gaps, correlations, the string order parameter, and identify the VBS point as a disorder point in the sense of classical statistical mechanics. It is a quantum remnant of the classical transition between a ground state with long-range N\'eel order and a ground state with incommensurate long-range order. \textcopyright{} 1996 The American Physical Society.
The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.