The Kitaev-AKLT model

TL;DR

The paper studies a spin-1 chain with Kitaev-like bond-directional couplings, blending Kitaev and AKLT ideas to obtain exact ground states via fractionalization of each spin-1 into two spin-1/2 partons. At special points, notably $\theta=\pi/4$, the Hamiltonian is a sum of bond-projectors and exhibits an exponentially degenerate ground-state manifold that can be written as MPSs with bond dimension four; the ground states are labeled by bond-conserved quantities $\{w_k\}$ and connected to a spin-1 cluster model. The authors further analyze the purely biquadratic limits $\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$, provide MPS representations for the solvable point $\theta=\frac{\pi}{4}$, and construct fractionalized variational ansatzes (uniform $w=+1$ and uniform $w=-1$) to capture ground-state physics away from the solvable points, including edge-state structure and perturbative stability. Overall, the work offers a framework for understanding spin-1 Kitaev-like chains, provides exact ground-state constructions, and presents variational methods and tensor-network representations that may extend to related lattices and perturbations.

Abstract

Inspired by the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, we present exact solutions for a spin-1 chain with Kitaev-like couplings. We consider an expanded Kitaev model with bilinear and biquadratic terms. At an exactly solvable point, the Hamiltonian can be reexpressed as a sum of projection operators. Unlike the AKLT model where projectors act on total spin, we project onto components of spin along the bond direction. This leads to exponential ground state degeneracy, expressed in terms of fractionalized spin-$\frac{1}{2}$ objects. Each ground state can be expressed concisely as a matrix product state. We construct a phase diagram by varying the relative strength of bilinear and biquadratic terms. The fractionalized states provide a qualitative picture for the spin-1 Kitaev model, yielding approximate forms for the ground state and low-lying excitations.

Figures (9)

• Figure 1: Fractionalized ground states at an exactly solvable point. Top: Bonds of the spin-1 Kitaev chain alternate between X and Y couplings. Bottom: The spin-1 moment at each site fractionalizes into two spin-$\frac{1}{2}$ objects or spinons, indicated as L (left) and R (right). On each bond, the spinons at the ends are placed in one of two valence bond wavefunctions. Finally, at every site, the L and R spinons are projected onto the local spin-1 space.
• Figure 2: Direct-product ground states. (a) Spins are placed in $\vert S_x = 0\rangle$ and $\vert S_y = 0\rangle$ states in alternating fashion. These two states are ground states when $Q$ is positive and $Q\geq \vert K\vert$. (b) All spins are placed in $\vert S_z = 0\rangle$ state, yielding the ground state when $K=0$ and $Q<0$.
• Figure 3: Matrix product state representations. (a) Fractionalized ground states at the exactly solvable point with $\theta=\frac{\pi}{4}$. The wavefunctions are not orthogonal to one another. Indices $\{ \chi\}$ represent the bond-state at each bond, while $\{ s\}$ represent the physical spin quantum numbers, $s_j=-1,0,1$. (b) Ground state of the generalized cluster model. Indices $\{ w\}$ represent bond conserved quantities. (c) Orthogonalized ground states at the exactly solvable point, obtained by contracting MPSs from (a) and (b). Matrices enclosed within dashed lines can each be contracted into a single matrix of bond-dimension four. Wavefunctions shown here are unnormalized.
• Figure 4: Ground state phase diagram of the spin-1 bilinear-biquadratic model. In the 'doubly degenerate' region, the ground states are the direct-product states of Fig. \ref{['fig.directproduct']}(a). The two phases are separated by exactly solvable points at $\theta = \frac{\pi}{4}$ and $\frac{3\pi}{4}$. A special point appears at $\theta=\frac{3\pi}{2}$, with a unique direct-product ground state where all spins are in the $S_z=0$ state, as shown in Fig. \ref{['fig.directproduct']}(b).
• Figure 5: (a) Overlap of the fractionalized state with all $w$'s set to $+1$ with the true ground state obtained from exact diagonalization. System size is plotted along the x-axis. The lines (from top to bottom) represent $\theta=40^\circ,30^\circ,20^\circ,10^\circ,0^\circ$. The lowest line in blue corresponds to the spin-1 Kitaev model with purely bilinear couplings. Overlaps approach unity as $\theta\rightarrow 45^\circ$. They decrease with increasing system size. (b) Overlap of a fractionalized state with the set of first excited states obtained from exact diaonalization. The fractionalized state is chosen with one $w$ set to -1 while all others are +1.