Topological and Trivial Valence-Bond Orders in Higher-Spin Kitaev Models

TL;DR

The paper addresses the existence and nature of valence-bond orders and their relation to topological order in higher-spin Kitaev models on the honeycomb lattice for spins $S \in \{1, 3/2, 2\}$. It deploys gradient-descent optimized iPEPS with enlarged unit cells to detect translational symmetry breaking and uses transfer-matrix spectra and symmetry-restricted calculations to probe topological order without imposing artificial virtual symmetries. The main results reveal three distinct bond-ordered ground states: plaquette order for $S=1$, topological dimer order for $S=3/2$, and non-topological dimer order for $S=2$, all with tripled unit cells. A theoretical analysis shows that half-integer spins realize $\mathbb{Z}_2$ topological order (toric-code-like) via locally conserved flux operators $W_p$, while integer spins stabilize a trivial dimer state; boundary MPS degeneracy signals topological sectors. The work clarifies the interplay between topological and symmetry-breaking orders and demonstrates improved tensor-network workflows for detecting such orders in frustrated magnets, with potential relevance to Kitaev-materials.

Abstract

We investigate the quantum phases of higher-spin Kitaev models using tensor network methods. Our results reveal distinct bond-ordered phases for spin-1, spin-$\tfrac{3}{2}$, and spin-2 models. In all cases, we find translational symmetry breaking with unit cells being tripled by forming valence-bond orders. However, these three phases are distinct, forming plaquette order, topological dimer order, and non-topological dimer order, respectively. Our findings are based on a cross-validation between variational two-dimensional tensor network calculations: an unrestricted exploration of symmetry-broken states versus the detection of symmetry breaking from cat-state behavior in symmetry-restricted states. The origin of different orders can also be understood from a theoretical analysis. Our work sheds light upon the interplay between topological and symmetry-breaking orders as well as their detection via tensor networks.

Figures (9)

• Figure 1: Ground-state bond-energy configurations in the Kitaev model for different spin values. Different colors represent spins along the $x$-, $y$-, and $z$-axes, while line thickness indicates the relative bond-energy strength. (a) Spin-$\tfrac{1}{2}$: No bond order, exhibiting uniform bond energies. The ground state is a gapless Kitaev spin liquid. (b) Spin-1: Plaquette order. Each site is connected to one weak and two strong bonds. (c) Spin-$\tfrac{3}{2}$: Topological dimer order. Each site connects to one strong and two weak bonds. Purple dashed lines mark entanglement between strong bonds, forming a kagome lattice. (d) Spin-2: trivial dimer order. Each site connects to one weak and two strong bonds, where the strong bonds form short-range entanglement. A limiting case is a configuration where one spin in a bond (e.g., within the black dashed box) only entangles with the other spin: $\tfrac{1}{\sqrt{2}}\left(|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle\right)$, where the up and down arrows represent the largest and smallest $S_z$ components, respectively.
• Figure 2: iPEPS ansatz and MPS boundary solutions in VUMPS contraction. (a) Left: $C_{3v}$-symmetric tensor for single-tensor iPEPS, the tensor on other sublattice is identical as a function of physical leg (dimension $d$) and virtual bonds x,y and z (dimension $D$). Right: A tripled-unit-cell structure with three independent tensors (ABC). (b) Boundary MPS as the fixed-point solutions of the iPEPS transfer matrix (appearing in the norm), where the physical legs of bra and ket have been contracted. The MPS is defined by the contraction of local tensors $A^{i}_\rho$ along the virtual legs with dimension $\chi$. (c) For $\mathbb{Z}_2$ topologically ordered states, the iPEPS transfer matrix exhibits a doubly degenerate fixed point structure, represented by boundary MPS with tensors $A_\rho$ and $A_{\rho^\prime}$ and associated with breaking of an emerging virtual $\mathbb{Z}_2$ symmetry. (d) The transfer matrix spectrum then contains trivial excitations, targeted by the ansatz in the upper panel, as well as domain wall excitations between the two fixed points, targeted by the lower ansatz.
• Figure 3: Average ground-state energy per site (left) and bond ratio (right) for higher-spin Kitaev models for (a) spin 1, (b) spin $\tfrac{3}{2}$, (c) spin 2. Our results are plotted as points on solid lines for unrestricted tripled-unit-cell iPEPS (see \ref{['fig: ipeps_unit_cell']}.), with bond dimension $D$ up to 12, and as a square box for the $C_{3v}$-symmetric PEPS with $D=8$. The contraction dimension $\chi$ for the environment is ramped to be large enough for convergence (up to 512). Previous (symmetric) iPEPS results obtained using simple-update [(S)TN-SU] PhysRevResearch.2.033318Jahromi2024 as well as DMRG results PhysRevB.102.121102 are also indicated. The bond ratio is defined as the ratio between strong and weak bond energies $|\langle S^{\gamma}_i S^{\gamma}_j\rangle|_{\max}/|\langle S^{\gamma'}_{i'} S^{\gamma'}_{j'}\rangle|_{\min}$.
• Figure 4: Spectra of inverse correlation lengths [see Eq. \ref{['eq:inversecorr']}] obtained from the transfer matrix for the single-tensor unit cell $C_{3v}$-restricted iPEPS for the spin-$1$ model (a), the tripled-unit-cell iPEPS for spin-1 (b), spin-$\tfrac{3}{2}$ (c) and spin-2 (d). The transverse momentum along the transverse matrix corresponds to a cut in the Brillouin zone (a), or in the three-fold reduced Brillouin zone (b-d), where $\tfrac{\pm 2\pi}{3}$ corresponds to (reduced) K and K' points. Only for spin-$\tfrac{3}{2}$ (b) did we find two degenerate boundary MPS and do we thus plot both a trivial and a domain wall spectrum, using the ansatz in Fig \ref{['fig: ipeps_unit_cell']}(d).
• Figure 5: The linear transfer matrix map: the optimal contraction order is $T_1 \rightarrow T_2 \rightarrow A \rightarrow A^\prime \rightarrow T_3 \rightarrow B \rightarrow B^\prime$, and the corresponding computational costs are $O(\chi^3D^4) \rightarrow O(\chi^2D^5d) \rightarrow O(\chi^2D^5d) \rightarrow O(\chi^3D^4) \rightarrow O(\chi^2D^5d) \rightarrow O(\chi^2D^5d)$.