The Heart of Entanglement: Chiral, Nematic, and Incommensurate Phases in the Kitaev-Gamma Ladder in a Field

Abstract

The bond-dependent Kitaev model on the honeycomb lattice with anyonic excitations has recently attracted considerable attention. However, in solid state materials other spin interactions are present, and among such additional interactions, the off-diagonal symmetric Gamma interaction, another type of bond-dependent term, has been particularly challenging to fully understand. A minimal Kitaev-Gamma (KG) model has been investigated by various numerical techniques under a magnetic field, but definite conclusions about field-induced spin liquids remain elusive and one reason may lie in the limited sizes of the two-dimensional geometry it is possible to access numerically. We therefore focus on the KG model defined on a two-leg ladder which is much more amenable to a complete study, and determine the entire phase diagram in the presence of a magnetic field along [111]-direction. Due to the competition between the interactions and the field, an extremely rich phase diagram emerges with fifteen distinct phases. Focusing near the antiferromagnetic Kitaev region, we identify nine different phases solely within this region: several incommensurate magnetically ordered phases, spin nematic, and two chiral phases with enhanced entanglement. Of particular interest is a highly entangled phase with staggered chirality with zero-net flux occurring at intermediate field, which along with its companion phases outline a heart-shaped region of high entanglement, the heart of entanglement. We compare our results for the ladder with a C3 symmetric cluster of the two-dimensional honeycomb lattice, and offer insights into possible spin liquids in the two-dimensional limit.

23 pages 28 figures

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• Insights into the Anisotropic Spin-$S$ Kitaev Chain
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• Spontaneous dimerization, spin-nematic order, and deconfined quantum critical point in a spin-1 Kitaev chain with tunable single-ion anisotropy
• Islands of Chiral Solitons in Integer Spin Kitaev Chains
• Origin of nonsymmorphic bosonization formulas in generalized antiferromagnetic Kitaev spin-$\frac{1}{2}$ chains from a renormalization-group perspective
• Electric field control of a quantum spin liquid in weak Mott insulators
• Left-left-right-right magnetic order in spin-1/2 Kitaev-Heisenberg chain
• Emergent magnetic order in the antiferromagnetic Kitaev model with a [111] field
• Emergent SU(2)${}_1$ conformal symmetry in the spin-1/2 Kitaev-Gamma chain with a Dzyaloshinskii-Moriya interaction
• Nonsymmorphic bosonization in one-dimensional generalized Kitaev spin-1/2 models
• Field-Induced Ordered Phases in Anisotropic Spin-1/2 Kitaev Chains
• Continuous phase transition from a chiral spin state to collinear magnetic order in a zigzag chain with Kitaev interactions
• Exact Ground States and Phase Diagram of the Quantum Compass Model under an in-plane Field
• Nonsymmorphic spin-space cubic groups and SU(2)${}_1$ conformal invariance in one-dimensional spin-1/2 models
• Spin and charge order in doped spin-orbit coupled Mott insulators
• Eleven Competing Phases in the Heisenberg-Gamma (J$\Gamma$) Ladder
• Tunable chiral spiral phases in a non-Hermitian Ising-Gamma spin chain
• Reviving Majorana zero modes in the spin-1/2 Kitaev ladder model
• Chiral spin liquid in a generalized Kitaev honeycomb model with ${\mathbb{Z}}_4$ 1-form symmetry
• Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal interactions
• Nesting instability of gapless U(1) spin liquids with spinon Fermi pockets in two dimensions
• Dynamical Response of the Kitaev Spin Liquid under Third-Nearest-Neighbor Heisenberg Interaction
• Fractional Wannier Orbitals and Tight-Binding Gauge Fields for Kitaev Honeycomb Superlattices with Flat Majorana Bands