Abstract
One-dimensional gapped phases that avoid any symmetry breaking have drawn enduring attention. In this paper, we study such phases in a bond-alternating spin-1 $K$-$\Gamma$ chain built of a Kitaev ($K$) interaction and an off-diagonal $\Gamma$ term. In the case of isotropic bond strength, a Haldane phase, which resembles the ground state of a spin-$1$ Heisenberg chain, is identified in a wide region. A gapped Kitaev phase situated at dominant ferromagnetic and antiferromagnetic Kitaev limits is also found. The Kitaev phase has extremely short-range spin correlations and is characterized by finite $\mathbb{Z}_2$-valued quantities on bonds. Its lowest entanglement spectrum is unique, in contrast to the Haldane phase whose entanglement spectrum is doubly degenerate. In addition, the Kitaev phase shows a double-peak structure in the specific heat at two different temperatures. In the pure Kitaev limit, the two peaks are representative of the development of short-range spin correlation at $T_h \simeq 0.5680$ and the freezing of $\mathbb{Z}_2$ quantities at $T_l \simeq 0.0562$, respectively. By considering bond anisotropy, regions of Haldane phase and Kitaev phase are enlarged, accompanied by the emergence of dimerized phases and three distinct magnetically ordered states.
Highlights
The Kitaev honeycomb model [1], consisting of bonddependent Ising couplings of spin-1/2 degrees of freedom, is a rare example which is exactly solvable and hosts a quantum spin liquid (QSL) ground state with fractionalized excitations, e.g., itinerant Majorana fermions and localized fluxes [2]
III we study the excitations of the Haldane phase and Kitaev phase in the isotropic K
We find that finite-size effects of the energy of the ground state and low-lying excited states are less pronounced in the periodic boundary condition (PBC), and the numerical result suggests that the Haldane phase possesses a nonzero excitation gap in the whole region
Summary
The Kitaev honeycomb model [1], consisting of bonddependent Ising couplings of spin-1/2 degrees of freedom, is a rare example which is exactly solvable and hosts a quantum spin liquid (QSL) ground state with fractionalized excitations, e.g., itinerant Majorana fermions and localized fluxes [2]. These excitations account for the doublepeak specific heat anomaly at two different energy scales [3]. We study the quantum phase diagram of a bond-alternating spin-1 K- chain using the density-matrix renormalization group (DMRG) method [40,41,42]. Further information about the specific heat in the spin-1/2 and spin-1 Kitaev chains is given in Appendixes A and B
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