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arXiv:2104.10725 · Physical Review Research

Unusual excitations and double-peak specific heat in a bond-alternating spin-1 K−Γ chain

Qiang Luo, Shijie Hu, Hae-Young Kee

Published Jul 13, 2021• 30 citations
arXivHTMLPDF

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 $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ chain built of a Kitaev ($K$) interaction and an off-diagonal $\mathrm{\ensuremath{\Gamma}}$ 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 ${T}_{h}\ensuremath{\simeq}0.5680$ and the freezing of ${\mathbb{Z}}_{2}$ quantities at ${T}_{l}\ensuremath{\simeq}0.0562$, 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.

DOI
10.1103/physrevresearch.3.033048

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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\ensuremath−\ensuremathΓK\ensuremath{-}\mathrm{\ensuremath{\Gamma}}K\ensuremath−\ensuremathΓ chain built of a Kitaev (KKK) interaction and an off-diagonal \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ 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 Z2{\mathbb{Z}}_{2}Z2​-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 Th\ensuremath≃0.5680{T}_{h}\ensuremath{\simeq}0.5680Th​\ensuremath≃0.5680 and the freezing of Z2{\mathbb{Z}}_{2}Z2​ quantities at Tl\ensuremath≃0.0562{T}_{l}\ensuremath{\simeq}0.0562Tl​\ensuremath≃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.
Recently, there has been a renewed interest in properties of the higher-spin Kitaev models, especially their low-dimensional analogs with additional interactions. These quasi-one-dimensional systems exhibit rich phase diagrams with symmetry-protected topological phases, Luttinger liquids, hidden order, and higher-rank magnetism. However, the nature of the pure spin-SSS Kitaev chains is not yet fully understood. Earlier works found a unique ground state with short-ranged correlations for S=1S=1S=1 and an intriguing double-peak structure in the heat capacity associated with an entropy plateau. To understand the low-energy excitations and thermodynamics for general SSS, we study the anisotropic spin-SSS Kitaev chain. Starting from the dimerized limit, we derive an effective low-energy Hamiltonian at finite anisotropy. For half-integer spins we find a trivial effective model, reflecting a nonlocal symmetry protecting the degeneracy, while for integer SSS we find interactions among the flux degrees of freedom that select a unique ground state. The effective model for integer spins is used to predict the low-energy excitations and thermodynamics, and we make a comparison with the semiclassical limit through linear spin wave theory. Finally, we speculate on the nature of the isotropic limit.
The quantum spin chain with bond-directional interactions can host fascinating phenomena. Here, the authors study such a spin chain, built of a Kitaev interaction and an off-diagonal exchange (dubbed the $\mathrm{\ensuremath{\Gamma}}$ term). This model possesses a self-dual relation and a mirror symmetry, and is found to exhibit a multicritical point, two topological phase transitions of distinct universality classes, three magnetically ordered states, and four disordered phases including the preeminent Haldane phase. These findings demonstrate a fertile playground to study quantum magnetism and topological phase transitions.
Bipartite entanglement, entanglement spectrum, and Schmidt gap in S=1 bond-alternative antiferromagnetic Heisenberg chain are investigated by the infinite time-evolving block decimation (iTEBD) method. The quantum phase transition (QPT) from the singlet-dimer phase to the Haldane phase can be detected by the singular behavior of bipartite entanglement, the sudden change of the entanglement spectrum, and the completely vanishing of the Schmidt gap. The critical point is determined to be around r c ≃ 0.587, and the second-order character of the QPT is verified. Doubly degenerate entanglement spectra of both even and odd bonds are observed in the Haldane phase, by which one can distinguish the Haldane phase from the singlet-dimer phase easily. Nearest-neighbor antiferromagnetic correlations and next-nearest-neighbor ferromagnetic correlations are found in the whole parameter region. At the critical massless point, although exponentially decaying antiferromagnetic correlation is observed, it approaches to a constant value finally. Therefore, long-range correlations exist and the correlation length becomes divergent at the critical point.
The ground state phase diagrams and thermodynamic properties of a spin-1 bond-alternating Heisenberg antiferromagnetic chain with a single-ion anisotropy in longitudinal and transverse magnetic fields are investigated jointly by means of the infinite time evolving block decimation, the linearized tensor renormalization group, and the density matrix renormalization group methods. It is found that in the magnetic field--bond alternating ratio plane six phases such as singlet dimer, Haldane, two Tomonaga-Luttinger liquid, 1/2 magnetization plateau, and spin-polarized phases are identified in a longitudinal field, while in a transverse field there are five phases including singlet dimer, Haldane, Z2{Z}_{2}Z2​ symmetry breaking, quasi-1/2 magnetization plateau, and quasi-spin-polarized phases. A reentrant behavior of the staggered magnetization in a transverse field is observed. The quantum critical behaviors in longitudinal and transverse magnetic fields are disclosed to fall into different universality classes with corresponding conformal field central charge c=1c=1c=1 and 1/21/21/2, respectively. The experimental data of the compound NTENP under both longitudinal and transverse magnetic fields are nicely fitted, and the Luttinger liquid behavior of low-temperature specific heat experimentally observed is also confirmed.
The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.
We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin SSS, we find three different symmetric Z2{Z}_{2}Z2​ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large SSS. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.