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arXiv:2012.03382 · Physical review. B./Physical review. B

Unveiling the phase diagram of a bond-alternating spin- 12 K−Γ chain

Qiang Luo, Jize Zhao, Xiaoqun Wang, Hae‐Young Kee

Published Apr 15, 2021• 39 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.103.144423

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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 $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ 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.

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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 \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ 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.
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 (KKK) 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 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 ${T}_{h}\ensuremath{\simeq}0.5680$ and the freezing of Z2{\mathbb{Z}}_{2}Z2​ 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.
In this work, we analyze the nonsymmorphic symmetry-group structures for a variety of generalized Kitaev spin chains and ladders with bond alternations, including Kitaev-Gamma chain, Kitaev-Heisenberg-Gamma chain, beyond-nearest-neighbor interactions, and two-leg spin ladders. The symmetry analysis is applied to determine the symmetry-breaking patterns of several magnetically ordered phases in the bond-alternating Kitaev-Gamma spin chains, as well as the dimerization order parameters for spontaneous dimerizations. Our work is useful in understanding the magnetic phases in related models and may provide guidance for the symmetry classifications of mean field solutions in further investigations.
Abstract We study the spin-1/2 Kitaev chain with additional XX and symmetric off-diagonal Gamma interactions. By combining the Jordan–Wigner transformation with density matrix renormalization group (DMRG) numerical simulations, we obtain the exact solution of the model and map out the phase diagram containing six distinct phases. The four gapped phases display ferromagnetic and antiferromagnetic magnetic orders along the (1,1,0)- and (1,-1,0)-spin directions, whereas in the gapless phases, the low-energy spectrum consists of two branches of helical Majorana fermions with unequal velocities. The transition lines separating different phases include deconfined quantum critical lines with dynamical critical exponent z = 1 and quadratic critical lines with z = 2. Our work reveals the rich interplay among symmetry, magnetic order, and quantum criticality in the Kitaev–XX–Gamma chain.
The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.
We investigate the ground-state phase diagram of the bond-alternating S=2S=2S=2 quantum spin chain with the XXZXXZXXZ and on-site anisotropies. For the on-site anisotropies, in addition to the popular D2∑j(Sjz)2D_2 \sum_j (S_j^z)^2D2​∑j​(Sjz​)2 term, we consider the D4∑j(Sjz)4D_4 \sum_j (S_j^z)^4D4​∑j​(Sjz​)4 term. Mainly we use the exact diagonalization and the level spectroscopy analysis. We show that the Haldane state, large-DDD state and the Dimer2 state belong to the same trivial phase, by finding the existence of adiabatic paths directly connecting these states without the quantum phase transition. Similarly, we show that the intermediate-DDD state and the Dimer1 state belong to the same symmetry protected topological phase.
Abstract Aligning with the ongoing search for quantum spin liquids (QSLs), identifying QSLs in Kitaev magnets has attracted significant research interest during the past decade. Nevertheless, it remains a significant challenge. One of the major difficulties is that Kitaev QSL is typically fragile to competing interactions such as off-diagonal exchanges, which are ubiquitous in real materials owing to spin-orbit coupling and crystal-field effects. This, in turn, generates many intriguing field-induced novel phases and the thermal Hall effect. In this review, we focus on the interplay between the Kitaev interaction and off-diagonal Γ and Γ′ exchange from a numerical perspective. The review discusses certain representative exotic phases such as Γ spin liquid, nematic ferromagnet, spin-flop phase, and distinct chiral spin states with spontaneous time-reversal symmetry breaking. It also presents quantum phase diagrams of anisotropic Kitaev-Γ chains that exhibit kaleidoscopy for both ordered and disordered phases.