PHYSICAL REVIEW B 102, 144437 (2020) Quantum phase transitions in the spin-1 Kitaev-Heisenberg chain Wen-Long You ,1,2 Gaoyong Sun ,1 Jie Ren ,3 Wing Chi Yu ,4 and Andrzej M. Oles´ 5,6 1College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China 2School of Physical Science and Technology, Soochow University, Suzhou, Jiangsu 215006, China 3Department of Physics, Changshu Institute of Technology, Changshu 215500, China 4Department of Physics, City University of Hong Kong, Kowloon, Hong Kong 5Institute of Theoretical Physics, Jagiellonian University, Profesora Stanisława Łojasiewicza 11, PL-30348 Kraków, Poland 6Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Received 20 August 2020; accepted 28 September 2020; published 23 October 2020) Recently, it has been proposed that higher-spin analogs of the Kitaev interactions K > 0 may also occur in a number of materials with strong Hund’s and spin-orbit coupling. In this work, we use Lanczos diagonalization and density matrix renormalization group methods to investigate numerically the S = 1 Kitaev-Heisenberg model. The ground-state phase diagram and quantum phase transitions are investigated by employing lo- cal and nonlocal spin correlations. We identify two ordered phases at negative Heisenberg coupling J < 0: z z > x x = a ferromagnetic phase with Si Si+1 0 and an intermediate left-left-right-right phase with Si Si+1 0. A quantum spin liquid is stable near the Kitaev limit, while a topological Haldane phase is found for J > 0. DOI: 10.1103/PhysRevB.102.144437 I. INTRODUCTION KH model was depicted using the density matrix renormaliza- tion group (DMRG) and exact diagonalization (ED) methods Kitaev-Heisenberg (KH) models were fostered by an en- [12]. Much attention has been paid to the Kitaev limit [13–20]. deavor of achieving the Kitaev physics in transition metal The two-spin correlation functions are found to be extremely oxides [1]. A continuing interest of bond-directional inter- short-ranged [21,22], indicating a QSL state. actions is motivated by topological quantum computing [2], Recently it was realized that an S =1 KH model could especially after Kitaev proposed an exactly solvable model of be designed by considering strong Hund’s coupling among frustrated quantum spins S = 1/2 on a two-dimensional (2D) e honeycomb lattice with bond-directional interactions [3]. The two electrons in g orbitals and strong spin-orbit coupling Kitaev model was initially treated as a mathematical model (SOC) at anion sites [23]. However, relatively little is known describing a topological quantum spin liquid (QSL) ground about the magnetic properties and particularly the elementary state (GS) and Majorana excitations, until Jackeli and Khali- excitation spectrum for higher S on the effect of Heisenberg ullin [4] demonstrated that the bond-directional interactions exchange in the Kitaev chain. It was realized long after Hal- could be realized in Mott insulators with strong spin-orbit dane’s pioneering work [24,25] that spin models with integer coupling. This innovative concept initiated intense theoreti- or half-odd integer S are qualitatively different. The Néel state cal and experimental search for the S = 1/2 Kitaev QSLs in is favored by the Heisenberg antiferromagnetic (AFM) term solid state materials [5]. It has been found that other inter- for half-integer spin S, while it cannot play a similar role when actions such as the isotropic Heisenberg and/or off-diagonal S is an integer. = exchange terms contribute [6–9], and real systems do not It is recognized that the GS of the S 1 Heisenberg anti- realize the QSL. ferromagnet belongs to the Haldane phase, which is separated The 2D model appears difficult to analyze and its phase from all excited states by a finite spin gap [26], and thus diagram has a QSL in the Kitaev limit [10,11], but even the two-spin correlation is quenched. The underlying physics of one-dimensional (1D) version of it has several interesting Haldane chains is fairly well understood both in theory and quantum phase transitions (QPTs) [12]. A spin-1/2 1D variant experiments. The Haldane phase of spin-1 XXZ AFM chains of KH model was defined on a chain, in which two types of was proposed in trapped ions systems [27]. For instance, a × nearest-neighbor Kitaev interactions sequentially switch be- hidden Z2 Z2 symmetry breaking takes place [28,29] and x x y y hence the string order parameters are nonzero in both x and tween Si Si+1 on odd and Si Si+1 on even bonds next to uniform Heisenberg interactions. The GS phase diagram of spin-1/2 z directions [30,31]. When the Kitaev interaction is taken into account, the spin chains only have a Z2 parity symmetry corresponding to the rotation of π around a given axis [32]. If the Z2 symmetry in the GS is broken, whether the string order Published by the American Physical Society under the terms of the parameter along the given axis becomes nonzero is unclear Creative Commons Attribution 4.0 International license. Further [33]. Therefore, it is also an interesting issue to explore the distribution of this work must maintain attribution to the author(s) existence of the string correlators in spin-1 chains with lower and the published article’s title, journal citation, and DOI. Open symmetries than the Heisenberg chain. The phase diagram access publication funded by the Max Planck Society. of the spin-1 generalized Kitaev chain (also dubbed as the 2469-9950/2020/102(14)/144437(6) 144437-1 Published by the American Physical Society YOU, SUN, REN, YU, AND OLES´ PHYSICAL REVIEW B 102, 144437 (2020) α = { x, y, z} (a) representation, Sbc i abc; i.e., S S S are given by FM LLRR QSL Haldane z 00 0 00i 0 −i 0 -1.0 -0.5 0.0 0.5 1.0 00−i , 000, i 00. (4) J 0 i 0 −i 00 000 (b) (c) N=12 α -0.57 0.2 N=20 { } N=40 Spin operators S j at site j in Eq. (1) obey the SU(2) algebra, β N=60 α, = δ c 0 S S i αβγ S N=100 [ i j ] ij j , with the totally antisymmetric tensor e 2 = + = -0.62 0.1 and (S j ) S(S 1) 2. First we consider the Kitaev limit in Eq. (1), i.e., J = 0. Then the global spin rotation SU(2) symmetry is not con- -0.67 0.0 served. We can write the spin operators in HˆK in terms 0.00 0.04 0.08 0.12 -10 1 ± ≡ x ± y of the ladder operators S j S j iSj , and one finds that 1/N J z, ± =± ± [S j S j ] S j , i.e., the Ising terms in Eq. (2) change the total pseudospin zth component at both odd x link and even FIG. 1. (a) Schematic phase diagram of the KH model with α iπSα y ± = e j K = 1; (b) finite-size scaling of the energy density for OBC and linkbyeither0or 2. A site parity operator is j ; {x,y,z} J = 0; (c) the energy gap for system sizes ranging from N = 12 to i.e., are given by the diagonal matrices that α = − α 2 xyz = N = 100. The blue and red symbols in (b) represent the GS energy satisfy j 1 2(S j ) and j j j I, where I is an iden- of Hˆ per site and per bond, respectively. The linear fits correspond tity matrix. The Hamiltonian in Eq. (2) has a global discrete K π to e (N ) =−0.3326/N − 0.6024 and e (N ) = 0.3136/N − 0.6034. symmetry with respect to rotation by an angle about the 0 0 α The green dashed line is e0(∞). x, y, z axes, i.e., j j , present in the dihedral group D2. x,y,z →− x,y,z The time-reversal symmetry, i.e., S j S j , and the x,y,z → x,y,z spatial-inversion symmetry, i.e., S j SN+1− j,arealso compass model in the literature) was also investigated [34]. respected. The GS properties and the low-energy excitations of spin-1 {α} Furthermore, one finds that all j matrices commute KH models are elusive and deserve a careful investigation. with each other. In addition, α commutes with Sα but anti- The purpose of this paper is twofold. First, we would like j j β α =β {α, β }={ π α , β }= to obtain the GS phase diagram and discuss the QPTs in the commutes with S j ( ), i.e., j S j exp(i S j ) S j 0. / 1D spin-1 KH model. Second, while some differences in the In this regard, the bond parity operators on odd even structure of the invariants between the models with half-odd bonds, integer and integer spins have been pointed out [35], the issue y y x x W2 j−1 = − , W2 j = + , (5) of whether there are systematic differences in the nature of 2 j 1 2 j 2 j 2 j 1 the low-energy spectrum is open [36]. The main result of our define the invariants of the Hamiltonian in Eq. (2) and eigen- study is that the GS of the spin-1 KH chain with periodic values of Wj are ±1. It can be verified that [Wj,Wk] = 0, boundary conditions (PBCs) changes from the QSL to the [Wj, HˆK] = 0. The GS of HˆK lies in the sector with all Wj = 1 left-left-right-right (LLRR) (Haldane) phase for J <0(J >0); which can be proved by applying the reflection positivity see Fig.