We provide new insights into the Abelian and non-Abelian chiral Kitaev spin liquids on the star lattice using the recently proposed loop gas (LG) and string gas states [Lee, Kaneko, Okubo, and Kawashima, Phys. Rev. Lett. 123, 087203 (2019)]. Those are compactly represented in the language of the tensor network. By optimizing only one or two variational parameters, accurate ansatze are found in the whole phase diagram of the Kitaev model on the star lattice. In particular, the variational energy of the LG state becomes exact (within machine precision) at two limits in the model, and the criticality at one of those is analytically derived from the LG feature. It reveals that the Abelian chiral spin liquids (CSLs) are well demonstrated by the short-ranged LG whereas the non-Abelian CSLs are adiabatically connected to the critical LG where the macroscopic loops appear. Furthermore, by constructing the minimally entangled states and exploiting their entanglement spectrum and entropy, we identify the nature of anyons and the chiral edge modes in the non-Abelian phase with the Ising conformal field theory.
We provide new insights into the Abelian and non-Abelian chiral Kitaev spin liquids on the star lattice using the recently proposed loop gas (LG) and string gas states [Lee, Kaneko, Okubo, and Kawashima, Phys. Rev. Lett. 123, 087203 (2019)]. Those are compactly represented in the language of the tensor network. By optimizing only one or two variational parameters, accurate ansatze are found in the whole phase diagram of the Kitaev model on the star lattice. In particular, the variational energy of the LG state becomes exact (within machine precision) at two limits in the model, and the criticality at one of those is analytically derived from the LG feature. It reveals that the Abelian chiral spin liquids (CSLs) are well demonstrated by the short-ranged LG whereas the non-Abelian CSLs are adiabatically connected to the critical LG where the macroscopic loops appear. Furthermore, by constructing the minimally entangled states and exploiting their entanglement spectrum and entropy, we identify the nature of anyons and the chiral edge modes in the non-Abelian phase with the Ising conformal field theory.
We propose a unified scheme to identify phase transitions out of the Z2 Abelian topological order, including the transition to a non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y. Lee, R. Kaneko, T. Okubo, and N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on the star lattice Kitaev model as an example, we compute the overlap of minimally entangled states through transfer matrices. We demonstrate that, similar to the anyon condensation, continuous deformation of a Z2-injective projected entangled-pair state (PEPS) also allows us to study the transition between Abelian and non-Abelian topological orders. We show that the charge and flux anyons defined in the Abelian phase transmute into the $\ensuremath{\sigma}$ anyon in the non-Abelian topological order. Furthermore, we show that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both the LG and SG states have infinite correlation length in the non-Abelian regime, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian.
Non-Abelian chiral spin liquids (CSLs) host non-Abelian anyon excitations and are promising platforms for topological quantum computation. However, realizing non-Abelian CSLs in short-range interacting spin systems remains elusive. Here, we study a spin-1 square-lattice model with the first- and second-neighbor bilinear (J1,J2) and biquadratic interactions (K1,K2), as well as the three-spin scalar chiral coupling Jχ. Inspired by the evidence of the Moore-Read state in exact diagonalization and infinite Projected Entangled Pair States calculations, we fix J1=1.0, J2/J1=0.623, K1/J1=−0.176, and obtain a quantum phase diagram for 0≤K2/J1≤0.4 and 0.3≤Jχ/J1≤0.6 by using the density matrix renormalization group (DMRG) calculations. We identify a stripe antiferromagnetic phase, a Néel antiferromagnetic phase, a magnetically ordered chiral spin state phase, and a non-Abelian CSL phase emerging near the boundaries of the ordered phases. The unbiased DMRG results provide strong evidence for the Moore-Read state, including the three topological sectors, the quantized spin Chern number C=1, and the level counting of low-lying entanglement spectra which is consistent with the chiral SU(2)2 conformal field theory. Our results may shed new light on searching for non-Abelian CSLs in other higher-spin (S>1/2) systems with short-range couplings.
We provide a framework for understanding the gapless Kitaev spin liquid (KSL) in the language of the tensor network (TN). Without introducing a Majorana fermion, most of the features of the KSL, including the symmetries, gauge structure, criticality, and vortex freeness, are explained in a compact TN representation. Our construction reveals a hidden string gas structure of the KSL. With only two variational parameters to adjust, we obtain an accurate KSL Ansatz with the bond dimension D=8 in a compact form, where the energy is about 0.007% higher than the exact one.
The infinite projected entangled simplex state (iPESS), a type of tensor network (TN) state, has been used successfully for simulating and characterizing {\it non-chiral} spin liquids on the kagome lattice. Here, we demonstrate that iPESS also provides a faithful representation of a {\it chiral} spin liquid (CSL) on the same lattice, namely the ground state of the spin-1/2 kagome Heisenberg antiferromagnet with a scalar chirality. By classifying local tensors according to SU(2) and point group symmetries, we construct a chiral ansatz breaking reflection P and time reversal T symmetries while preserving PT. The variational TN states are shown to host, for bond dimension D≥8, a chiral gapless entanglement spectrum following SU(2)1 conformal field theory. The correlation function shows a small weight long-range tail complying with the prediction of the TN bulk-edge correspondence. %{\color{red}With more constraints included, the chiral ansatz is reduced to a non-chiral one which preserves full point group symmetries due to an emergent {\it tensor conservation law} and is of relevance to the ground state at the Heisenberg point. Lastly, by simulations in the complete ansatz family we discuss the transition from the non-chiral spin liquid to the CSL induced by the scalar chirality term.} We identify a non-chiral manifold spanned by only a subset of symmetric tensors where a new emergent {\it tensor conservation law} is realized. This allows us to both probe the stability of the non-chiral spin liquid and discuss its transition to CSL induced by a scalar chirality term.
The star lattice hosts the Yao-Kivelson chiral spin liquid, an exactly solvable Kitaev spin liquid that breaks time reversal spontaneously and is non-Abelian or Abelian depending on the ratio of its two couplings. A density-matrix renormalization group study found that an added Heisenberg exchange leaves it a wide stability window for ferromagnetic but only a narrow one for antiferromagnetic Kitaev coupling. Using projective symmetry groups (PSGs) of fermionic partons, we show that the chirality is not a peculiarity of the solvable point: whenever the π spin rotations act on the partons as in the exact solution (spin-gauge locking), a time-reversal-symmetric mean-field state has no amplitude on the triangles, at any range, so a Kitaev-type spin liquid on this lattice must break time reversal. The Abelian and non-Abelian liquids of both signs share a single chiral PSG, within which their transition is a band inversion. The Klein duality, which reverses the Heisenberg coupling, permutes the PSGs and leaves exactly the locked ones unchanged; a Z2 spin-liquid phase mapped onto itself is therefore locked and, if it spans a connected range of couplings, contains a Kitaev point. Unprojected mean-field theory in Kitaev's Majorana representation treats both signs alike. Exact second-order perturbation theory about the Kitaev points locates the asymmetry: the triangles respond identically for both signs, whereas on the bonds between triangles the two off-diagonal Heisenberg terms create the same four visons and interfere constructively for antiferromagnetic and almost destructively for ferromagnetic coupling, admixing a low-energy four-vison state 82 times more strongly on the antiferromagnetic side. The PSG results rest on a classification of 352Z2 PSGs with spin-orbit-coupled amplitudes, which also identifies the fermionic partners of the known Schwinger-boson spin liquids.
The infinite projected entangled simplex state (iPESS), a type of tensor network (TN) state, has been used successfully for simulating and characterizing nonchiral spin liquids on the kagome lattice. Here, we demonstrate that iPESS also provides a faithful representation of a chiral spin liquid (CSL) on the same lattice, namely the ground state of the spin-1/2 kagome Heisenberg antiferromagnet with a scalar chirality. By classifying local tensors according to SU(2) and point group symmetries, we construct a chiral ansatz breaking reflection P and time reversal T symmetries while preserving PT. The variational TN states are shown to host, for bond dimension $D\ensuremath{\ge}8$, a chiral gapless entanglement spectrum following SU(2)1 conformal field theory. The correlation function shows a small weight long-range tail complying with the prediction of the TN bulk-edge correspondence. We identify a nonchiral manifold spanned by only a subset of symmetric tensors where a new emergent tensor conservation law is realized. This allows us to both probe the stability of the nonchiral spin liquid and discuss its transition to CSL induced by a scalar chirality term.