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.
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.
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 study the 3D Kitaev and Kitaev-Heisenberg models, respectively, on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph projected entangled-pair states (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high temperature leads to fractionalization of spins to Majorana fermions and gauge fields that occurs in two separate temperature regimes, leaving their fingerprint on specific heat as a double-peak feature as well as on other quantities such as the thermal entropy, spin-spin correlations, and bond entropy. Using the Majorana representation of the Kitaev model, we further show that the low-temperature thermal transition to the Kitaev quantum spin liquid (QSL) phase is associated with the nontrivial Majorana band topology and the presence of Weyl nodes, which manifests itself via nonvanishing Chern number and finite thermal Hall conductivity. Beyond the pure Kitaev limit, we study the 3D Kitaev-Heisenberg (KH) model on the hyperoctagon lattice and extract the full phase diagram for different Heisenberg couplings. We further explore the thermodynamic properties of the magnetically-ordered regions in the KH model and show that, in contrast to the QSL phase, here the thermal phase transition follows the standard Landau symmetry-breaking theory.
The spin-1/2 Kitaev model offers the exactly solvable example of quantum spin liquids.Possible material realizations of the spin-1/2 Kitaev systems and the prospect of using the Majorana fermion excitations for quantum computations have revolutionized quantum spin liquids research.Recently it has been suggested that higher-spin, especially spin-1, Kitaev exchange interactions can be realized in a variety of materials.Numerical computations on small clusters indicate that the ground state of the spin-1 Kitaev model may also be a quantum spin liquid.On the other hand, the nature of the ground state remains elusive since the spin-1 model is not exactly solvable in contrast to the spin-1/2 model.In this work, using the quantum-entanglement based tensor network approach, we construct an explicit ground-state wave function for the spin-1 Kitaev model, which is written only in terms of physical spin operators.We establish the existence of distinct topological sectors on a torus by constructing the minimally entangled states in the degenerate ground-state manifold and evaluating topological entanglement entropy.Our results suggest that the ground state of the spin-1 Kitaev model is a gapped quantum spin liquid with Z 2 gauge structure and Abelian quasiparticles.We explain the subtle differences between the spin-1/2 and spin-1 Kitaev quantum spin liquids.
The Kitaev honeycomb model is a paradigm of exactly solvable models, showing nontrivial physical properties such as topological quantum order, Abelian and non-Abelian anyons, and chirality. Its solution is one of the most beautiful examples of the interplay of different mathematical techniques in condensed matter physics. In this paper, we show how to derive a tensor network (TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in particular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TN structure made of fermionic unitary operators, corresponding to the unitary quantum circuit building up the many-body quantum state. In our derivation we review how the different ``solution ingredients'' of the Kitaev honeycomb model can be accounted for in the TN language, namely, Jordan-Wigner transformation, braidings of Majorana modes, fermionic Fourier transformation, and Bogoliubov transformation. The TN built in this way allows for a clear understanding of several properties of the model. In particular, we show how the fidelity diagram is straightforward both at zero temperature and at finite temperature in the vortex-free sector. We also show how the properties of two-point correlation functions follow easily. Finally, we also discuss the pros and cons of contracting of our 3d TN down to a 2d projected entangled pair state (PEPS) with finite bond dimension. The results in this paper can be extended to generalizations of the Kitaev model, e.g., to other lattices, spins, and dimensions.
While most of the features of the Kitaev honeycomb model are already clarified from the combined Z2 plaquette and Majorana fermion picture, we show that retrieving the exact quantum many-body form gives a clue to fix the remaining issues on its spin liquid property. We provide an exact two-dimensional tensor network description of a full set of solutions from a random product state, by sequential quantum-number projection and Gutzwiller projection, both expressed as local matrix-product operators of dimension D=2. The ground and excited states share the same bond dimension yet exhibit distinct Schmidt spectra, distinguishing their area-law and volume-law entanglement. We demonstrate that the topological entanglement entropy emerges {\it for all eigenstates only after} the Gutzwiller projection onto the physical spin Hilbert space, where the gauge constraint removes exactly half of the Schmidt states. This establishes that excited states are likewise topologically ordered. We further show that there exists a projection-induced collapse of the wave function into a single gauge sector in the Parton mean-field states that is known to give a good description of the spin liquids. The demonstrations on the S=1/2 kagome and triangular Heisenberg models suggest that the present perspective is applied to a wider class of models hosting Z2 spin liquids. Our finding visualizes a bulk-boundary correspondence with the Wilson-loop characterization of topological order.
In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Γ) couplings. We investigate the quantum (S=1/2) K-J-Γ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z_{2} state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.