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Register a new userAbelian and non-Abelian chiral spin liquids in a compact tensor network representation
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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 (SG) states [H.-Y. Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)]. Those are compactly represented in the language of 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 CSLs are well demonstrated by the short-ranged LG while 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.
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We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids - spin-analogues of fractional non-Abelian quantum Hall states- with gapped bulk and gapless chiral edge excitations described by the SU(2)$_n$ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of a generalized spin-$n/2$ ladders with multi-spin exchange interaction which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)$_n$ gapless edge excitations.
Guided by the recent discovery of SU($2$)$_1$ and SU($3$)$_1$ chiral spin liquids on the square lattice, we propose a family of generic time-reversal symmetry breaking SU($N$)-symmetric models, of arbitrary $Nge 2$, in the fundamental representation, with short-range interactions extending at most to triangular units. The evidence for Abelian chiral spin liquid (CSL) phases in such models is obtained via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small clusters are carried out up to $N=10$, revealing (in some range of the Hamiltonian parameters) a bulk gap and ground-state degeneracy on periodic clusters as well as linear dispersing chiral modes on the edge of open systems, whose level counting is in full agreement with SU($N$)$_1$ Wess-Zumino-Witten conformal field theory predictions. Using an SU($N$)-symmetric version of iDMRG for $N=2,3$ and $4$ to compute entanglement spectra on (infinitely-long) cylinders in all topological sectors, we provide additional unambiguous signatures of the SU($N$)$_1$ character of the chiral liquids. An SU($4$)-symmetric chiral PEPS is shown to provide a good variational ansatz of the $N=4$ ground state, constructed in a manner similar to its $N=2$ and $N=3$ analogs. The entanglement spectra in all topological sectors of an infinitely long cylinder reveal specific features of the chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Results for the correlation lengths suggest some form of long-range correlations in SU($N$) chiral PEPS, which nevertheless do not preclude an accurate representation of the gapped SU($N$) CSL phases. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups.
We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. The CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSLs separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number $pm 1$) CSL obey non-Abelian statistics.
We extend the scope of Kitaev spin liquids to non-Archimedean lattices. For the pentaheptite lattice, which results from the proliferation of Stone-Wales defects on the honeycomb lattice, we find an exactly solvable non-Abelian chiral spin liquid with spontaneous time reversal symmetry breaking due to lattice loops of odd length. Our findings call for potential extensions of exact results for Kitaev models which are based on reflection positivity, which is not fulfilled by the pentaheptite lattice. We further elaborate on potential realizations of our chiral spin liquid proposal in strained $alpha$-RuCl$_3$.
We study $S=1$ spin liquid states on the kagome lattice constructed by Gutzwiller-projected $p_x+ip_y$ superconductors. We show that the obtained spin liquids are either non-Abelian or Abelian topological phases, depending on the topology of the fermionic mean-field state. By calculating the modular matrices $S$ and $T$, we confirm that projected topological superconductors are non-Abelian chiral spin liquid (NACSL). The chiral central charge and the spin Hall conductance we obtained agree very well with the $SO(3)_1$ (or, equivalently, $SU(2)_2$) field theory predictions. We propose a local Hamiltonian which may stabilize the NACSL. From a variational study we observe a topological phase transition from the NACSL to the $Z_2$ Abelian spin liquid.
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