We consider the effect of interplane coupling on the energetics of the relative sign of statistics of anyons in adjacent planes. We offer qualitative arguments as well as numerical results which support the idea that opposing chirality (i.e., ``antiferromagnetic'' order of the statistics) is favored over a wide range of interplane coupling V--including both attractive and repulsive interactions between planes. We argue that our results apply to the normal (nonsuperfluid) phase in the thermodynamic limit.
Despite its deceptive simplicity, few concepts have more fundamental implications than chirality, from the therapeutic activity of drugs to the fundamental forces of nature. In magnetic materials, chirality gives rise to unconventional phenomena such as the anomalous Hall effect and multiferroicity, taking an enhanced flavour in the so-called spin-liquid phases where magnetic disorder prevails. Kagome systems sit at the crossroad of these ideas. Motivated by the recent synthesis of rare-earth kagome materials and the progresses in optical-lattice experiments, we bring together an entire network of spin liquids with anisotropic and Dzyaloshinskii-Moriya interactions. This network revolves around the Ising antiferromagnet and ends on (ferromagnetic) chiral spin liquids with spontaneously broken time-reversal symmetry. As for the celebrated Heisenberg antiferromagnet, it now belongs to a triad of equivalently disordered phases. The present work provides a unifying theory of kagome spin liquids with time-reversal invariant nearest-neighbour Hamiltonians.
Ordering of frustrated S=1/2 and 1 XY and Heisenberg spin chains with the competing nearest- and next-nearest-neighbor antiferromagnetic couplings is studied by exact diagonalization and density-matrix renormalization-group methods. It is found that the S=1 XY chain exhibits both gapless and gapped `chiral' phases characterized by the spontaneous breaking of parity, in which the long-range order parameter is a chirality,
κi=SixSi+1y−SiySi+1x, whereas the spin correlation decays either algebraically or exponentially. Such chiral phases are not realized in the S=1/2 XY chain nor in the Heisenberg chains.
Chiral spin liquids (CSLs) in three dimensions and thermal phase transitions to paramagnet are studied by unbiased Monte Carlo simulations. For an extension of the Kitaev model to a three-dimensional tricoordinate network dubbed the hypernonagon lattice, we derive low-energy effective models in two different anisotropic limits. We show that the effective interactions between the emergent
Z2 degrees of freedom called fluxes are unfrustrated in one limit, while highly frustrated in the other. In both cases, we find a first-order phase transition to the CSL, where both time-reversal and parity symmetries are spontaneously broken. In the frustrated case, however, the CSL state is highly exotic---the flux configuration is subextensively degenerate while showing a directional order with broken
C3 rotational symmetry. Our results provide two contrasting archetypes of CSLs in three dimensions, both of which allow approximation-free simulation for investigating the thermodynamics.
We study the nearest-neighbor
XXZ Heisenberg quantum antiferromagnet on the kagome lattice. Here we consider the effects of several perturbations: (a) a chirality term, (b) a Dzyaloshinski-Moriya term, and (c) a ring-exchange type term on the bowties of the kagome lattice, and inquire if they can support chiral spin liquids as ground states. The method used to study these Hamiltonians is a flux attachment transformation that maps the spins on the lattice to fermions coupled to a Chern-Simons gauge field on the kagome lattice. This transformation requires us to consistently define a Chern-Simons term on the kagome lattice. We find that the chirality term leads to a chiral spin liquid even in the absence of an uniform magnetic field, with an effective spin Hall conductance of ${\ensuremath{\sigma}}_{xy}^{s}=\frac{1}{2}$ in the regime of
XY anisotropy. The Dzyaloshinski-Moriya term also leads a similar chiral spin liquid but only when this term is not too strong. An external magnetic field also has the possibility of giving rise to additional plateaus which also behave like chiral spin liquids in the
XY regime. Finally, we consider the effects of a ring-exchange term and find that, provided its coupling constant is large enough, it may trigger a phase transition into a chiral spin liquid by the spontaneous breaking of time-reversal invariance.
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
352 Z2 PSGs with spin-orbit-coupled amplitudes, which also identifies the fermionic partners of the known Schwinger-boson spin liquids.