We identify the gapless-point geometry of the honeycomb lattice model, where its three bond classes carry mutually anticommuting Hermitian-unitary matrix hoppings. For the chosen 8 × 8 Jordan-Wigner realization, we demonstrate that the exact zero condition detQ(k)=0⟺∑ie2ik⋅δi=0 yields eight inequivalent gapless momenta within the first Brillouin zone. Each zero point hosts four two-component low-energy cones with projected velocity v=3TA/(2ℏ). We further construct a two-stage mass mechanism: NNN mixing achieves 2/4 pairing selection within the test supercell domain, while the on-site marker mass 21(3−χ) has its spectrum restricted to {1,1,1,2,2} in units of εE. The ratio of 20:1 originates entirely from the traceless involution χ (full-channel parity with the exact symmetry {H,χ⊗I}=0 of the hopping Hamiltonian), giving 2=(N+1)/(N−1) at channel number N=3 ([12]). The analytical results are complemented by exact diagonalization and supercell folding checks.
An exactly solvable lattice with periodic inverse-square singularities exhibits two physically distinct mechanisms of spectral-gap closure. The model arises from the generalized trigonometric Pöschl-Teller potential in the coupling range where both local solutions at each singularity are square-integrable. A transmitting, current-preserving self-adjoint matching condition connects neighboring cells and generates a nontrivial Bloch spectrum. For a generic lattice with two inequivalent singular sublattices, the spectrum contains two alternating families of gaps, and the band edges reproduce the scale-invariant spectra of the associated finite singular system. One gap family closes when the two sublattices become equivalent. In this symmetric limit, however, the primitive period halves, so the apparent band crossing results from Brillouin-zone folding. A different family closes at a critical but genuinely asymmetric configuration. Because the primitive period is preserved, adjacent bands then meet at the center of the original Brillouin zone in a genuine massless Dirac-like cone. Near either critical limit, a small finite gap acts as an effective Dirac mass: the dispersion remains quadratic near a band edge but crosses over to a massive Dirac-like form beyond that narrow region. The band curvatures define effective electron and hole masses. Exact Bloch eigenfunctions further reveal that isospectral lattices can display different single-particle probability densities. This model therefore provides an analytically controlled one-dimensional setting in which conventional quadratic carriers, massive Dirac-like band dispersions, and a tunable massless contact coexist within the same singular lattice.
Motivated by two-dimensional bilayer systems we numerically study an anti-ferromagnetic spin-
1/2 Heisenberg model on the honeycomb lattice with a nearest-neighbour exchange coupling
J1, and a next-nearest-neighbour exchange coupling
J2′ for the B sublattice sites only. Using infinite Projected Entangled-Pair States (iPEPS), variational uniform matrix-product states on cylinders, and exact diagonalization, we find an extended symmetric regime
0.4≲J2′/J1≲0.6, in which local observables show no magnetic, valence-bond, or chiral spin order. At
J2′/J1=0.5, the PEPS correlation length grows systematically with bond dimension, and an inverse-correlation-length extrapolation favors a vanishing limit. We also find that various observables scale algebraically with the finite bond-dimension-induced correlation length, which points to a gapless spin liquid ground state. We propose a
Z2 Dirac spin liquid parton state, with Dirac points that are protected by translation, time-reversal and three-fold rotation symmetry, as a promising candidate state to explain the numerical results. We also discuss the possibility that the symmetric ground state is a featureless, short-range entangled state with a small gap.
Dirac points lie at the heart of many fascinating phenomena in condensed matter physics, from massless electrons in graphene to the emergence of conducting edge states in topological insulators. At a Dirac point, two energy bands intersect linearly and the particles behave as relativistic Dirac fermions. In solids, the rigid structure of the material sets the mass and velocity of the particles, as well as their interactions. A different, highly flexible approach is to create model systems using fermionic atoms trapped in an optical lattice, a method which so far has only been applied to explore simple lattice structures. In my talk I will report on the creation of Dirac points with adjustable properties in a tunable honeycomb optical lattice [1]. Using momentum-resolved interband transitions, we observe a minimum band gap inside the Brillouin zone at the position of the Dirac points. We exploit the unique tunability of our lattice potential to adjust the effective mass of the Dirac fermions by breaking the inversion symmetry of the lattice. Moreover, changing the lattice anisotropy allows us to move the position of the Dirac points inside the Brillouin zone. When increasing the anisotropy beyond a critical limit, the two Dirac points merge and annihilate each other. We map out this topological transition in lattice parameter space and find excellent agreement with ab initio calculations. Our results not only pave the way to model materials where the topology of the band structure plays a crucial role, but also provide the possibility to explore many-body phases resulting from the interplay of complex lattice geometries with interactions.
We construct two superconducting pairing models on doped Kitaev honeycomb, whose microscopic kernels are derived rather than assumed. Both are driven by a bond-local glue neutral between the even- and odd-parity pairing channels; the chirality--parity linkage is set by bond-star geometry, not by the microscopic glue. (i) Model~1(odd-parity sector): the kernel follows from the flux-sector selection rule of the pure model, fixing its component-diagonal, nearest-neighbor separable, rank-one-per-bond form, with
wα=JK2χα∝rαmα, where
mα follows from the Hellmann--Feynman theorem and
rα≃0.24 is computed numerically. The spin-vertex state is a unitary triplet with a bond-locked anisotropic
d vector; its gap is a Gram sum of bond sines, so nodal directions are fixed by geometry. (ii) Model~2(even): the kernel is fixed by the
C3 little-group selection rule of a zone-corner chiral phonon, up to a Cooper-channel extension whose momentum constraint we state explicitly. The charge-vertex pairing is a
d+id singlet: the parity decomposition
f±=g±+p± projects onto
g±, so chirality is slaved to the phonon polarization, with
C=±4 and
κxy/T=2κ0. Reversing the pump helicity reverses the Chern number, Kerr rotation, and thermal Hall sign. (iii) General theorem: on odd-fold bond stars, the angular-momentum ladder of a single-star chiral form factor alternates in parity. Every chiral state selected by such a bond-local form factor is a geometrically enforced singlet--triplet composite with universal amplitude ratios (
n=3:
d⊕p;
n=5:
g⊕p or
d⊕f); coherent doubled-star combinations can restore parity purity. The Chern number of the mixed state follows the dominant on-shell component across a topological boundary. The spin glue drives
C=±2; the charge kernel
C=±4.
We show how to construct fully symmetric states without topological order on a honeycomb lattice for
S=21 spins using the language of projected entangled pair states. An explicit example is given for the virtual bond dimension
D=4. Four distinct classes differing by lattice quantum numbers are found by applying the systematic classification scheme introduced by two of the authors [S. Jiang and Y. Ran, Phys. Rev. B 92, 104414 (2015)]. Lack of topological degeneracy or other conventional forms of symmetry breaking in the proposed wave functions are checked by numerical calculations of the entanglement entropy and various correlation functions. Exponential decay of all correlation functions measured are strongly indicative of the energy gap for the putative parent Hamiltonian of the state. Our work provides the first explicit realization of a featureless quantum state for spin-
21 particles on a honeycomb lattice.
By combining the density matrix renormalization group (DMRG) method with Gutzwiller projected wave functions, we study the SU(4) symmetric spin-orbital model on the honeycomb lattice. We find that the ground states can be well described by a Gutzwiller projected $\ensuremath{\pi}$-flux state with Dirac-type gapless excitations at one quarter filling. Although these Dirac points are gapped by emergent gauge fluxes on finite cylinders, they govern the critical behavior in the thermodynamic limit. By inserting a $\ensuremath{\theta}=\ensuremath{\pi}$ spin flux to twist the boundary condition, we can shift the gapless sector to the ground state, which provides compelling evidence for the presence of a gapless Dirac spin-orbital liquid.