Spin analogs of superconductivity and integer quantum Hall effect in an array of spin chains

Motivated by the successful idea of using weakly coupled quantum electronic wires to realize the quantum Hall effects and the quantum spin Hall effects, we theoretically study two systems composed of weakly coupled quantum spin chains within the mean-field approximations, which can exhibit spin analogs of superconductivity and the integer quantum Hall effect. First, a certain bilayer of two arrays of interacting spin chains is mapped, via the Jordan-Wigner transformation, to an attractive Hubbard model that exhibits fermionic superconductivity, which corresponds to spin superconductivity in the original spin Hamiltonian. Secondly, an array of spin-orbit-coupled spin chains in the presence of a suitable external magnetic field is transformed to an array of quantum wires that exhibits the integer quantum Hall effect, which translates into its spin analog in the spin Hamiltonian. The resultant spin superconductivity and spin integer quantum Hall effect can be characterized by their ability to transport spin without any resistance.

Figure 1

Schematic of a bilayer of two arrays of weakly coupled spin chains (shown as the solid lines indexed by $m$), each of which can be represented by a one-dimensional system of (spinless) Jordan-Wigner fermions. The top and bottom layer indices serve as the pseudospin up and down for the fermions, respectively. The green box represents a pseudospin-singlet Cooper pair of two fermions established by an Ising interlayer interaction.

Figure 2

(a) Schematic of an array of spin-orbit-coupled spin-1/2 spin chains, which can support chiral edge modes of the Jordan-Wigner fermions. The coupling of four spins (colored by yellow) illustrates the interchain interaction $\mathcal{O}$ [Eq. (2)]. (b) A schematic plot showing how the interchain interaction gives rise to chiral edge modes with the gapped bulk. At the external magnetic field $h$ corresponding to the filling factor $\nu =1$, the JW fermion can flow in the left direction on the top chain (colored by blue) and in the right direction on the bottom chain (colored by red) in (a), which are represented by the blue (left) and red (right) dots in (b), respectively. The particle current in the JW representation corresponds to the spin current polarized along the $z$ axis.