Composite spin liquid in a correlated topological insulator: Spin liquid without spin-charge separation

In this paper, we present an alternative type of insulator, namely composite spin liquid, which can be regarded as a short-range B-type topological spin-density wave as proposed by J. He et al. [Phys. Rev. B 84, 035127 (2011)]. Composite spin liquid is a topological ordered state beyond the classification of traditional spin liquid states. The elementary excitations are the “composite electrons” with both spin and charge degrees of freedom, together with topological spin texture. This topological state supports the chiral edge mode but no topological degeneracy.

Figure 1

The dispersion of electrons for $t^{\prime }=0.0288t$ when $U=0.$ We can see clearly that at the high symmetry points, the energy gap is zero and the dispersions look like Dirac cones.

Figure 2

The illustration of the relationship between AF order and the quantum disordered state.

Figure 3

The phase diagram. There are seven phases: QAH, band insulator, A-TSDW, B-TSDW, chiral spin liquid, composite spin liquid, and trivial AF-SDW. The regions of chiral spin liquid and composite spin liquid are the quantum disordered regions of $\alpha >4\pi$.

Figure 4

The dimensionless coupling constant $\alpha =g\Lambda$ for the case of the parameter as $t^{\prime }=0.1t.$ For the region with $\alpha >4\pi$, the ground state is chiral spin liquid (yellow region).

Figure 5

The dimensionless coupling constant $\alpha =g\Lambda$ for the case of the parameter as $t^{\prime }=0.033t.$ For the regions with $\alpha >4\pi$, the ground states are spin liquid states—chiral spin liquid (yellow region) or composite spin liquid (green region).

Figure 6

The spin order parameter ${\mathcal{M}}_0$ for the case of the parameter as $t^{\prime }=0.033t$. Yellow region denotes chiral spin liquid and green region denotes composite spin liquid, of which ${\mathcal{M}}_0=0$.

Figure 7

The spin gap $m_s$ for the case of the parameter as $t^{\prime }=0.033t.$ Yellow region denotes chiral spin liquid and green region denotes composite spin liquid, of which $m_s\ne 0$.

Figure 8

The dimensionless coupling constant $\alpha =g\Lambda$ for the case of the parameter as $t^{\prime }=0.0288t.$ For the region with $\alpha >4\pi$, the ground state is composite spin liquid (green region).

Figure 9

The spin order parameter ${\mathcal{M}}_0$ for the case of the parameter as $t^{\prime }=0.0288t$. The green region denotes composite spin liquid, of which ${\mathcal{M}}_0=0$.

Figure 10

The spin gap $m_s$ for the case of the parameter as $t^{\prime }=0.0288t.$ The green region denotes composite spin liquid, of which $m_s\ne 0$.

Figure 11

The dimensionless coupling constant $\alpha =g\Lambda$ for the case of the parameter as $t^{\prime }=0.025t.$ For the region with $\alpha >4\pi$, the ground state is composite spin liquid (green region).

Figure 12

The dimensionless coupling constant $\alpha =g\Lambda$ for the case of the parameter as $t^{\prime }=0.02t.$ We can see that the dimensionless coupling constant $\alpha$ is always smaller than ${\alpha }_c=4\pi .$ That mean there does not exist a quantum disordered region at all.

Figure 13

The illustration of the relationship between A-TSDW and chiral spin liquid.

Figure 14

The illustration of the edge state of chiral spin liquid. There exists a “spin” chiral edge state (the lines with arrows) and a “charge” chiral edge state (the lines with dots).

Figure 15

The illustration of the relationship between B-TSDW and composite spin liquid.

Figure 16

The charge energy gap for the case of $t^{\prime }=0.033t$, $\varepsilon =0.15t$: the charge carrier is a composite electron. In composite SL, the charge energy gap is that of spin gap, $m_s$; in B-TSDW, the charge energy gap is that of a skyrmion and an antiskyrmion, ${\Delta }_c$. The energy gap of fermion quasiparticles is very big (see inset).

Figure 17

The illustration of the edge state of composite spin liquid. There exists a single chiral edge mode.

Figure 18

The illustration of composite spin liquid, of which the elementary excitations are the “composite electrons” that are $S=1/2$ charge $\pm e$ fermions with trapping a topological spin texture.

Figure 19

The illustration of the fractional quantum Hall state. Small blue balls with single arrows denote the anyonic excitations with $\pm e/3$ charge.

Figure 20

The illustration of quantum spin liquid. The blue arrows denote an $S=1/2$ chargeless spinon.