# Fractional Chern Insulators and the W∞ algebra

The basic theoretical puzzle posed by FCIs is to provide a microscopic description of their physics. A number of theoretical works have attempted to bridge the gap between FCIs and FQHE including adiabatic continuity of Wannier functions [8–9], the extended Hamiltonian formalism [10], parton/slave particle treatments [11] and attempts to characterize interactions using pseudopotentials [12–13], among others. While these approaches provide a number of useful insights into the problem, they generically don’t distinguish between different single-particle flat band Hamiltonians; however, numerical work [14–15] has found that single-particle effects can strongly influence the stability of FCI phases. In an effort to better understand these single-particle effects, our group has taken several important steps towards the development of a unified theoretical treatment of FQH-FCI phenomena.

In the context of the usual FQHE, the algebra of density operators projected to the lowest Landau level closes at all orders. As noted by Girvin, MacDonald and Platzman (GMP) [16], this plays an important role in the explanation of the stability of the FQHE using the single mode approximation, which is the ansatz that the most relevant low-energy neutral excitations of the FQHE are collective density waves (magnetorotons) generated by the action of projected density operators on the ground state. In [3], we noted that an isomorphism of algebras would lead to identical low energy physics, and thus would explain the stability of FCI states. By examining the algebra of Chern band projected density operators, we found that this algebra closes at long wavelengths and for constant Berry curvature, whereupon it is isomorphic to the W∞ algebra proposed by GMP. For Hamiltonians projected to a single Chern band, this provides a route to replicating lowest Landau level physics on the lattice.

### References

[8] X.-L. Qi, Phys. Rev. Lett. **107**, 126803 (2011).

[9] Y.-L. Wu, N. Regnault, and B. A. Bernevig, Phys. Rev. B **86**, 085129 (2012).

[10] G. Murthy and R. Shankar, Phys. Rev. B **86**, 195146 (2012).

[11] J. McGreevy, B. Swingle, and K.-A. Tran, Phys. Rev. B **85**, 125105 (2012).

[12] C. H. Lee, R. Thomale, and X.-L. Qi, Phys. Rev. B **88**, 035101 (2013).

[13] Y.-L. Wu, N. Regnault, and B. A. Bernevig, Phys. Rev. Lett. **110**, 106802 (2013).

[14] D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nat. Commun. **2**, 389 (2011).

[15] Y.-L. Wu, B. A. Bernevig, and N. Regnault, Phys. Rev. B **85**, 075116 (2012).

[16] S. Girvin, A. H. MacDonald, and P. Platzman, Phys. Rev. B **33**, 2481 (1986).