Hall viscosity of manifestly modular invariant composite fermion wave functions

The Hall viscosity has been proposed to be robust within a topological phase [1]. We perform a numerical calculation of the Hall viscosity of general fractional quantum Hall states by deforming the modular parameter $\tau$ on a torus. For this purpose, we follow the approach in [2] to construct composite fermion wave functions on a torus in terms of Haldane’s modified sigma function [3] to achieve a manifestly modular invariant form. (Numerical calculations show that the wave functions in this new form are identical to those in Ref. [2], therefore proving modular invariance for the latter wave functions as well.) We compare our Hall viscosities with the orbital spins for various fractional quantum Hall states in a broad region of the modular parameter $\tau$.
[1] J. E. Avron, R. Seiler and P. G. Zograf, Phys. Rev. Lett. 75, 697(1995); N. Read, arXiv: 0807.3107; N. Read, Phys. Rev. B 79, 045308(2009); N. Read and E. H. Rezayi, Phys. Rev. B 84, 085316(2011).
[2] S. Pu, Y-H. Wu and J. K. Jain, Phys. Rev. B 96, 195302(2017).
[3] F. D. M. Haldane, J. Math. Phys. 59, 071901(2018)