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The original form of the Simon-Lee Hamiltonian is not gauge invariant. The fully gauge invariant form of the off-diagonal Bogoliubov-deGennes operator in a d$_{x^2-y^2}$ superconductor is given below (we thank Nick Read and Jinwu Ye for illuminating discussions on gauge invariance of effective Hamiltonians). Detailed derivation appears in the Appendix A of
cond-mat/0007296

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