Hidden Fermi Liquidity and Topological Criticality in the Finite Temperature Kitaev Model

The fate of exotic spin liquid states with fractionalized excitations at finite temperature ($T$) is of great interest, since signatures of fractionalization manifest in finite-temperature ($T$) dynamics in real systems, above the tiny magnetic ordering scales. Here, we study a Jordan-Wigner fermionized Kitaev spin liquid at finite $T$ employing combined Exact diagonalization and Monte Carlo simulation methods. We uncover $(i)$ checkerboard or stripy-ordered flux crystals depending on density of flux, and $(ii)$ establish, surprisingly, that: $(a)$ the finite-$T$ version of the $T=0$ transition from a gapless to gapped phases in the Kitaev model is a Mott transition of the fermions, belonging to the two-dimensional Ising universality class. These transitions correspond to a topological transition between a string condensate and a dilute closed string state $(b)$ the Mott "insulator" phase is a precise realization of Laughlin's gossamer (here, p-wave) superconductor (g-SC), and $(c)$ the Kitaev Toric Code phase (TC) is a {\it fully} Gutzwiller-projected p-wave SC. These findings establish the finite-$T$ QSL phases in the $d = 2$ to be {\it hidden} Fermi liquid(s) of neutral fermions.