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The Kalmeyer-Laughlin chiral spin liquids (CSL) and the Z2 spin liquids aretwo of the simplest topologically ordered states. Here I develop a theory of adirect quantum phase transition between them. Each CSL is characterized by aninteger n and is topologically equivalent to the 1/2n Laughlin fractionalquantum Hall (FQH) state. Depending on the parity of n, the transition from theCSL is either to a "twisted" version of the Z2 spin liquid, the "doubledsemion" model, or the conventional Z_2 spin liquid ("toric code"). In thepresence of SU(2) spin symmetry, the triplet gap remains open through thetransition and only singlet operators acquire algebraic correlations. Anessential observation is that the CSL/Laughlin FQH states can be understood interms of bosonic integer quantum Hall (BIQH) states of Schwinger bosons orvortices, respectively. I propose several novel many-body wave functions thatcan interpolate through the transition.

Transitions Between Chiral Spin Liquids and Z2 Spin Liquids