Quantum valence bond ice theory for proton-driven quantum spin-dipole liquids

We present a theory of a hybrid quantum liquid state, the quantum spin-dipole liquid (QSDL), in a hydrogen-bonded electron system, by combining a quantum proton ice and Anderson's resonating valence bond spin liquid theory, motivated by the recent experimental discovery of a quantum spin liquid with proton fluctuations in κ-H_{3}(Cat-EDT-TTF)_{2} (a.k.a. H-Cat). In our theory, an electron spin liquid and a proton dipole liquid are realized simultaneously in the ground state called quantum valence bond ice. In this state, neither of them can be established independently of the other. Analytical and numerical calculations reveal that this state has a large entanglement entropy between spins and dipoles, which is far beyond the (crude) Born-Oppenheimer approximation. We also examine the stability of QSDL with respect to perturbations and discuss implications for experiments in H-Cat and its deuterated analog D-Cat.