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

## TL;DR

This paper introduces a hybrid quantum liquid state called quantum spin-dipole liquid (QSDL) in hydrogen-bonded systems, combining proton ice and spin liquid theories, with implications for experimental materials like H-Cat.

## Contribution

It develops a theoretical framework for a coupled spin and dipole quantum liquid state, revealing strong entanglement and challenging the Born-Oppenheimer approximation.

## Key findings

- QSDL exhibits large spin-dipole entanglement.
- The state is stable against certain perturbations.
- Implications for experimental realization in H-Cat and D-Cat.

## Abstract

We present a theory of a hybrid quantum liquid state, $\textit{quantum spin-dipole liquid}$ (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 $\kappa$-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 $\textit{quantum valence bond ice}$. 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.