Is it possible a state be a Quantum Spin Liquid and has long range order at the same time? Quantum spin liquid ([QSL])1 is usually defined as a kind of phase that
(1) no long-range order,
(2) has long-range entanglement and
(3) hosts emergent gauge structures or fractionalized excitations.
I am wondering is there any principle ensures that properties(2) and (3) imply (1)? Or actually we can have a phase that satisfis (2) and (3) but has long-range order at the same time?
Thanks for any comments or answers.
 A: The characteristics of QSLs are very much an active area of research. I am not aware of any argument which would guarantee that (2) + (3) implies (1). However, in systems with half-integer spin, it is well accepted that (1) implies (2) which in turn often leads to (3). Here I paraphrase the argument from Section 5.1 of the review article by Savary and Balents (Rep. Prog. Phys. 80 (2017) 016502):

*

*The Lieb-Schultz-Mattis theorem (and extensions/generalisations thereof) implies that in the thermodynamic limit, a system with half-integer spin and an $SU(2)$ invariant Hamiltonian must have either a gapless excitation or a ground state degeneracy.

*If one can realise a state of this system with no spontaneous symmetry breaking (no long-range order), then it cannot be a topologically trivial quantum paramagnet (the "typical" magnetically disordered state), since this is fully gapped and has no ground state degeneracy.

The authors point out that on a torus, the case with ground state degeneracy implies topological order (the toric code is the typical example of a highly-entangled QSL with emergent anyon excitations). The other interesting point is that if there is a gapless excitation in the disordered phase, it is not a Goldstone mode, and so must exist due to non-trivial topology.
I hope this at least partially answers your question.
