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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.

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  • $\begingroup$ As a trivial counter-example, consider a two-layer stack of a QSL and a symmetry-breaking phase of matter. Even if one couples the two (without driving a quantum phase transition), all three properties you mention will be robust. $\endgroup$
    – Ruben Verresen
    Commented Aug 15, 2020 at 16:41
  • $\begingroup$ @Ruben Verresen Sure you can construct such states. But naively I think this kind of state can not be low energy states of a physically relevant Hamiltonian (especially short-range interacting models). Moreover, my question is more about whether the QSL itself can have long-range order rather than a QSL weakly coupled with a different symmetry breaking state. $\endgroup$
    – mr.no
    Commented Aug 16, 2020 at 3:37
  • $\begingroup$ @RubenVerresen you are totally correct, there is no reason a QSL material could not also host non-magnetic long-range order. I think despite the lack of specificity in the statement (1), OP is interested in the presence or absence of magnetic LRO. $\endgroup$
    – user167506
    Commented Aug 17, 2020 at 3:55

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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):

  1. 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.
  2. 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.

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