Are Resonating Valence-Bond States Long-Range Entangled?

Whether a resonating valence-bond (RVB) state is long-range entangled depends on which RVB state you mean. RVB is a broad construction: spins form singlet pairs, and the wavefunction may be a superposition of many different singlet coverings. That description alone does not determine whether the state has topological order.

The key distinction is between having entangled pairs and belonging to a phase with long-range entanglement. A state can contain many entangled singlets yet be transformable into a simple product state by local operations. By contrast, intrinsic topological order cannot be removed by a finite-depth circuit of local operations, even if the circuit is allowed to change the state without preserving any symmetry.

What “long-range entangled” means here

In condensed matter, the phrase commonly refers to entanglement that is not confined to local, removable clusters. For a gapped ground state, a useful distinction is:

• A short-range-entangled state can be converted to a product state by a finite-depth local unitary circuit, if no symmetry constraint is imposed.
• An intrinsically long-range-entangled state cannot. Topologically ordered spin liquids are standard examples.

This is a statement about the quantum state as a whole, not about whether individual valence bonds are long or short. A nearest-neighbor singlet is entangled across its two spins, but a lattice filled with a fixed pattern of such singlets can still be short-range entangled.

Symmetry-protected topological states are a related case: they can be disentangled if symmetry is ignored, but not by circuits that preserve the relevant symmetry. They are usually classed as short-range entangled in the intrinsic, symmetry-free sense.

A fixed dimer covering is not long-range entangled

Consider four spins paired into two singlets, with each pair on neighboring sites:

$$|\psi\rangle = |s_{12}\rangle |s_{34}\rangle, \qquad |s_{ij}\rangle = \frac{|\uparrow_i\downarrow_j\rangle-|\downarrow_i\uparrow_j\rangle}{\sqrt{2}}.$$

Each pair is entangled. But the two singlets are independent, and local operations on each pair can map its singlet to a fixed product state. Repeating that operation throughout a fixed dimer covering gives a product state after a single layer of local gates. Thus this dimer state is short-range entangled, even though it is not a product state in the original spin basis.

A superposition of coverings changes the problem. The overlaps and relative signs between coverings matter, and a sufficiently coherent superposition can have topological order. “Resonating” does not guarantee that it does.

When an RVB state is long-range entangled

A well-known example is a gapped, short-bond RVB spin liquid on a non-bipartite lattice such as the triangular lattice. In its topologically ordered phase, the state has Z₂ topological order. Its properties include distinct topological sectors on a torus and fractionalized excitations, such as spinons and visons. These features cannot be accounted for by a fixed arrangement of local singlets, and the state is intrinsically long-range entangled.

One diagnostic is the entanglement entropy of a large region. In two spatial dimensions, a topologically ordered ground state typically has

$$S(A) = \alpha L - \gamma + \cdots,$$

where $L$ is the boundary length, the first term is a non-universal boundary contribution, and the constant term $\gamma$ is the topological entanglement entropy. For an ideal Z₂ topological phase, $\gamma = \ln 2$ when entropy is measured using natural logarithms. This diagnostic needs care with finite sizes and geometry; it is not enough to identify a phase from one small-system number.

RVB states can also be critical

A classic nearest-neighbor RVB wavefunction on the square lattice is critical: its dimer correlations decay algebraically. It is not the same kind of gapped Z₂ topological spin liquid as the triangular-lattice example. Criticality and topological order are different properties, so “RVB” does not by itself tell you which one is present.

Terminology can vary for gapless states. In the strict finite-depth-circuit sense, algebraic correlations are incompatible with a state made from a product by a finite-depth local circuit: such a circuit acting on a product state can generate correlations only over a bounded distance. But authors often reserve “long-range entangled” in discussions of spin liquids for intrinsic topological order, especially in the gapped setting. It is therefore clearer to say whether an RVB state is gapped and topologically ordered, critical, or short-range entangled, rather than relying on the label alone.

Common mistakes

• Equating singlet entanglement with long-range entanglement. A single singlet is entangled, but a fixed collection of local singlets is locally removable.
• Assuming every superposition of coverings is topological. The amplitudes, signs, lattice, and correlations determine the phase.
• Treating short-range bonds as proof of short-range entanglement. A superposition of short bonds can still encode topological order.
• Calling every critical state a topological spin liquid. Algebraic correlations do not establish topological order; check for the phase’s defining properties.

So the precise answer is conditional: some RVB states are intrinsically long-range entangled, notably those realizing a gapped topological spin liquid; others are short-range entangled or critical.