Quantum Dimer Models and Height/Gauge Descriptions

Quantum dimer models impose a hard local covering constraint, turning apparently simple bond variables into an emergent electric field or height. The mapping depends on lattice bipartiteness and background charge. The Rokhsar–Kivelson point is a special solvable point, not generic evidence for a stable liquid on every lattice.

Required background. Compact U(1) gauge fields supplies electric flux and monopoles.

Helpful background. Gapped spin liquids supplies the distinction between liquid and valence-bond order; quantum simulation of QFT supplies implementation and validation criteria.

Hard constraint and gauge dictionary

Let $n_{i\mu}=0,1$ mark a dimer on the link from site $i$ in direction $\mu$, with exactly one dimer touching each site:

$$\sum_{\mu}\big(n_{i\mu}+n_{i-\hat\mu,\mu}\big)=1.$$

On a bipartite lattice with sublattice sign $\eta_i=\pm1$, define an oriented electric field $E_{i\mu}=\eta_i n_{i\mu}$. The constraint becomes a Gauss law with staggered background charge, $\nabla\cdot E=\eta_i$. Local plaquette flips preserve it. Monomers violate it and act as gauge charges.

In two spatial dimensions, a divergence-free fluctuation can be represented by a dual scalar height. Dimer correlations become derivatives or vertex operators of the height; lattice symmetry determines which locking cosines are allowed. Proliferation of these vertex operators produces crystalline dimer order, while a rough height describes critical correlations Moessner and Raman 2011, §§ 2–3.

Rokhsar–Kivelson Hamiltonian

For each flippable plaquette $p$ with two parallel coverings $|h_p\rangle$ and $|v_p\rangle$,

$$H=\sum_p\left[-t\big(|h_p\rangle\langle v_p|+|v_p\rangle\langle h_p|\big) +v\big(|h_p\rangle\langle h_p|+|v_p\rangle\langle v_p|\big)\right].$$

At $v=t$, it is a sum of projectors and the equal-amplitude superposition in each connected winding sector has zero energy Rokhsar and Kivelson 1988. On the square lattice this point is critical and nearby locking generally yields valence-bond order. On nonbipartite lattices such as the triangular lattice, a finite Z2 liquid regime can occur. Lattice geometry is therefore not a detail.

Diagnostics and limitations

Measure winding-sector splittings, monomer confinement, dimer structure factors, height stiffness, and vison correlations. Frozen configurations and disconnected sectors can complicate finite-size sampling. A nearly equal-amplitude state does not establish deconfinement unless monomers, visons, and topology sectors behave consistently.

The mapping retains the constrained dimer Hilbert space. A microscopic magnet may contain longer bonds, spinful excitations, or constraint violations; deriving their scales is required before using the dimer theory quantitatively.

Exercise

At the Rokhsar–Kivelson point $v=t$, show that one plaquette term is positive semidefinite.

Solution

The term is $t(|h\rangle-|v\rangle)(\langle h|-\langle v|)$, an outer product with nonnegative expectation value. It annihilates the equal-amplitude local combination $|h\rangle+|v\rangle$.

References

• Roderich Moessner and Kumar S. Raman, “Quantum Dimer Models,” in Introduction to Frustrated Magnetism, Springer, Berlin and Heidelberg (2011) 437–479, doi:10.1007/978-3-642-10589-0_17.
• Daniel S. Rokhsar and Steven A. Kivelson, “Superconductivity and the Quantum Hard-Core Dimer Gas,” Physical Review Letters 61 (1988) 2376–2379, doi:10.1103/PhysRevLett.61.2376.