Defect mediated quantum melting of charge ordered insulators

TL;DR

The paper addresses how two-dimensional WMIs at fractional filling transition into gapped QCLs with topological order. It develops a defect-proliferation framework that treats WMI defects as precursors to QCL anyons, deriving dual vortex theories and PSG constraints to distinguish possible proximate topological orders. A key result is that defect-mediated melting from the minimal fermionic WMI to the minimal TO is allowed only for odd $q$, while even $q$ forbids a direct route and requires a higher-fold WMI (e.g., $2q$) to access the minimal TO; from the QCL side, odd $q$ yields a $Z_q$ SET and even $q$ yields a $Z_{2q}$ SET, with confinement driven by neutral anyon condensation. Together these findings constrain the possible intermediate states between WMIs and metallic or topological liquids, with implications for moiré materials and related platforms, and point to experimental tests via STM and transport in systems exhibiting Wigner-Mott physics.

Abstract

Two-dimensional (2d) electronic systems on a lattice at fractional filling $ν= p/q$ exhibit a competition between charge ordered insulators, called Wigner-Mott insulators (WMIs), at large Coulomb repulsion and Fermi-liquid metals at large electronic kinetic energy. When those two energy scales are roughly equal, insulating states that restore the lattice translation symmetry, which we call quantum charge liquids (QCLs), may emerge. When gapped, these QCLs must exhibit topological order. In this work, we show that the allowed topological ordered phases that are proximate to the WMI strongly depend on the charge ordering in the WMI. In particular, we show that when $q$ is even, no direct transition exists between a WMI with the smallest allowed unit cell size from filling constraints, i.e., the "minimal" WMI, and the topological order with the smallest ground state degeneracy on a torus allowed by filling constraints, i.e., the "minimal" TO. Furthermore, we describe the quantum melting transition of the WMIs to the proximate QCLs in terms of the proliferation of the topological defects of the WMIs. The field theory of this transition in terms of the topological defects reveals their role as precursors to the anyon excitations in the QCLs.

Figures (4)

• Figure 1: Caricatures of the four inequivalent stripe orders on the square lattice at $1/2$ filling. We draw the charge density wave pattern as being bond centered for convenience. Each pattern is related to the subsequent one by ${R}_{\pi/2}$. An example unit cell is drawn in magenta in each sketch.
• Figure 2: Caricatures of the two inequivalent vortex configurations on the square lattice at 1/2 filling. The blue lines are domain walls. The right picture is obtained from the left picture by acting with $T_x$. Note that the right vortex has the same stripe order in quadrants 1 and 3 but opposite stripe orders in quadrants 2 and 4. Thus, the right vortex has opposite vorticity under $\mathbb{Z}_4$. Each vortex has an empty half unit cell at the vortex core.
• Figure 3: Caricature of the defect configuration $\phi_+^\dagger \phi_-$, dressed with the appropriate Wilson lines, at 1/2 filling that, when proliferated, results in a $\mathbb{Z}_2$ TO. The defect has vorticity $4\pi$ and has a core consisting of a unit cell.
• Figure 4: A caricature of the defect $\phi_\ell(x) \exp{-i\int^x a_\ell}$, dressed by an operator such that each domain has vertical stripes, for the trimer model at filling $1/3$. The domain walls are blue, with charge $+1$ under the stripe order parameter group $\mathbb{Z}_3$, and the junction center is an empty magenta box. We represent the bosons as living in trimers, drawn in cyan, to show explicitly the empty site at the junction center. Some vertical trimers are necessary along domain walls to ensure that the filling fraction remains $1/3$ outside of the junction center. As before, the charge density wave pattern and shapes of the domain walls are model specific.