From Quantum Dimers to the $π$-flux Toric Code via Deconfined Multicriticality

Abstract

Two-dimensional Rokhsar-Kivelson (RK) dimer models on bipartite lattices are generally limited to translation-symmetry-broken dimer crystals. We introduce a tensor-product regularisation of the dimer Hilbert space that yields a qubit Hamiltonian interpolating from the RK model to the $π$-flux toric code, thereby accessing a deconfined $\mathbb{Z}_2$ topological liquid. In this framework, the $\mathbb{Z}_2$ liquid descends from a multicritical $U(1)$ spin liquid through condensation of a charge-2 Higgs field, thus avoiding confinement. Using iDMRG together with low-energy field theory, we determine a phase diagram containing two continuous quantum phase transitions -- a $3\mathrm{D}$ XY$^{\ast}$ transition between the $\mathbb{Z}_2$ liquid and the columnar/plaquette-VBS, and a quantum Lifshitz transition between two dimer crystals -- alongside a first-order transition between the staggered crystal and the $\mathbb{Z}_2$ liquid. Our field theory suggests a deconfined multicritical point described by an Abelian Higgs model with dynamical critical exponent, $z=2$, where the three transitions meet, highlighting the interplay of fractionalisation and emergent gauge fluctuations.

Figures (10)

• Figure 1: Schematic phase diagram of the model defined by the Hamiltonian in Eq. \ref{['eq_gauge_invariant_H_deconf_full']} exhibiting three major phases: $\mathbb{Z}_2$ topologically ordered (TC$_\pi$), c/p-VBS and s-VBS, separated by three phase boundaries -- (i) 3D $XY^{*}$ critical line between the c/p-VBS and the $\mathbb{Z}_2$ liquid, (ii) a quantum Lifshitz transition between the two VBSs, and (iii) first-order transition between the s-VBS and $\mathbb{Z}_2$ liquid. All these phase transitions meet at a multicritical point. The yellow shaded region corresponds to a small region of possible incommensurate VBS phase with a finite tilt expected from the field theory. The question mark ("?") for the tilted VBS indicates that the present numerics do not provide conclusive evidence for this phase.
• Figure 2: Mapping between dimers and Ising variables (Eq. \ref{['eq_dimer_mapping']}) : The red bonds with curly (dotted) lines represent the presence (absence) of dimers (Eq. \ref{['eq_gaussdimer']}) residing on a square lattice. The black curly (solid) bonds represent Ising variable $Z=-1(Z=+1)$ on bonds of the dual lattice.
• Figure 3: Different dimer crystals: (a) staggered, (b) (vertical) columnar, and (c) plaquette VBSs.
• Figure 4: iDMRG scans of the phase diagram of Eq. \ref{['eq_gauge_invariant_H_deconf_full']} in the $\Gamma/J-\Omega/J$ plane for an infinite cylinder of circumference, $L_y=4$: (a) correlation length, $\xi$, (b) bipartite von-Neumann entanglement entropy, $S$, (c) s-VBS order parameter $O_{stag}$ (Eq. \ref{['eq_ostag']}), (d) c/p-VBS order parameter $O_{col/plaq}$ (Eq. \ref{['eq_ocol']}). We take $\kappa=10$, $J=1$ and the iDMRG bond dimension, $\chi=300$. The red dots indicate the points where the c/p-VBS order parameter vanishes, while the yellow dots denote the locations of discontinuities in the s-VBS order parameter. These points are obtained from various cuts discussed below.
• Figure 5: iDMRG data along the $\Omega = 0$ line for an infinite cylinder with circumference $L_y = 4$, $\kappa = 10$ and $J = 1$ (bond dimension $\chi = 250$): (a) Correlation length $\xi$ and bipartite von-Neuman entanglement entropy, $S$. (b) c/p-VBS order parameter $O_{\mathrm{col/plaq}}$ (Eq. \ref{['eq_ocol']}), s-VBS order parameter $O_{\mathrm{stag}}$ (Eq. \ref{['eq_ostag']}), and the mean value of the star operator $O_{\mathrm{star}}$.