Monomer-dimer tensor-network basis for qubit-regularized lattice gauge theories

Traditional $\mathrm{SU}(N)$ lattice gauge theories (LGTs) can be formulated using an orthonormal basis constructed from the irreducible representations (irreps) $V_{\lambda}$ of the $\mathrm{SU}(N)$ gauge symmetry. On a lattice, the elements of this basis are tensor networks comprising dimer tensors on the links labeled by a set of irreps $\{\lambda_\ell\}$ and monomer tensors on sites labeled by $\{\lambda_s\}$. These tensors naturally define a local site Hilbert space, $\mathcal{H}^g_s$, on which gauge transformations act. Gauss's law introduces an additional index $\alpha_s = 1, 2, \dots, \mathcal{D}(\mathcal{H}_s^g)$ that labels an orthonormal basis of the gauge-invariant subspace of $\mathcal{H}^g_s$. This monomer-dimer tensor-network (MDTN) basis, $\left| \{\lambda_s\},\{\lambda_\ell\},\{\alpha_s\}\right\rangle$, of the physical Hilbert space enables the construction of new qubit-regularized $\mathrm{SU}(N)$ gauge theories that are free of sign problems while preserving key features of traditional LGTs. Here, we investigate finite-temperature confinement-deconfinement transitions in a simple qubit-regularized $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ gauge theory in $d=2$ and $d=3$ spatial dimensions, formulated using the MDTN basis, and show that they reproduce the universal results of traditional LGTs at these transitions. Additionally, in $d=1$, we demonstrate using a plaquette chain that the string tension at zero temperature can be continuously tuned to zero by adjusting a model parameter that plays the role of the gauge coupling in traditional LGTs.