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

TL;DR

This work builds a monomer-dimer tensor-network (MDTN) basis for SU(N) lattice gauge theories, enabling qubit-regularized, sign-problem-free Hamiltonians that preserve gauge invariance. By restricting irreps and formulating H_Q = ∑_ℓ Ê_ℓ − δ ∑_P (Û_P + h.c.), it simultaneously studies classical (δ = 0) and quantum (δ > 0) regimes; finite-temperature confinement-deconfinement universality is demonstrated in d = 2 and d = 3 for SU(2) and SU(3) using classical loop algorithms, with Ising and Potts universality in 2D and Ising-like behavior in 3D, while a 1D SU(2) chain shows that the string tension σ can be tuned to zero as δ → ∞. The paper also provides large-δ perturbation theory for the 1D chain, yielding E_0 and σ expansions that identify 1/δ as a relevant gauge-coupling-like parameter and establish a deconfined critical point at δ = ∞. Overall, MDTN offers a scalable framework to explore qubit-regularized gauge theories in higher dimensions, potentially enabling continuum Yang–Mills behavior from finite-dimensional local spaces.

Abstract

Traditional $\mathrm{SU}(N)$ lattice gauge theories (LGTs) can be formulated using an orthonormal basis constructed from the irreducible representations (irreps) $V_λ$ 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 $\{λ_\ell\}$ and monomer tensors on sites labeled by $\{λ_s\}$. 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 $α_s = 1, 2, \dots, \mathcal{D}(\mathcal{H}_s^g)$ that labels an orthonormal basis of the gauge-invariant subspace of $\mathcal{H}^g_s$. This monomer-dimer tensor-network (MDTN) basis, $\left| \{λ_s\},\{λ_\ell\},\{α_s\}\right\rangle$, 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.

Figures (11)

• Figure 1: A pictorial representation of the two kinds of tensors that form the basis. The top shows the oriented dimer-tensor associated with the links, while the bottom shows the monomer-tensor at sites.
• Figure 2: A pictorial representation of ${{\cal H}_s^g}$ involving four dimer-tensors and a monomer-tensor. In this illustration, ${{\cal H}_s^g}=V_{\bar{\lambda}_1}\otimes V_{\lambda_2}\otimes V_{\bar{\lambda}_3}\otimes V_{\lambda_4} \otimes V_{\lambda_s}$ and the physical Hilbert space is obtained by projecting on to ${\mathrm{SU}}(N)$ singlets of ${{\cal H}_s^g}$. The various singlets are labeled by $\alpha_s = 1,2,\dots,{{\cal D}({\cal H}_s^g)}$.
• Figure 3: Illustration of a configuration in a qubit-regularized ${\mathrm{SU}}(2)$ pure gauge theory within the scheme, with nondynamical source matter fields located at sites $x$ and $y$. Blue circles represent $\lambda=1$ (singlets), while red circles denote $\lambda=2$ (doublets). Monomer tensors in the trivial are not shown, whereas those in the doublet are displayed at sites $x$ and $y$. On all sites, ${\cal D}_s({\cal H}_s^g) = 1$, except at sites $x$ and $y$, where ${\cal D}_s({\cal H}_s^g) = 2$. The label $\alpha_s$ is suppressed in this illustration.
• Figure 4: Illustration of a configuration in a qubit-regularized ${\mathrm{SU}}(3)$ pure gauge theory within the scheme, with nondynamical source matter fields located at sites $x$ and $y$. Blue circles represent $\lambda=1$ (singlets), while red and yellow circles denote $\lambda=3$ (triplets) and $\lambda=\bar{3}$ (anti-triplets), respectively. Monomer tensors in the trivial are not shown, whereas those in the triplet and anti-triplet are displayed at sites $x$ and $y$. On all sites, ${\cal D}_s({\cal H}_s^g) = 1$, except at site $x$, where ${\cal D}_s({\cal H}_s^g) = 2$. The label $\alpha_s$ is suppressed in this illustration.
• Figure 5: Illustration of the action of ${\hat{\cal U}_P}$. Here, we assume that the states $\left| {\{\lambda_s\},\{\lambda_\ell\},\{\alpha_s\}}\right\rangle$ are restricted according to the scheme, where $\lambda^+$ denotes the irrep obtained by adding a box to the Young tableau of $\lambda$, while $\lambda^-$ represents the obtained by removing a box, following the cyclic property of modulo $N$-boxes. The orientation of the plaquette, shown on the left, sets the convention for how the change. Notably, this construction preserves the $N$-ality. Thus, every plaquette action generates allowed states in the physical Hilbert space, although the singlet spaces and their dimensions before and after the action may differ. The complete definition of how singlet spaces are mapped is encoded in the coefficients $c({\{\lambda_s\},\{\lambda_\ell\},\{\alpha_s\}},\{\alpha_s'\},P)$, introduced in the definition of ${\hat{\cal U}_P}$ in \ref{['eq:chupdef']}.