Vacuum structure of gapped QCD$_2$ theories from the infinite Hamiltonian lattice

TL;DR

This work investigates the vacuum structure of gapped (1+1)D QCD-like theories with massless fermions by marrying continuum coset/TQFT data with Hamiltonian lattice descriptions realized through infinite MPS methods. The authors formulate a lattice decay rule and use uniform MPS with link-enhanced MPOs to identify lattice vacua that adiabatically connect to continuum degenerate vacua, even when lattice regularizations break non-invertible symmetries. Across four theories, they confirm continuum predictions for vacuum degeneracy, chiral condensate ratios, and particle degeneracies, and provide analytic strong-coupling expansions as consistency checks. The results demonstrate the power of infinite-lattice tensor networks to probe IR-topological structures in gauge theories and suggest avenues for extending these methods to broader gauge groups and higher dimensions, as well as exploring soliton spectra and symmetry restoration mechanisms.

Abstract

Gapped two-dimensional gauge theories with massless fermions generically have rich vacuum structures consisting of many degenerate vacua related by the action of topological line operators. The algebra of such operators has been used to calculate ratios of vacuum expectation values of local operators and to predict nontrivial particle-soliton degeneracies. In this paper, we use recently-developed tensor network methods to study several examples of such theories via their Hamiltonian lattice descriptions. Our lattice results agree with all previously-made predictions. Furthermore, we identify the lattice strong-coupling states that can be adiabatically continued to the degenerate vacua in the continuum limit. We conjecture a procedure, referred to as a lattice decay rule, for how this identification works in general. This rule allows us to compute the continuum vacuum degeneracy by studying the lattice Hamiltonian in the strong-coupling limit.

Figures (15)

• Figure 1: Difference in energy density between the lattice vacua as a function of the dimensionless lattice spacing, showing that all states become degenerate vacua in the continuum limit. Here $\varepsilon_{\{\bm{r}_1,\bm{r}_2\}}$ is the energy density of the lattice vacuum that is adiabatically connected to the strong-coupling state with link representations $\{\bm{r}_1,\bm{r}_2\}$. Lattice vacua that are related to those plotted by exact lattice symmetries (one-site translation and $G$ outer-automorphism) are omitted (see color coding in Table \ref{['tab:lattice_vacua']}).
• Figure 2: The probability of finding representations $\{\bm r_1,\bm r_2\}$ on consecutive links in the two $p=0$ lattice vacua of the $\mathop{\mathrm{SU}}\nolimits(2)+\psi_{\bm 5}$ theory. The gray dashed line represents all representations not explicitly plotted.
• Figure 3: The elementary lattice decays in the $\mathop{\mathrm{SU}}\nolimits(2)+\psi_{\bm{5}}$ theory. States are labeled by $\{\bm{r}_1,\bm{r}_2\}$, with colored circles indicating their flux tube sectors. The states identified by the lattice decay rule are marked with double boxes and colored according to the scheme in Table \ref{['tab:lattice_vacua']}.
• Figure 4: The elementary lattice decay pathways in the $\mathop{\mathrm{SU}}\nolimits(2)\times\mathop{\mathrm{SU}}\nolimits(2)+\psi_{(\bm{3},\bm{3})}$ theory. States are labeled by $\{\bm{r}_1,\bm{r}_2\}$, with colored circles indicating their flux tube sectors. The states identified by the lattice decay rule are marked with double boxes and colored according to the scheme in Table \ref{['tab:lattice_vacua']}.
• Figure 5: The elementary lattice decay pathways in the $\mathop{\mathrm{SU}}\nolimits(2)\times\mathop{\mathrm{SU}}\nolimits(2)+\psi_{(\bm{2},\bm{4})}$ theory. States are labeled by $\{\bm{r}_1,\bm{r}_2\}$, with colored circles indicating their flux tube sectors. The states identified by the lattice decay rule are marked with double boxes and colored according to the scheme in Table \ref{['tab:lattice_vacua']}.