Infinite matrix product states for $(1+1)$-dimensional gauge theories

TL;DR

This paper introduces link-enhanced matrix product operators (LEMPOs) and symmetric matrix product states (MPS) to form a local, translation-invariant lattice Hamiltonian framework for (abelian and non-abelian) gauge theories in (1+1) dimensions, enabling accurate infinite-lattice simulations. By encoding gauge fields on virtual bonds via the MPS symmetry constraints and combining physical and link operators into a LEMPO, the authors implement gauge-theory Hamiltonians without gauge fixing that breaks translation invariance, and without introducing hard cutoffs on gauge degrees of freedom. They validate the approach with massless and massive Schwinger model calculations and SU($N_c$) adjoint QCD$_2$ (N_c = 2,3), achieving precise continuum extrapolations, detailed spectra, and string tensions, including interesting features like SUSY points and flux-tube sector dependence. The framework opens pathways to real-time dynamics, higher dimensions, and investigations of novel vacuum structures and symmetries, making it a powerful tool for non-perturbative gauge theory studies using tensor networks.

Abstract

We present a matrix product operator construction that allows us to represent the lattice Hamiltonians of (abelian or non-abelian) gauge theories in a local and manifestly translation-invariant form. In particular, we use symmetric matrix product states and introduce link-enhanced matrix product operators (LEMPOs) that can act on both the physical and virtual spaces of the matrix product states. This construction allows us to study Hamiltonian lattice gauge theories on infinite lattices. As examples, we show how to implement this method to study the massless and massive one-flavor Schwinger model and adjoint QCD$_2$.

Figures (11)

• Figure 1: The spatial arrangement of the degrees of freedom in the Schwinger model Hamiltonian on the lattice.
• Figure 2: The lattice strong-coupling expansions \ref{['eq:schwinger_sc']}, truncated at various orders in $y \equiv \frac{1}{(ga)^4}$ (green, red, and blue solid lines), compared with our lattice data (orange dots). Padé approximants to the strong-coupling expansions (black solid line) agree rather well with the lattice data even as $a\to 0$.
• Figure 3: Precision lattice estimates of the lightest particle mass and the chiral condensate in the massless Schwinger model. We extrapolate to $a\to 0$ using an ensemble of polynomial fits in powers of $g^2 a^2$ (shown in light gray), with degrees between 2 and 5 and using varying subsets of the data, and estimate errors from the variation in these fits. The results agree very well with the exact values; see \ref{['eq:schwinger_numbers']} and Table \ref{['tab:schwinger_masses']}.
• Figure 4: The spectrum of the Schwinger model as a function of the electron mass $m$, estimated using the quasiparticle ansatz with uniform matrix product states. As $\theta$ approaches $\pi$, the density of states in the $m>m_\text{crit} \approx 0.33g$ region grows, and at $\theta = \pi$ the spectrum becomes a continuum; this can be interpreted as the continuum of two-body states formed from solitons interpolating between the two degenerate vacua in this region.
• Figure 5: The energy density difference $\Delta \varepsilon(m, \theta)$ in the Schwinger model, defined in \ref{['eq:schwinger_sigma']}. The black dashed line is the weak-coupling expression \ref{['eq:schwinger_sc_wc']} for $\theta = \pi$. The lattice data also agrees well with the weak-coupling expansion at other values of $\theta$, as well as the small-mass expansion \ref{['eq:schwinger_sigma_lowmass']}, but we do not show these approximations in this figure in order to avoid clutter.