Gauss law, minimal coupling and fermionic PEPS for lattice gauge theories

Abstract

In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced. We look at the Gauss law from two different points of view: for the gauge field, it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.

Lattice regularisation and entanglement structure of the Gross-Neveu model

“…dimensions [21,26,28,34,43], most of the tensor network effort has so far been invested in QFTs in (1+1) dimensions, and in particular the λφ 4 model [8,9,17,18,20,22,24] and the Schwinger model, i.e. (1+1)-dimensional quantum electrodynamics, (as well as nonabelian generalizations thereof) [19, 23, 25, 27, 29-31, 33, 35-39, 41, 44].…”

Variational Monte Carlo simulation with tensor networks of a pureZ3gauge theory in(2+1)DEmonts1, Bañuls2, Cirac3 et al. 2020Phys. Rev. DSelf Cite330140View full textAdd to dashboardCite

“…The fiducial operator AðxÞ which creates the modes out of the vacuum has to be Gaussian, and be invariant under transformation generated by G 0 ðxÞ. Hence, it is given by [22,24] AðxÞ ¼ exp…”

“…For a detailed overview of the gauging procedure in terms of PEPS operators, i.e., in graphical notation, we refer to Refs. [21,22,24]. In order to create more than a product state, we project the virtual, fermionic modes adjacent to each link onto maximally entangled states.…”

“…In the case of the magnetic energy, the first term on the right-hand side of (27) vanishes since the gauge field has no explicit dependence on the parameters. It remains to calculate the expression ∂ ∂α jΨðGÞj 2 since we know the form of jΨðGÞj 2 from the evaluation of the transition probability (24) already. Using Jacobi's formula…”

“…In these works, the global symmetry of a matter-only PEPS is lifted to a local one by introducing a gauge field, in a way analogous to minimal coupling. The latter gauging method has been used for the construction of gauged Gaussian fermionic PEPSs [22,23], where the matter state to be gauged is a free (Gaussian) fermionic state, in a manner analogous to minimal coupling of a Hamiltonian [24]. The restriction to this subclass of PEPSs enables the efficient contraction of the states with Monte Carlo techniques [25].…”