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.
Highlights
We introduce a staggered fermionic Hamiltonian, where one sublattice corresponds to particles and another to anti-particles; if one tunes the hopping coefficients in a slightly different way, which was not done here for the sake of simplicity, a Dirac model is obtained in the continuum limit [22]: H f = M (−1)x ψ† (x) ψ (x) + ε ψ† (x) ψ x + ei + h.c. , (5)
To make the symmetry local, we introduce on each link = (x, i) of the lattice a new Hilbert space for the gauge field
Why does that work? Because every Trotter step involves the dynamics of a single link Hamiltonian, just like H2 we considered above (27), and its gauging gives rise to simple, one dimensional Gauss laws generated by operators such as those of (30) rather than the ambiguous, vector Gauss laws of the general case (14)
Summary
That provide the standard model’s description of interactions, are fascinating. Constructing a quantum simulator or a tensor network state involves an interesting challenge: one has to reconstruct a physical model from elementary building blocks In this process, one faces the most fundamental elements of a theory, which is decomposed to its smallest ingredients. The gauge symmetry is formulated, in both classical and quantum gauge theories, by means of the Gauss law, an equation – or a set of equations, one per space point – that relates the gauge field and the matter fields. In lattice formulations with continuous time, which do not put time and space on an equal footing, it cannot be obtained as an equation of motion either As it is a constraint, it means we can try to solve it, and if we manage to do that, we plug it into the other equations of motion to reduce the number of degrees of freedom. The lecture reviews the works [13,14,15,16,17,18,19,20]
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