Abstract
The entropic order parameters measure in a universal geometric way the statistics of non-local operators responsible for generalized symmetries. In this article, we compute entropic order parameters in weakly coupled gauge theories. To perform this computation, the natural route of evaluating expectation values of physical (smeared) non-local operators is prevented by known difficulties in constructing suitable smeared Wilson loops. We circumvent this problem by studying the smeared non-local class operators in the enlarged non-gauge invariant Hilbert space. This provides a generic approach for smeared operators in gauge theories and explicit formulas at weak coupling. In this approach, the Wilson and ’t Hooft loops are labeled by the full weight and co-weight lattices respectively. We study generic Lie groups and discuss couplings with matter fields. Smeared magnetic operators, as opposed to the usual infinitely thin ones, have expectation values that approach one at weak coupling. The corresponding entropic order parameter saturates to its maximum topological value, except for an exponentially small correction, which we compute. On the other hand, smeared ’t Hooft loops and their entropic disorder parameter are exponentially small. We verify that both behaviors match the certainty relation for the relative entropies. In particular, we find upper and lower bounds (that differ by a factor of 2) for the exact coefficient of the linear perimeter law for thin loops at weak coupling. This coefficient is unphysical/non-universal for line operators. We end with some comments regarding the RG flows of entropic parameters through perturbative beta functions.
Highlights
A new perspective on generalized symmetries in quantum field theory (QFT) emerges by analyzing basic properties of the way observables are attached to spacetime regions [1].Given a region R with associated algebra A(R), generated by local degrees of freedom, causality takes the form A(R) ⊂ A(R ), (1.1)where R refers to the causal complement of the region R, i.e., the points spatially separated from R, and A is the commutant of A, i.e., the set of operators that commute with A
We find that Wilson and ’t Hooft loops are labeled by the weight and co-weight lattice of the gauge group, respectively
The probability distribution of the vacuum composed with the conditional expectation has the same form at these positions that it has at the origin due to the enlarged symmetry and we find pE∆x ∼ |Z1 | e− (22πc2)2 ω∨·( ij ω(i)Mi−j 1ω(j))·ω∨
Summary
A new perspective on generalized symmetries in quantum field theory (QFT) emerges by analyzing basic properties of the way observables are attached to spacetime regions [1]. Since the output of the analysis is some coupling-dependent functions valid at weak coupling, we end up briefly discussing how they run with the renormalization group (RG) flow, by using the known results from beta functions in pure gauge theories This gives important information on what can or can not be expected about the general behavior of entropic order parameters with the RG flow. The a and b operators do not commute with each other, and acting with the a operators we can produce endomorphisms of the maximal algebra of the region R containing the b operators, and vice-versa This means the non-local operators a’s and b’s associated with R and R respectively, can be considered as “topological operators” effecting a generalized symmetry on their respective complementary regions R and R. The representations of an Abelian group form another Abelian group, known as the Pontryagin dual
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
