Z2lattice Gerbe theory

Abstract

Discretized formulations of 2-form abelian [1][2][3][4][5][6][7] and non-abelian [8][9][10][11] gauge fields on ddimensional hypercubic lattices have been discussed in the past by various authors and most recently in [12]. In this note we recall that the Hamiltonian of a Z2 variant of such theories is one of the family of generalized Ising models originally considered by Wegner [13]. For such "Z2 lattice gerbe theories" general arguments can be used to show that a phase transition for Wilson surfaces will occur for d > 3 between volume and area scaling behaviour. In 3d the model is equivalent under duality to an infinite coupling model and no transition is seen, whereas in 4d the model is dual to the 4d Ising model and displays a continuous transition. In 5d the Z2 lattice gerbe theory is self-dual in the presence of an external field and in 6d it is self-dual in zero external field. The study of the lattice discretization of 1-form gauge fields, lattice gauge theory, is now a mature subject. Higher form gauge fields on the lattice have received less attention, although 2-form gauge fields or gerbes have also proved to be of considerable interest. In string theory interest in higher form fields dates back to the work of Kalb and Ramond [14] and they have also arisen more recently in M-theory, particularly with regard to describing M5-branes. Abelian gerbes suffice to describe the six-dimensional low energy effective theory arising from a single M5-brane [15,16], but multiple M5-branes require a non-abelian theory. This has proved difficult to formulate in the continuum and motivated the authors of [12] to revisit both abelian and non-abelian gerbe theories on Euclidean hypercubic lattices in the manner of the Wilsonian formulation of lattice gauge theory [17][18][19][20][21][22]. The theory of abelian 2-form (and also higher nform) fields on a lattice had already been explored from varying perspectives in [1][2][3][4][5][6][7] and non-abelian lattice theories were also proposed in [8][9][10][11]. Later, rather similar ideas were pursued in [23][24][25]. In the lattice gerbe theories the gauge variables lived on the faces of the cubes of a hypercubic lattice and the Hamiltonian (i.e. action in field theoretic language) was given by the product of face variables round elementary cubes, which were called cubets in [12], summed over the lattice.We take a step further back here and consider what might be learnt from the simplest Hamiltonians that maintain the 2-form gauge invariance, which are constructed from Z 2 spins and which we call Z 2 lattice gerbe theories, borrowing the nomenclature of [12]. Many of the characteristic features of a lattice gauge theory are already discernible in the simplest Z 2 variant, so a reasonable expectation is that such a Z 2 lattice gerbe theory might offer a similar degree of insight into the behaviour of more general, including non-abelian, lattice gerbe theories.

Fracton topological order, generalized lattice gauge theory, and duality

“…In contrast to our construction, applying the standard gauging procedure to any of the spin models shown in Table I by introducing a gauge field on the links of the cubic lattice would result in a Hamiltonian with conventional Z 2 topological order. Our procedure is also distinct from discretizations of "higher-form" gauge theories, in which interactions between (n − 1)-form matter fields are mediated by an n-form gauge field [34,35]. The subsystem symmetry of the classical spin system (1) has now been promoted to a local spin-flip symmetry of the Hamiltonian (3).…”

Model of fractionalization of Faraday lines in compact electrodynamics

“…(18). Thus, our model is also an example of a lattice Gerbe theory [6,7] emerging as an effective field theory description of the CQED phase with fractionalized Faraday lines, similar to how lattice gauge theories can emerge as descriptions of fractionalized phases of bosons with short-ranged interactions.…”

“…4 are in agreement with this. In general dimension, adding a gapped rank-1 fieldã μ (r)-i.e., introducing only a small "line dynamics" parameter κ-will not affect the rank-2 confinement/deconfinement transition [6,7]; however, it will destroy the higher-rank generalization of the Wilson loop diagnostics [6], similarly to what happens when adding a dynamical matter field to a lattice gauge theory [12]. In the rank-2 deconfined phase at large λ and small κ, theã μ (r) fields represent true gapped line excitations-fractionalized Faraday lines carrying 1/N of the elementary electric field strength.…”

“…The model works by energetically binding together multiple monopoles, and we also study it numerically in (3+1) dimensions using Monte Carlo. We will also derive microscopically an effective description of this phase in terms of fractionalized Faraday lines coupled to a rank-2 Z N field ("Gerbe field" [6,7]) in the rank-2 deconfined phase, and we will show the ground state degeneracy of…”

β‐Carotene Revisited by Transient Absorption and Stimulated Raman Spectroscopy

“…[39] For 1 b 2 one tunes betweenasingle exponentiala nd aG aussianf unction, respectively;i nt his range the function of Equation (2) is often referred to as "compressed exponential" (CE). It can be described by ac ontinuous distribution of Gaussians; [40] examples are given in Refs.…”