Towards classification of Fracton phases: the multipole algebra

We present an effective field theory approach to the Fracton phases. The approach is based the notion of a multipole algebra. It is an extension of space(-time) symmetries of a charge-conserving matter that includes global symmetries responsible for the conservation of various components of the multipole moments of the charge density. We explain how to construct field theories invariant under the action of the algebra. These field theories generally break rotational invariance and exhibit anisotropic scaling. We further explain how to partially gauge the multipole algebra. Such gauging makes the symmetries responsible for the conservation of multipole moments local, while keeping rotation and translations symmetries global. It is shown that upon such gauging one finds the symmetric tensor gauge theories, as well as the generalized gauge theories discussed recently in the literature. The outcome of the gauging procedure depends on the choice of the multipole algebra. In particular, we show how to construct an effective theory for the $U(1)$ version of the Haah code based on the principles of symmetry and provide a two dimensional example with operators supported on a Sierpinski triangle. We show that upon condensation of charged excitations Fracton phases of both types as well as various SPTs emerge. Finally, the relation between the present approach and the formalism based on polynomials over finite fields is discussed.Comment: 17 pages, 8 figure

Bimetric Theory of Fractional Quantum Hall States

We present a bimetric low-energy effective theory of fractional quantum Hall (FQH) states that describes the topological properties and a gapped collective excitation, known as Girvin-Macdonald-Platzman (GMP) mode. The theory consist of a topological Chern-Simons action, coupled to a symmetric rank two tensor, and an action \`a la bimetric gravity, describing the gapped dynamics of the spin-$2$ GMP mode. The theory is formulated in curved ambient space and is spatially covariant, which allows to restrict the form of the effective action and the values of phenomenological coefficients. Using the bimetric theory we calculate the projected static structure factor up to the $k^6$ order in the momentum expansion. To provide further support for the theory, we derive the long wave limit of the GMP algebra, the dispersion relation of the GMP mode, and the Hall viscosity of FQH states. We also comment on the possible applications to fractional Chern insulators, where closely related structures arise. Finally, it is shown that the familiar FQH observables acquire a curious geometric interpretation within the bimetric formalism.Comment: 14 pages, v2: Acknowledgments updated, v3: A few presentation improvements, Published versio

Entanglement Entropy in 2D Non-abelian Pure Gauge Theory

We compute the Entanglement Entropy (EE) of a bipartition in 2D pure non-abelian $U(N)$ gauge theory. We obtain a general expression for EE on an arbitrary Riemann surface. We find that due to area-preserving diffeomorphism symmetry EE does not depend on the size of the subsystem, but only on the number of disjoint intervals defining the bipartition. In the strong coupling limit on a torus we find that the scaling of the EE at small temperature is given by $S(T) - S(0) = O\left(\frac{m_{gap}}{T}e^{-\frac{m_{gap}}{T}}\right)$, which is similar to the scaling for the matter fields recently derived in literature. In the large $N$ limit we compute all of the Renyi entropies and identify the Douglas-Kazakov phase transition.Comment: 6 page