Synthetic photonic materials are an emerging platform for exploring the interface between microscopic quantum dynamics and macroscopic material properties [1][2][3][4][5]. Photons experiencing a Lorentz force develop handedness, providing opportunities to study quantum Hall physics and topological quantum science [6][7][8]. Here we present an experimental realization of a magnetic field for continuum photons. We trap optical photons in a multimode ring resonator to make a two-dimensional gas of massive bosons, and then employ a non-planar geometry to induce an image rotation on each round-trip [9]. This results in photonic Coriolis/Lorentz and centrifugal forces and so realizes the Fock-Darwin Hamiltonian for photons in a magnetic field and harmonic trap [10]. Using spatialand energy-resolved spectroscopy, we track the resulting photonic eigenstates as radial trapping is reduced, finally observing a photonic Landau level at degeneracy. To circumvent the challenge of trap instability at the centrifugal limit [10,11], we constrain the photons to move on a cone. Spectroscopic probes demonstrate flat space (zero curvature) away from the cone tip. At the cone tip, we observe that spatial curvature increases the local density of states, and we measure fractional state number excess consistent with the Wen-Zee theory, providing an experimental test of this theory of electrons in both a magnetic field and curved space [12][13][14][15]. This work opens the door to exploration of the interplay of geometry and topology, and in conjunction with Rydberg electromagnetically induced transparency, enables studies of photonic fractional quantum Hall fluids [16,17] and direct detection of anyons [18,19].The Lorentz force on a charged particle moving in a magnetic field leads to the unique topological features of quantum Hall systems, including precisely quantized Hall conductance, topologically protected edge transport, and, in the presence of interactions, the predicted anyonic and non-abelian braiding statistics that form the basis of topological quantum computing [20]. To controllably explore the emergence of these phenomena, efforts have recently focused on realizing synthetic materials in artificial magnetic fields, and in particular, upon implementations for cold atoms and photons. Successful photonic implementations have employed lattices with engineered tunneling [6,[21][22][23][24]. However, it is desirable to realize artificial magnetic fields in the simpler case of a continuum (lattice-free) material [7,25,26], where strong interactions are more easily accessible and the theory maps more directly to fractional quantum Hall systems. In this work, we develop a new approach and demonstrate the first continuum synthetic magnetic field for light.To achieve photonic Landau levels we harness the powerful analogy between photons in a near-degenerate multimode cavity and massive, trapped 2d particles [27,28]. Owing to mirror curvature, the transverse dynamics of a running wave resonator are equivalent to those of a 2D quantum h...

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.

We compute electromagnetic, gravitational and mixed linear response functions of twodimensional free fermions in external quantizing magnetic field at an integer filling factor. The results are presented in the form of the effective action and as an expansion of currents and stresses in wave-vectors and frequencies of the probing electromagnetic and metric fields. We identify the terms in linear response functions coming from geometric Chern-Simons, Wen-Zee, and gravitational Chern-Simons terms in effective action. We derive the expressions for Hall conductivity, Hall viscosity and find the current and charge density responses to the spatial curvature as well as stresses caused by inhomogeneous electromagnetic fields.