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We explore the dynamics of artificial one-and two-dimensional Ising-like quantum antiferromagnets with different lattice geometries by using a Rydberg quantum simulator of up to 36 spins in which we dynamically tune the parameters of the Hamiltonian. We observe a region in parameter space with antiferromagnetic (AF) ordering, albeit with only finite-range correlations. We study systematically the influence of the ramp speeds on the correlations and their growth in time. We observe a delay in their build-up associated to the finite speed of propagation of correlations in a system with short-range interactions. We obtain a good agreement between experimental data and numerical simulations taking into account experimental imperfections measured at the single particle level. Finally, we develop an analytical model, based on a short-time expansion of the evolution operator, which captures the observed spatial structure of the correlations, and their build-up in time.

The low-energy spectra of many body systems on a torus, of finite size L, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for 2+1D systems. Using a combination of analytical and numerical techniques, we accurately calculate and analyse the low-energy torus spectrum at an Ising critical point which provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times 1/L. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition, in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides qualitative insight into the operator content of the critical field theory.PACS numbers: 05.30. Rt, 11.25.Hf, 75.10.Jm, 75.40.Mg1 Introduction -Quantum critical points continue to attract tremendous attention in condensed matter, statistical mechanics and quantum field theory alike. Recent highlights include the discovery of quantum critical points which lie beyond the Ginzburg-Landau paradigm [1, 2], the striking success of the conformal bootstrap program for Wilson-Fisher fixed points [3], and the intimate connection between entanglement quantities and universal data of the critical quantum field theory [4][5][6][7][8].A surprisingly little explored aspect in this regard is the finite (spatial) volume spectrum on numerically easily accessible geometries, such as the Hamiltonian spectrum on a 2D spatial torus at the quantum critical point [9]. In the realm of 1+1D conformal critical points there exists a celebrated mapping between the spectrum of scaling dimensions of the field theory in R 2 and the Hamiltonian spectrum on a circle (spacetime cylinder: S 1 × R) [10]. This result is routinely used to perform accurate numerical spectroscopy of conformal critical points using a variety of numerical methods [11,12]. In higher dimensions the situation is less favorable: Cardy has shown [13] that the corresponding conformal map can be generalized to a map between R d and S d−1 × R. While numerical simulations in this so-called radial quantization geometry have been attempted at several occasions [14][15][16][17][18], this numerical approach remains very challenging due to the curved geometry, which is inherently difficult to regularize in numerical simulations.Although low-energy spectra on different toroidal configurations have been discussed in the context of some specific field theories (in Euclidean spacetime) [19][20][21][22][23], our understanding of critical energy spectra is rather limited beyond free theories [24][25][26][27][28]. This is due to the absence of a known relation between the scaling dimensions of the field theory and the torus energy spectra.In this Letter we present a combined numerical and analytical study of the Hamiltonian torus energy spectrum of the 3D Ising conformal field theory (CFT), and show that it is ac-

We reconstruct the equilibrium phase diagram of quantum square ice, realized by the transverse-field Ising model on the checkerboard lattice, using a combination of quantum Monte Carlo, degenerate perturbation theory and gauge mean-field theory. The extensive ground-state degeneracy of classical square ice is lifted by the transverse field, leading to two distinct order-by-disorder phases, a plaquette valence-bond solid for low field, and a canted Néel state for stronger fields. These two states appear via a highly non-linear effect of quantum fluctuations, and they can be identified with the phases of a lattice gauge theory (quantum link model) emerging as the effective Hamiltonian of the system within degenerate perturbation theory up to the 8th order. The plaquette valence-bond solid melts at a very low temperature, above which the system displays a thermally induced quantum Coulomb phase, supporting deconfined spinons. Introduction. Kinematically constrained systems represent a central theme of statistical physics and condensed matter, as they often represent the effective low-energy description of fundamental lattice many-body Hamiltonians. Prominent examples are to be found in models of frustrated magnetism (e.g. frustrated Ising models [1], quantum dimer models [2]) and of ice physics and its magnetic (spin-ice) analogs [3]. In such models the energy is typically minimized by an exponentially degenerate manifold of states satisfying a local constraint -the so-called ice rule. A fundamental insight is gained when recognizing that the ice rule can be cast in the form of a Gauss law for an emergent electric field -corresponding, in the case of spin models, to the orientation of one of the spin components. When these systems are endowed with quantum dynamics, their effective Hamiltonian describing quantum fluctuations within the constrained manifold takes the form of a quantum lattice gauge theory (LGT). A deconfined phase of the LGT -namely a phase in which the gauge field is not able to bind charges -corresponds to a novel phase supporting fractionalized excitations in the original spin model. Important examples thereof are represented by the deconfined phase of the Z 2 LGT in dimensions d = 2, 3 [4], corresponding to the so-called Z 2 spin liquid in the magnetic context; and the deconfined phase of the d = 3 compact lattice quantum electrodynamics (QED) [5], corresponding to the so-called U(1) (or Coulomb) spin liquid [6]. The former represents a strong candidate for the ground-state of frustrated S = 1/2 Heisenberg antiferromagnets (e.g. on the Kagomé lattice [7]) while the latter is expected to be realized as the ground state of 3d quantum spin ice [8].