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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 study the finite-size spectrum of the O(N ) symmetric Wilson-Fisher conformal field theory (CFT) on the d = 2 spatial-dimension torus using the expansion in = 3 − d. This is done by deriving a set of universal effective Hamiltonians describing fluctuations of the zero momentum modes. The effective Hamiltonians take the form of N -dimensional quantum anharmonic oscillators, which are shown to be strongly coupled at the critical point for small . The low-energy spectrum is solved numerically for N = 1, 2, 3, 4. Using exact diagonalization (ED), we also numerically study explicit lattice models known to be in the O(2) and O(3) universality class, obtaining estimates of the low-lying critical spectrum. The analytic and numerical results show excellent agreement and the critical low energy torus spectra are qualitatively different among the studied CFTs, identifying them as a useful fingerprint for detecting the universality class of a quantum critical point.