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 describe the quantum phase transition in the N -state chiral clock model in spatial dimension d = 1. With couplings chosen to preserve time-reversal and spatial inversion symmetries, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For such couplings and N = 3, the clock model is expected to have a direct phase transition from a gapped phase with a broken global Z N symmetry, to a gapped phase with the Z N symmetry restored. The transition has dynamical critical exponent z = 1, and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in d = 1, involving the onset of a single boson condensate in the background of a higher-dimensional N -boson condensate.We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in 2 − d, with 4 − N chosen to be of order 2 − d. At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for N = 3, finding good evidence for a direct phase transition, and obtain estimates for z and the correlation length exponent ν.arXiv:1808.07056v2 [cond-mat.str-el] 19 Oct 2018 42 4. I (M ) 4 44 C. Renormalization constants for the Z N dilute Bose gas 45 D. The self-dual phase boundary in the chiral clock model 46 E. Analysis as λ → 0 48