Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories

Abstract

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-

Quantum robustness and phase transitions of the 3D Toric Code in a field

“…It belongs to the (3 + 1)D-Ising universality class and has mean-field critical exponents. The corresponding quantum criticality in the dual picture, i.e., for the 3D toric code in a uniform magnetic field h z , is then (3 + 1)D-Ising* [90]. Duality transformation for h = (0, h y , 0).…”

Anisotropic layer construction of anisotropic fracton models

“…Obviously, the ground state becomes a fully polarized state with a trivial topological order in the limit of large h X or h Z , and thus there will be a phase transition between the Z 2 and trivial topological orders for some h X or h Z . In fact, these models have been studied previously [72][73][74][75][76]; the corresponding phase transitions are conjectured to be in the (2+1)-d Ising * universality class-the Wilson-Fisher fixed point of a real scalar field theory coupled with a Z 2 gauge field [77].…”

Quantum robustness of fracton phases

“…Apart from understanding the role of thermal fluctuations on fracton codes, it is important and interesting to investigate the breakdown of fracton topological order at zero temperature. For the more conventional topological models like the toric code, color codes, or stringnet models very rich phase diagrams with quantum critical behavior have been found when adding perturbations like an external magnetic field [8,[53][54][55][56][57][58][59][60][61][62][63][64][65]. This is widely unexplored for three-dimensional fracton order, which is the main motivation for this work.…”