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From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators

Aritra Banerjee, Arpan Bhattacharyya, Priya Drashni, Srinidhi Pawar

TL;DR

This work probes the contraction from 2d CFTs to BMS$_3$/Conformal Carrollian theories using chaos diagnostics, by modeling a free scalar and mapping the CFT vacuum to a BMS vacuum via a two-mode squeezed state as $\epsilon\to0$. It computes circuit complexity through covariance-matrix methods and information-geometric tools (Fubini-Study metric and Berry curvature), showing a per-mode divergence $\mathcal{C}_k\sim \frac{1}{\sqrt{2}}\operatorname{arccosh}\big(\frac{1+\epsilon^2}{2\epsilon}\big)$ at the BMS point, and a geometric picture with an infinite-length geodesic in the state manifold. Time evolution under the $\epsilon$-dependent Hamiltonian reveals OTOCs that remain oscillatory for $0<\epsilon\le1$ but become polynomial in time as $\epsilon\to0$, and Krylov complexity shows $t^2$-scaling at the BMS point, indicating suppressed chaos. Overall, the paper uncovers a quantum-phase-like transition at the BMS limit, linking flat-space holography, vacuum structure, and information-theoretic markers, and lays groundwork for exploring higher-dimensional BMS theories and their chaotic properties.

Abstract

We probe the contraction from $2d$ relativistic CFTs to theories with Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, using diagnostics of quantum chaos. Starting from an Ultrarelativistic limit on a relativistic scalar field theory and following through at the quantum level using an oscillator representation of states, one can show the CFT$_2$ vacuum evolves smoothly into a BMS$_3$ vacuum in the form of a squeezed state. Computing circuit complexity of this transmutation using the covariance matrix approach shows clear divergences when the BMS point is hit or equivalently when the target state becomes a boundary state. We also find similar behaviour of the circuit complexity calculated from methods of information geometry. Furthermore, we discuss the hamiltonian evolution of the system and investigate Out-of-time-ordered correlators (OTOCs) and operator growth complexity, both of which turn out to scale polynomially with time at the BMS point.

From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators

TL;DR

This work probes the contraction from 2d CFTs to BMS/Conformal Carrollian theories using chaos diagnostics, by modeling a free scalar and mapping the CFT vacuum to a BMS vacuum via a two-mode squeezed state as . It computes circuit complexity through covariance-matrix methods and information-geometric tools (Fubini-Study metric and Berry curvature), showing a per-mode divergence at the BMS point, and a geometric picture with an infinite-length geodesic in the state manifold. Time evolution under the -dependent Hamiltonian reveals OTOCs that remain oscillatory for but become polynomial in time as , and Krylov complexity shows -scaling at the BMS point, indicating suppressed chaos. Overall, the paper uncovers a quantum-phase-like transition at the BMS limit, linking flat-space holography, vacuum structure, and information-theoretic markers, and lays groundwork for exploring higher-dimensional BMS theories and their chaotic properties.

Abstract

We probe the contraction from relativistic CFTs to theories with Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, using diagnostics of quantum chaos. Starting from an Ultrarelativistic limit on a relativistic scalar field theory and following through at the quantum level using an oscillator representation of states, one can show the CFT vacuum evolves smoothly into a BMS vacuum in the form of a squeezed state. Computing circuit complexity of this transmutation using the covariance matrix approach shows clear divergences when the BMS point is hit or equivalently when the target state becomes a boundary state. We also find similar behaviour of the circuit complexity calculated from methods of information geometry. Furthermore, we discuss the hamiltonian evolution of the system and investigate Out-of-time-ordered correlators (OTOCs) and operator growth complexity, both of which turn out to scale polynomially with time at the BMS point.
Paper Structure (17 sections, 96 equations, 2 figures)

This paper contains 17 sections, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Revisiting BMS$_3$ invariant scalar field
  3. The intrinsic model
  4. Canonical quantization
  5. Limiting perspective
  6. 'Position space' representation of the vacuum
  7. Circuit complexity: From CFT to BMS
  8. Circuit complexity: A brief Introduction
  9. Behaviour of circuit complexity as a function of $\epsilon$
  10. Information Geometry
  11. Fubini-Study metric
  12. Berry Curvature
  13. Hamiltonian Evolution
  14. Diagonalisation
  15. Out-of-time-ordered correlators
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Complexity as function of flow parameter $\epsilon.$$\epsilon=1$ and $\epsilon=0$ corresponds to the CFT and BMS point respectively. It clearly diverges at the point $\epsilon=0.$ We have rescaled the complexity by a factor of $\sqrt{2}.$
  • Figure 2: Fubini-Study distance as function of flow parameter $\epsilon.$$\epsilon=1$ and $\epsilon=0$ corresponds to the CFT and BMS point respectively.