Many-Body Tunneling Drives Spontaneous Oscillation in Time Crystals at Critical Temperature

The quest to create systems that defy conventional expectations of equilibrium has led researchers to explore time crystals, phases of matter exhibiting spontaneous oscillations without requiring external energy input. Ximo Wang, Qiwei Han, and Zhenqi Bai, all from Shanxi University, alongside Hongyan Fan and Yichi Zhang, demonstrate a pathway to realising these perpetually moving states in strongly interacting many-body systems. Their work establishes a theoretical framework, utilising concepts from AdS/CFT duality, to understand how cooperative quantum tunnelling enables continuous time crystals to emerge and oscillate spontaneously, particularly within systems like three-dimensional optical lattices. This research is significant because it identifies a universal scaling law governing these time-crystalline behaviours at a critical temperature, offering insights into the fundamental properties of these exotic phases and potentially paving the way for their practical realisation

Time Crystals and AdS Black Hole Duality

This document details a theoretical investigation into time crystals, exploring connections between their properties and those of charged black holes in Anti-de Sitter (AdS) space. The research aims to establish a relationship between the fundamental characteristics of a time crystal, such as interaction strength and tunneling rates, and its overall temperature, linking microscopic details to macroscopic behaviour. This work utilises concepts from both condensed matter physics and general relativity to achieve this connection, representing a novel approach to understanding non-equilibrium systems. Time crystals, a recently discovered phase of matter, exhibit spontaneous breaking of time-translation symmetry, meaning they oscillate even in their ground state, unlike conventional crystals which break spatial symmetry. Establishing a link between these exotic systems and black holes, objects defined by extreme gravitational fields, offers a new perspective on both, potentially revealing deeper underlying principles governing their behaviour.

Calculations begin with the Reissner-Nordström (RN) AdS metric, a well-established solution describing a charged black hole in this specific spacetime, and determine the black hole’s horizon radius and temperature using established principles of general relativity, establishing a foundation for comparison with the time crystal’s properties. The RN AdS metric describes the spacetime geometry around a black hole possessing both mass and electric charge, existing within an AdS space, a spacetime with constant negative curvature. The Hawking temperature, calculated from the metric, represents the black hole’s effective temperature due to quantum effects, and crucially, depends on the black hole’s charge and the AdS radius. A key achievement is the derivation of a scaling law connecting the time crystal’s temperature to its microscopic parameters, specifically the mass of scalar particles mediating interactions within the time crystal and the scaling of the horizon radius with that mass. This involves relationships defining the chemical potential of the time crystal, which represents the energy required to add a particle to the system, and incorporating fundamental constants like Planck’s constant ($\hbar$) and the speed of light ($c$). Through careful substitution and simplification, the researchers arrive at equations quantitatively relating the time crystal’s temperature to its interaction strength, tunneling rate, and system size, demonstrating a surprising correspondence between these seemingly disparate systems.

The research also delves into the microscopic origins of this behaviour through a perturbative calculation, aiming to justify the effective Hamiltonian used in the scaling analysis. Starting with a Hamiltonian describing the time crystal, researchers apply the Schrieffer-Wolff transformation, a technique used to eliminate high-energy degrees of freedom from a quantum system, allowing focus on the essential physics governing the low-energy behaviour. This transformation effectively ‘integrates out’ the high-energy states, simplifying the Hamiltonian and revealing the dominant interactions. Through perturbation theory, they calculate corrections to the Hamiltonian, revealing a term proportional to the square of the tunneling amplitude ($\Gamma$) divided by the interaction strength ($U$), representing the effect of cooperative tunneling and providing a microscopic justification for the effective Hamiltonian used in the scaling analysis. The tunneling amplitude describes the probability of a particle escaping from a potential well, and its cooperative nature arises from the collective behaviour of particles within the time crystal. This term demonstrates that the time crystal’s temperature is not simply determined by external factors, but also by the intrinsic properties of its constituent particles and their interactions. Ultimately, this work utilises a dual description, linking a strongly correlated system, the time crystal, to a gravitational description involving the black hole, a concept rooted in the AdS/CFT correspondence, a conjecture proposing a duality between gravitational theories in AdS space and conformal field theories on its boundary. The derived scaling laws provide a quantitative connection between microscopic and macroscopic properties, while the perturbative calculations offer a microscopic justification for the effective Hamiltonian. While requiring a strong background in condensed matter physics, general relativity, and quantum field theory for full comprehension, it offers valuable insights into the fascinating physics of time crystals and their connection to black hole physics, potentially opening new avenues for exploring non-equilibrium phenomena and the fundamental nature of time itself.