Thermal Transitions Reveal Singularities and Changes in Quantum Systems

Shaun D. Hampton and colleagues at the Korea Institute for Advanced Study compute the perturbative expansion of thermal free energy to high orders, extracting key data that characterise the transition between the bounce and the sphaleron, critical phenomena in quantum mechanics involving metastable vacuum decay. The calculations determine the transition temperature and order, alongside the decay rate, including the one-loop prefactor, solely from perturbative calculations, avoiding the need for semiclassical approximations. The findings reveal how the behaviour of Borel singularities reflects the nature of the transition, smoothly joining for second-order transitions and exhibiting a distinct kink for first-order ones.

High-order thermal expansion reveals precise on-shell action and transition characteristics

A perturbative expansion of thermal free energy was computed to an unprecedented 250th order, a substantial improvement over previous methods limited to around 20th order expansions. This advance enabled the extraction of Borel singularity data, (A, b, S), as functions of temperature, detailing both second-order and first-order transitions between a bounce and a sphaleron. The bounce represents a classical solution describing the nucleation of a true vacuum within the false vacuum, while the sphaleron is a saddle-point solution mediating transitions between topologically distinct vacuum states. Previously, such high-order calculations were computationally prohibitive, requiring significant advances in computational algorithms and resources. The location of the Borel singularity, denoted as A, accurately reproduces the on-shell action of the dominant saddle, differing by less than 0.03% across the temperature range studied, demonstrating remarkable precision. The on-shell action is a crucial quantity in determining the probability of quantum tunnelling from the false vacuum to the true vacuum.

The characteristic exponent, ‘b’, shifts between values of 0 and 1/2 during the transition between decay mechanisms, directly reflecting the presence or absence of zero modes, essentially free vibrational modes, in the system. Zero modes indicate directions in the configuration space where the action is stationary, contributing to the overall decay probability. The Stokes constant, ‘S’, consistently matched the one-loop determinant, a complex calculation representing quantum corrections around the saddle point, confirming the accuracy of the prefactor in the decay rate calculation. The one-loop determinant accounts for the quantum fluctuations around the classical solution, significantly influencing the overall decay rate. Detailed analysis reveals that the perturbative expansion of thermal free energy can determine transition temperature and order without relying on traditional semiclassical approximations, although the method’s precision becomes limited when the two leading theoretical possibilities become very similar, hindering its immediate application to more complex systems. Semiclassical approximations, while computationally simpler, often introduce inaccuracies by neglecting quantum effects.

Analytic continuation via Borel resurgence reveals unstable quantum system decay pathways

Borel resurgence, a mathematical technique for extracting hidden information from seemingly incomplete calculations, proved central to this work. It allowed the circumvention of approximations previously relied upon in calculating unstable quantum system decay, much like reconstructing a blurry image by focusing on its sharpest parts. Generating a wealth of perturbative data involved computing the thermal free energy to very high orders, but directly interpreting this data proved challenging due to its inherent incompleteness. Perturbative series, while powerful, often suffer from divergence issues, limiting their direct applicability. Borel resurgence provides a means to analytically continue these series, extending their range of validity and revealing hidden singularities.

The method facilitated the analytic continuation of data, effectively filling gaps and revealing the system’s behaviour, specifically the characteristics of the transition between decay pathways. Computations were performed for two models exhibiting both first and second-order transitions, with data generated to the 250th order using a thermal trace involving up to 100 energy levels. The thermal trace involves summing over the contributions from various energy states at a given temperature, providing a complete description of the system’s thermal behaviour. The location of a Borel singularity reproduced the on-shell action of the dominant saddle on both sides of the transition, joining smoothly in the second-order case and developing a kink in the first-order case. This behaviour provides a clear diagnostic for distinguishing between different types of phase transitions.

This approach determines the transition temperature and decay rate without semiclassical approximations. Extending these findings to more realistic, complex physical scenarios presents a significant challenge, despite the current reliance on computations performed for specific, simplified models representing first and second-order transitions. These simplified models allow for a controlled investigation of the underlying physics, but their applicability to more complex systems needs further investigation. Nevertheless, acknowledging these calculations are presently limited to relatively simple theoretical models does not diminish their importance. The ability to accurately calculate decay rates from first principles, without relying on approximations, represents a significant step forward in our understanding of vacuum stability.

A method for calculating vacuum decay rates, the probability a universe will end in a lower energy state, has been demonstrated, without relying on approximations that often obscure the true physics. Vacuum decay is a fundamental concept in cosmology, with implications for the long-term stability of the universe. The Korea Institute for Advanced Study team has established a novel method for characterising thermal transitions between unstable quantum states, bypassing reliance on approximations previously essential for modelling vacuum decay. Computing the thermal free energy to an unprecedented order allowed data detailing both second-order and first-order transitions to be extracted, revealing how the behaviour of key mathematical values, known as Borel singularities, reflects the nature of the change occurring. This approach provides a clearer picture of how universes might transition between energy states, refining calculations of vacuum decay and the stability of the universe. The precise determination of the decay rate is crucial for assessing the longevity of our universe and understanding its ultimate fate.

The research successfully calculated vacuum decay rates without the need for approximations commonly used in cosmological modelling. This is important because it offers a more accurate understanding of how universes might transition between different energy states and assesses the stability of the universe. By computing the thermal free energy to a high order for both first and second-order transitions, researchers characterised key mathematical values that reflect the nature of these changes. The authors note that extending these findings to more complex physical scenarios remains a significant challenge.