Core Concepts

The appearance of infinite permutation cycles in interacting bosonic systems is a necessary but not sufficient condition for Bose-Einstein condensation. The formation of infinite cycles is accompanied by singularities in thermodynamic quantities, which can be as strong as a one-sided divergence of the isothermal compressibility in three and four dimensions.

Abstract

The content examines the circumstances under which infinite permutation cycles can arise in systems of interacting bosons. It starts by recalling the basic facts about the ideal Bose gas, where infinite cycles appear simultaneously with Bose-Einstein condensation (BEC) as a consequence of saturation.

For interacting atoms, the occurrence of infinite cycles and BEC may not be related in the same way. The author demonstrates that the appearance of infinite cycles is always accompanied by a singularity in thermodynamic quantities, which can be as strong as a one-sided divergence of the isothermal compressibility in three and four dimensions.

The analysis is based on the path integral representation of the partition function. The author introduces several probability densities to decompose the partial density carried by cycles of length n. By taking the thermodynamic limit, an equation is derived that relates the density to the chemical potential and the properties of the finite cycles.

The author then discusses the conditions under which infinite cycles can be avoided or appear, even in one and two dimensions, depending on the long-range behavior of the pair potential. For repulsive long-range interactions in three dimensions, it is conjectured that infinite cycles and BEC are completely absent. Conversely, attractive long-range tails can lead to the appearance of infinite cycles in one and two dimensions, even without BEC.

The transition from finite to infinite cycles is considered a phase transition in its own right, reminiscent of the percolation transition. The author suggests that superfluidity may be more closely related to the presence of infinite cycles rather than BEC, as supported by experimental observations in helium liquids.

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by Andras Suto at **arxiv.org** 10-01-2024

Deeper Inquiries

The presence or absence of infinite cycles in interacting bosonic systems has profound implications for their thermodynamic properties and phase behavior. Infinite cycles are indicative of a saturation phenomenon where a significant fraction of particles occupies these cycles, leading to macroscopic occupancy of zero-momentum states. This is crucial for understanding the nature of Bose-Einstein condensation (BEC), as the emergence of infinite cycles is a necessary condition for BEC to occur. However, the relationship is nuanced; the existence of infinite cycles does not guarantee BEC, especially in the presence of interactions.
When infinite cycles are present, they can lead to singularities in thermodynamic quantities, such as a one-sided divergence in the isothermal compressibility. This divergence indicates a phase transition-like behavior, suggesting that the system undergoes significant changes in response to variations in density or temperature. Conversely, the absence of infinite cycles, particularly in three dimensions under long-range repulsive interactions, can prevent BEC entirely, indicating a fundamentally different phase behavior. This absence can also imply that the system remains in a normal phase, where particles are distributed among finite cycles, leading to different statistical mechanics and thermodynamic responses.
Moreover, the presence of infinite cycles can affect the stability of the system, influencing the critical temperature for phase transitions and the nature of excitations within the system. The interplay between finite and infinite cycles can lead to complex phase diagrams, where regions of stability and instability are delineated by the density and interaction strength. Thus, understanding infinite cycles is essential for a comprehensive grasp of the thermodynamic landscape of interacting bosonic systems.

The insights gained from the analysis of infinite cycles in interacting bosonic systems, particularly regarding long-range interactions, can significantly inform the study of other many-body quantum systems, including fermionic systems and those with mixed statistics. In fermionic systems, the Pauli exclusion principle introduces different statistical behavior compared to bosons. However, the concepts of cycle formation and the impact of interactions remain relevant. For instance, the presence of long-range interactions in fermionic systems could lead to similar phenomena, such as the formation of collective excitations or the emergence of new phases that are not present in short-range interaction scenarios.
Additionally, the framework developed for bosonic systems can be adapted to explore the effects of long-range interactions in fermionic systems, particularly in contexts like superconductivity, where pairing interactions can lead to collective behavior reminiscent of bosonic systems. The analysis of infinite cycles could provide a new perspective on the nature of Cooper pairs and their role in the superfluid phase of fermionic systems.
For systems with mixed statistics, such as anyons in two-dimensional systems, the insights regarding the interplay between interaction strength and cycle formation can lead to a deeper understanding of how these systems behave under varying conditions. The presence of long-range attractive or repulsive interactions could influence the statistics of the particles, potentially leading to novel phases or transitions that are not captured by traditional models.
Overall, the exploration of infinite cycles and their implications in bosonic systems opens avenues for investigating similar phenomena in fermionic and mixed-statistics systems, enriching the theoretical framework of many-body quantum physics.

Yes, the framework developed in the analysis of infinite cycles in interacting bosonic systems can indeed be extended to explore the connection between infinite cycles and superfluid properties in a more rigorous and quantitative manner. Superfluidity is characterized by the ability of a fluid to flow without viscosity, a phenomenon closely related to the macroscopic occupation of low-energy states, akin to the formation of infinite cycles in bosonic systems.
To establish a quantitative connection, one could employ the path integral formalism and statistical mechanics techniques used in the analysis of infinite cycles. By examining the behavior of the one-particle reduced density matrix and its eigenvalues, one can derive conditions under which superfluidity emerges in relation to the presence of infinite cycles. The critical role of long-range interactions, as highlighted in the context of bosonic systems, can also be investigated in superfluid systems, particularly in how they influence the density of states and the distribution of particles among various energy levels.
Furthermore, the singularities associated with the transition from finite to infinite cycles can be correlated with the onset of superfluidity, providing a framework to study the critical temperature and other thermodynamic properties. The analysis could also incorporate the effects of external fields and boundary conditions, which are crucial in real-world superfluid systems.
By extending the framework to include superfluid properties, researchers can gain insights into the mechanisms that govern superfluidity, potentially leading to new predictions about the behavior of superfluid systems under various conditions. This rigorous approach could also facilitate comparisons between different types of superfluids, such as those arising from bosonic versus fermionic systems, enriching our understanding of quantum fluids in general.

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