Ordering in statistical systems on the way to the thermodynamic limit

TL;DR

This work introduces order indices, defined as $\omega(A) = \frac{\log ||A||}{\log |\mathrm{Tr}A|}$ for trace-class operators, to quantify how order emerges and grows as a finite statistical system approaches the thermodynamic limit. By applying these indices to first- and second-order reduced density operators and to correlation operators, the authors analyze Bose-Einstein condensation, superconducting pairing, magnetic ordering, and crystal-like density ordering within mean-field frameworks, including Bogolubov shifts and HF-Bogolubov approximations. The results show that diagonal order (e.g., crystal order) and pairing order produce distinct asymptotic indices (up to 1 for diagonal, up to 1/2 for pairing) in the thermodynamic limit, while single-particle order can saturate at 1 under suitable conditions; interactions tend to slow the growth of order for finite systems. Overall, the order-index framework provides a rigorous, size-dependent measure of preordering that connects finite-size physics to the thermodynamic limit and offers a unified lens for analyzing emergent order in diverse quantum many-body systems.

Abstract

It is well known that the mathematically accurate description of ordering and related symmetry breaking in statistical systems requires to consider the thermodynamic limit. But the order does not appear from nowhere, and yet before the thermodynamic limit is reached, there should exist some kind of preordering that appears and grows in the process of increasing the system size. The quantitative description of growing order, under the growing system size, is developed by introducing the notion of {\it order indices}. The rigorous proof of the phase transition existence is a separate difficult problem that is not the topic of the present paper. We illustrate the approach resorting to several models in the mean-field approximation, which makes it possible to demonstrate the notion of order indices for finite systems in a clear way. We show how the order grows on the way to the thermodynamic limit for Bose-Einstein condensation, arising superconductivity, magnetization, and crystallization phenomena.

Figures (5)

• Figure 1: Condensate fraction $n_0$ (dashed-dotted line), fraction of uncondensed particles $n_1$ (dotted line), anomalous average $\sigma$ (dashed line), and dimensionless sound velocity $s$ (solid line) as functions of the gas parameter $\gamma$, at zero temperature.
• Figure 2: Appearance of order at Bose-Einstein condensation under increasing system size. First order index $\omega(\hat{\rho}_1)$ as a function of $\ln N$ for different gas parameters $\gamma=0.1$ (solid line), $\gamma=0.5$ (dashed line), and $\gamma =1$ (dashed-dotted line).
• Figure 3: Appearance of order at Bose-Einstein condensation under increasing system size. Second order index $\omega(\hat{\rho}_2)$ as a function of $\ln N$ for different gas parameters $\gamma=0.1$ (solid line), $\gamma=0.5$ (dashed line), and $\gamma=1$ (dashed-dotted line).
• Figure 4: Appearance of order at superconduction transition under increasing system size. Second order index $\omega(\hat{\rho}_2)$ as a function of $\ln N$ for different fractions of pair-correlated particles $n_{cor}=0.1$ (solid line), $n_{cor}= 0.5$ (dashed line), and $n_{cor}=1$ (dashed-dotted line).
• Figure 5: Appearance of order at magnetic transition under increasing system size. First order index $\omega(\hat{C}_1)$ as a function of $\ln N$ for different magnetizations $M \equiv (1/S) \langle S_i^z \rangle$, with $M=0.01$ (solid line), $M= 0.1$ (dashed line), and $M=0.5$ (dashed-dotted line).