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Journal of Statistical Mechanics Theory and Experiment

Exact magnetic properties of two-dimensional spin-1 Cooper pairs

Ze Cheng

Published Feb 1, 2023• 1 citation
Publisher

Abstract

Abstract Exact magnetic properties of two-dimensional (2D) spin-1 Cooper pairs are investigated within the framework of quantum statistical mechanics. We derive an exact analytical expression of the magnetization of 2D spin-1 Cooper pairs, which involves a q-digamma function and a q-Pochhammer symbol in mathematics. The magnetization density m(T) is a function of temperature T, Landé factor g, number density σ, and magnetic field B. Firstly, we examine the evolution of the magnetization density m(T) with the Landé factor g. m(T) is negative when g < 1, and changes gradually to positive after g c . Here, g c = 1 is the critical value of the Landé factor g, at which m ( T ) = 0 . Secondly, we examine the evolution of the magnetization density m(T) with the temperature T. When g < 1, there are the two asymptotic lines for the magnetization density m(T). The low-temperature asymptotic line is m ( T ) = − 387.5 nC s−1 and the high-temperature asymptotic line is m ( T ) = 0 . Finally, we examine the evolution of the magnetization density m(T) with the number density σ. No matter how large the Landé factor g is, there is a lower cutoff σ c of the number density σ. The lower cutoff number density is about σ c = 10 12 cm−2. If σ < σ c , the Cooper pair system does not support a solution of the reduced chemical potential x. As the number density σ is increased, the Bose–Einstein condensation of 2D Cooper pairs is strengthened.

DOI
10.1088/1742-5468/acac74

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Abstract Bose–Einstein condensation of two-dimensional spin-1 Cooper pairs in a uniform magnetic field is investigated within the framework of quantum statistical mechanics. We derive an exact analytical solution of two-dimensional spin-1 Cooper pairs in a uniform magnetic field, which involves a q -digamma function in mathematics. In two dimensions, it is found that the critical temperature and condensate fraction are monotonically increasing functions of magnetic field B , Landé factor g , and number density σ . Our investigation in two dimensions provides an example of the magnetic-field-tuned insulator-superconductor quantum phase transition.
The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential \ensuremath{\mu} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where \ensuremath{\mu} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where \ensuremath{\mu} diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where V is fixed, class (a) is realized for electron number density N smaller than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​, which is an increasing function of V, and class (b) is realized for N larger than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​. If N\ensuremath{\gg}Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​ in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential \ensuremath{\mu}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=Vcr{\mathit{V}}_{\mathrm{cr}}Vcr​(N) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.
Observed crossover events between different power functions of absolute temperature occurring below about ~1 K in the temperature dependence of the heat capacity or the thermal conductivity of the conventional superconductors are identified as transitions from Maxwell-Boltzmann to Bose-Einstein (BE) statistics of the Cooper-pairs. Because of the low mass of the Cooper pairs of 2me (with me as mass of the electron) and their high density, the BE-condensation temperature, TBE, of the Cooper-pairs is about five orders of magnitude higher than for the dilute alkali atom condensates. The condensation temperature TBE turns out to be proportional to the superconducting transition temperature TSC. From the observed TBE-values it is possible to calculate the density of the Cooper pairs. Assuming that the Cooper pairs form a dense gas of bosons, the diameter of the Cooper-pair orbitals turns out to be equal to the London penetration depth. As a conclusion, due to the large orbital diamagnetism of the Cooper-pairs pairs, only one layer of Cooper-pairs, next to the inner surface of the sample, is sufficient to shield an applied external magnetic field completely.
Motivated by recent experiments on low-dimensional frustrated quantum magnets with competing nearest-neighbor exchange coupling J 1 and next nearest-neighbor exchange coupling J 2 we investigate the magnetic susceptibility of two-dimensional J 1 - J 2 Heisenberg models with arbitrary spin quantum number s . We use exact diagonalization and high- temperature expansion up to order 10 to analyze the influence of the frustration strength J 2 / J 1 and the spin quantum number s on the position and the height of the maximum of the susceptibility. The derived theoretical data can be used to get information on the ratio J 2 / J 1 by comparing with susceptibility measurements on corresponding magnetic compounds.
Our experimental scheme is based on a barrel optical microresonator filled with a dye solution. The barrel mirror provides a confining potential, a chemical potential, and an effective mass for a photon, making the system formally equivalent to a one-dimensional gas of harmonically trapped, number-conserving, and massive bosons. Within the framework of quantum statistical mechanics, we propose an exact analytical solution to the problem of Bose-Einstein condensation in harmonically trapped, one-dimensional, and ideal photons. It is found that the photon number of vapor is characterized by an analytical function, which involves a qqq-digamma function in mathematics. The numerical calculation of the analytical solution gives many interesting results. In the thermodynamic limit, the analytical expressions of the critical temperature and the condensate fraction are derived. We find that the spectral radiance of a one-dimensional barrel cavity has a sharp peak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.
We start this article by recalling the main exact results previously obtained for describing the static magnetic properties of 2d square lattices composed of classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg couplings) [4c]. Near the critical temperature TC= 0 K we give criterions allowing to directly determine the magnetic phases characterizing 2d magnetic compounds described by our microscopic model. We show that there are three distinct regimes: The Renormalized Classical Regime (RCR), the Quantum Disordered Regime (QDR) and the Quantum Critical Regime (QCR). The static properties give a good image of the spin arrangement and thus remain a good starting point for the study of dynamic properties. As a result, inside each of the three regimes, we may derive the corresponding dynamic behaviors. For practical reasons that we shall detail, we restrict our study to the RCR and QCR cases. An experimental test is given for illustrating this theoretical work. We notably show that it allows one to derive the static correlation length which is an important tool for understanding both static and dynamic properties.