Thermodynamic Properties and Persistent Currents of Harmonic Oscillator Under AB-Flux Field in a Point-Like Defect with Inverse Square Potential

In this analysis, we study the eigenvalue solution of the non-relativistic particles confined by the Aharonov-Bohm (AB) flux field in the presence of potential the superposition of a harmonic oscillator plus inverse square potential with constant term in the background of a point-like global monopole. Afterwards, we study the thermodynamic properties of the quantum system at finite temperatures T≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\ne 0$$\end{document} and calculate the vibrational free energy, mean energy, specific heat capacity, and the entropy by using the partition function Z(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(\beta )$$\end{document}. These quantities are then analyzed and show the influences of the topological defect with flux field and potential. We also see that the energy eigenvalue depends on the geometric quantum phase, and thus an electromagnetic analogue of the Aharonov-Bohm effect is observed. This dependence of the eigenvalue gives rise to a persistent current and we analyze the effects of the topological defect with potential on it.


Introduction
The exact and approximate eigenvalue solutions of the wave equations have significant because the total wave function contains all the necessary information of a quantum system under investigation. The investigations of the topological defect in a quantum system have significant because its presence changes the geometric properties of a space-time and hence, changes the physical properties of the quantum 1 3 system. The eigenvalue solutions of the quantum system and the thermodynamic functions are influenced by the topological defect and modified the results compared to Minkowski flat space. The wave equations in two-, three-, and four-as well as higher dimensions in the background of the flat space (Minkowski space-time) have been studied in the literature. In these investigations, interaction potential of various kinds, such as linear confining potential, Coulomb potential, pseudoharmonic potential, Mie-type potential, Cornell-type potential and its generalization. In addition, a few exponential-type potential of various kinds (Yukawa, Hulthen, Morse, Rosen-Morse, Deng-Fan, and Eckart potential) have also been considered in the quantum systems. The analytical eigenvalue solutions of the wave equations have been obtained using different techniques or methods, such as the Nikiforov-Uvarov method and its functional analysis, the exact quantization method, WKB-approximation, the Laplace's transform and a few more. It is worth mentioning that only a few potential models give us the exact eigenvalue solutions of the wave equations. In the case of exponential-type potentials, a suitable approximation scheme, the most common one is the Greene-Aldrich scheme and its improved version, the Pekeris scheme have been employed (see, Refs. [1][2][3][4][5][6] and related references therein). Many authors have investigated the quantum systems in the presence of external uniform magnetic and quantum flux fields with these potentials. The quantum flux field in the quantum system shifted the eigenvalue solutions that shows an analogue to the Aharonov-Bohm effect. The thermodynamic and magnetic properties of the quantum system with these fields have been studied in the literature [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].
The relativistic wave equations in the background of curved space-time, such as the Gödel-type metric, Som-Raychaudhuri space-time and other space-times, have been studied. Many authors investigated the wave equations in these curved spacetimes with topological defects, such as cosmic string. In the literature, the non-relativistic wave equation with different potentials in the background of the topological defects has been studied only in a few works. The non-relativistic wave equation in the background of the topological defects produced by a cosmic string with magnetic and flux fields has been investigated (see, Ref. [18] and related references therein). Furthermore, the quantum systems in the background of the topological defect produced a point-like global monopole with different potentials have been studied only in Refs. [19][20][21][22][23][24][25][26][27].
In this analysis, we choose the following potential form given by [25] where the first term is the harmonic oscillator with 2 = 1 2 M 2 , and −2 is a parameter which caracterizes the potential strength. This potential the superposition of a harmonic oscillator plus inverse square potential have been used by many authors in the literature [28][29][30][31][32].
We present the exact analytical eigenvalue solution of the non-relativistic wave equation under the influence of quantum flux field with the above potential in a point-like global monopole. Afterwards, we study the thermodynamic functions and calculate the partition function Z using the energy eigenvalue expression. Other thermodynamic functions, such as the vibrational free energy, mean energy, entropy and the specific heat capacity of the quantum system at finite temperatures T ≠ 0 , are presented and analyze the effects of the topological defect with this potential. We see that the energy eigenvalue depends on the geometric quantum phase and this dependence of the eigenvalue on it gives rise to a persistent current. We see that this persistent current is influenced by the topological defect of a point-like global monopole and gets shifted compared to the flat space result. This paper is organized as follows: in Sect. 2, we present the eigenvalue solution of the non-relativistic wave equation in the presence of the Aharonov-Bohm flux field in a point-like global monopole space-time with potential; in Sect. 3, we study the thermodynamic properties of the quantum system at finite temperature and present the expression of different functions; and in Sect. 4, we present our results. We have used the natural units c = 1 = ℏ.

Eigenvalue Solution of Harmonic Oscillator in a Point-Like Defect with Inverse square Potential Under AB-Flux Field
In this section, we review the study of quantum motions of the non-relativistic particles under the influence of a quantum flux field in a topological defect geometry with potential (harmonic oscillator plus inverse square potential) [25]. Then, we use the energy eigenvalue expression to calculate the partition function and other thermodynamic properties of the quantum system at finite temperature T ≠ 0. A static and spherically symmetric space-time describing a point-like global monopole in the spherical coordinates (t, r, , ) is given by [19][20][21][22][23][24][25] where < 1 represents the topological defect parameter, and i, j = 1, 2, 3 . Here (x 1 = r, x 2 = , x 3 = ) with 0 ≤ r < ∞, 0 ≤ < , and 0 ≤ < 2 are the spatial coordinates.
The time-dependent Schrödinger wave equation taking into account the effects of background curvature R of the geometry coupled non-minimally with the field is described by the wave equation [19][20][21][22][23][24][25] where M is the rest mass of the particles.
By the method of separation of the variables, one can express the total wave function (t, r, , ) in terms of variables. Suppose, a possible total wave function in terms of a radial wave function (r) as (t, r, , functions, and l, m are, respectively, the orbital and magnetic moment quantum numbers. For the present investigation, we choose the electromagnetic three-vector potential A as A r = 0 = A , A = AB 2 r sin Refs. [23][24][25] where, AB = 0 and 0 = 2 e −1 . Thereby, expressing the wave Eq. (3) in the space-time background (2) and substituting the electromagnetic three-vector potential A and the wave function , we arrive at the following radial equation for (r): where l → l � = (l − ) due to the presence of the magnetic flux field in the quantum system. Substituting the potential (1) in Eq. (7), we obtain the following equation: where we have defined Let us perform a change of variable via x = ( 2 ) 1∕2 r 2 in Eq. (5), we obtain the following second-order differential equation: The above equation can be solved using the parametric Nikiforov-Uvarov method [23][24][25]33]. Using this NU-method, one can obtain the following expression of the energy E n,l given by The above eigenvalue can be expressed as Journal of Low Temperature Physics (2023) 211: [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] where we have substituted 2 = 1 2 M 2 , and is the oscillator frequency. Equation (10) is the energy eigenvalue of a harmonic oscillator under the influence of the quantum flux field with an inverse square plus constant potential in the background of the topological defect produced by a point-like global monopole. This eigenvalue solution gets modified by the topological defect of the geometry compared to the flat space result with inverse square potential and the flux field.
In the next section, we will study the thermodynamic functions of the harmonic oscillator at finite temperature T ≠ 0 , and analyze the effects of the topological defect, the quantum flux field, and inverse square potential on them. We use the above energy expression and calculate the partition function as well as other thermodynamic quantities. In addition, we also calculate the persistent currents expression because the energy eigenvalue depends on the geometric quantum phase.

Thermodynamic Functions and Persistent Currents of the Quantum System
As stated in the introduction, many authors have investigated the thermodynamic properties of a quantum system. Most of the investigations have been carried out in the flat space background and only a handful works in the background of the topological defects. Therefore, studies of the thermodynamic functions for the above quantum mechanical system have some significance because we will show later on that these functions are influenced by the topological defects and the quantum flux field. The thermodynamic properties of a quantum system can be obtained from the partition function Z given by [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]34] where = 1 T , with and T are the Boltzmann constant and the absolute temperature, respectively.
Using the energy expression (10) in (11), we obtain the following expression of the partition function given by which means that the partition function Z of an harmonic oscillator depends on the potential strength −2 , the topological defect characterized by the parameter , and the magnetic flux . One can see that the topological defect of a point-like global monopole modified the partition function compared to the flat space result. We have (10) E n,l = a + 2 b n, n = 0, 1, 2, ...,  (Fig. 1). We have seen the partition function decreases with increasing the values of for fixed values of the other parameters. This decrease in the partition function shifted upward with increasing the magnetic flux (Fig. 1a), decreasing the oscillator frequency (Fig. 1b), increasing the absolute temperature T (Fig. 1c), and decreasing the potential strength −2 (Fig. 1d).
Other thermodynamic functions, such as the vibrational free energy F, mean energy U, entropy S, and the specific heat capacity C of harmonic oscillator, can be derived from the partition function (12) as follows.
Free Energy F It is defined in terms of partition function Z( ) as Using the partition function expression (12), we obtain the expression of the vibrational free energy given by which depends on the topological defects characterized by the parameter , the potential strength −2 , and the magnetic flux. We have plotted a few graphs of it with the topological defect parameter for various values of other parameters (Fig. 2). One can see that this vibrational free energy increases with increasing the topological defect parameter for fixed values of other parameters. The increasing level shifted upward with decreasing the magnetic flux (Fig. 2a), increasing the potential strength −2 (Fig. 2b), increasing the oscillator frequency (Fig. 2c), and decreasing the absolute temperature T but increasing the magnetic flux (Fig. 2d).

Mean Energy
The mean energy U in terms of the partition function Z is defined by Using the partition function expression (12), we obtain the following expression of the mean energy given by which depends on topological defects characterized by the parameter , the potential strength −2 , and the magnetic flux. We have plotted a few graphs of the free energy with the topological defects parameter for various values of other parameters (Fig. 3). One can see that the mean energy increases with increasing the topological defect parameter for fixed values of the other parameters. This increase in the mean energy shifted upward with decreasing the magnetic flux (Fig. 3a), decreasing the magnetic flux but increasing the absolute temperature T (Fig. 3b), increasing both the magnetic flux and the oscillator frequency (Fig. 3c), and decreasing the absolute temperature T but increasing the oscillator frequency (Fig. 3d).

Specific heat capacity C
The specific heat capacity in terms of the partition function is defined by Using the expression (12), we obtain the following expression of the specific heat capacity of a harmonic oscillator which depends only on the topological defects characterized by the parameter , and the magnetic flux . We have plotted a few graphs of the specific heat capacity with the topological defects parameter for various values of other parameter (Fig. 4). We see that the specific heat capacity decreases with increasing the values of , and the oscillator frequency for fixed values of other parameter. This decreasing level of the specific heat capacity shifted downward with increasing the oscillator frequency (Fig. 4a), decreasing the absolute temperature T (Fig. 4b), decreasing the absolute temperature T but increasing the oscillator frequency (Fig. 4c), and decreasing the absolute temperature T but increasing the topological defect parameter (Fig. 4d).

Entropy S
The entropy S of a system in terms of the partition function is defined by Using the expression (12), we obtain the following expression of the entropy of a harmonic oscillator given by which depends on the topological defects characterized by the parameter , and the magnetic flux . We have plotted a few graphs of the free energy with the topological defect parameter for various values of other parameters (Fig. 5). Here also, we see that the entropy of the system decreases with increasing the topological defect parameter , and the oscillator frequency for fixed values of other parameters. The decreasing level of the entropy shifted downward with increasing the oscillator frequency (Fig. 5a), decreasing the absolute temperature T (Fig. 5b), decreasing the absolute temperature T but increasing the oscillator frequency (Fig. 5c), and increasing the topological defect parameter (Fig. 5d). From expressions of the last two thermodynamic functions, we see that the specific heat capacity and the entropy of the harmonic oscillator is greater in a point-like global monopole compared to the results in the flat space background. We show these differences by plotting graphs (Fig. 6), where the dotted curve represents the specific heat capacity in the flat space and curve one in a point-like global monopole for fixed value of , . Similarly, we plot graphs for the entropy (Fig. 7), where the dotted curve represents in the flat space and curve one in a point-like global monopole for fixed value of , .
Using the energy eigenvalue expression (9) and substituting 2 = 1 2 M 2 , we obtain the following expression of the BY-relation which depends on the topological defect characterized by the parameter , the oscillator frequency , the magnetic flux field AB , and the potential strength −2 . We have plotted a few graphs of this showing the influences of the topological defect keeping fixed the other parameters (Fig. 6).
In the absence of the topological defect, that is → 1 , the space-time geometry under consideration becomes Minkowski flat space, and therefore, the Byers-Yang relation becomes which depends on the potential strength −2 , and the oscillator frequency with the flux field.

Conclusions
In this paper, we studied the thermodynamic properties and persistent currents of a harmonic oscillator under the influence of the Aharonov-Bohm flux field with inverse square potential in the background of a point-like global monopole. We first presented the eigenvalue solution of the harmonic oscillator and have shown that this eigenvalue solution is influenced by the topological defect of a pointlike global monopole, and gets modified compared to the flat space result. We used this energy eigenvalue expression to calculate the partition function Z( ) and other thermodynamic functions, such as the vibrational free energy, mean energy, specific heat capacity, and entropy of the quantum system at finite temperature T ≠ 0.
The partition function of the harmonic oscillator is given by the expression Eq. (12), the vibrational free energy Eq. (14), and the mean energy Eq. (16). We have shown that these functions depend on the topological defect of the pointlike global monopole characterized by the parameter , the potential strength −2 , the magnetic flux field AB , and the absolute temperature T. We plotted a few graphs of these functions with the topological defect parameter and have seen that the partition function decreases with increasing the values of ( Fig. 1), and the vibrational free energy (Fig. 2) and the mean energy (Fig. 3) increases with increasing the values of .
Other thermodynamic functions of the harmonic oscillator, such as the specific heat capacity is given by the expression Eq. (18) and the entropy Eq. (20). We have seen that these two functions depend on the topological defect of the point-like global monopole characterized by the parameter , and the absolute temperature T. A few graphs of the specific heat capacity per Boltzmann constant ( C∕ ) (Fig. 4) and the entropy per Boltzmann constant ( S∕ ) (Fig. 5) are generated and have shown that these two decrease with increasing the values of , and the oscillator frequency . We have also shown that these two functions are more in the point-like global monopole compared to the flat space results. The differences have been shown by plotting graphs (Figs. 6 and 7) in the flat space and the point-like global monopole. Thus, there is no disagreement with the Dulong-Petit law and the second law of thermodynamics.
Since the presented energy eigenvalue depends on the geometric quantum phase and thus, this dependence of the eigenvalue gives rise to a persistent current, and we have presented this by the expression Eq. (22). One can see that the persistent current is influenced by the topological defect characterized by the Fig. 8 Plots of I n,l with topological defect parameter for different values of other parameters. Here, we fixed the parameters l = 1 = M , 0 = 1 parameter , and the potential strength −2 , and gets modified compared to the flat space result. We plotted graph of I n,l that decreases with increasing the values of (Fig. 8). In all generated figures, the units of various physical parameters have been chosen in a system of units where c = 1 = ℏ = G.