Properties of an $\alpha$-$T_{3}$ Aharonov-Bohm quantum ring: Interplay of Rashba spin-orbit coupling and topological defect

Without screw dislocation ($k\eta=0$): We exclude the effects of topological defect, or what we call as screw discolation ($k\eta=0$) to begin with. Fig. 2(a) and (b) display the energies as a function of the ring radius, $R$, for various values of $\alpha$ at a fixed $\lambda_{R}$ value. One can easily verify the results of the $\alpha$-$T_{3}$ quantum ring without the RSOC term [Mijanur, ] by setting $\lambda_{R}=0$ ($k\eta=0$ anyway) in Eq. (9). In this case, we have considered a particular value for the RSOC, namely, $\lambda_{R}=0.5t$ corresponding to $\alpha=0.5$ and $1$, and plotted only the $m=-1$, $0$, and $1$ bands represented by red, green, and blue curves, respectively. When $\lambda_{R}=0$, the system exhibits three bands, with one being a flat band. However, with a non-zero $\lambda_{R}$ the original three bands split into six spin dependent bands, including two non-dispersive flat bands and four dispersive bands as described by Eq. (9). From Fig. 2, it is evident that all the energy branches have a 1/$R$ dependence and approach $E\to 0$ for very large radii, irrespective of the value of $\alpha$. Additionally, the dispersive bands remain non-degenerate, in contrast to the case of the pseudospin-1 $\alpha$-$T_{3}$ QR without SOC. Moreover, the dispersive $\uparrow$-spin and $\downarrow$-spin bands split as well. Specifically, for $m=0$, the energies are given by,

$$E_{2}=\frac{\kappa\epsilon}{2}\sqrt{1+\frac{\lambda_{R}^{2}}{t^{2}}}$$

(10)

and

$$E_{3}=\frac{\kappa\epsilon}{2}\sqrt{1+\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}}.$$

(11)

It can be observed that the $\uparrow$-spin energy band, $E_{2}$ is independent of $\alpha$, while the $\downarrow$-spin band, $E_{3}$ has a dependency on $\alpha$. This is an interesting result, since the $\uparrow$-spin band energies are insensitive to whether we are talking about graphene or the dice lattice. Consequently, the splitting between the $m=0$ bands (green curves in Fig. 2) increases with increasing values of $\alpha$. Whereas, the splitting between the bands with $m=-1$ (red curves in Fig. 2) and $m=1$ (blue curves in Fig. 2) decreases as $\alpha$ increases. Furthermore, the energy splitting decreases with increase of $|m|$ values for all values of $\alpha$. In addition to that, the energy splitting increases with an increase of $\lambda_{R}$. An intriguing observation is that for $\alpha=1$, i.e., the dice lattice, the energies are given by,

$$E_{2}=\frac{\kappa\epsilon}{2}\sqrt{(1+4m^{2})(1+\frac{\lambda_{R}^{2}}{t^{2}})}$$

(12)

and

$$E_{3}=\frac{\kappa\epsilon}{2}\sqrt{1+4m^{2}+4m\frac{\lambda_{R}^{2}}{t^{2}}}.$$

(13)

Thus, $E_{2}$ is a even function of $m$, making it two-fold degenerate corresponding to $m=\pm 1,\pm 2,\pm 3,....$ etc. On the other hand, the $E_{3}$ band is an odd function of $m$, resulting in it being non-degenerate as illustrated in Figs. 2(b).

With screw dislocation ($k\eta\neq 0$): The scenario involving a topological defect arises when we implement the transformation via $m^{\prime}\to(m-k\eta)$ in the results of the Euclidean model. Consequently, upon examining Eq. (9), it becomes apparent that, unlike the situation where $k\eta=0$, all the energy levels for both $\uparrow$-spin and $\downarrow$-spin states, including the $m=0$ bands, become dependent on the parameters $\alpha$ and $\lambda_{R}$. Hence, the splitting between the spin-split bands hinges on the presence of dislocations in the system, the parameter $\alpha$, the strength of the Rashba coupling, $\lambda_{R}$, as dictated by Eq. (9). Furthermore, for $\alpha=1$, it is worth noting that $E_{2}$ and $E_{3}$ are non-degenerate, in contrast to the previous case. These findings are illustrated in Figs. 2(c) and (d). Additionally, the quantum number $m$ undergoes a shift that depends on $\eta$, which is associated with the Burgers vector. This shift is a manifestation of the Aharonov-Bohm effect, akin to what is observed in the context of a one-dimensional quantum ring penetrated by a magnetic flux [Fuhr, ]. The energy levels as a function of the strength of the screw dislocation are presented in Fig. 3 for two distinct cases, namely, $\alpha=0.5$ and $\alpha=1$ (dice lattice). Moreover, Eq. (9) makes it clear that the dispersive energy levels exhibit a hyperbolic dependence on the screw dislocation. It is important to note that the energy spectrum depends on the Burgers vector, $b^{z}=2\pi\eta$, as well as the radius, $R$ of the ring, and hence associated with the geometry of the system.

Additionally, there are energy extrema, with maxima in the valence band and minima in the conduction band, that pertain to different spin bands. The positions of these extrema are determined by the strength of dislocation as follows. For the $\uparrow$-spin bands, the extrema are found at

$$k\eta=m-\frac{1}{2}\frac{1-\alpha^{2}}{1+\alpha^{2}}$$

(14)

which is independent of $\lambda_{R}$. For the dice case ($\alpha=1$), the extrema occur at $k\eta=m$. However, for the $\downarrow$-spin bands, the extrema occurs at

$$k\eta=m+\frac{1}{2}\frac{\frac{\lambda_{R}^{2}}{t^{2}}+\frac{1-\alpha^{2}}{1+\alpha^{2}}}{1+\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}}.$$

(15)

The expression above shows a dependency on $\lambda_{R}$ and hence we shall observe an interplay between the dislocation and RSOC. In fact, this interplay gives rise to interesting consequences as we shall see later. Furthermore, for each value of $m$, a band crossing point exists for $\alpha<1$. However, in the case of $\alpha=1$, no band crossings occur, instead, the bands touch each other at certain specific $k\eta$ values, depending upon the parameter $\lambda_{R}$.

Now let us discuss the case when the $\alpha$-$T_{3}$ ring is threaded by a perpendicular magnetic field ${\bf B}=B_{0}\hat{z}$. The only non-zero component of the vector potential is $A_{\theta}=B\rho/2$. Notice that, in the non-Euclidean metric of the dislocation, the vector potential that produces the uniform magnetic field, is identical to the flat space (Euclidean) potential vector. The spectrum of the system is modified by the field flux as follows,

	$\displaystyle E_{1}(\Phi)$	$\displaystyle=0$		(16)
	$\displaystyle E_{2}(\Phi)$	$\displaystyle=\kappa\frac{\epsilon}{2}\biggl{\{}\bigg{[}1+4\big{(}m-k\eta+\frac{\Phi}{\Phi_{0}}\big{)}^{2}-4\big{(}m-k\eta+\frac{\Phi}{\Phi_{0}}\big{)}$
		$\displaystyle\frac{1-\alpha^{2}}{1+\alpha^{2}}\bigg{]}\bigg{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\bigg{)}\biggl{\}}^{\frac{1}{2}}$
	$\displaystyle E_{3}(\Phi)$	$\displaystyle=\kappa\frac{\epsilon}{2}\biggl{\{}\Big{[}1+4\big{(}m-k\eta+\frac{\Phi}{\Phi_{0}}\big{)}^{2}\Big{]}\bigg{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}\bigg{)}+$
		$\displaystyle 4\big{(}m-k\eta+\frac{\Phi}{\Phi_{0}}\big{)}\bigg{(}\frac{\lambda_{R}^{2}}{t^{2}}+\frac{1-\alpha^{2}}{1+\alpha^{2}}\bigg{)}\biggl{\}}^{\frac{1}{2}}$

where $\beta=\Phi/\Phi_{0}$ with $\Phi=\pi R^{2}B_{0}$ is magnetic flux through the ring and $\Phi_{0}$ is the usual flux quantum ($=h/e$). The Zeeman coupling has been neglected at small enough values of the field. The addition of a magnetic field, represented by a U(1) minimal coupling with flux $\Phi$ threading the ring, breaks the time reversal symmetry allowing for the emergence of persistent charge currents [Butt, ] which we shall discuss later.

Without screw dislocation ($k\eta=0$): In Fig. 4, we show the dependence of a few energy levels on the ring radius, $R$, considering $B_{0}=5$T for the two aforementioned cases i.e., $\alpha=0.5$ and $1$ with $\lambda_{R}=0.5t$. Each level exhibits a non-monotonic behaviour as a function of the radius $R$. The energy levels attain an extremum (minimum for conduction band and maximum for valence band) at a particular value of $R$. However, the positions of these extrema depend on the values of $m$, $\alpha$ and $\lambda_{R}$ explicitly. In the limit of small $R$, all the energy levels vary inversely with $R$. On the other hand, the energy scales as, $E\sim|R|$ in limit of large $R$. Additionally, for a fixed magnetic field and for large $m$, the locations of the extrema points for different $m$ depend on $R$ as $R\propto\sqrt{|m|}$ irrespective of $\alpha$. Thus, the concept of large radii differs for different values of $m$. Consequently, for negative values of $m$, the extrema points of the energy exhibit a scaling behaviour, namely, $E_{min}\propto 1/\sqrt{|m|}$, resulting in a diminishing of the spectral gap with increasing $|m|$. Conversely, for positive values of $m$, the energy extrema scales as, $E_{min}\propto\sqrt{m}$. Furthermore, from Eq. (16), it is evident that in presence of a magnetic field, the energy splitting between the bands of the $m=0$ level as well as the $m\neq 0$ levels decreases with the increase in the values of the parameters $\alpha$ and $\lambda_{R}$. Again, from Eq. (16) it is noted that for $\alpha<1$, there are two points where the spin bands cross each other as a function of $R$ for $m=0$ and negative values of $m$, i.e., $m=-1,-2,-3,...$ etc. bands, whereas there is only one band crossing point for the possitive values, namely, $m=1,2,3...$. These crossings obey the following condition,

$$\frac{\lambda_{R}^{2}}{t^{2}}\Big{[}1+4\big{(}m+\frac{\Phi}{\Phi_{0}}\big{)}^{2}-4\big{(}m+\frac{\Phi}{\Phi_{0}}\big{)}\Big{]}-\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}\Big{[}1+4\big{(}m+\frac{\Phi}{\Phi_{0}}\big{)}^{2}+4\big{(}m+\frac{\Phi}{\Phi_{0}}\big{)}\Big{]}-8\big{(}m+\frac{\Phi}{\Phi_{0}}\big{)}\frac{1-\alpha^{2}}{1+\alpha^{2}}=0.$$

(17)

Eq. (17) can be checked against the plot shown in Fig. 4(a). Now, for the dice lattice case ($\alpha=1$), the above mentioned condition requires $m+\frac{\Phi}{\Phi_{0}}=\frac{1}{2}$, which implies that along the radius $R$, the energy band crossing points occur at $R=\sqrt{2}l_{0}\sqrt{(1/2-m)}$, where $l_{0}=\sqrt{\hbar/(eB_{0})}$ is the magnetic length. Consequently, there is only one band crossing point for $m=0$ and $m=-1,-2,-3,..$ etc. bands. Furthermore, band crossing is prohibited for positive values of $m$ as there are no real values of $R$ for $m>0$, indicating that the corresponding spin bands do not cross each other and hence there will not be any degeneracy as illustrated in Fig. 4(b) by the blue curves.

The energy levels as a function of the external magnetic flux ($\Phi/\Phi_{0}$) are depicted in Figs. 5(a) and (b) for a quantum ring with $R=10$ $nm$, considering two cases, namely, $\alpha=0.5$ and $\alpha=1$ with $\lambda_{R}=0.5$ in unit of $t$. The curves are represented by red, green, and blue colors corresponding to $m=-1$, $m=0$, and $m=1$, respectively. The magnetic field dependence of the energy spectra becomes evident when we rewrite Eq. (16) as,

$$E^{2}_{2}-\frac{\epsilon^{2}}{4}\Big{[}1+4\big{(}m+\frac{\Phi}{\Phi_{0}}\big{)}^{2}-4\big{(}m+\frac{\Phi}{\Phi_{0}}\big{)}\frac{1-\alpha^{2}}{1+\alpha^{2}}\Big{]}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\Big{)}=0$$

(18)

and

	$\displaystyle E^{2}_{3}-\frac{\epsilon^{2}}{4}\Big{[}\Big{\{}1+4\big{(}m+\beta\big{)}^{2}\Big{\}}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}\Big{)}+$		(19)
	$\displaystyle 4\big{(}m+\beta\big{)}\Big{(}\frac{\lambda_{R}^{2}}{t^{2}}+\frac{1-\alpha^{2}}{1+\alpha^{2}}\Big{)}\Big{]}=0.$

Thus, the energies display a hyperbolic dependence on the applied magnetic field, exhibiting extrema at the flux values given by,

$$\frac{\Phi}{\Phi_{0}}=-m+\frac{1}{2}\frac{1-\alpha^{2}}{1+\alpha^{2}}$$

(20)

for $\uparrow$-spin band $E_{2}$, the extrema points are independent of the strength of the Rashba coupling, but depends on the values of $m$ and the parameter $\alpha$. For the dice lattice ($\alpha=1$), the extrema occur at $\Phi/\Phi_{0}=-m$. However, the extrema for the $\downarrow$-spin band $E_{3}$ occur at

$$\frac{\Phi}{\Phi_{0}}=-m-\frac{1}{2}\frac{\frac{\lambda_{R}^{2}}{t^{2}}+\frac{1-\alpha^{2}}{1+\alpha^{2}}}{1+\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}},$$

(21)

showing a dependency on the strength of Rashba SOC, $\alpha$ and $m$. For the dice lattice, the extrema are obtained at $\Phi/\Phi_{0}=-m-\frac{1}{2}\frac{\lambda_{R}^{2}}{t^{2}}$. The energy gaps at the extrema points, that is, the minimum values of the gap are given by,

	$\displaystyle\Delta E_{2}$	$\displaystyle=\frac{2\epsilon\alpha}{1+\alpha^{2}}\sqrt{1+\frac{\lambda_{R}^{2}}{t^{2}}},$		(22)
	$\displaystyle\Delta E_{3}$	$\displaystyle=\frac{2\epsilon\alpha}{1+\alpha^{2}}\sqrt{\frac{1-\frac{\lambda_{R}^{4}}{t^{4}}}{1+\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}}}.$

Therefore, it is observed that for a fixed value of Rashba coupling, the energy gaps for both the spin bands increase with increase in $\alpha$. However, the minimum energy gap for both the spin bands is independent of $m$. Also the $\downarrow$-spin bands, namely, $E_{3}$ have lower energies than the $\uparrow$-spin bands ($E_{2}$). The $\uparrow$-spin $E_{2}$ and $\downarrow$-spin $E_{3}$ bands are illustrated in the Fig. 5(a). For the dice lattice case, and for $\lambda_{R}=0.5t$, the energy gaps are obtained as, $\Delta E_{2}\approx 74$ $meV$ and $\Delta E_{3}\approx 64$ $meV$ which can be verified from the Fig. 5(b). Furthermore, from Fig. 5 it is evident that $E_{2}(m)\neq E_{2}(-m)$ and $E_{3}(m)\neq E_{3}(-m)$, indicating the existence of finite spin currents.

With screw dislocation ($k\eta\neq 0$): Let us now delve into the impact of a topological defect, specifically a screw dislocation, on the electronic spectra of the $\alpha$-$T_{3}$ QR. Referring to Equation (16), we find that all of the previously discussed characteristics remain unaltered, except we may consider that $m$ as an effective quantum number, let us call it as $m^{\prime}$ with $m^{\prime}\to(m-k\eta)$ which has a shift in the $m$ values by an amount of $k\eta$. The results are visually represented in Figs. 4(c) and (d), while considering $\alpha$ values of $0.5$ and $1$, with the parameter $k\eta$ set to a modest value, say $k\eta=0.3$. Again, for the case of $\alpha=1$, the band crossing points emerge at $R=\sqrt{2}l_{0}\sqrt{\{1/2-(m-k\eta)\}}$, signifying a shift to the right as depicted in Fig. 4(d).

Furthermore, the conditions for the extremal points defined in Eqs. (20) and (21) are also modified due to $m^{\prime}\to(m-k\eta)$, resulting in a displacement of these extremal points by an amount equal to $k\eta$ in the positive direction, as illustrated in Figs. 5(c)-(f). It is important to note that the energy gaps at these extremal points, as per Eq. 22, are not influenced by the presence of the topological defect ($k\eta$), their dependencies solely rely on the parameters $\alpha$ and $\lambda_{R}$. To demonstrate this, we have considered two values of $\lambda_{R}$, namely, $\lambda_{R}=0.5t$ and $\lambda_{R}=0.8t$. The energy gap values for the $\alpha=1$ case are presented in the respective figures (see Fig. 5).

In this particular scenario, we are dealing with a one-dimensional quantum ring in the presence of a screw dislocation and Aharonov–Bohm flux. The energy spectrum displays a parabolic dependence on the Burgers vector, similar to its dependence on the magnetic flux. Upon analysing our findings, we can infer that a particle within a space featuring a topological defect behaves similarly to a particle in a Euclidean space in the presence of an effective magnetic flux traversing the ring. This effective flux is a combination of two contributions, the first is of a topological nature stemming from the topological defect, while the other is due to the magnetic flux $\Phi$. By adjusting the magnetic flux as an external fine-tuning parameter to counterbalance the topological contribution introduced by the defect, we can nullify the Aharonov–Bohm effect in the ring. In such cases, the energy spectrum resembles that of a particle moving in a quantum ring within a space devoid of any topological defect. This is a very important result. It provides a clue how the effects of a topological defect can be totally or partially compensated by an external filed with regard to the observation of the AB effect. Further consequences on the spin and charge currents are elucidated below.

The charge persistent current in the low-energy state can be calculated using the linear response formula, $j_{Q}=-\sum_{m,\kappa}\frac{\partial E}{\partial\Phi}$, where the sum refers to all (and only) the occupied states (for the valence band ($\kappa=-1$)) and the $m$ values are chosen carefully to perform the summation. Since the current is periodic in $\Phi/\Phi_{0}$ with a period of 1 (that is $\Phi=\Phi_{0}$), we restrict the discussion to the window $-1\leq\Phi/\Phi_{0}\leq 1$. The analytical form for the charge current is obtained as,

	$\displaystyle j_{Q}^{\kappa}$	$\displaystyle=-\frac{\epsilon^{2}\kappa}{2\Phi_{0}}\sum_{m}\frac{\big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\big{)}\Big{[}2\big{(}m-k\eta+\frac{\Phi}{\Phi_{0}}\big{)}-\frac{1-\alpha^{2}}{1+\alpha^{2}}\Big{]}}{E_{2}(\Phi)}$		(23)
		$\displaystyle-\frac{\epsilon^{2}\kappa}{2\Phi_{0}}\sum_{m}\frac{2\big{(}m-k\eta+\frac{\Phi}{\Phi_{0}}\big{)}\big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\frac{1-\alpha^{2}}{1+\alpha^{2}}\big{)}+\big{(}\frac{\lambda_{R}^{2}}{t^{2}}+\frac{1-\alpha^{2}}{1+\alpha^{2}}\big{)}}{E_{3}(\Phi)}.$

The spin branches closest to the Fermi energy exhibit non-monotonic behaviour, resulting in two distinct contributions to the charge current coming from the $\uparrow$-spin and $\downarrow$-spin components. Since we are calculating the current contributions arising from the low-energy states, it is clear from Fig. 5 that for a certain range of $\Phi/\Phi_{0}$, only one energy state labelled by a particular value of $m$ is present. Hence, the sum in Eq. (23) comprises of only one value of $m$. Furthermore, based on the observations in Fig. 5, we can discern that the low-energy states when $\alpha=0.5$ encompass both the spin bands. Consequently, in our current calculations, we considered contributions from both the spin branches. In contrast, for the $\alpha=1$ scenario, the low-energy state exclusively comprises the $\downarrow$-spin branches, and thus, we only have accounted for the contributions from the $\downarrow$-spin to compute the persistent current. The outcomes of these calculations are depicted in Fig. 6 for both $\alpha=0.5$ and $\alpha=1$, considering various combinations of $\lambda_{R}$ and $k\eta$, all the while maintaining a fixed ring radius of $R=10$ $nm$. The asymmetric spectral features between the two spin branches allows for the possibility of a net spin currents, as we shall see below. For all values of $\alpha$, the persistent currents oscillate periodically with $\Phi/\Phi_{0}$, with a periodicity of $\Phi/\Phi_{0}=1$. Figs. 6(a) and (b) illustrate that the persistent currents can be tuned by adjusting the parameter $\alpha$ for a fixed value of the Rashba coupling ($\lambda_{R}$). Moreover, the charge persistent currents can be manipulated via $\lambda_{R}$ for a fixed $\alpha$, (see Fig. 6(a), (c) and (e)) since the Rashba parameter can be controlled by a gate voltage. Further, it is worth noting the presence of a kink in the current profile when $\alpha<1$. This kink arises because different spin bands contribute to the current, as indicated by the distinctions denoted by $\uparrow$ and $\downarrow$ for their respective spin bands. However, this kink phenomenon is absent for the dice lattice ($\alpha=1$), as evident in Figure 6(b), (d), and (f). The reason being, in this case, only the $\downarrow$-spin bands contribute to the current, as mentioned earlier.

Furthermore, the topological defect plays a pivotal role in the behaviour of the charge current. As illustrated in Figure 6, we can observe that, for a fixed value of $\lambda_{R}$, the screw dislocation induces a phase shift in the current, regardless of the specific value of $\alpha$. For example, every point of the current profile is shifted by the same amount as the strength of the topological defect as shown in the middle panel of Fig. 6. However, the topological defect does not influence the current profile itself. A desired shift in the current profile may be achieved via the controllable parameter $k\eta$. Further, the Rashba coupling, $\lambda_{R}$ plays a role as well. The depth in the kink of the current profile decreases with the increasing $\lambda_{R}$ (see bottom panel of Fig. 6). This can be understood from the lower panel of Figure 5, as the increase in $\lambda_{R}$ results in a decrease in the $\uparrow$-spin contribution in the low-energy state within a certain range of $\Phi/\Phi_{0}$. However, the overall oscillation period remains unaltered.

In summary, we can manipulate the persistent current profile by fine-tuning the Rashba coupling, and we have the ability to shift the phase of the current to suit our specific applications by adjusting the strength of the topological defect.

We shall now study equilibrium spin currents. In contrast to the formalism for obtaining the charge current, one can obtain the spin currents by accounting for distinct velocities for different spin branches. Thus, we define equilibrium spin current as,

$$j_{S}=j_{Q}(\uparrow)-j_{Q}(\downarrow).$$

(24)

We have calculated the equilibrium spin currents following the procedure discussed earlier. The peculiar separation of the spin branches results in differences of the velocities between the two spin projections, giving rise to a spin current, as shown in Fig. 7. The figure illustrates a significant spin current for small values of the flux, which can be attributed to the large charge current originating from a single spin branch.

The striking feature is that the magnitude as well as the pattern of the spin currents depend upon the parameters $\alpha$ and the strength of the Rashba coupling ($\lambda_{R}$). We present results for $\alpha=0.5$, and $\alpha=1$ with different values of $\lambda_{R}$ and topological defect $k\eta$ for a ring of radius, $R=10$ $nm$. The presence of the Rashba coupling breaks inversion symmetry (in addition to the $\sigma_{z}$ symmetry) in the plane even for small $\lambda_{R}$. The symmetry breaking determines the spin labelling of the energy branches that take part in yielding the spin currents. Additionally, the spin currents exhibit periodic behaviour with $\Phi/\Phi_{0}$, with a periodicity equal to one flux quantum, irrespective of $\alpha$. Similar to the case of charge persistent current, we also notice a phase shift introduced by the topological defect. The magnitude of this phase shift precisely corresponds to the strength of the defect.

These findings underscore the potential of $\alpha$-$T_{3}$ quantum rings as key components for spintronic applications, where we can manipulate the performance of the devices by adjusting both the Rashba coupling and the topological defect.

Having studied the transport features, it may be of interest to explore the thermodynamic properties such as, specific heat, bulk modulus, compressibility, etc. Specifically, to the best of our knowledge these quantities have never been explored in the presence of Rashba coupling and topological defects in the context of a QR. However, we shall restrict ourselves in studying the thermodynamic potentials, entropy, internal energy, and the specific heat in the following.

In this section, we will study the thermodynamic properties of an AB $\alpha$-$T_{3}$ ring with Rashba coupling, which is in contact with a thermal reservoir at a finite temperature. These properties encompass fundamental thermodynamic quantities, specifically the Helmholtz free energy, the internal energy, entropy, and heat capacity. We shall exclusively focus on stationary states characterized by positive energies ($E>0$) that constitute our thermodynamic ensemble. Given the strict exclusion of particle-particle interactions, the excitation of negative-energy states and the issue of pair production are excluded for the discussions [Gran, ; Rach, ]. Consequently, the partition function involves a summation solely over positive-energy states, simplifying the analysis considerably. We commence by assessing the corresponding partition function as,

$$Z=\sum_{m=0}^{\infty}(e^{-\beta E_{2}^{+}}+e^{-\beta E_{3}^{+}})$$

(25)

where $\beta=\frac{1}{k_{B}T}$, $k_{B}$ is the Boltzmann constant and $T$ is the equilibrium temperature. Rearranging the expressions in Eq. (16) and using Eq. (25), we obtain that for a one-fermion confined in the $\alpha$-$T_{3}$ AB ring the partition function is,

$$Z=\sum_{m=0}^{\infty}e^{-\beta\sqrt{A_{1}m^{2}+B_{1}m+C_{1}}}+\sum_{m=0}^{\infty}e^{-\beta\sqrt{A_{2}m^{2}+B_{2}m+C_{2}}}$$

(26)

where the quantities are,

	$\displaystyle A_{1}$	$\displaystyle=\epsilon^{2}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\Big{)}$		(27)
	$\displaystyle B_{1}$	$\displaystyle=\epsilon^{2}\Big{(}2\frac{\Phi}{\Phi_{0}}-2k\eta-\cos 2\xi\Big{)}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\Big{)}$
	$\displaystyle C_{1}$	$\displaystyle=\epsilon^{2}\Big{\{}\frac{1}{4}+\big{(}k\eta-\frac{\Phi}{\Phi_{0}}\big{)}^{2}+\big{(}k\eta-\frac{\Phi}{\Phi_{0}}\big{)}\cos 2\xi\Big{\}}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\Big{)}$
	$\displaystyle A_{2}$	$\displaystyle=\epsilon^{2}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\cos 2\xi\Big{)}$
	$\displaystyle B_{2}$	$\displaystyle=\epsilon^{2}\Big{\{}2\big{(}\frac{\Phi}{\Phi_{0}}-k\eta\big{)}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\cos 2\xi\Big{)}+\Big{(}\frac{\lambda_{R}^{2}}{t^{2}}+\cos 2\xi\Big{)}\Big{\}}$
	$\displaystyle C_{2}$	$\displaystyle=\epsilon^{2}\Big{[}\Big{\{}\frac{1}{4}+\big{(}k\eta-\frac{\Phi}{\Phi_{0}}\big{)}^{2}\Big{\}}\Big{(}1+\frac{\lambda_{R}^{2}}{t^{2}}\cos 2\xi\Big{)}+\big{(}\frac{\Phi}{\Phi_{0}}-k\eta\big{)}$
		$\displaystyle\Big{(}\frac{\lambda_{R}^{2}}{t^{2}}+\cos 2\xi\Big{)}\Big{]}.$

Since Eq. (26) cannot be computed in a closed form, we assume that the AB ring is submitted to a strong magnetic field ($\Phi\gg\Phi_{0}$). Therefore, the expression under the summation takes the approximate form $e^{-\beta\sqrt{B_{1/2}m+C_{1/2}}}$, which is a monotonically decreasing function and the associated integral,

$$\int_{0}^{\infty}e^{-\beta\sqrt{Bx+C}}dx=\frac{2}{B\beta^{2}}(1+\beta\sqrt{C})e^{-\beta\sqrt{C}},$$

(28)

is finite. Thus, the form ensures the convergence of the series. To evaluate the partition function (26) at the strong-field limit we will use the Euler-Maclaurin sum formula given by [Grei, ; Oliv, ],

$$\sum_{n=0}^{\infty}f(n)=\frac{1}{2}f(0)+\int_{0}^{\infty}f(x)dx-\sum_{p=1}^{\infty}\frac{1}{(2p)!}B_{2P}f^{(2p-1)}(0),$$

(29)

where $B_{2p}$ are the Bernoulli numbers, $B_{2}=1/6$, $B_{4}=-1/30$,…. They are defined through the series

$$\frac{x}{e^{x}-1}=\sum_{n=0}^{\infty}B_{n}\frac{x^{n}}{n!}.$$

(30)

Therefore, Eq. (29) can be rewritten simply as,

$$\sum_{n=0}^{\infty}f(n)\simeq\frac{1}{2}f(0)+\int_{0}^{\infty}f(x)dx-\frac{1}{12}f^{\prime}(0)+\frac{1}{720}f^{\prime\prime\prime}(0)-...,$$

(31)

According to the expression above, we can write the partition function (26) explicitly as,

$$\begin{split}Z\simeq e^{-\beta\sqrt{C_{1}}}\Big{[}\frac{2}{B_{1}\beta^{2}}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+\frac{1}{2}+\Big{(}\frac{B_{1}}{24\sqrt{C_{1}}}-\frac{B_{1}^{3}}{720\sqrt{C_{1}^{5}}}\Big{)}\beta+\frac{1}{90}\Big{(}\frac{A_{1}}{2C_{1}}-\frac{B_{1}^{2}}{8C_{1}^{2}}\Big{)}\beta^{2}-\mathcal{O}(\beta^{3})\Big{]}\\ +e^{-\beta\sqrt{C_{2}}}\Big{[}\frac{2}{B_{2}\beta^{2}}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}+\frac{1}{2}+\Big{(}\frac{B_{2}}{24\sqrt{C_{2}}}-\frac{B_{2}^{3}}{720\sqrt{C_{2}^{5}}}\Big{)}\beta+\frac{1}{90}\Big{(}\frac{A_{2}}{2C_{2}}-\frac{B_{2}^{2}}{8C_{2}^{2}}\Big{)}\beta^{2}-\mathcal{O}(\beta^{3})\Big{]}\end{split}$$

(32)

where $f(n)=e^{-\beta\sqrt{B_{1/2}n+C_{1/2}}}$ and $\mathcal{O}(\beta^{3})$ represents terms involving high order in $\beta$ which will be neglected henceforth. Notice that when one considers the high-temperatures regime ($\beta\ll 1$), Eq. (32) becomes,

$$Z\simeq\frac{2}{B_{1}\beta^{2}}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+\frac{2}{B_{2}\beta^{2}}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}.$$

(33)

The stability of graphene at high temperature [Kim, ], provides motivation to study the thermodynamic properties the $\alpha$-$T_{3}$ AB ring. Indeed, the total partition function $Z_{N}$ for $N$ fermions can be approximated by,

$$Z_{N}\simeq\Bigg{[}\frac{2}{B_{1}\beta^{2}}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+\frac{2}{B_{2}\beta^{2}}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}\Bigg{]}^{N}.$$

(34)

Therefore, using the partition function in Eq. (34), we can now determine all related thermodynamic quantities. Indeed, after some algebra we obtain the Helmholtz free energy $F$, the internal energy $U$, the entropy $S$ and the heat capacity $C_{V}$ as follows:

$$F=-\frac{1}{\beta}\mathrm{ln}Z_{N}=-\frac{N}{\beta}\mathrm{ln}\Bigg{[}\frac{2}{B_{1}\beta^{2}}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+\frac{2}{B_{2}\beta^{2}}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}\Bigg{]}.$$

(35)

$$U=-\frac{\partial}{\partial\beta}\mathrm{ln}Z_{N}=N\frac{B_{2}\Big{(}2+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}2+\beta\sqrt{C_{2}}\Big{)}}{\beta\Big{[}B_{2}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}\Big{]}}.$$

(36)

$$S=k_{B}\beta^{2}\frac{\partial F}{\partial\beta}=Nk_{B}\Bigg{[}\mathrm{ln}\Big{\{}\frac{2\Big{(}1+\beta\sqrt{C_{1}}\Big{)}}{B_{1}\beta^{2}}+\frac{2\Big{(}1+\beta\sqrt{C_{2}}\Big{)}}{B_{2}\beta^{2}}\Big{\}}+\frac{B_{2}\Big{(}2+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}2+\beta\sqrt{C_{2}}\Big{)}}{B_{2}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}}\Bigg{]}.$$

(37)

$$C_{V}=-k_{B}\beta^{2}\frac{\partial U}{\partial\beta}=Nk_{B}\Bigg{[}\frac{2\Big{\{}B_{2}\Big{(}3+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}3+\beta\sqrt{C_{2}}\Big{)}\Big{\}}}{\Big{\{}B_{2}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}\Big{\}}}-\frac{\Big{\{}B_{2}\Big{(}2+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}2+\beta\sqrt{C_{2}}\Big{)}\Big{\}}^{2}}{\Big{\{}B_{2}\Big{(}1+\beta\sqrt{C_{1}}\Big{)}+B_{1}\Big{(}1+\beta\sqrt{C_{2}}\Big{)}\Big{\}}^{2}}\Bigg{]}.$$

(38)