# More current with less particles due to power-law hopping

###### Abstract

We show that the simple model of an ordered one-dimensional non-interacting system with power-law hopping has extremely interesting transport properties. We restrict ourselves to the case where the power-law decay exponent , so that the thermodynamic limit is well-defined. We explore the quantum phase-diagram of this model in terms of the zero temperature Drude weight, which can be analytically calculated. Most interestingly, we reveal that for , there is a phase where the Drude weight diverges as filling fraction goes to zero. Thus, in this regime, counter intuitively, reducing number of particles increases transport and is maximum for a sub-extensive number of particles. This behavior is immune to adding interactions. Measurement of persistent current due to a flux through a mesoscopic ring with power-law hopping will give an experimental signature of this phase. This behavior survives up to a finite temperature for a finite system. At higher temperatures, a crossover is seen. The maximum persistent current shows a power-law decay at high temperatures. This is in contrast with short ranged systems, where the persistent current decays exponentially with temperature.

### .1 Introduction

Quantum systems with long-range interactions present a platform with a lot of rich and interesting physics which differ greatly from the corresponding short-ranged systems. Recent controlled experimental realization of long-range quantum spin systems on several platforms Ryabtsev et al. (2010); Béguin et al. (2013); Browaeys et al. (2016); Korenblit et al. (2012); Jurcevic et al. (2017); Neyenhuis et al. (2017); Britton et al. (2012); Richerme et al. (2014); Jurcevic et al. (2014); Zhang et al. (2017a); de Paz et al. (2013); Yan et al. (2013); Álvarez et al. (2015); Hung et al. (2016); Choi et al. (2017); Zhang et al. (2017b); Ni et al. (2008); Moses et al. (2016); Smale et al. (2018) and demonstration of exotic physics such as time crystals Choi et al. (2017); Zhang et al. (2017b), prethermalizationNeyenhuis et al. (2017), dynamical phase transitionsJurcevic et al. (2017); Zhang et al. (2017a); Smale et al. (2018) and environment assisted transport Maier et al. (2019) have sparked intense research activities in such systems. Theoretical investigations are going on in the exotic physics mentioned above Žunkovič et al. (2018); Halimeh and Zauner-Stauber (2017); Gong and Duan (2013); Mori (2018); Lerose et al. (2019); Dutta and Bhattacharjee (2001); Dutta and Dutta (2017); Lerose et al. (2018a), and also on exploring Lieb-Robinson boundsHauke and Tagliacozzo (2013); Eisert et al. (2013); Gong et al. (2014); Foss-Feig et al. (2015); Cevolani et al. (2015, 2016); Frérot et al. (2018); Tran et al. (2018); Luitz and Bar Lev (2019), entanglement dynamicsKoffel et al. (2012); Vodola et al. (2014); Frérot et al. (2017); Schachenmayer et al. (2013); Buyskikh et al. (2016); Roy and Sharma (2018); Gong et al. (2017), and localization propertiesLevitov (1989, 1990); Mirlin et al. (1996); Burin and Maksimov (1989); de Moura et al. (2005a); Rodríguez et al. (2000); Deng et al. (2018); Celardo et al. (2016); Nosov et al. (2019); Gopalakrishnan (2017); Deng et al. (2018); Biddle et al. (2011); Balagurov et al. (2004); Malyshev et al. (2004); Xiong and Zhang (2003); de Brito et al. (2004); de Moura et al. (2005b); de Moura and Lyra (1998); Cao et al. (2017); Yao et al. (2014a); Burin (2015a, b); Tikhonov and Mirlin (2018); Yao et al. (2014b); De Tomasi (2019a); Singh et al. (2017); De Tomasi (2019b); Sabyasachi Nag (2019); Russomanno et al. (2017); Lerose et al. (2018b); Sciolla and Biroli (2011) of long-range systems.

The localization properties of long-range disordered systems have been in limelight recently. In one and two dimensions, short-range non-interacting (quadratic Hamiltonian) systems with random on-site disorder show Anderson localization. It was shown in early works that in presence of long-range hopping that decays as a power-law, it is possible to have delocalized states in low dimensionsLevitov (1989, 1990); Mirlin et al. (1996); Burin and Maksimov (1989); de Moura et al. (2005a); Rodríguez et al. (2000). Until recently, it was commonly believed that long-range hopping in disordered systems in low-dimensions have a delocalizing effect. But very recent works have have shown that long-range correlated hopping can also lead to localizationNosov et al. (2019). Further works have explored the effect of power-law hopping on quasi-periodic systemsDeng et al. (2018); Celardo et al. (2016); Gopalakrishnan (2017); Deng et al. (2018). Another interesting direction has been on the fate of many-body localization in presence of long-range hopping. Several very recent works explore this directionYao et al. (2014a); Burin (2015a, b); Tikhonov and Mirlin (2018); Yao et al. (2014b); De Tomasi (2019a); Singh et al. (2017); De Tomasi (2019b); Sabyasachi Nag (2019); Russomanno et al. (2017); Lerose et al. (2018b, a); Sciolla and Biroli (2011).

The common theme that all these works point to is that low-dimensional long-range systems have very interesting transport properties. However, surprisingly, there is an extreme lack of works directly exploring transport coefficients of long-range systems. In Ref.Gutman et al. (2016), the energy transport in an Anderson insulator in presence of power-law interactions was explored. In this work, we directly explore transport properties of a one-dimensional fermionic ordered system with power-law hopping. We confine ourselves to the case where power-law hopping exponent , so that the thermodynamic limit is well-defined. We characterize transport by calculating the zero temperature Drude weight. Our most interesting result is that, is a phase where the Drude weight diverges as filling fraction goes to zero. Thus transport in such systems increase on reducing the number of particles and is maximum for a sub-extensive number of particles. This rather peculiar behavior stems from a non-analyticity of the energy dispersion relation of the non-interacting model. Being a statement about zero-filling, this is generically true for any ordered system in presence of any interaction. As an experimental signature of this phase, we propose the measurement of persistent current through a mesoscopic ring with power-law hopping. Persistent current has been previously measured in mesoscopic metal ringsBluhm et al. (2009); Castellanos-Beltran et al. (2013); Lévy et al. (1990); Chandrasekhar et al. (1991). For mesoscopic ordered systems with power-law hopping, in the phase with , the persistent current will be greatly enhanced on reducing the number of particles, which is the tell-tale signature of this phase. The enhancement of peristent current on reducing the number of particles will be seen up to a finite-temperature for finite size systems. We also show that persistent current in such systems is exponentially larger than that in short range systems. The maximum persistent current decays as a power-law with temperature, whereas in short-range systems, it is known to decay exponentially.

The paper is organized as follows. In Sec. B, the definitions of Drude weight and persistent current are introduced. The long-range hopping model and the existence of thermodynamic limit of that model are shown in Sec. C. Our main results concerning the zero temperature Drude weight are given in Sec.D. In Sec. E, we calculate persistent current for zero and finite temperature. In Sec. F, we conclude by summarizing our results and discussing further works.

### .2 Definitions: The Drude weight and the persistent current

The Drude weight characterizes the strength of the zero frequency peak in electronic conductivity. The electronic conductivity of an isolated system in the thermodynamic limit is given by the Green-Kubo formula. At finite frequency, the Green-Kubo formula is given by

(1) |

Here is the Drude weight, and is the regular part of conductivity. In a perfect conductor showing ballistic transport, is non-zero and conductivity diverges at zero frequency. Thus, for ballistic transport at zero frequency, the Drude weight is the only relevant quantity. This is the case for non-interacting ordered systems, and for integrable systems in general.

The Green-Kubo formula for conductivity is based on the linear response of the system to an electric field. However, there is another way of driving current in a system. Imagine a one-dimensional electronic system with periodic boundary conditions. Putting a magnetic flux through the ring will drive a persistent current through the system. Let be the thermodynamic potential of the system. Then, the persistent current BÃ¼ttiker et al. (1983); Cheung et al. (1988, 1989); Montambaux et al. (1990); Bouzerar et al. (1994) through the system is given by

(2) |

where is the temperature. Here and henceforth we have set Boltzmann constant . At zero temperature, the expression for persistent current reduces to

(3) |

where is the ground state energy of the system.

The connection between these two different ways of driving transport was given by Kohn in his seminal work Kohn (1964). He showed that, at zero temperature,

(4) |

where, is the flux at which the ground state energy is minimum and is the system size. For practical purposes, this gives an alternative, easier way to evaluate the zero temperature Drude weight for one dimensional systems. In the following, we will look at the Drude weight of a one-dimensional ordered systems with power-law hopping using this procedure. We will also look at the persistent currents in such systems.

### .3 The model and existence of the thermodynamic limit

We consider a one-dimensional ordered non-interacting system with periodic boundary conditions, with a flux threaded through the ring. The flux effectively modifies the hopping between sites at a distance by a Pierls’ phase, . Here is the flux quantum given by =, where , , and are velocity of light, Planck’s constant and electronic charge. Henceforth, we will set . The resulting Hamiltonian is given by

(5) |

Here is the fermionic annihilation operator. The system has power-law hopping with exponent and strength . The long-range power-law hopping has a hard cut-off at , where the least integer greater than or equal to . This cut-off is required to uniquely define the periodic boundary condition. The important point here is that the cut-off scales with system-size.

This model can be easily diagonalized via a Fourier transform. The single particle eigenvalues and eigenvectors of the system are given by

(6) |

where, for even number of system size and for odd number of system size. Since the single particle eigenstates are all completely delocalized, transport is ballistic. So, as discussed above, transport is entirely characterized by the Drude weight. The Drude weight is defined only in the thermodynamic limit. Since this is a long-range system, existence of the thermodynamic limit is not obvious. It can be checked as follows that the thermodynamic limit is well-defined for . (In general, thermodynamic limit is well-defined for , where is the dimension of the system.)

For thermodynamic limit to exist, the summation in must converge for . By standard convergence tests, this is guaranteed if the series converges. This is the hyperharmonic series, which converges for . Thus, the Hamiltonian is well-defined in the thermodynamic limit for . However this is not the only requirement for existence of the thermodynamic limit. At zero temperature, one has to further check that the ground state energy is extensive. For a given number of particles , the many-body ground state energy is given by summing over up to the Fermi level. This can be done analytically to obtain,

(7) |

for odd and for even , and is the filling fraction. This relation always holds true for . To check the extensivity of , we note that the terms in the summation where decay as . So, for large , the main contribution to the sum comes from small . Thus, we can truncate the upper limit of the sum to , where is independent of . For such small , we have So, upon taking the limit , such that , we have

(8) |

where

(9) |

is the generalized Clausen function of order . Thus, for large and given filling , and hence is indeed extensive for all values of . So, the thermodynamic limit is well-defined for . Since we are interested in calculating the Drude weight, we confine ourselves to .

### .4 The Drude weight: finite-size oscillations and divergence

Armed with the expression for ground state energy, we now proceed to calculate the zero temperature Drude weight from Eq. 4. It can be checked that minimum value of occurs at . Using Eq. .3, we can straightforwardly write down the expression for

(10) |

The summation in the above expression can be broken into two parts: a) contribution from small , (b) contribution from terms where . The contribution from small can be approximately calculated from the first line of Eq. .3. The contribution from terms with is calculated by defining . The resulting expression, for large , is

(11) |

where is a finite number independent of . The above expression shows that shows oscillations with . The period of the oscillation is given by that for in the summand in Eq. 11. So the period of oscillation is . The amplitude of the oscillations decay as , for . Thus, in the thermodynamic limit, we get the analytical expression for the Drude weight

(12) |

Fig. 1(a) shows the oscillations of with for half-filling, for which the period of oscillation is system-sizes. Fig. 1(b) shows the decay of with , showing the approach is . This establishes the correctness of the approximate expression in Eq. 11.

So, we have found that the Drude weight is given in terms of the generalized Clausen function . Now, the following result can be shown for (see Appendix. A),

(13) |

Thus, for , diverges at small as . But, from the definition of , we see that . So, is discontinuous at for . This means that for , the Drude weight diverges as , is discontinuous at . In fact, as shown in Fig. 2(a) for , in this regime, decreases monotonically with for . This reveals the extremely counter-intuitive result that describes a phase where transport increases with decrease in the number of particles, and is maximum for a sub-extensive number of particles. For , goes to zero at small , and hence there is no discontinuity at . As shown in Fig. 2(a) for , for , shows the more intuitive behavior of having a maximum at a finite value of , i.e, for an extensive number of particles. Thus, at , we get a quantum phase transition, with the Drude weight at zero temperature showing markedly different behavior on either sides of it. This makes the ‘critcal’ point of the phase transition. At , depends on . is known in closed form, . So we get the following simple expression for the Drude weight for ,

(14) |

Thus in the ‘critical’ case, the discontinuity at and the monotonic decrease with survives, but, as . So even for at this ‘critical’ point, transport increases with decrease in the number of particles. The corresponding plot is shown in Fig. 2(a). In Fig. 2(b), the value of where is maximum, is plotted against . The quantum phase transition at is extremely clear from this plot. The results for the nearest neighbour case () are also shown in the plots for comparison. For nearest neighbour case, increases monotonically with increase in number of particles, and is maximum at half-filling due to particle-hole symmetry (which is not present in the long-range model). These observations can also be directly checked from Eq. 10 for (Appendix B).

The origin of the divergence of at at for can be traced to a divergence of group velocity in the system in this regime. In the thermodynamic limit, defining , at zero flux, the dispersion formula can be written from Eq. 6 as,

(15) |

The group velocity is given by the first derivative of ,

(16) |

Thus, the behavior of with is the same as that of with . This immediately shows that

(17) |

Since is the bottom of the band, it corresponds to filling fraction going to zero. So, for , as the number of particle decreases, the group velocity of the particles increases, leading to a maximum transport for a sub-extensive number of particles. For , , and this effect is not seen.

An intuitive picture of why increasing particles in a system with long-range hopping can reduce transport can be given as follows. Transport depends on the number of particles as well as the velocity of the particles. With a few number of particles, there are a lot of empty sites in the system. If the particles can hop to really long distances, this can lead to very high velocity, but of few particles. However, the fermionic nature of the particles inhibits the long range hopping as the number of particles increases. So, there is a trade-off between the number of particles and the range of hopping which dictates the filling at which there will be maximum transport. For , we find that the range of hopping is such that the velocity actually diverges for few particles. So increasing the number of particles only reduces the velocity, thereby reducing transport.

It is interesting to note that the dispersion relation for this system is actually non-analytic at for all values of . This can be seen as follows. Let be an arbitrary integer. Then, it can be easily checked that,

(18) |

This shows the non-analyticity of at for all values of . Thus, actually, for any integer , is a phase marked by divergence of all derivatives of of higher order than , and the integers and are the phase boundaries. This behavior can be seen in the Drude weight at filling fraction going to zero,

(19) |

However, the divergence of higher derivatives of does not affect the transport behavior as spectacularly as divergence of itself, which happens for (see Fig. 3).

In the limit of filling fraction going to zero, interactions, however strong, play negligible role. So, the divergence of Drude weight for at is immune to interactions. We now check this explicitly for two different interacting systems. The two different models of interaction we consider are

(20) |

where . The first of the two above interaction Hamiltonians describe nearest neighbour interactions, while the second describe long-range interactions with power-law decay. For simplicity, we have chosen the power-law exponent to be same as that in the non-interacting part. Our main point is to check that the divergence of Drude weight for a sub-extensive number of particles survives in presence of interactions. So we fix the number of particles at and plot vs in presence of above interactions, and compare with the non-interacting model. The plots are shown in Fig. 4. We have kept interactions strong, . As expected, with increase in the effect of interactions is lost, and for both the interacting models approach that of the non-interacting one. Thus, as is also clear from the plots, that even in presence of interactions, for , diverges, for , saturates to , and for , decays to zero, in the limit of zero filling fraction. So, increase of transport with decrease in number of particles is a completely generic behavior for ordered systems with power-law hopping in the phase . Having shown this, in the following we return to the non-interacting system and address the more pertinent question of experimental significance.

Although, increase in imply increase of transport, the experimental implication of this is not immediately clear in terms of electronic conductivity. This is because, any non-zero value of gives infinite electronic conductivity. So, in terms of electronic conductivity, the meaning of a divergence of is not obvious. A more direct experimental implication can be given in terms of the persistent current, which we explore in the next section.

### .5 The persistent current

#### .5.1 Zero temperature

Directly from the definitions of the and , it is clear that system size behavior of is same as that of . Since, for perfect conductors, goes to a constant with system-size, decays as . (This is one of the ways of distinguishing a perfect conductor from a superconductor, where there can be persistent currents up to macroscopic length scales.) So observation of persistent current requires mesoscopic systems. The relation between at zero temperature and precisely means that the physics described in previous sections can be experimentally explored by measuring persistent currents in mesoscopic rings with long-range hopping.

To see this explicitly, let us write down the expression for persistent current at zero temperature from Eqs. 2 and .3,

(21) | |||

for odd number of particles and for even number of particles. Exactly as we have done for the Drude weight, the summation in the above equation can be broken in two parts corresponding to small and . For large , the expression becomes

(22) |

It is evident that behavior of with system-size is same as that of . We see from above expression that at finite filling , as expected for ballistic system. Also as expected from relation with , there are oscillations with system size with a period of , the amplitude of which decay as . On the other hand, noting that for a sub-extensive number of particles, and using Eq. 13, we see that

(23) |

This shows that goes to zero in the thermodynamic limit even as , for all values of , which is the expected behavior for non-superconducting systems. Further, remembering that at finite filling, we see that, with fixed, for , is enhanced on reducing the number of particles. This, as discussed above, is the hallmark of this phase. This is true for any value of where is non-zero. This also shows that for , the maximum persistent current decays exponentially with . Exactly at , the persistent current at all filling. But, from Eq. 22, it is seen that the maximum persistent current still occurs at . The variation of the magnitude of persistent current with is shown for three values of in Fig. 5 for , at a chosen value of the flux. In the regime , the enhancement of persistent current for a sub-extensive number of particles for is very clear. The oscillations with in Fig. 5 have period , as can be seen from Eq. 22. These oscillations are suppressed with system size as . The behavior of the maximum current is shown with system size in the inset for various values of . As explained above, it decays as for , and has for .

So far we have shown that the phase showing the counter-intuitive feature of maximum transport at a sub-extensive number of particles is well captured by persistent current at zero temperature. In the next subsection, we check whether the effect of temperature.

#### .5.2 Finite temperature

The effect of having maximum current at a sub-extensive number of particles is essentially due to large level-spacings near the bottom of the band for . This means, for a finite system, the effect will survive as long as the temperature is small enough to resolve the energy level spacing at the bottom of the band. Thus, phase showing ‘more current with less particles’ can be experimentally observed in the temperature regime

(24) |

where is the energy difference between the lowest single-particle level and the next one. It can be shown that, . So, for a finite system, the effect can be observed from persistent current measurements up to a finite temperature. At higher temperatures, the level-spacing cannot be resolved, and the effect will be lost. This means, in the thermodynamic limit, there is no phase transition as a function of at any finite temperature. This is typical for quantum phase transitions. Let us now explicitly check this.

We assume a grand-canonical ensemble with inverse temperature and chemical potential . The persistent current and the filling fraction at a finite temperature and chemical potential is given by,

(25) |

where is the Fermi distribution function. Unlike the zero temperature case, it is hard to make analytical progress in the finite-temperature case. So we numerically explore the behavior of persistent current at finite temperatures. In Fig. 6, we show the behavior of with at finite temperatures for , and a chosen value of flux. It is clear that at low temperature, the maximum persistent current occurs at small . But at higher temperatures, the maximum persistent current occurs at a finite filling fraction. To further investigate this effect, we calculate which is defined as the filling fraction at which the persistent current is maximum. The dependence of on various parameters is shown in the top panel of Fig. 7. The variation of with is shown in Fig. 7(a) for , at various temperatures. At low temperature, (), shows a jump with from to a finite value. From the plot, it is seen that the transition point is shifter slightly from . This is due to finite-size effect. At higher temperature (), smoothly increases with and approaches a constant. So there is no phase transition, but instead a crossover. In Fig. 7(a), we show scaling of with . It is clear from the figure that for , as long as Eq. 24 is satisfied. Thus, current is maximum for a sub-extensive number of particles. As system-size is increased further, tends to a constant, showing that current is maximum for an extensive number of particles. However, for a given and given system size , does not show a sharp change with increase in temperature, but rises smoothly Fig. 7(c) from being of to a finite value.

In the bottom panel of Fig. 7 we show the behavior of the maximum persistent current with various parameters. Fig. 7(d) shows plot of with for various temperatures. As long as Eq. 24 is satisfied, decays exponentially with with the same rate as at zero temperature. After that, decays exponentially, but at a faster rate, until it reaches close to the value corresponding to the nearest neighbour model. Then it saturates to the value corresponding to the nearest neighbour model (). These effects are seen in Fig. 7(d) for . On the other hand, for , only the intermediate regime of exponential decay is seen for the choice of parameters. Fig. 7(e) shows plot of with for various values of temperature and . As long as Eq. 24 is satisfied, . After that, it still decays as a power-law, but with a higher power-law exponent. This exponent depends on temperature. This is in contrast with the behavior for nearest neighbour case (), where it is known that the persistent current at a finite temperature decays exponentially with system size. Fig. 7(f) shows plot of with for various values of for . Interestingly, it shows that decays as a power-law with power close to . This is in stark contrast with that of the nearest neighbour model, where it is known that decays exponentially with temperature. This is consistent with Fig. 7(d), which also shows that the maximum persistent current for the power-law hopping case is exponentially larger than that of the nearest neighbour case ().

Thus, we have theoretically demonstrated that the effect of maximum current at a sub-extensive number of particles due to power-law hopping can be experimentally observed by measuring persistent current in a mesoscopic ring. Further, the maximum persistent current at a finite temperature will be exponentially increased due to power-law hopping.

### .6 Conclusion

In this manuscript, we have investigated the transport properties of 1D ordered fermionic systems with power-law hopping, with power-law exponent . Our most interesting result is that describes a phase where the Drude weight diverges as filling fraction goes to zero. In other words, for , transport is maximum for a sub-extensive number of particles. Since, for a sub-extensive number of particles there is no effect of interactions, this is generically true for 1D ordered systems with power-law hopping. We have also explicitly checked this for two different models of interaction. This rather counter-intuitive effect occurs because, for , the group velocity for the non-interacting system diverges at the bottom of the band. We have further shown that measurement of persistent current in a mesoscopic ring with power-law hopping will give experimental signature of this effect. For a mesoscopic system, this effect survives at a finite temperature. In fact, persistent current at a finite temperature is exponentially enhanced due to power-law hopping, as compared to that of nearest neighbour hopping. This observation can have interesting potential applications.

Our results in this manuscript opens a plethora of questions. This is more so because of an extreme lack of investigations related to transport properties of long-range systems. We have shown here that Drude weight diverges for a sub-extensive number of particles for generic ordered systems with power-law hopping. One interesting question is the effect of interactions on transport properties of systems with power-law hopping at a finite filling. This is extremely challenging to investigate because many numerical techniques fail for long-range systems. The combined effect of having finite-size effects up to large systems, the non-sparseness of the Hamiltonian matrix, the presence of long-range entanglement makes finite-size scaling from exact-diagonalization also difficult. We hope to rise to such challenges in future works.

Another further interesting question is the effect of disorder or a quasi-periodic potential. In fact, it is known that there will be algebraic localization in such casesDeng et al. (2018). Understanding transport properties of such systems in the light of our results is of interest. Further, in this work, we have shown the consequence of non-analyticity of system’s energy dispersion on the isolated system transport properties in thermodynamic limit. Yet another extremely interesting question is the consequence of the non-analyticity in an open system transport set-up, i.e, where the central system is connected to two leads at different chemical potentials. Detailed investigations in both these directions are currently on-going and will be published in subsequent works.

### .7 Acknowledgements

M.S would like to acknowledge University Grants Commission (UGC) of India for her research fellowship.

## Appendix A

Here we look at the behavior of as small , which is the most important mathematical result for governing our main result. The summation in the definition of can be broken into two parts, one for , and the other for and higher, where is an arbitrary number greater than . Then, we have

(26) |

where, in the second term we have used . The summation in the second term is convergent, so in the limit , this goes to zero for . For small , the first term can be evaluated as

(27) |

This shows that

(28) |

This is the result used in main text.

## Appendix B

Here we check the directly from Eq. 10. To check this, we rigorously convert the sum in Eq. 10 into an integral by setting , and taking large limit. Then we have,

(29) |

For a single electron, , which tends to zero in the thermodynamic limit. For , from above equation,

(30) |

Thus, we have,

(31) |

Noting that for a sub-extensive number of particles, , this is exactly consistent with the behavior from obtained from Eq. 12 and Eq. 13.

## Appendix C Nearest neighbour persistent current

Here we show that the persistent current for a non-interacting system with nearest neighbour hopping decays exponentially with temperature and with system size. This is shown by plotting the maximum persistent current with temperature and with system-size in a log-linear plot in Fig. 8.

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