The Pauli principle, normal modes and superfluidity: the emergence of collective organizational phenomena

The Pauli principle, normal modes and superfluidity: the emergence of collective organizational phenomena

D. K. Watson
University of Oklahoma
Homer L. Dodge Department of Physics and Astronomy
Norman, OK 73019
July 15, 2019
Abstract

Understanding the emergence of collective organizational phenomena is a major goal in many fields of physics from condensed matter to cosmology. Using a recently introduced manybody perturbation formalism for fermions, we propose a mechanism for the emergence of collective behavior, specifically superfluidity, driven by quantum statistics and the enforcement of the Pauli principle through the selection of normal modes. The method, which is called symmetry invariant perturbation theory (SPT), uses group theory and graphical techniques to solve the manybody Schrodinger equation through first order exactly. The solution at first order defines collective coordinates in terms of five N-body normal modes, identified as breathing, center of mass, single particle angular excitation, single particle radial excitation and phonon. A correspondence is established “on paper” that enforces the Pauli principle through the assignment of specific normal mode quantum numbers. Applied in the unitary regime, this normal mode assignment yields occupation only in an extremely low frequency N-body phonon mode at ultralow temperatures. A single particle radial excitation mode at a much higher frequency creates a gap that stabilizes the superfluidity at low temperatures. Coupled with the corresponding values for the frequencies at unitarity obtained by this manybody calculation, we obtain good agreement with experimental thermodynamic results including the lambda transition in the specific heat. Our results suggest that the emergence of collective behavior in macroscopic systems is driven by the Pauli principle and its selection of the correct collective coordinates in the form of N-body normal modes.

I Introduction

Collective behavior of large systems of particles has long been an area of great interest. Collections of particles can behave quite differently from the complex motion of isolated particles, often acquiring qualitatively simple forms of behavior. As pointed out by Anderson in his treatise, “More is Different”, “the whole becomes not only more than but very different from the sum of its parts”anderson (). The appearance of magnetism, zero-viscosity in superfluids, and zero resistivity in superconducting metals are all examples of simple behaviors that arise, not from detailed microscopic forces, but from the emergence of collective organizational phenomena. These phenomena depend on powerful and general principles of organization that are not well understood, but have the potential to reveal fundamental insights into the collective behaviors of large systems. These principles can supersede the difference between classical and quantum physics and effect cooperative macroscopic behavior that differs drastically from the expected individual microscopic behavior (e.g. electron repulsion). Elucidating the dynamics behind these principles of organization remains an important challenge in many fields of physics.

Studying quantum systems of identical particles such as the ultracold gases can reveal the influence of organizational principles that are due to quantum statistics, which for fermions means the Pauli exclusion principle. The Pauli principle can provide an effective repulsion that is dependent on particle statistics as opposed to interparticle interactions. In certain regimes, such as the unitarity regime for ultracold gases, the Pauli principle can dominate the physical interaction and control the dynamics. When this effect is dominant, systems exhibit collective behavior that is universal as found in trapped Fermi superfluids at unitarity. Universal behavior is also seen in the quark-gluon plasmas of the early universe, high temperature superconductivity, and neutron stars. Knowledge of the thermodynamics of the unitary gas thus has consequences for understanding the equation of state in these other regimes at vastly different scales. Using cooling and trapping techniques, the atomic physics community has been able to study finite size systems of ultracold gases that exhibit universal properties. Such systems with sufficiently strong interactions behave identically on a scale given by the average particle separation, independent of the details of the short range interaction.

Understanding the dynamics behind this large scale organization has been an elusive goal. The present paper seeks to address this goal by proposing a simple, straightforward description of the connection between the Pauli principle, the normal modes of a macroscopic system of identical particles and the emergence of superfluidity. This description is based on the symmetry invariant perturbation method (SPT) and its exact first order solutions which are the N-body normal modes. Much of the work in this approach, which is equivalent to the work in any fully interacting manybody calculation, has been done “on paper” using group theory and graphical techniques. The Pauli principle is also applied “on paper” by imposing restrictions on the normal mode quantum numbers at first order in the perturbation. This not only enforces the Pauli principle with trivial numerical cost, but does so in a way that can directly reveal large scale collective behavior through the N-body normal modes. The well-known phonon behavior of superfluids is seen to result from enforcing the Pauli principle at ultracold temperatures.

To test this understanding of the dynamics, we use the SPT formalism to calculate thermodynamic properties in the unitary regime including energy, entropy, and heat capacity. We obtain good agreement with experiment. In particular, the lambda transition in the specific heat is clearly seen and agrees well with experimental results. The good agreement with experiment supports the validity of this simple description of the dynamics behind the emergence of collective behavior. Driven by the enforcement of the Pauli principle, phonon normal modes and a single particle radial excitation normal mode that becomes occupied as the temperature increases, create a gapped system that has the correct thermodynamic behavior for this superfluid regime.

Ii The Partition Function

In a recent paperpartition (), we developed an approach for the determination of the partition function for strongly-interacting identical fermions in ultracold regimes and applied it to a model system of harmonically-confined, harmonically-interacting fermions, successfully calculating various thermodynamic quantitiespartition ().

In the present study, we now apply this approach to the determination of the partition function and several thermodynamic quantities for the ultracold, strongly interacting, confined fermion systems in the unitary regime which have been extensively studied both experimentally and theoreticallyzwierlein2 (); regal (); jin1 (); jin2 (); jin3 (); jin (); kinast1 (); thomas (); thomas1 (); thomas2 (); thomas3 (); thomas4 (); horikoshi (); grimm1 (); grimm2 (); jochim (); ketterle1 (); nascimbene (); hu1 (); hu2 (); hu3 (); hu4 (); hu5 (); bulgac (); burovski1 (); burovski2 (); nishida1 (); nishida2 (); nishida3 (); leggett (); strinati (); levin (); randeria1 (); randeria2 (); randeria3 (); haussmann1 (); haussmann2 (); haussmann3 (). Strongly-interacting systems are particularly challenging due to the exponential scaling of complexity which for conventional methods scales as a function of particle number, . Accurate partition functions can require millions of energy levels depending on, among other things, the temperature. To date, determining the full energy spectrum of systems with four or more particles remains a challengedaily ().

The SPT method circumvents these daunting numerical demands in several ways FGpaper (); energy (); matrix_method (); paperI (); laingdensity (); JMPpaper (); test (); toth (); rearrangeprl (); prl (); harmoniumpra (); partition (). Specifically, we are able to rearrange the numerical work into analytic building blocks that allow a formulation that does not scale with . These analytic building blocks have been calculated and stored previously minimizing the work needed for new calculations. The Pauli principle is applied “on paper” resulting in trivial numerical demands compared to conventional methods that explicitly enforce the antisymmetry of the manybody wave function. This method was recently successfully used to calculate ground state energies in the unitary regime where results comparable in accuracy to benchmark Monte Carlo results for were obtained in a few seconds of computer timeprl (). We also performed an explicit test of our method of enforcing the Pauli principle harmoniumpra ().

The first order SPT solution yields a harmonic spectrum with five frequencies belonging to the five N-body normal modes. Similar to the confined ideal gas with its harmonic spectrum, the full SPT spectrum is known for this manybody problem. Determining the partition function for systems with a completely known spectrum still presents a non-trivial problem due to the difficulty of determining the degeneracies of the manybody states and enforcing the correct symmetry on those states as previous studies on confined ideal gases readily revealborrmann (); schmidt1 (); schmidt2 (); butts (); toms (); schneider ().

In this paper, as demonstrated in our earlier model studypartition (), we use a conceptually different approach to the determination of the partition function. The Pauli principle is applied very simply through a trivial normal mode quantum number assignment for each term in the sum of states without ever obtaining the actual wave function. The degeneracy of each energy level is a natural result of doing a straightforward partitioning of the number of energy quanta among all particles into different normal mode assignments according to the Pauli principle and collecting the statistics. Finally, the full excitation spectrum is known through first order.

Iii Application: Unitary Fermi Gas

We assume an -body system of fermions, with spin up and spin down fermions such that , confined by a spherically symmetric harmonic potential with frequency . For the unitary regime, we replace the actual atom-atom potential by an attractive square well potential of radius and a potential depth adjusted so the s-wave scattering length, is infiniteprl ().

We apply this method to a Fermi gas in the unitary regime, using the full formalism, defining symmetry coordinates from the internal displacement coordinatesFGpaper (); energy () and using the FG methoddcw () to solve for the five normal coordinates and their frequencies, . The roots, , are highly degenerate due to the symmetry, resulting in a reduction to five distinct roots that correspond to five irreducible representations of WDC () and yield five normal modes, labelled by FGpaper (). The normal modes are phonon, i.e. compressional modes; has single particle radial behavior; shows single particle angular behavior; is a center-of-mass motion, and is a symmetric breathing motion. The energy through first order in : FGpaper ()

(1)

gives the full spectrum of excited states through the assignment of the normal mode quantum numbers that enforce the Pauli principle. The possible assignments are found by relating the normal mode states to the states of the confining potential, a spherically symmetric three dimensional harmonic oscillator for which the restrictions imposed by antisymmetry are known. These two series of states can be related in the double limit , where both representations are valid. Two conditions resultprl (); harmoniumpra ():

(2)

where the radial and orbital angular momentum quantum numbers of the three dimensional harmonic oscillator, and , respectively, satisfy , with , the energy level quanta of the ith particle defined by: .

These equations determine a set of possible normal mode states that are consistent with an antisymmmetric wave function from the known set of permissible harmonic oscillator configurations.

The SPT energies are obtained from Eq. (1) with the normal mode quanta determined from Eq. (2) to ensure antisymmetry. We choose quanta that correspond to the lowest values of the normal mode frequencies to yield the lowest energy for each excited state. This results in occupation in , the phonon mode, and in , a single particle radial mode, which have the lowest angular and radial frequencies respectively. The conditions are:

(3)

Iv Manybody “pairing” in phonon normal modes: The transition from Fermi to Bose statistics

At first order in the SPT method, the five normal modes include an extremely low frequency, highly degenerate phonon mode. This phonon mode provides a manybody wave function resulting in cooperative, coherent behavior of the fermions at extremely low temperatures. This model does not describe pairing between individual pairs of fermions, but rather sets up a picture of a manybody coherent wave with fermions in the highly degenerate, lowest frequency mode, each fermion in synced motion with many other fermions making it impossible to determine which fermion is paired with which. This type of synced manybody motion in real space is dictated by the Pauli principle at ultralow temperatures where this is the only mode with nonzero quanta. This “manybody pairing” is a precursor to the two-body pairing (in real space) and allows a natural transition from Fermi statistics to Bose statistics as individual particles form pairs. Analogously, the single particle radial excitation normal mode does not describe excitation out of an individual pair of fermions, but rather the excitation of a single particle out of the synced motion of the manybody phonon mode.

V Thermodynamic Results

We determined the following thermodynamic quantities: energy, , entropy, , and heat capacity, :

(4)

with the canonical partition function, the jth manybody energy, its degeneracy and the temperature (). Fig. 1 shows our SPT results for the energy in units of , where is the Fermi energy, compared to experiment and theoryhu2 (); hu3 (); nascimbene () as a function of . Our approach which does an explicit summation included energies corresponding to energy quanta up through 110. In Fig. 2 we compare our SPT results for E(S), i.e. the energy vs. the entropy with experimentzwierlein2 (); nascimbene (); jin1 (); thomas (); thomas1 (). Finally, in Fig. 3, we compare our SPT results for the heat capacity, in units of to previous resultsthomas1 (); kinast1 (); hu3 ().

Good agreement is obtained for all three thermodynamic quantities with existing experimental and theoretical results. Our calculations, which have no adjustable parameters, are the first-order results of SPT theory. The calculations do show finite effects, i.e. fluctuations, as we varied that presumably would even out as increases. As expected, below our critical temperature , the results converge fairly rapidly, while above , more and more states become thermally accessible making convergence challenging. As is increased, the number of states needed in the partition function rises quite rapidly and the corresponding degeneracies also become quite large, straining current desktop capabilities. Converging the results for the particle numbers and temperatures used in this paper was tractable on a desktop with a few hours of time. The increase in resources needed as and/or the temperature increases is severe, but is not exponential. This is expected since we are not obtaining the explicitly antisymmetrized wave functions for each state in a degenerate level. The graph of the energy vs. temperature shows good agreement at very low temperatures with other theoretical calculations. Experimental results do not yet reach these very low temperatures.

We looked at particle numbers between and . The results in the graphs are for particle number for Figs. 1 and  2 and for for the heat capacity in Fig. 3. For E(S), we agree well with several experimentalzwierlein2 (); jin1 (); thomas (); thomas1 (); nascimbene () and theoretical results including NSRhu1 (); hu2 (), Monte Carlobulgac () and a field theoretic approachhaussmann3 (). In Fig. 2, we show the comparison with the experimental results. While the energies and entropies settled into good values quickly as increased above ten; for the heat capacity, the lambda transition is barely visible at , and the critical temperature is lower , although we can see the convergence toward the Boltzmann value of for a confined gas for this low . As increases, the lambda transition becomes quite sharp and trends to higher . At shown in Fig. 3, we can converge the lambda peak, but not the higher temperature behavior toward with current desktop resources. We estimate the critical temperature to be . This compares well with some results in the literature: nascimbene (), burovski2 (), hu2 (); hu5 (); haussmann3 (), but is smaller than other reported results: kinast1 (); hu5 (); bulgac (), 0.29thomas (); hu5 ().

Figure 1: The universal thermodynamic function E(T). Our SPT results for 12 particles are compared to experimental: ENSnascimbene () and theoretical results: NSRhu2 (); hu3 () and GGhu2 (); hu3 (). The comparison values in all the figures were extracted directly from graphs in the literature.
Figure 2: E(S)for a trapped Fermi gas at unitarity. Our SPT results for 12 particles are compared with experimental data: ENSnascimbene (), JILAjin1 (), Dukethomas (); thomas1 (), MITzwierlein2 ().
Figure 3: Heat capacity vs temperature for a trapped Fermi gas at unitarity. Our SPT results for 36 particles are compared to experimental: Duke1thomas1 (), Duke2kinast1 () and theoretical results: NSRhu3 () as a function of temperature.

Vi Conclusions.

Understanding the dynamics behind the emergence of collective organizational phenomena has been a goal both experimentally and theoretically in multiple fields of physics. In regimes exhibiting collective behavior, simple behavior can emerge from the complexity of the microscopic world due to the influence of organizational phenomena, replacing microscopic uncertainty with large scale certainty.

From the quark-gluon plasmas of the early universe to ultra cold Fermi superfluids, the Pauli principle drives the behavior of mesoscopic and macroscopic Fermi systems, underpinning the emergence of collective states and superseding forces at the microscopic scale. Our work has revealed that at ultralow temperatures, the Pauli principle is responsible for selecting only a very low energy phonon mode in which fermions behave in a cooperative manner that minimizes the energy. At higher temperatures, a single particle radial excitation normal mode becomes occupied. This mode has a much higher frequency than the phonon mode providing a gapped spectrum that stabilizes the superfluid behavior.

The remaining three normal modes, single excitation angular, breathing, and center of mass, have higher frequencies which will become accessible one by one depending on their magnitudes as the temperature increases. The exact values of these magnitudes will, of course, be influenced by the physics of the particular system.

These five single particle and collective modes could also underlie the very successful collective and individual-particle motion description of nuclear dynamics developed by Bohr and Mottelsonbohr (); bohr1 () in the 1950’s which remains an important paradigm in nuclear physics.

Our picture of the physics in the unitary regime is simple, but it is based on a complex, fully-interacting solution of a manybody Hamiltonian through first order. This simple physical picture emerges from the decision to use group theory to solve the first-order harmonic equation using the FG method which yields normal mode solutions. For regimes with simple emergent behavior, this approach provides a natural link between manybody complexity and large scale simplicity.

The Pauli principle is known to have a profound effect on energy levels as well as quantum statistics particularly at low temperatures. This study has now offered evidence that the Pauli principle has a profound effect on the emergence of collective organizational phenomena, acting as a powerful driving force in the emergence of collective organizational states of matter.

Vii Acknowledgments

We thank H. Hu and M. Zwierlein for providing some detailed results. Support by the National Science Foundation through Grant No. PHY-1607544 is gratefully acknowledged.

References

  • (1) P.W. Anderson, Science 177, 393(1972).
  • (2) D.K. Watson, Phys. Rev. A 96, 033601(2017).
  • (3) M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Science, 335(6068), 563-567 (2012).
  • (4) C.A. Regal, M Greiner, and D.S. Jin, Phys. Rev. Lett. 92, 040403, 2004.
  • (5) J. T. Stewart, J.P. Gaebler, C.A. Regal and D.S. Jin, Phys. Rev. Lett. 97, 220406 (2006).
  • (6) J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati, Nat. Phys. 6, 569 (2010).
  • (7) J.T. Stewart, J.P. Gaebier, and D.S. Jin, Nature, 454(7205):744-747(2008).
  • (8) J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Phys. Rev. Lett. 104, 235301 (2010).
  • (9) J.Kinast, A. Turlapov, J.E. Thomas, Q. Chen, J. Stajic, K. Levin, Science 307, 729 (2010).
  • (10) L. Luo, B. Clancy, J. Joseph, J. Kinast, and J. E. Thomas, Phys. Rev. Lett., 98,080402 (2007).
  • (11) L. Luo, J.E. Thomas, J. Low Temp. Phys. 154, 1 (2009).
  • (12) C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka, T. Schafer, and J. E. Thomas, Science 331,58(2011).
  • (13) K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Science 298, 2179 (2002).
  • (14) A. Turlapov, J. Kinast, B. Clancy, Le Luo, J. Joseph, and J. Thomas, J. Low Temp. Phys. 150(3), 567-576 (2008).
  • (15) M. Horikoshi et al. Science 327, 442(2010).
  • (16) M. K. Tey, L. A. Sidorenkov, E. R. S. Guajardo, R. Grimm, M. J. H. Ku, M. W.Zwierlein, Y.-H. Hou, L. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 110(5), 055303(2013).
  • (17) L. A. Sidorenkov, M. K. Tey, R. Grimm, Y.-H. Hou, L. Pitaevskii, and S. Stringari, Nature, 498(7452), 78-81 (2014).
  • (18) A. Schirotzek, Y. I. Shin, C. H. Schunck, and W. Ketterle, Phys. Rev. Lett. 101(14), 140403 (2008).
  • (19) S. Nascimbene et al. Nature(London) 463, 1057(2010).
  • (20) F. Serwane, G. Zurn, T. Lompe, T.B. Ottenstein, A.N. Wenz, S. Jochim, Science 332, 336-338 (2011).
  • (21) H. Hu, P.D. Drummond, and X.-J. Liu, Nat. Phys. 3, 469 (2007).
  • (22) H. Hu, X.-J. Liu, and P.D. Drummond, New Journal of Physics 12, 063038 (2010).
  • (23) H. Hu, X.-J. Liu, and P.D. Drummond, Phys. Rev. A 73, 023617 (2006).
  • (24) H. Hu, X.-J. Liu, and P.D. Drummond, Europhys. Lett. 74(4), 574-580 (2006).
  • (25) H. Hu, X.-J. Liu, and P.D. Drummond, Phys. Rev. A 77, 061605(R) (2008).
  • (26) A. Bulgac, J.E. Drut, and P. Magierski, Phys. Rev. Lett 99, 120401 (2007).
  • (27) E. Burovski, E. Kozik, N. Prokof’er, B. Svistunov and M. Troyer, Phys. Rev. Lett. 101, 090402 (2008).
  • (28) E. Burovski, N. Prokof’er, B. Svistunov and M. Troyer, New J. of Phys. 8, 153 (2006).
  • (29) Y. Nishida and D. T. Son, Phys. Rev. Lett. 97, 050403 (2006).
  • (30) Y. Nishida and D. T. Son, Phys. Rev. A 75, 063617 (2007).
  • (31) Y. Nishida and D. T. Son, In W. Zwerger, editor, The BCS-BEC Crossover and the Unitary Fermi Gas, Lecture Notes in Physics, Springer, 2011.
  • (32) A. J. Leggett, In Modern Trends in the Theory of Condensed Matter, Proceedings of the XVIth Karpacz Winter School of Theoretical Physics, Karpacz, Poland, pages 13-27, Springer-Verlag, Berlin, 1980.
  • (33) Q. Chen, J. Stajic, S. Tan, and K. Levin, Phys. Rep. 412(1), 1- 88 (2005).
  • (34) A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett. 93(10), 100404 (2004).
  • (35) C. A. R. Sa de Melo, M. Randeria, and J. R. Engelbrecht, Phys. Rev. Lett. 71, 3202-3205 (1993).
  • (36) J. R. Engelbrecht, M. Randeria, and C. A. R. Sa de Melo, Phys. Rev. B 55(22), 15153 (1997).
  • (37) R. B. Diener, R. Sensarma, and M. Randeria, Phys. Rev. A 77, 023626 (2008).
  • (38) R. Haussmann, Phys. Rev. B 49(18), 12975 (1994).
  • (39) R. Haussmann, W. Rantner, S. Cerrito, and W. Zwerger, Phys. Rev. A 75(2), 023610 (2007).
  • (40) R Haussmann and W Zwerger, Phys. Rev. A 78, 063602 (2008).
  • (41) K.M. Daily and D. Blume, Phys. Rev. A 81, 053615 (2010).
  • (42) D.K. Watson, Phys. Rev. A 92, 013628 (2015)
  • (43) D.K. Watson, Phys. Rev. A 93, 023622 (2016).
  • (44) B.A. McKinney, M. Dunn, D.K. Watson, and J.G. Loeser, Ann. Phys. (NY) 310, 56 (2003).
  • (45) B.A. McKinney, M. Dunn, D.K. Watson, Phys. Rev. A 69, 053611 (2004).
  • (46) M. Dunn, T.C. Germann, C.A. Traynor, D.Z. Goodson, J.D. MorganIII, D.K. Watson, and D. R. Herschbach, J. Chem. Phys. 101 5987 (1994).
  • (47) M. Dunn, D.K. Watson, and J.G. Loeser, Ann. Phys. (NY), 321, 1939 (2006).
  • (48) W.B. Laing, M. Dunn, and D.K. Watson, Phys. Rev. A 74, 063605 (2006).
  • (49) W.B. Laing, M. Dunn, and D.K. Watson, J. of Math. Phys. 50, 062105 (2009).
  • (50) W.B. Laing, D.W. Kelle, M. Dunn, and D.K. Watson, J Phys A 42, 205307 (2009).
  • (51) M. Dunn, W.B. Laing, D. Toth, and D.K. Watson, Phys. Rev A 80, 062108 (2009).
  • (52) D.K. Watson and M. Dunn, Phys. Rev. Lett. 105, 020402 (2010).
  • (53) P. Borrmann and G. Franke, J. Chem. Phys. 98, 2484(1993).
  • (54) H.-J. Schmidt and J. Schnack, Physica A, 260, 479(1998)).
  • (55) J.-J. Schmidt and J. Schnack, Physica A 265, 584(1999).
  • (56) D. A. Butts and D. S. Rokhaar, Phys. REv. A 55, 4346(1997).
  • (57) D. J. Toms, Annal of Physics 320, 487 (2005).
  • (58) J. Schneider and H. Wallis, Phys. Rev. A 57, 1253(1998).
  • (59) E.B. Wilson, Jr., J.C. Decius, P.C. Cross, Molecular vibrations: The theory of infrared and raman vibrational spectra. McGraw- Hill, New York, 1955.
  • (60) See for example reference dcw, , Appendix XII, p. 347.
  • (61) A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 1 (Benjamin, New York, 1969).
  • (62) A. Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 16(1952); A. Bohr and B. Mottelson, “Nuclear Structure”, Vol 2 (Reading, MA: W.A. Benjamin, Inc.).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
191465
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description