3362:
2986:
4990:, which indicates that there are in fact electron states between the Landau levels. These states are localized in, for example, impurities of the material where they are trapped in orbits so they can not contribute to the conductivity. That is why the resistivity remains constant in between Landau levels. Again if the magnetic field decreases, one gets the classical result in which the resistivity is proportional to the magnetic field.
7168:
5036:
3828:
5069:, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious
4220:
7596:
5073:. In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away. Also, the experiments control the filling factor and not the Fermi energy. If this diagram is plotted as a function of filling factor, all the features are completely washed away, hence, it has very little to do with the actual Hall physics.
7620:
3106:
7632:
7608:
3357:{\displaystyle N_{B}={\frac {\phi }{\phi _{0}}}={\frac {BA}{BL_{y}\Delta x_{k}}}={\frac {A}{2\pi l_{B}^{2}}}{\begin{array}{lcr}&l_{B}&\\&=&\\&&\end{array}}{\frac {AeB}{2\pi \hbar }}{\begin{array}{lcr}&\omega _{\rm {c}}&\\&=&\\&&\end{array}}{\frac {m^{*}\omega _{\rm {c}}A}{2\pi \hbar }}}
801:
4605:
672:. Other models to explain the fractional quantum Hall effect also exists. Currently it is considered an open research problem because no single, confirmed and agreed list of fractional quantum numbers exists, neither a single agreed model to explain all of them, although there are such claims in the scope of
4116:
is not an integer. It happens to be an integer when there is an exact number of filled Landau levels. Instead, it becomes a non-integer when the top level is not fully occupied. In actual experiments, one varies the magnetic field and fixes electron density (and not the Fermi energy!) or varies the
4600:
which become more relevant as the magnetic field increases. Obviously, the height of the peaks are larger as the magnetic field increases since the density of states increases with the field, so there are more carriers which contribute to the resistivity. It is interesting to notice that if the
5107:
responsible for the transverse induced voltage and the Hall effect, one may look at the
Coulomb potential difference in the Bohr atom as the induced single atom Hall voltage and the periodic electron motion on a circle as a Hall current. Defining the single atom Hall current as a rate a single
4351:
lies in between Landau levels where there are no states available for carriers, so the conductivity becomes zero (it is considered that the magnetic field is big enough so that there is no overlap between Landau levels, otherwise there would be few electrons and the conductivity would be
4190:, by increasing the magnetic field, the Landau levels move up in energy and the number of states in each level grow, so fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level (
831:
In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. To determine the values of the energy levels the SchrĂśdinger equation must be solved.
4536:
2258:
5323:
3443:
Note the dependency of the density of states with the magnetic field. The larger the magnetic field is, the more states are in each Landau level. As a consequence, there is more confinement in the system since fewer energy levels are occupied.
1117:
806:
805:
802:
807:
156:
4930:
692:
and
Lincoln Derick were able to manufacture the first silicon dioxide field effect transistors at Bell Labs, the first transistors in which drain and source were adjacent at the surface. Subsequently, a team demonstrated a working
5018:
can be created. This process can be expressed through a metaphor of photons bouncing between multiple mirrors. By shooting the light across multiple mirrors, the photons are routed and gain additional phase proportional to their
3528:
815:
changes and the corresponding position on a graph of hall coefficient and magnetic field|Illustrative only. The levels spread out with increasing field. Between the levels the quantum hall effect is seen. DOS is the density of
1515:
804:
4091:
5401:
3621:. However, if a large magnetic field is applied, the energies split into two levels due to the magnetic moment associated with the alignment of the spin with the magnetic field. The difference in the energies is
5076:
Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the
3437:
839:. The system considered is an electron gas that is free to move in the x and y directions, but is tightly confined in the z direction. Then, a magnetic field is applied in the z direction and according to the
2638:
426:
is more complicated and still considered an open research problem. Its existence relies fundamentally on electronâelectron interactions. In 1988, it was proposed that there was a quantum Hall effect without
410:
The striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. Since the electron density remains constant when the
1244:
4417:
667:
is still considered an open research problem. The fractional quantum Hall effect can be also understood as an integer quantum Hall effect, although not of electrons but of chargeâflux composites known as
2559:
1196:
To solve this equation it is possible to separate it into two equations since the magnetic field just affects the movement along x and y axes. The total energy becomes then, the sum of two contributions
3573:
5081:. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called
3676:
4790:
3977:
3939:
2104:
2678:
1832:
2047:
1387:
7082:
4843:
6253:
K. S. Novoselov; Z. Jiang; Y. Zhang; S. V. Morozov; H. L. Stormer; U. Zeitler; J. C. Maan; G. S. Boebinger; P. Kim; A. K. Geim (2007). "Room-temperature quantum Hall effect in graphene".
5194:
2162:
2154:
1708:
3861:
2912:
4296:
890:
4409:
727:
researchers
Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. In 1978, the
579:
was defined for use in resistance calibrations worldwide. On 16 November 2018, the 26th meeting of the
General Conference on Weights and Measures decided to fix exact values of
5205:
3619:
2844:
4568:
2980:
1171:
500:
4188:
3751:
2303:
1764:
1560:
2778:
1191:
1142:
971:
4666:
4636:
415:
is in a clean spectral gap, this situation corresponds to one where the Fermi level is an energy with a finite density of states, though these states are localized (see
4214:
2705:
3822:
5146:
4968:
4701:
3901:
3881:
2587:
916:
2494:
At zero field, the density of states per unit surface for the two-dimensional electron gas taking into account degeneration due to spin is independent of the energy
644:. Exact quantization in full generality is not completely understood but it has been explained as a very subtle manifestation of the combination of the principle of
4593:
is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the conductivity will have a maximum .
4257:
3034:
2484:
2457:
2430:
2403:
963:
803:
4988:
4721:
4591:
4345:
4155:
4135:
4114:
4001:
1419:
758:, who connected the quantized conductivity to the quantized charge transport in a Thouless charge pump. Most integer quantum Hall experiments are now performed on
5813:
5749:
3722:
76:
4851:
2801:
2329:
7075:
5126:
4372:). Consequently, the resistivity becomes zero too (At very high magnetic fields it is proven that longitudinal conductivity and resistivity are proportional).
4370:
4316:
3795:
3775:
3696:
3098:
3074:
3054:
3007:
2932:
2864:
2756:
2736:
2373:
2349:
1728:
1640:
1620:
1600:
1580:
936:
502:) has the important property of being exceedingly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of
5486:
3450:
6951:
5770:
1424:
7068:
4009:
6944:
731:
researchers Jun-ichi
Wakabayashi and Shinji Kawaji subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs.
3100:
can be calculated from the ratio between the total magnetic flux that passes through the sample and the magnetic flux corresponding to a state.
6940:
7020:
7002:
6882:
D. Hsieh; D. Qian; L. Wray; Y. Xia; Y. S. Hor; R. J. Cava; M. Z. Hasan (2008). "A topological Dirac insulator in a quantum spin Hall phase".
6053:
6028:
5670:
59:
5334:
5199:
and the induced Hall voltage as a difference between the hydrogen nucleus
Coulomb potential at the electron orbital point and at infinity:
3373:
4668:, of a two-dimensional electron gas as a function of magnetic field. Both vertical axes were divided by the quantum unit of conductance
5945:
4531:{\displaystyle \sigma ={\frac {1}{\det \rho }}{\begin{pmatrix}\rho _{yy}&-\rho _{xy}\\-\rho _{yx}&\rho _{xx}\end{pmatrix}}\;.}
2592:
7612:
7026:
A. Baumgartner; T. Ihn; K. Ensslin; K. Maranowski; A. Gossard (2007). "Quantum Hall effect transition in scanning gate experiments".
1200:
7663:
7600:
6380:
548:, the discoverer of exact quantization. The quantum Hall effect also provides an extremely precise independent determination of the
2500:
3537:
1766:
therefore commutes with the
Hamiltonian. By substituting this Ansatz into the SchrĂśdinger equation one gets the one-dimensional
5441:
5078:
3624:
664:
423:
5057:. A striking model of much interest in this context is the AzbelâHarperâHofstadter model whose quantum phase diagram is the
4601:
magnetic field is very small, the longitudinal resistivity is a constant which means that the classical result is reached.
7636:
7581:
7192:
5446:
4970:
if the filling factor is an integer. In experiments, however, plateaus are observed for whole plateaus of filling values
7127:
835:
Since the system is subjected to a magnetic field, it has to be introduced as an electromagnetic vector potential in the
431:. This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of the
7668:
6396:
6142:
Jun-ichi
Wakabayashi; Shinji Kawaji (1978). "Hall effect in silicon MOS inversion layers under strong magnetic fields".
3944:
3906:
3531:
698:
560:
4735:
2055:
7683:
4935:
One concludes then, that the transverse resistivity is a multiple of the inverse of the so-called conductance quantum
3582:, for each state available in the Landau levels it corresponds to two electrons, one electron with each value for the
6107:
Tsuneya Ando; Yukio
Matsumoto; Yasutada Uemura (1975). "Theory of Hall effect in a two-dimensional electron system".
4117:
electron density and fixes the magnetic field. Both cases correspond to a continuous variation of the filling factor
2643:
2253:{\displaystyle \varepsilon _{xy}\equiv \varepsilon _{n_{x}}=\hbar \omega _{\rm {c}}\left(n_{x}+{\frac {1}{2}}\right)}
1840:
1773:
1252:
765:, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in
591:(the elementary charge), superseding the 1990 conventional value with an exact permanent value (intrinsic standard)
7372:
7281:
5046:
5153:
4795:
4327:
It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. When
652:). The integer quantum Hall effect instead is considered a solved research problem and understood in the scope of
7321:
7296:
5451:
5039:
5020:
7245:
6796:
M. F. L. Golterman; K. Jansen; D. B. Kaplan (1993). "ChernâSimons currents and chiral fermions on the lattice".
5328:
One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von
Klitzing constant as
2109:
1645:
7688:
7491:
7235:
7091:
5481:
5436:
5419:
432:
401:. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether
6179:"New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance"
677:
657:
5614:"Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the 'Parity Anomaly'"
3834:
7678:
7337:
5781:
3824:
is proportional to the magnetic field so, the larger the magnetic field is, the more relevant is the split.
836:
553:
549:
4262:
2869:
846:
5318:{\displaystyle U=V_{\text{C}}(\infty )-V_{\text{C}}(r)=0-V_{\text{C}}(r)={\frac {e}{4\pi \epsilon _{0}r}}}
751:
720:
7388:
7367:
7301:
5881:
5095:
The value of the von Klitzing constant may be obtained already on the level of a single atom within the
4378:
2707:
the time between scattering events. Commonly it is assumed that the precise shape of Landau levels is a
1602:
directions, the solution of the SchrĂśdinger equation can be chosen to be the product of a plane wave in
529:
416:
5559:
Hansson, T.H. (April 2017). "Quantum Hall physics: Hierarchies and conformal field theory techniques".
2809:
4544:
3588:
2937:
435:
which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.
7673:
7658:
7357:
7132:
7035:
6970:
6901:
6862:
6815:
6762:
6693:
6626:
6553:
6488:
6433:
6327:
6272:
6227:
6190:
6151:
6116:
5910:
5699:
5625:
5578:
5520:
5423:
5058:
4003:
as the ratio between the density of states in a 2DEG and the density of states in the Landau levels.
1147:
1112:{\displaystyle \left\{{\frac {1}{2m^{*}}}\left^{2}+V(z)\right\}\psi (x,y,z)=\varepsilon \psi (x,y,z)}
728:
451:
7057:
and V. B. Timofeev, Quantum Hall Effect, Sov. Phys. â Semiconductors v. 20, pp. 617â647 (1986).
4160:
3727:
2263:
1733:
1520:
7311:
7291:
7276:
6375:. 6.4 Uniform magnetic Field; 6.5 Magnetic Field in a Narrow Channel, 6.6 The Quantum Hall Effect.
5461:
2712:
2352:
1767:
724:
708:" voltage controls the number of charge carriers in this layer. This allows researchers to explore
428:
398:
7619:
5692:
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
2761:
1176:
1125:
653:
7458:
7448:
7197:
6925:
6891:
6831:
6805:
6778:
6752:
6725:
6683:
6652:
6616:
6585:
6543:
6512:
6457:
6423:
6390:
6353:
6296:
6262:
6080:
5863:
5594:
5568:
5456:
5083:
5066:
755:
735:
673:
669:
545:
7286:
5687:
4641:
4611:
6477:"Realizing effective magnetic field for photons by controlling the phase of dynamic modulation"
4223:
Occupation of Landau levels in a magnetic field neglecting the spin splitting, showing how the
4193:
3983:
In order to get the number of occupied Landau levels, one defines the so-called filling factor
2683:
7566:
7531:
7438:
7362:
7016:
6998:
6917:
6717:
6709:
6644:
6577:
6569:
6504:
6449:
6376:
6345:
6288:
6049:
6024:
5941:
5855:
5723:
5715:
5666:
5643:
5538:
3831:
Density of states in a magnetic field, neglecting spin splitting. (a)The states in each range
3804:
3798:
2985:
649:
194:
39:
5131:
4938:
4671:
3886:
3866:
2572:
895:
7536:
7428:
7408:
7403:
7398:
7393:
7250:
7230:
7187:
7152:
7122:
7043:
6978:
6909:
6870:
6823:
6770:
6701:
6634:
6561:
6496:
6441:
6335:
6280:
6235:
6198:
6159:
6124:
6089:
6016:
5989:
5918:
5847:
5707:
5633:
5586:
5528:
759:
645:
184:
151:{\displaystyle R_{xy}={\frac {V_{\text{Hall}}}{I_{\text{channel}}}}={\frac {h}{e^{2}\nu }},}
6670:
Dutt, Avik; Lin, Qian; Yuan, Luqi; Minkov, Momchil; Xiao, Meng; Fan, Shanhui (2020-01-03).
4925:{\displaystyle \rho _{xy}={\frac {h}{\nu e^{2}}}\Rightarrow \sigma =\nu {\frac {e^{2}}{h}}}
4230:
3012:
2462:
2435:
2408:
2381:
1421:
is considered as an infinite well. Thus the solutions for the z direction are the energies
941:
7561:
7182:
7099:
7054:
6530:
Schine, Nathan; Ryou, Albert; Gromov, Andrey; Sommer, Ariel; Simon, Jonathan (June 2016).
5509:"The quantum Hall effect continues to reveal its secrets to mathematicians and physicists"
5070:
5054:
5035:
5007:
4973:
4706:
4576:
4330:
4140:
4120:
4099:
3986:
1395:
762:
709:
204:
7516:
6604:
3701:
2803:-plane if the vector potential was differently chosen one should find circular symmetry.
7039:
6974:
6905:
6866:
6819:
6766:
6697:
6630:
6557:
6492:
6437:
6331:
6276:
6231:
6194:
6155:
6120:
5914:
5703:
5629:
5582:
5524:
4596:
This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ called
2783:
2311:
7624:
7571:
7453:
7342:
7167:
5977:
5111:
5062:
5024:
5015:
5011:
4355:
4301:
3827:
3780:
3760:
3681:
3583:
3083:
3059:
3039:
2992:
2917:
2849:
2741:
2721:
2358:
2334:
1713:
1625:
1605:
1585:
1565:
921:
743:
55:
6671:
6605:"Haldane quantum Hall effect for light in a dynamically modulated array of resonators"
6411:
5835:
3523:{\textstyle n_{B}={\frac {\hbar \omega _{\rm {c}}}{2}}{\frac {m^{*}}{\pi \hbar ^{2}}}}
3277:
3217:
2378:
From the expression for the Landau levels one notices that the energy depends only on
754:. A link between exact quantization and gauge invariance was subsequently proposed by
524:
to nearly one part in a billion. It has allowed for the definition of a new practical
7652:
7496:
7478:
7463:
7443:
7347:
7316:
7147:
6827:
6774:
6729:
6461:
5867:
5598:
5104:
3754:
826:
713:
525:
6782:
6516:
6357:
6300:
4219:
7433:
7255:
7157:
7008:
6929:
6835:
6656:
6589:
4348:
840:
17:
6476:
5741:
2565:
As the field is turned on, the density of states collapses from the constant to a
6992:
5590:
3036:
with the 1st wave functions corresponding to an infinite well confinement in the
7553:
7271:
7240:
7220:
5962:
5805:
5638:
5613:
5466:
5050:
4224:
1510:{\textstyle \varepsilon _{z}={\frac {n_{z}^{2}\pi ^{2}\hbar ^{2}}{2m^{*}L^{2}}}}
689:
412:
391:
209:
171:
51:
43:
7047:
6743:
D. B. Kaplan (1992). "A Method for simulating chiral fermions on the lattice".
6445:
6203:
6178:
6020:
5533:
5508:
4845:
one finds out the quantization of the transverse resistivity and conductivity:
4227:
moves to maintain a constant density of electrons. The fields are in the ratio
7468:
7306:
7142:
6874:
6314:
Tsukazaki, A.; Ohtomo, A.; Kita, T.; Ohno, Y.; Ohno, H.; Kawasaki, M. (2007).
5471:
5096:
4086:{\displaystyle \nu ={\frac {n_{\rm {2D}}}{n_{B}}}={\frac {hn_{\rm {2D}}}{eB}}}
2566:
746:
and Gerhard Dorda, made the unexpected discovery that the Hall resistance was
705:
6713:
6672:"A single photonic cavity with two independent physical synthetic dimensions"
6648:
6573:
6531:
6508:
6500:
6453:
6239:
6008:
5922:
5719:
7526:
7352:
7137:
6705:
6639:
6340:
6315:
6284:
5851:
5100:
770:
7060:
6921:
6721:
6581:
6349:
6292:
5859:
5727:
5711:
5647:
5542:
3797:
when it is antiparallel. This fact called spin splitting implies that the
6428:
6267:
6163:
6093:
6075:
5476:
3579:
2708:
766:
704:
In a MOSFET, conduction electrons travel in a thin surface layer, and a "
6913:
6853:
D. R. Yennie (1987). "Integral quantum Hall effect for nonspecialists".
6810:
6757:
6565:
6128:
5901:
R. B. Laughlin (1981). "Quantized Hall conductivity in two dimensions".
5487:
Coulomb potential between two current loops embedded in a magnetic field
4541:
If the longitudinal resistivity is zero and transversal is finite, then
2934:
an integer, one gets that each parabolic potential is placed at a value
7576:
7543:
7521:
7501:
5099:
while looking at it as a single-electron Hall effect. While during the
4570:. Thus both the longitudinal conductivity and resistivity become zero.
2718:
Another feature is that the wave functions form parallel strips in the
739:
6983:
6958:
6048:. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg. p. 321.
5993:
5978:"Surface Protection and Selective Masking during Diffusion in Silicon"
5663:
Quantum Hall Effects: Recent Theoretical and Experimental Developments
7511:
7506:
7112:
6994:
Quantum Hall Effects - Field Theoretical Approach and Related Topics.
6076:"Focus: LandmarksâAccidental Discovery Leads to Calibration Standard"
5396:{\displaystyle R_{\text{Bohr}}(n)={\frac {U}{I}}=n{\frac {h}{e^{2}}}}
5003:
694:
2640:. At finite temperature, however, the Landau levels acquire a width
6688:
6621:
6548:
5573:
3530:
it is clear that each Landau level contains as many states as in a
3432:{\displaystyle n_{B}={\frac {m^{*}\omega _{\rm {c}}}{2\pi \hbar }}}
7486:
7107:
6896:
5034:
776:
6412:"Analogs of quantum-Hall-effect edge states in photonic crystals"
5406:
which for the Bohr atom is linear but not inverse in the integer
738:, working at the high magnetic field laboratory in Grenoble with
5742:"2022 CODATA Value: conventional value of von Klitzing constant"
4999:
4604:
3883:-function Landau level. (b) Landau levels have a non-zero width
2633:{\displaystyle \Delta \varepsilon _{xy}=\hbar \omega _{\rm {c}}}
773:
47:
7064:
1239:{\displaystyle \varepsilon =\varepsilon _{z}+\varepsilon _{xy}}
750:
quantized. For this finding, von Klitzing was awarded the 1985
7117:
5061:
shown in the figure. The vertical axis is the strength of the
6218:
D. J. Thouless (1983). "Quantization of particle transport".
5418:
Relativistic examples of the integer quantum Hall effect and
5103:
on a circular orbit the centrifugal force is balanced by the
1144:
is the canonical momentum, which is replaced by the operator
5045:
The integers that appear in the Hall effect are examples of
2780:. Since there is nothing special about any direction in the
2554:{\displaystyle n_{\rm {2D}}={\frac {m^{*}}{\pi \hbar ^{2}}}}
3568:{\displaystyle \Delta \varepsilon =\hbar \omega _{\rm {c}}}
4998:
The quantum Hall effect, in addition to being observed in
4732:
From the classical relation of the transverse resistivity
1710:. This is because the vector potential does not depend on
918:. Thus the SchrĂśdinger equation for a particle of charge
640:
The fractional quantum Hall effect is considered part of
397:
is roughly but not exactly equal to the filling factor of
6957:
Avron, Joseph E.; Osadchy, Daniel; Seiler, Ruedi (2003).
3671:{\textstyle \Delta E=\pm {\frac {1}{2}}g\mu _{\rm {B}}B}
2589:
functions, corresponding to the Landau levels separated
2375:, which are the wavefunctions of a harmonic oscillator.
769:
at temperatures as high as room temperature, and in the
6475:
Fang, Kejie; Yu, Zongfu; Fan, Shanhui (November 2012).
5810:
The NIST Reference on Constants, Units, and Uncertainty
5746:
The NIST Reference on Constants, Units, and Uncertainty
699:
electron behavior in a nearly ideal two-dimensional gas
4798:
4738:
4447:
3627:
3591:
3453:
2872:
2646:
2112:
2058:
1776:
1427:
6316:"Quantum Hall effect in polar oxide heterostructures"
5337:
5208:
5156:
5134:
5114:
5091:
Bohr atom interpretation of the von Klitzing constant
5014:
and coupling phases or on-site phases, an artificial
4976:
4941:
4854:
4709:
4674:
4644:
4614:
4579:
4547:
4420:
4381:
4358:
4333:
4304:
4265:
4233:
4196:
4163:
4143:
4123:
4102:
4012:
3989:
3947:
3909:
3889:
3869:
3837:
3807:
3783:
3763:
3730:
3704:
3684:
3540:
3376:
3109:
3086:
3062:
3042:
3015:
2995:
2940:
2920:
2852:
2846:
and applying the periodic boundary conditions in the
2812:
2786:
2764:
2744:
2724:
2686:
2595:
2575:
2503:
2465:
2438:
2411:
2384:
2361:
2337:
2314:
2266:
2165:
1843:
1736:
1716:
1648:
1628:
1608:
1588:
1568:
1523:
1398:
1255:
1203:
1179:
1150:
1128:
974:
944:
924:
898:
849:
454:
79:
5128:
is making Kepler revolutions with angular frequency
3777:
is taken when the spin is parallel to the field and
723:
of the Hall conductance was originally predicted by
697:
at Bell Labs 1960. This enabled physicists to study
7552:
7477:
7421:
7381:
7330:
7264:
7213:
7206:
7175:
7098:
811:Animated graph showing filling of Landau levels as
6939:, K. von Klitzing, PoincarĂŠ Seminar (Paris-2004).
5395:
5317:
5188:
5140:
5120:
4982:
4962:
4924:
4837:
4785:{\textstyle \rho _{xy}={\frac {B}{en_{\rm {2D}}}}}
4784:
4715:
4695:
4660:
4630:
4585:
4562:
4530:
4403:
4364:
4339:
4310:
4290:
4251:
4216:) and this is called the magnetic quantum limit.
4208:
4182:
4149:
4129:
4108:
4085:
3995:
3972:{\displaystyle \hbar \omega _{\rm {c}}>\Gamma }
3971:
3934:{\displaystyle \hbar \omega _{\rm {c}}<\Gamma }
3933:
3895:
3875:
3855:
3816:
3789:
3769:
3745:
3716:
3690:
3670:
3613:
3567:
3522:
3431:
3356:
3092:
3068:
3048:
3028:
3001:
2974:
2926:
2906:
2858:
2838:
2795:
2772:
2750:
2730:
2699:
2672:
2632:
2581:
2553:
2478:
2451:
2424:
2397:
2367:
2343:
2331:plane are given by the product of a plane wave in
2323:
2297:
2252:
2148:
2099:{\textstyle \omega _{\rm {c}}={\frac {eB}{m^{*}}}}
2098:
2041:
1826:
1758:
1722:
1702:
1634:
1614:
1594:
1574:
1554:
1509:
1413:
1381:
1238:
1185:
1165:
1136:
1111:
957:
930:
910:
884:
494:
150:
70:exhibits steps that take on the quantized values
6952:Quantum Hall Effect Observed at Room Temperature
6603:Minkov, Momchil; Savona, Vincenzo (2016-02-20).
5896:
5894:
5554:
5552:
4608:Longitudinal and transverse (Hall) resistivity,
4548:
4433:
3367:Thus the density of states per unit surface is
2673:{\textstyle \Gamma ={\frac {\hbar }{\tau _{i}}}}
1827:{\textstyle x_{k_{y}}={\frac {\hbar k_{y}}{eB}}}
6959:"A Topological Look at the Quantum Hall Effect"
3801:for each level is reduced by a half. Note that
3080:The number of states for each Landau Level and
2042:{\displaystyle \leftu(x)=\varepsilon _{xy}u(x)}
532:, based on the resistance quantum given by the
1562:and the wavefunctions are sinusoidal. For the
1382:{\displaystyle \leftu(z)=\varepsilon _{z}u(z)}
7076:
5049:. They are known in mathematics as the first
4838:{\textstyle n_{\rm {2D}}=\nu {\frac {eB}{h}}}
8:
6177:K. v. Klitzing; G. Dorda; M. Pepper (1980).
5189:{\displaystyle I={\frac {\omega e}{2\pi }},}
5023:. This creates an effect like they are in a
3076:-direction there are travelling plane waves.
2308:And the wavefunctions for the motion in the
1246:. The corresponding equations in z axis is:
6410:Raghu, S.; Haldane, F. D. M. (2008-09-23).
5010:, but through the manipulation of discrete
4703:(units are misleading). The filling factor
3903:in a more realistic picture and overlap if
2149:{\textstyle l_{B}^{2}={\frac {\hbar }{eB}}}
7210:
7166:
7083:
7069:
7061:
6069:
6067:
6065:
5806:"2022 CODATA Value: von Klitzing constant"
4524:
2106:is defined as the cyclotron frequency and
1703:{\displaystyle \psi _{xy}=u(x)e^{ik_{y}y}}
552:, a quantity of fundamental importance in
448:The quantization of the Hall conductance (
6982:
6895:
6809:
6756:
6687:
6638:
6620:
6547:
6427:
6339:
6266:
6202:
6013:Technical Memorandum of Bell Laboratories
5688:"Developments in the quantum Hall effect"
5637:
5572:
5532:
5385:
5376:
5360:
5342:
5336:
5303:
5287:
5269:
5241:
5219:
5207:
5163:
5155:
5133:
5113:
4975:
4952:
4946:
4940:
4911:
4905:
4884:
4871:
4859:
4853:
4820:
4804:
4803:
4797:
4769:
4768:
4755:
4743:
4737:
4708:
4685:
4679:
4673:
4649:
4643:
4619:
4613:
4578:
4546:
4507:
4492:
4472:
4454:
4442:
4427:
4419:
4392:
4380:
4357:
4332:
4303:
4278:
4264:
4232:
4195:
4168:
4162:
4142:
4122:
4101:
4062:
4061:
4051:
4040:
4026:
4025:
4019:
4011:
3988:
3956:
3955:
3946:
3918:
3917:
3908:
3888:
3868:
3846:
3845:
3836:
3806:
3782:
3762:
3736:
3735:
3729:
3703:
3683:
3658:
3657:
3640:
3626:
3601:
3590:
3558:
3557:
3539:
3511:
3497:
3491:
3478:
3477:
3467:
3458:
3452:
3408:
3407:
3397:
3390:
3381:
3375:
3330:
3329:
3319:
3312:
3286:
3285:
3276:
3250:
3225:
3216:
3207:
3202:
3186:
3174:
3161:
3143:
3132:
3123:
3114:
3108:
3085:
3061:
3041:
3020:
3014:
2994:
2963:
2958:
2945:
2939:
2919:
2893:
2879:
2871:
2851:
2830:
2817:
2811:
2785:
2765:
2763:
2743:
2723:
2691:
2685:
2662:
2653:
2645:
2623:
2622:
2603:
2594:
2574:
2542:
2528:
2522:
2509:
2508:
2502:
2470:
2464:
2443:
2437:
2416:
2410:
2389:
2383:
2360:
2336:
2313:
2271:
2265:
2235:
2226:
2210:
2209:
2191:
2186:
2170:
2164:
2131:
2122:
2117:
2111:
2088:
2074:
2064:
2063:
2057:
2018:
1988:
1978:
1968:
1963:
1944:
1938:
1937:
1927:
1913:
1901:
1887:
1881:
1872:
1858:
1852:
1842:
1807:
1797:
1786:
1781:
1775:
1750:
1739:
1738:
1735:
1715:
1689:
1681:
1653:
1647:
1627:
1622:-direction with some unknown function of
1607:
1587:
1567:
1528:
1522:
1498:
1488:
1473:
1463:
1453:
1448:
1441:
1432:
1426:
1397:
1361:
1313:
1299:
1293:
1284:
1270:
1264:
1254:
1227:
1214:
1202:
1178:
1149:
1129:
1127:
1029:
1019:
1008:
993:
980:
973:
949:
943:
923:
897:
850:
848:
483:
474:
459:
453:
407:is an integer or fraction, respectively.
133:
123:
112:
102:
96:
84:
78:
6009:"Silicon-Silicon Dioxide Surface Device"
4603:
4218:
3826:
3698:a factor which depends on the material (
2984:
843:the electromagnetic vector potential is
799:
5499:
3948:
3910:
3856:{\displaystyle \hbar \omega _{\rm {c}}}
3838:
3550:
3508:
3470:
3423:
3348:
3270:
2655:
2615:
2539:
2355:attenuated by the gaussian function in
2202:
2156:the magnetic length. The energies are:
2133:
1855:
1800:
1470:
1267:
1157:
6388:
5982:Journal of the Electrochemical Society
3941:. (c) The levels become distinct when
2907:{\textstyle k={\frac {2\pi }{L_{y}}}j}
7013:Perspectives in Quantum Hall Effects.
6532:"Synthetic Landau levels for photons"
4723:is displayed for the last 4 plateaus.
4291:{\displaystyle \nu =4,{\frac {8}{3}}}
885:{\displaystyle \mathbf {A} =(0,Bx,0)}
7:
7607:
6046:History of Semiconductor Engineering
2738:-direction spaced equally along the
712:by operating high-purity MOSFETs at
648:together with another symmetry (see
7631:
742:-based MOSFET samples developed by
6997:World Scientific, Singapore 2008,
5686:von Klitzing, Klaus (2005-09-15).
5665:(3rd ed.). World Scientific.
5228:
4808:
4773:
4404:{\displaystyle \sigma =\rho ^{-1}}
4066:
4030:
3966:
3957:
3928:
3919:
3890:
3847:
3808:
3737:
3659:
3628:
3578:Given the fact that electrons are
3559:
3541:
3479:
3409:
3331:
3287:
3167:
2647:
2624:
2596:
2513:
2211:
2065:
1939:
1894:
1884:
1306:
1296:
1160:
25:
5976:Frosch, C. J.; Derick, L (1957).
5936:Jainendra, Jain (19 April 2012).
5836:"In Praise of Exact Quantization"
3614:{\textstyle s=\pm {\frac {1}{2}}}
3447:Rewriting the last expression as
2839:{\displaystyle L_{x}\times L_{y}}
1392:To simplify things, the solution
27:Electromagnetic effect in physics
7630:
7618:
7606:
7595:
7594:
5000:two-dimensional electron systems
4563:{\displaystyle \det \rho \neq 0}
2975:{\displaystyle x_{k}=l_{B}^{2}k}
2766:
1130:
1020:
1009:
851:
48:two-dimensional electron systems
6937:25 years of Quantum Hall Effect
5065:and the horizontal axis is the
2989:Parabolic potentials along the
1166:{\displaystyle -i\hbar \nabla }
495:{\displaystyle G_{xy}=1/R_{xy}}
444:Electrical resistance standards
6074:Lindley, David (15 May 2015).
5940:. Cambridge University Press.
5442:Fractional quantum Hall effect
5354:
5348:
5281:
5275:
5253:
5247:
5231:
5225:
5079:fractional quantum Hall effect
5002:, can be observed in photons.
4893:
4598:Shubnikovâde Haas oscillations
4183:{\displaystyle n_{B}\propto B}
4096:In general the filling factor
3746:{\displaystyle \mu _{\rm {B}}}
2298:{\displaystyle n_{x}=1,2,3...}
2036:
2030:
2008:
2002:
1985:
1950:
1759:{\displaystyle {\hat {p}}_{y}}
1744:
1674:
1668:
1555:{\displaystyle n_{z}=1,2,3...}
1408:
1402:
1376:
1370:
1351:
1345:
1334:
1328:
1106:
1088:
1076:
1058:
1047:
1041:
879:
858:
665:fractional quantum Hall effect
424:fractional quantum Hall effect
1:
7193:Spontaneous symmetry breaking
5882:"Haldane nobel prize Lecture"
5447:Quantum anomalous Hall effect
2806:Given a sample of dimensions
6828:10.1016/0370-2693(93)90692-B
6775:10.1016/0370-2693(92)91112-M
6373:The physics of low-dimension
5591:10.1103/RevModPhys.89.025005
4994:Photonic quantum Hall effect
2773:{\displaystyle \mathbf {A} }
1186:{\displaystyle \varepsilon }
1137:{\displaystyle \mathbf {p} }
892:and the scalar potential is
215:can take on either integer (
5639:10.1103/PhysRevLett.61.2015
5053:and are closely related to
5047:topological quantum numbers
796:Integer quantum Hall effect
36:integer quantum Hall effect
7705:
7373:Spin gapless semiconductor
7282:Nearly free electron model
7048:10.1103/PhysRevB.76.085316
7015:Wiley-VCH, Weinheim 2004,
6446:10.1103/PhysRevA.78.033834
6204:10.1103/PhysRevLett.45.494
6021:10.1142/9789814503464_0076
5534:10.1038/d41586-020-02230-7
5031:Topological classification
4661:{\displaystyle \rho _{xy}}
4631:{\displaystyle \rho _{xx}}
2758:-axis, along the lines of
1730:and the momentum operator
824:
585:(the Planck constant) and
7590:
7322:Density functional theory
7297:electronic band structure
7164:
6950:Magnet Lab Press Release
6875:10.1103/RevModPhys.59.781
5612:F. D. M. Haldane (1988).
5561:Reviews of Modern Physics
5452:Quantum cellular automata
4209:{\displaystyle \nu <1}
2700:{\displaystyle \tau _{i}}
7664:Condensed matter physics
7492:Bogoliubov quasiparticle
7236:Quantum spin Hall effect
7128:BoseâEinstein condensate
7092:Condensed matter physics
6501:10.1038/nphoton.2012.236
6395:: CS1 maint: location (
6240:10.1103/PhysRevB.27.6083
5923:10.1103/PhysRevB.23.5632
5507:Editorial (2020-07-29).
5482:Quantum spin Hall effect
5437:Quantum Hall transitions
5422:arise in the context of
5420:quantum spin Hall effect
5006:do not possess inherent
4323:Longitudinal resistivity
4157:to be an integer. Since
3817:{\displaystyle \Delta E}
3724:for free electrons) and
678:ChernâSimons Lagrangians
658:ChernâSimons Lagrangians
433:quantum spin Hall effect
6706:10.1126/science.aaz3071
6640:10.1364/OPTICA.3.000200
6341:10.1126/science.1137430
6285:10.1126/science.1137201
5852:10.1126/science.1194123
5771:"26th CGPM Resolutions"
5661:Ezawa, Zyun F. (2013).
5618:Physical Review Letters
5141:{\displaystyle \omega }
4963:{\displaystyle e^{2}/h}
4696:{\displaystyle e^{2}/h}
3896:{\displaystyle \Gamma }
3876:{\displaystyle \delta }
2582:{\displaystyle \delta }
2432:. States with the same
911:{\displaystyle \phi =0}
554:quantum electrodynamics
550:fine-structure constant
5834:Franz, Marcel (2010).
5712:10.1098/rsta.2005.1640
5397:
5319:
5190:
5142:
5122:
5042:
5040:Hofstadter's butterfly
4984:
4964:
4926:
4839:
4786:
4728:Transverse resistivity
4724:
4717:
4697:
4662:
4632:
4587:
4564:
4532:
4405:
4375:With the conductivity
4366:
4341:
4319:
4312:
4292:
4253:
4210:
4184:
4151:
4137:and one cannot expect
4131:
4110:
4087:
3997:
3980:
3973:
3935:
3897:
3877:
3857:
3818:
3791:
3771:
3747:
3718:
3692:
3672:
3615:
3569:
3524:
3433:
3358:
3094:
3077:
3070:
3050:
3030:
3003:
2976:
2928:
2908:
2860:
2840:
2797:
2774:
2752:
2732:
2701:
2674:
2634:
2583:
2555:
2480:
2453:
2426:
2399:
2369:
2345:
2325:
2299:
2254:
2150:
2100:
2043:
1828:
1760:
1724:
1704:
1636:
1616:
1596:
1576:
1556:
1511:
1415:
1383:
1240:
1187:
1167:
1138:
1113:
959:
932:
912:
886:
817:
752:Nobel Prize in Physics
544:. This is named after
496:
152:
7368:Topological insulator
7302:Anderson localization
5963:"Quantum Hall Effect"
5398:
5320:
5191:
5143:
5123:
5038:
4985:
4965:
4927:
4840:
4787:
4718:
4698:
4663:
4633:
4607:
4588:
4565:
4533:
4406:
4367:
4342:
4313:
4293:
4254:
4252:{\displaystyle 2:3:4}
4222:
4211:
4185:
4152:
4132:
4111:
4088:
3998:
3974:
3936:
3898:
3878:
3858:
3830:
3819:
3792:
3772:
3748:
3719:
3693:
3673:
3616:
3570:
3525:
3434:
3359:
3095:
3071:
3051:
3031:
3029:{\displaystyle x_{k}}
3004:
2988:
2977:
2929:
2909:
2861:
2841:
2798:
2775:
2753:
2733:
2702:
2675:
2635:
2584:
2556:
2481:
2479:{\displaystyle k_{y}}
2454:
2452:{\displaystyle n_{x}}
2427:
2425:{\displaystyle k_{y}}
2400:
2398:{\displaystyle n_{x}}
2370:
2346:
2326:
2300:
2255:
2151:
2101:
2044:
1829:
1770:equation centered at
1761:
1725:
1705:
1637:
1617:
1597:
1577:
1557:
1512:
1416:
1384:
1241:
1193:is the total energy.
1188:
1168:
1139:
1114:
960:
958:{\displaystyle m^{*}}
933:
913:
887:
810:
534:von Klitzing constant
530:electrical resistance
497:
417:Anderson localization
153:
46:which is observed in
7246:AharonovâBohm effect
7133:Fermionic condensate
6164:10.1143/JPSJ.44.1839
6094:10.1103/physics.8.46
5424:lattice gauge theory
5414:Relativistic analogs
5335:
5206:
5154:
5132:
5112:
5059:Hofstadter butterfly
4983:{\displaystyle \nu }
4974:
4939:
4852:
4796:
4736:
4716:{\displaystyle \nu }
4707:
4672:
4642:
4612:
4586:{\displaystyle \nu }
4577:
4545:
4418:
4379:
4356:
4340:{\displaystyle \nu }
4331:
4302:
4263:
4231:
4194:
4161:
4150:{\displaystyle \nu }
4141:
4130:{\displaystyle \nu }
4121:
4109:{\displaystyle \nu }
4100:
4010:
3996:{\displaystyle \nu }
3987:
3945:
3907:
3887:
3867:
3863:are squeezed into a
3835:
3805:
3781:
3761:
3728:
3702:
3682:
3625:
3589:
3538:
3451:
3374:
3107:
3084:
3060:
3040:
3013:
2993:
2938:
2918:
2870:
2850:
2810:
2784:
2762:
2742:
2722:
2684:
2644:
2593:
2573:
2569:, a series of Dirac
2501:
2463:
2436:
2409:
2382:
2359:
2335:
2312:
2264:
2163:
2110:
2056:
1841:
1774:
1734:
1714:
1646:
1626:
1606:
1586:
1566:
1521:
1425:
1414:{\displaystyle V(z)}
1396:
1253:
1201:
1177:
1148:
1126:
972:
942:
922:
896:
847:
837:SchrĂśdinger equation
729:Gakushuin University
452:
77:
58:, in which the Hall
7669:Quantum electronics
7637:Physics WikiProject
7312:tight binding model
7292:Fermi liquid theory
7277:Free electron model
7226:Quantum Hall effect
7207:Electrons in solids
7040:2007PhRvB..76h5316B
6975:2003PhT....56h..38A
6914:10.1038/nature06843
6906:2008Natur.452..970H
6867:1987RvMP...59..781Y
6820:1993PhLB..301..219G
6767:1992PhLB..288..342K
6698:2020Sci...367...59D
6631:2016Optic...3..200M
6566:10.1038/nature17943
6558:2016Natur.534..671S
6493:2012NaPho...6..782F
6438:2008PhRvA..78c3834R
6332:2007Sci...315.1388T
6277:2007Sci...315.1379N
6232:1983PhRvB..27.6083T
6195:1980PhRvL..45..494K
6156:1978JPSJ...44.1839W
6129:10.1143/JPSJ.39.279
6121:1975JPSJ...39..279A
5915:1981PhRvB..23.5632L
5704:2005RSPTA.363.2203V
5698:(1834): 2203â2219.
5630:1988PhRvL..61.2015H
5583:2017RvMP...89b5005H
5525:2020Natur.583..659.
5462:Conductance Quantum
4347:is an integer, the
3717:{\displaystyle g=2}
3212:
2968:
2353:Hermite polynomials
2127:
1973:
1949:
1768:harmonic oscillator
1458:
965:in this system is:
938:and effective mass
725:University of Tokyo
32:quantum Hall effect
18:Quantum Hall Effect
7684:Mesoscopic physics
7198:Critical phenomena
6044:Lojek, Bo (2007).
6007:KAHNG, D. (1961).
5938:Composite Fermions
5457:Composite fermions
5393:
5315:
5186:
5138:
5118:
5084:composite fermions
5067:chemical potential
5043:
5012:optical resonators
4980:
4960:
4922:
4835:
4782:
4725:
4713:
4693:
4658:
4628:
4583:
4560:
4528:
4518:
4401:
4362:
4337:
4320:
4308:
4288:
4249:
4206:
4180:
4147:
4127:
4106:
4083:
3993:
3981:
3969:
3931:
3893:
3873:
3853:
3814:
3787:
3767:
3743:
3714:
3688:
3668:
3611:
3565:
3520:
3429:
3354:
3310:
3248:
3198:
3090:
3078:
3066:
3056:direction. In the
3046:
3026:
3009:-axis centered at
2999:
2972:
2954:
2924:
2904:
2856:
2836:
2796:{\displaystyle xy}
2793:
2770:
2748:
2728:
2697:
2670:
2630:
2579:
2551:
2476:
2449:
2422:
2395:
2365:
2341:
2324:{\displaystyle xy}
2321:
2295:
2250:
2146:
2113:
2096:
2039:
1959:
1933:
1824:
1756:
1720:
1700:
1632:
1612:
1592:
1572:
1552:
1507:
1444:
1411:
1379:
1236:
1183:
1163:
1134:
1109:
955:
928:
908:
882:
818:
736:Klaus von Klitzing
674:composite fermions
670:composite fermions
642:exact quantization
561:conventional value
546:Klaus von Klitzing
492:
148:
7646:
7645:
7532:Exciton-polariton
7417:
7416:
7389:Thermoelectricity
7021:978-0-471-11216-7
7007:Sankar D. Sarma,
7003:978-981-270-032-2
6984:10.1063/1.1611351
6890:(7190): 970â974.
6542:(7609): 671â675.
6416:Physical Review A
6326:(5817): 1388â91.
6226:(10): 6083â6087.
6144:J. Phys. Soc. Jpn
6109:J. Phys. Soc. Jpn
6055:978-3-540-34258-8
6030:978-981-02-0209-5
5994:10.1149/1.2428650
5909:(10): 5632â5633.
5846:(5992): 639â640.
5672:978-981-4360-75-3
5624:(18): 2015â2018.
5567:(25005): 025005.
5391:
5368:
5345:
5313:
5272:
5244:
5222:
5181:
5121:{\displaystyle e}
4920:
4891:
4833:
4792:and substituting
4780:
4440:
4365:{\displaystyle 0}
4311:{\displaystyle 2}
4286:
4081:
4046:
3799:density of states
3790:{\displaystyle -}
3770:{\displaystyle +}
3691:{\displaystyle g}
3648:
3609:
3518:
3489:
3427:
3352:
3274:
3214:
3181:
3138:
3093:{\displaystyle k}
3069:{\displaystyle y}
3049:{\displaystyle z}
3002:{\displaystyle x}
2927:{\displaystyle j}
2899:
2859:{\displaystyle y}
2751:{\displaystyle x}
2731:{\displaystyle y}
2668:
2549:
2490:Density of states
2368:{\displaystyle x}
2344:{\displaystyle y}
2243:
2144:
2094:
1921:
1908:
1879:
1822:
1747:
1723:{\displaystyle y}
1635:{\displaystyle x}
1615:{\displaystyle y}
1595:{\displaystyle y}
1575:{\displaystyle x}
1505:
1320:
1291:
1000:
931:{\displaystyle q}
808:
559:In 1990, a fixed
222:) or fractional (
195:elementary charge
143:
118:
115:
105:
50:subjected to low
16:(Redirected from
7696:
7634:
7633:
7622:
7610:
7609:
7598:
7597:
7537:Phonon polariton
7429:Amorphous magnet
7409:Electrostriction
7404:Flexoelectricity
7399:Ferroelectricity
7394:Piezoelectricity
7251:Josephson effect
7231:Spin Hall effect
7211:
7188:Phase transition
7170:
7153:Luttinger liquid
7100:States of matter
7085:
7078:
7071:
7062:
7051:
6988:
6986:
6933:
6899:
6878:
6840:
6839:
6813:
6804:(2â3): 219â223.
6793:
6787:
6786:
6760:
6751:(3â4): 342â347.
6740:
6734:
6733:
6691:
6667:
6661:
6660:
6642:
6624:
6600:
6594:
6593:
6551:
6527:
6521:
6520:
6481:Nature Photonics
6472:
6466:
6465:
6431:
6429:cond-mat/0602501
6407:
6401:
6400:
6394:
6386:
6368:
6362:
6361:
6343:
6311:
6305:
6304:
6270:
6268:cond-mat/0702408
6250:
6244:
6243:
6215:
6209:
6208:
6206:
6174:
6168:
6167:
6139:
6133:
6132:
6104:
6098:
6097:
6071:
6060:
6059:
6041:
6035:
6034:
6004:
5998:
5997:
5973:
5967:
5966:
5958:
5952:
5951:
5933:
5927:
5926:
5898:
5889:
5888:
5886:
5878:
5872:
5871:
5831:
5825:
5824:
5822:
5821:
5802:
5796:
5795:
5793:
5792:
5786:
5780:. Archived from
5775:
5767:
5761:
5760:
5758:
5757:
5738:
5732:
5731:
5683:
5677:
5676:
5658:
5652:
5651:
5641:
5609:
5603:
5602:
5576:
5556:
5547:
5546:
5536:
5504:
5402:
5400:
5399:
5394:
5392:
5390:
5389:
5377:
5369:
5361:
5347:
5346:
5343:
5324:
5322:
5321:
5316:
5314:
5312:
5308:
5307:
5288:
5274:
5273:
5270:
5246:
5245:
5242:
5224:
5223:
5220:
5195:
5193:
5192:
5187:
5182:
5180:
5172:
5164:
5147:
5145:
5144:
5139:
5127:
5125:
5124:
5119:
5108:electron charge
5101:cyclotron motion
5021:angular momentum
4989:
4987:
4986:
4981:
4969:
4967:
4966:
4961:
4956:
4951:
4950:
4931:
4929:
4928:
4923:
4921:
4916:
4915:
4906:
4892:
4890:
4889:
4888:
4872:
4867:
4866:
4844:
4842:
4841:
4836:
4834:
4829:
4821:
4813:
4812:
4811:
4791:
4789:
4788:
4783:
4781:
4779:
4778:
4777:
4776:
4756:
4751:
4750:
4722:
4720:
4719:
4714:
4702:
4700:
4699:
4694:
4689:
4684:
4683:
4667:
4665:
4664:
4659:
4657:
4656:
4637:
4635:
4634:
4629:
4627:
4626:
4592:
4590:
4589:
4584:
4569:
4567:
4566:
4561:
4537:
4535:
4534:
4529:
4523:
4522:
4515:
4514:
4500:
4499:
4480:
4479:
4462:
4461:
4441:
4439:
4428:
4410:
4408:
4407:
4402:
4400:
4399:
4371:
4369:
4368:
4363:
4346:
4344:
4343:
4338:
4317:
4315:
4314:
4309:
4297:
4295:
4294:
4289:
4287:
4279:
4258:
4256:
4255:
4250:
4215:
4213:
4212:
4207:
4189:
4187:
4186:
4181:
4173:
4172:
4156:
4154:
4153:
4148:
4136:
4134:
4133:
4128:
4115:
4113:
4112:
4107:
4092:
4090:
4089:
4084:
4082:
4080:
4072:
4071:
4070:
4069:
4052:
4047:
4045:
4044:
4035:
4034:
4033:
4020:
4002:
4000:
3999:
3994:
3978:
3976:
3975:
3970:
3962:
3961:
3960:
3940:
3938:
3937:
3932:
3924:
3923:
3922:
3902:
3900:
3899:
3894:
3882:
3880:
3879:
3874:
3862:
3860:
3859:
3854:
3852:
3851:
3850:
3823:
3821:
3820:
3815:
3796:
3794:
3793:
3788:
3776:
3774:
3773:
3768:
3752:
3750:
3749:
3744:
3742:
3741:
3740:
3723:
3721:
3720:
3715:
3697:
3695:
3694:
3689:
3677:
3675:
3674:
3669:
3664:
3663:
3662:
3649:
3641:
3620:
3618:
3617:
3612:
3610:
3602:
3574:
3572:
3571:
3566:
3564:
3563:
3562:
3529:
3527:
3526:
3521:
3519:
3517:
3516:
3515:
3502:
3501:
3492:
3490:
3485:
3484:
3483:
3482:
3468:
3463:
3462:
3438:
3436:
3435:
3430:
3428:
3426:
3415:
3414:
3413:
3412:
3402:
3401:
3391:
3386:
3385:
3363:
3361:
3360:
3355:
3353:
3351:
3340:
3336:
3335:
3334:
3324:
3323:
3313:
3311:
3308:
3307:
3306:
3303:
3297:
3294:
3292:
3291:
3290:
3279:
3275:
3273:
3262:
3251:
3249:
3246:
3245:
3244:
3241:
3235:
3232:
3230:
3229:
3219:
3215:
3213:
3211:
3206:
3187:
3182:
3180:
3179:
3178:
3166:
3165:
3152:
3144:
3139:
3137:
3136:
3124:
3119:
3118:
3099:
3097:
3096:
3091:
3075:
3073:
3072:
3067:
3055:
3053:
3052:
3047:
3035:
3033:
3032:
3027:
3025:
3024:
3008:
3006:
3005:
3000:
2981:
2979:
2978:
2973:
2967:
2962:
2950:
2949:
2933:
2931:
2930:
2925:
2913:
2911:
2910:
2905:
2900:
2898:
2897:
2888:
2880:
2865:
2863:
2862:
2857:
2845:
2843:
2842:
2837:
2835:
2834:
2822:
2821:
2802:
2800:
2799:
2794:
2779:
2777:
2776:
2771:
2769:
2757:
2755:
2754:
2749:
2737:
2735:
2734:
2729:
2706:
2704:
2703:
2698:
2696:
2695:
2679:
2677:
2676:
2671:
2669:
2667:
2666:
2654:
2639:
2637:
2636:
2631:
2629:
2628:
2627:
2611:
2610:
2588:
2586:
2585:
2580:
2560:
2558:
2557:
2552:
2550:
2548:
2547:
2546:
2533:
2532:
2523:
2518:
2517:
2516:
2486:are degenerate.
2485:
2483:
2482:
2477:
2475:
2474:
2458:
2456:
2455:
2450:
2448:
2447:
2431:
2429:
2428:
2423:
2421:
2420:
2404:
2402:
2401:
2396:
2394:
2393:
2374:
2372:
2371:
2366:
2350:
2348:
2347:
2342:
2330:
2328:
2327:
2322:
2304:
2302:
2301:
2296:
2276:
2275:
2259:
2257:
2256:
2251:
2249:
2245:
2244:
2236:
2231:
2230:
2216:
2215:
2214:
2198:
2197:
2196:
2195:
2178:
2177:
2155:
2153:
2152:
2147:
2145:
2143:
2132:
2126:
2121:
2105:
2103:
2102:
2097:
2095:
2093:
2092:
2083:
2075:
2070:
2069:
2068:
2048:
2046:
2045:
2040:
2026:
2025:
1998:
1994:
1993:
1992:
1983:
1982:
1972:
1967:
1948:
1943:
1942:
1932:
1931:
1922:
1914:
1909:
1907:
1906:
1905:
1892:
1891:
1882:
1880:
1878:
1877:
1876:
1863:
1862:
1853:
1833:
1831:
1830:
1825:
1823:
1821:
1813:
1812:
1811:
1798:
1793:
1792:
1791:
1790:
1765:
1763:
1762:
1757:
1755:
1754:
1749:
1748:
1740:
1729:
1727:
1726:
1721:
1709:
1707:
1706:
1701:
1699:
1698:
1694:
1693:
1661:
1660:
1641:
1639:
1638:
1633:
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1581:
1579:
1578:
1573:
1561:
1559:
1558:
1553:
1533:
1532:
1516:
1514:
1513:
1508:
1506:
1504:
1503:
1502:
1493:
1492:
1479:
1478:
1477:
1468:
1467:
1457:
1452:
1442:
1437:
1436:
1420:
1418:
1417:
1412:
1388:
1386:
1385:
1380:
1366:
1365:
1341:
1337:
1321:
1319:
1318:
1317:
1304:
1303:
1294:
1292:
1290:
1289:
1288:
1275:
1274:
1265:
1245:
1243:
1242:
1237:
1235:
1234:
1219:
1218:
1192:
1190:
1189:
1184:
1172:
1170:
1169:
1164:
1143:
1141:
1140:
1135:
1133:
1118:
1116:
1115:
1110:
1054:
1050:
1034:
1033:
1028:
1024:
1023:
1012:
1001:
999:
998:
997:
981:
964:
962:
961:
956:
954:
953:
937:
935:
934:
929:
917:
915:
914:
909:
891:
889:
888:
883:
854:
809:
763:heterostructures
760:gallium arsenide
676:and Non Abelian
646:gauge invariance
631:
630:
628:
624:
618:
616:
615:
610:
607:
590:
584:
578:
577:
575:
543:
523:
522:
520:
519:
514:
511:
501:
499:
498:
493:
491:
490:
478:
467:
466:
406:
396:
389:) values. Here,
388:
386:
384:
383:
380:
377:
370:
368:
367:
364:
361:
354:
352:
351:
348:
345:
338:
336:
335:
332:
329:
322:
320:
319:
316:
313:
306:
304:
303:
300:
297:
290:
288:
287:
284:
281:
274:
272:
271:
268:
265:
258:
256:
255:
252:
249:
242:
240:
239:
236:
233:
221:
214:
202:
192:
182:
169:
157:
155:
154:
149:
144:
142:
138:
137:
124:
119:
117:
116:
113:
107:
106:
103:
97:
92:
91:
69:
21:
7704:
7703:
7699:
7698:
7697:
7695:
7694:
7693:
7689:1980 in science
7649:
7648:
7647:
7642:
7586:
7567:Granular matter
7562:Amorphous solid
7548:
7473:
7459:Antiferromagnet
7449:Superparamagnet
7422:Magnetic phases
7413:
7377:
7326:
7287:Bloch's theorem
7260:
7202:
7183:Order parameter
7176:Phase phenomena
7171:
7162:
7094:
7089:
7025:
6991:Zyun F. Ezawa:
6956:
6881:
6852:
6849:
6847:Further reading
6844:
6843:
6811:hep-lat/9209003
6798:Physics Letters
6795:
6794:
6790:
6758:hep-lat/9206013
6745:Physics Letters
6742:
6741:
6737:
6682:(6473): 59â64.
6669:
6668:
6664:
6602:
6601:
6597:
6529:
6528:
6524:
6487:(11): 782â787.
6474:
6473:
6469:
6409:
6408:
6404:
6387:
6383:
6370:
6369:
6365:
6313:
6312:
6308:
6252:
6251:
6247:
6217:
6216:
6212:
6183:Phys. Rev. Lett
6176:
6175:
6171:
6141:
6140:
6136:
6106:
6105:
6101:
6073:
6072:
6063:
6056:
6043:
6042:
6038:
6031:
6006:
6005:
6001:
5975:
5974:
5970:
5960:
5959:
5955:
5948:
5935:
5934:
5930:
5900:
5899:
5892:
5884:
5880:
5879:
5875:
5833:
5832:
5828:
5819:
5817:
5804:
5803:
5799:
5790:
5788:
5784:
5773:
5769:
5768:
5764:
5755:
5753:
5740:
5739:
5735:
5685:
5684:
5680:
5673:
5660:
5659:
5655:
5611:
5610:
5606:
5558:
5557:
5550:
5506:
5505:
5501:
5496:
5491:
5432:
5416:
5381:
5338:
5333:
5332:
5299:
5292:
5265:
5237:
5215:
5204:
5203:
5173:
5165:
5152:
5151:
5130:
5129:
5110:
5109:
5093:
5071:self-similarity
5033:
5008:electric charge
4996:
4972:
4971:
4942:
4937:
4936:
4907:
4880:
4876:
4855:
4850:
4849:
4822:
4799:
4794:
4793:
4764:
4760:
4739:
4734:
4733:
4730:
4705:
4704:
4675:
4670:
4669:
4645:
4640:
4639:
4615:
4610:
4609:
4575:
4574:
4543:
4542:
4517:
4516:
4503:
4501:
4488:
4482:
4481:
4468:
4463:
4450:
4443:
4432:
4416:
4415:
4388:
4377:
4376:
4354:
4353:
4329:
4328:
4325:
4300:
4299:
4261:
4260:
4229:
4228:
4192:
4191:
4164:
4159:
4158:
4139:
4138:
4119:
4118:
4098:
4097:
4073:
4057:
4053:
4036:
4021:
4008:
4007:
3985:
3984:
3951:
3943:
3942:
3913:
3905:
3904:
3885:
3884:
3865:
3864:
3841:
3833:
3832:
3803:
3802:
3779:
3778:
3759:
3758:
3731:
3726:
3725:
3700:
3699:
3680:
3679:
3653:
3623:
3622:
3587:
3586:
3553:
3536:
3535:
3507:
3503:
3493:
3473:
3469:
3454:
3449:
3448:
3416:
3403:
3393:
3392:
3377:
3372:
3371:
3341:
3325:
3315:
3314:
3309:
3304:
3302:
3295:
3293:
3281:
3263:
3252:
3247:
3242:
3240:
3233:
3231:
3221:
3191:
3170:
3157:
3153:
3145:
3128:
3110:
3105:
3104:
3082:
3081:
3058:
3057:
3038:
3037:
3016:
3011:
3010:
2991:
2990:
2941:
2936:
2935:
2916:
2915:
2889:
2881:
2868:
2867:
2848:
2847:
2826:
2813:
2808:
2807:
2782:
2781:
2760:
2759:
2740:
2739:
2720:
2719:
2687:
2682:
2681:
2658:
2642:
2641:
2618:
2599:
2591:
2590:
2571:
2570:
2538:
2534:
2524:
2504:
2499:
2498:
2492:
2466:
2461:
2460:
2439:
2434:
2433:
2412:
2407:
2406:
2385:
2380:
2379:
2357:
2356:
2333:
2332:
2310:
2309:
2267:
2262:
2261:
2222:
2221:
2217:
2205:
2187:
2182:
2166:
2161:
2160:
2136:
2108:
2107:
2084:
2076:
2059:
2054:
2053:
2014:
1984:
1974:
1923:
1897:
1893:
1883:
1868:
1864:
1854:
1848:
1844:
1839:
1838:
1814:
1803:
1799:
1782:
1777:
1772:
1771:
1737:
1732:
1731:
1712:
1711:
1685:
1677:
1649:
1644:
1643:
1624:
1623:
1604:
1603:
1584:
1583:
1564:
1563:
1524:
1519:
1518:
1494:
1484:
1480:
1469:
1459:
1443:
1428:
1423:
1422:
1394:
1393:
1357:
1309:
1305:
1295:
1280:
1276:
1266:
1260:
1256:
1251:
1250:
1223:
1210:
1199:
1198:
1175:
1174:
1146:
1145:
1124:
1123:
1007:
1003:
1002:
989:
985:
979:
975:
970:
969:
945:
940:
939:
920:
919:
894:
893:
845:
844:
829:
823:
800:
798:
791:
784:
756:Robert Laughlin
710:quantum effects
686:
638:
636:Research status
626:
622:
620:
611:
608:
603:
602:
600:
598:
592:
586:
580:
573:
571:
569:
563:
542:
536:
515:
512:
507:
506:
504:
503:
479:
455:
450:
449:
446:
441:
402:
390:
381:
378:
375:
374:
372:
365:
362:
359:
358:
356:
349:
346:
343:
342:
340:
333:
330:
327:
326:
324:
317:
314:
311:
310:
308:
301:
298:
295:
294:
292:
285:
282:
279:
278:
276:
269:
266:
263:
262:
260:
253:
250:
247:
246:
244:
237:
234:
231:
230:
228:
223:
216:
208:
205:Planck constant
198:
188:
183:is the channel
181:
175:
168:
162:
129:
128:
108:
98:
80:
75:
74:
68:
62:
56:magnetic fields
42:version of the
28:
23:
22:
15:
12:
11:
5:
7702:
7700:
7692:
7691:
7686:
7681:
7679:Quantum phases
7676:
7671:
7666:
7661:
7651:
7650:
7644:
7643:
7641:
7640:
7628:
7625:Physics Portal
7616:
7604:
7591:
7588:
7587:
7585:
7584:
7579:
7574:
7572:Liquid crystal
7569:
7564:
7558:
7556:
7550:
7549:
7547:
7546:
7541:
7540:
7539:
7534:
7524:
7519:
7514:
7509:
7504:
7499:
7494:
7489:
7483:
7481:
7479:Quasiparticles
7475:
7474:
7472:
7471:
7466:
7461:
7456:
7451:
7446:
7441:
7439:Superdiamagnet
7436:
7431:
7425:
7423:
7419:
7418:
7415:
7414:
7412:
7411:
7406:
7401:
7396:
7391:
7385:
7383:
7379:
7378:
7376:
7375:
7370:
7365:
7363:Superconductor
7360:
7355:
7350:
7345:
7343:Mott insulator
7340:
7334:
7332:
7328:
7327:
7325:
7324:
7319:
7314:
7309:
7304:
7299:
7294:
7289:
7284:
7279:
7274:
7268:
7266:
7262:
7261:
7259:
7258:
7253:
7248:
7243:
7238:
7233:
7228:
7223:
7217:
7215:
7208:
7204:
7203:
7201:
7200:
7195:
7190:
7185:
7179:
7177:
7173:
7172:
7165:
7163:
7161:
7160:
7155:
7150:
7145:
7140:
7135:
7130:
7125:
7120:
7115:
7110:
7104:
7102:
7096:
7095:
7090:
7088:
7087:
7080:
7073:
7065:
7059:
7058:
7052:
7023:
7005:
6989:
6954:
6948:
6934:
6879:
6861:(3): 781â824.
6855:Rev. Mod. Phys
6848:
6845:
6842:
6841:
6788:
6735:
6662:
6595:
6522:
6467:
6402:
6381:
6363:
6306:
6261:(5817): 1379.
6245:
6210:
6189:(6): 494â497.
6169:
6134:
6115:(2): 279â288.
6099:
6061:
6054:
6036:
6029:
5999:
5968:
5953:
5947:978-1107404250
5946:
5928:
5890:
5873:
5826:
5797:
5762:
5733:
5678:
5671:
5653:
5604:
5548:
5498:
5497:
5495:
5492:
5490:
5489:
5484:
5479:
5474:
5469:
5464:
5459:
5454:
5449:
5444:
5439:
5433:
5431:
5428:
5415:
5412:
5404:
5403:
5388:
5384:
5380:
5375:
5372:
5367:
5364:
5359:
5356:
5353:
5350:
5341:
5326:
5325:
5311:
5306:
5302:
5298:
5295:
5291:
5286:
5283:
5280:
5277:
5268:
5264:
5261:
5258:
5255:
5252:
5249:
5240:
5236:
5233:
5230:
5227:
5218:
5214:
5211:
5197:
5196:
5185:
5179:
5176:
5171:
5168:
5162:
5159:
5137:
5117:
5092:
5089:
5063:magnetic field
5032:
5029:
5025:magnetic field
5016:magnetic field
4995:
4992:
4979:
4959:
4955:
4949:
4945:
4933:
4932:
4919:
4914:
4910:
4904:
4901:
4898:
4895:
4887:
4883:
4879:
4875:
4870:
4865:
4862:
4858:
4832:
4828:
4825:
4819:
4816:
4810:
4807:
4802:
4775:
4772:
4767:
4763:
4759:
4754:
4749:
4746:
4742:
4729:
4726:
4712:
4692:
4688:
4682:
4678:
4655:
4652:
4648:
4625:
4622:
4618:
4582:
4573:Instead, when
4559:
4556:
4553:
4550:
4539:
4538:
4527:
4521:
4513:
4510:
4506:
4502:
4498:
4495:
4491:
4487:
4484:
4483:
4478:
4475:
4471:
4467:
4464:
4460:
4457:
4453:
4449:
4448:
4446:
4438:
4435:
4431:
4426:
4423:
4398:
4395:
4391:
4387:
4384:
4361:
4352:approximately
4336:
4324:
4321:
4307:
4285:
4282:
4277:
4274:
4271:
4268:
4248:
4245:
4242:
4239:
4236:
4205:
4202:
4199:
4179:
4176:
4171:
4167:
4146:
4126:
4105:
4094:
4093:
4079:
4076:
4068:
4065:
4060:
4056:
4050:
4043:
4039:
4032:
4029:
4024:
4018:
4015:
3992:
3968:
3965:
3959:
3954:
3950:
3930:
3927:
3921:
3916:
3912:
3892:
3872:
3849:
3844:
3840:
3813:
3810:
3786:
3766:
3739:
3734:
3713:
3710:
3707:
3687:
3667:
3661:
3656:
3652:
3647:
3644:
3639:
3636:
3633:
3630:
3608:
3605:
3600:
3597:
3594:
3561:
3556:
3552:
3549:
3546:
3543:
3514:
3510:
3506:
3500:
3496:
3488:
3481:
3476:
3472:
3466:
3461:
3457:
3441:
3440:
3425:
3422:
3419:
3411:
3406:
3400:
3396:
3389:
3384:
3380:
3365:
3364:
3350:
3347:
3344:
3339:
3333:
3328:
3322:
3318:
3305:
3301:
3298:
3296:
3289:
3284:
3280:
3278:
3272:
3269:
3266:
3261:
3258:
3255:
3243:
3239:
3236:
3234:
3228:
3224:
3220:
3218:
3210:
3205:
3201:
3197:
3194:
3190:
3185:
3177:
3173:
3169:
3164:
3160:
3156:
3151:
3148:
3142:
3135:
3131:
3127:
3122:
3117:
3113:
3089:
3065:
3045:
3023:
3019:
2998:
2971:
2966:
2961:
2957:
2953:
2948:
2944:
2923:
2903:
2896:
2892:
2887:
2884:
2878:
2875:
2855:
2833:
2829:
2825:
2820:
2816:
2792:
2789:
2768:
2747:
2727:
2694:
2690:
2665:
2661:
2657:
2652:
2649:
2626:
2621:
2617:
2614:
2609:
2606:
2602:
2598:
2578:
2563:
2562:
2545:
2541:
2537:
2531:
2527:
2521:
2515:
2512:
2507:
2491:
2488:
2473:
2469:
2459:but different
2446:
2442:
2419:
2415:
2392:
2388:
2364:
2340:
2320:
2317:
2306:
2305:
2294:
2291:
2288:
2285:
2282:
2279:
2274:
2270:
2260:,
2248:
2242:
2239:
2234:
2229:
2225:
2220:
2213:
2208:
2204:
2201:
2194:
2190:
2185:
2181:
2176:
2173:
2169:
2142:
2139:
2135:
2130:
2125:
2120:
2116:
2091:
2087:
2082:
2079:
2073:
2067:
2062:
2050:
2049:
2038:
2035:
2032:
2029:
2024:
2021:
2017:
2013:
2010:
2007:
2004:
2001:
1997:
1991:
1987:
1981:
1977:
1971:
1966:
1962:
1958:
1955:
1952:
1947:
1941:
1936:
1930:
1926:
1920:
1917:
1912:
1904:
1900:
1896:
1890:
1886:
1875:
1871:
1867:
1861:
1857:
1851:
1847:
1820:
1817:
1810:
1806:
1802:
1796:
1789:
1785:
1780:
1753:
1746:
1743:
1719:
1697:
1692:
1688:
1684:
1680:
1676:
1673:
1670:
1667:
1664:
1659:
1656:
1652:
1631:
1611:
1591:
1571:
1551:
1548:
1545:
1542:
1539:
1536:
1531:
1527:
1501:
1497:
1491:
1487:
1483:
1476:
1472:
1466:
1462:
1456:
1451:
1447:
1440:
1435:
1431:
1410:
1407:
1404:
1401:
1390:
1389:
1378:
1375:
1372:
1369:
1364:
1360:
1356:
1353:
1350:
1347:
1344:
1340:
1336:
1333:
1330:
1327:
1324:
1316:
1312:
1308:
1302:
1298:
1287:
1283:
1279:
1273:
1269:
1263:
1259:
1233:
1230:
1226:
1222:
1217:
1213:
1209:
1206:
1182:
1162:
1159:
1156:
1153:
1132:
1120:
1119:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1053:
1049:
1046:
1043:
1040:
1037:
1032:
1027:
1022:
1018:
1015:
1011:
1006:
996:
992:
988:
984:
978:
952:
948:
927:
907:
904:
901:
881:
878:
875:
872:
869:
866:
863:
860:
857:
853:
825:Main article:
822:
819:
797:
794:
786:
780:
744:Michael Pepper
716:temperatures.
685:
682:
637:
634:
596:
567:
540:
489:
486:
482:
477:
473:
470:
465:
462:
458:
445:
442:
440:
437:
207:. The divisor
179:
166:
159:
158:
147:
141:
136:
132:
127:
122:
111:
101:
95:
90:
87:
83:
66:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7701:
7690:
7687:
7685:
7682:
7680:
7677:
7675:
7672:
7670:
7667:
7665:
7662:
7660:
7657:
7656:
7654:
7639:
7638:
7629:
7627:
7626:
7621:
7617:
7615:
7614:
7605:
7603:
7602:
7593:
7592:
7589:
7583:
7580:
7578:
7575:
7573:
7570:
7568:
7565:
7563:
7560:
7559:
7557:
7555:
7551:
7545:
7542:
7538:
7535:
7533:
7530:
7529:
7528:
7525:
7523:
7520:
7518:
7515:
7513:
7510:
7508:
7505:
7503:
7500:
7498:
7495:
7493:
7490:
7488:
7485:
7484:
7482:
7480:
7476:
7470:
7467:
7465:
7462:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7437:
7435:
7432:
7430:
7427:
7426:
7424:
7420:
7410:
7407:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7386:
7384:
7380:
7374:
7371:
7369:
7366:
7364:
7361:
7359:
7356:
7354:
7351:
7349:
7348:Semiconductor
7346:
7344:
7341:
7339:
7336:
7335:
7333:
7329:
7323:
7320:
7318:
7317:Hubbard model
7315:
7313:
7310:
7308:
7305:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7278:
7275:
7273:
7270:
7269:
7267:
7263:
7257:
7254:
7252:
7249:
7247:
7244:
7242:
7239:
7237:
7234:
7232:
7229:
7227:
7224:
7222:
7219:
7218:
7216:
7212:
7209:
7205:
7199:
7196:
7194:
7191:
7189:
7186:
7184:
7181:
7180:
7178:
7174:
7169:
7159:
7156:
7154:
7151:
7149:
7146:
7144:
7141:
7139:
7136:
7134:
7131:
7129:
7126:
7124:
7121:
7119:
7116:
7114:
7111:
7109:
7106:
7105:
7103:
7101:
7097:
7093:
7086:
7081:
7079:
7074:
7072:
7067:
7066:
7063:
7056:
7053:
7049:
7045:
7041:
7037:
7034:(8): 085316.
7033:
7029:
7024:
7022:
7018:
7014:
7010:
7006:
7004:
7000:
6996:
6995:
6990:
6985:
6980:
6976:
6972:
6968:
6964:
6963:Physics Today
6960:
6955:
6953:
6949:
6946:
6942:
6938:
6935:
6931:
6927:
6923:
6919:
6915:
6911:
6907:
6903:
6898:
6893:
6889:
6885:
6880:
6876:
6872:
6868:
6864:
6860:
6856:
6851:
6850:
6846:
6837:
6833:
6829:
6825:
6821:
6817:
6812:
6807:
6803:
6799:
6792:
6789:
6784:
6780:
6776:
6772:
6768:
6764:
6759:
6754:
6750:
6746:
6739:
6736:
6731:
6727:
6723:
6719:
6715:
6711:
6707:
6703:
6699:
6695:
6690:
6685:
6681:
6677:
6673:
6666:
6663:
6658:
6654:
6650:
6646:
6641:
6636:
6632:
6628:
6623:
6618:
6614:
6610:
6606:
6599:
6596:
6591:
6587:
6583:
6579:
6575:
6571:
6567:
6563:
6559:
6555:
6550:
6545:
6541:
6537:
6533:
6526:
6523:
6518:
6514:
6510:
6506:
6502:
6498:
6494:
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6435:
6430:
6425:
6422:(3): 033834.
6421:
6417:
6413:
6406:
6403:
6398:
6392:
6384:
6382:9780511819070
6378:
6374:
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6364:
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6000:
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5987:
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5979:
5972:
5969:
5964:
5961:Tong, David.
5957:
5954:
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5943:
5939:
5932:
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5920:
5916:
5912:
5908:
5904:
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5811:
5807:
5801:
5798:
5787:on 2018-11-19
5783:
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5555:
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5519:(7818): 659.
5518:
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5150:
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5135:
5115:
5106:
5105:Lorentz force
5102:
5098:
5090:
5088:
5086:
5085:
5080:
5074:
5072:
5068:
5064:
5060:
5056:
5055:Berry's phase
5052:
5051:Chern numbers
5048:
5041:
5037:
5030:
5028:
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5022:
5017:
5013:
5009:
5005:
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3990:
3963:
3952:
3925:
3914:
3870:
3842:
3829:
3825:
3811:
3800:
3784:
3764:
3756:
3755:Bohr magneton
3732:
3711:
3708:
3705:
3685:
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2206:
2199:
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2171:
2167:
2159:
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2140:
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2128:
2123:
2118:
2114:
2089:
2085:
2080:
2077:
2071:
2060:
2033:
2027:
2022:
2019:
2015:
2011:
2005:
1999:
1995:
1989:
1979:
1975:
1969:
1964:
1960:
1956:
1953:
1945:
1934:
1928:
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1103:
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1064:
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1055:
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1038:
1035:
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1016:
1013:
1004:
994:
990:
986:
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976:
968:
967:
966:
950:
946:
925:
905:
902:
899:
876:
873:
870:
867:
864:
861:
855:
842:
838:
833:
828:
827:Landau levels
821:Landau levels
820:
814:
795:
793:
790:
783:
778:
775:
772:
768:
764:
761:
757:
753:
749:
745:
741:
737:
732:
730:
726:
722:
717:
715:
714:liquid helium
711:
707:
702:
700:
696:
691:
683:
681:
679:
675:
671:
666:
661:
659:
655:
651:
647:
643:
635:
633:
614:
606:
595:
589:
583:
566:
562:
557:
555:
551:
547:
539:
535:
531:
527:
518:
510:
487:
484:
480:
475:
471:
468:
463:
460:
456:
443:
438:
436:
434:
430:
429:Landau levels
425:
420:
418:
414:
408:
405:
400:
399:Landau levels
395:
394:
226:
220:= 1, 2, 3,...
219:
213:
212:
206:
201:
196:
191:
186:
178:
173:
165:
145:
139:
134:
130:
125:
120:
109:
99:
93:
88:
85:
81:
73:
72:
71:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
7635:
7623:
7611:
7599:
7517:Pines' demon
7256:Kondo effect
7225:
7158:Time crystal
7055:E. I. Rashba
7031:
7028:Phys. Rev. B
7027:
7012:
7009:Aron Pinczuk
6993:
6966:
6962:
6936:
6887:
6883:
6858:
6854:
6801:
6797:
6791:
6748:
6744:
6738:
6679:
6675:
6665:
6612:
6608:
6598:
6539:
6535:
6525:
6484:
6480:
6470:
6419:
6415:
6405:
6372:
6371:Davies J.H.
6366:
6323:
6319:
6309:
6258:
6254:
6248:
6223:
6220:Phys. Rev. B
6219:
6213:
6186:
6182:
6172:
6147:
6143:
6137:
6112:
6108:
6102:
6085:
6079:
6045:
6039:
6012:
6002:
5985:
5981:
5971:
5956:
5937:
5931:
5906:
5903:Phys. Rev. B
5902:
5876:
5843:
5839:
5829:
5818:. Retrieved
5809:
5800:
5789:. Retrieved
5782:the original
5777:
5765:
5754:. Retrieved
5745:
5736:
5695:
5691:
5681:
5662:
5656:
5621:
5617:
5607:
5564:
5560:
5516:
5512:
5502:
5417:
5407:
5405:
5327:
5198:
5094:
5082:
5075:
5044:
4997:
4934:
4731:
4597:
4595:
4572:
4540:
4374:
4349:Fermi energy
4326:
4095:
3982:
3577:
3446:
3442:
3366:
3079:
2805:
2717:
2564:
2493:
2377:
2307:
2051:
1391:
1195:
1121:
841:Landau gauge
834:
830:
812:
788:
781:
747:
733:
721:quantization
719:The integer
718:
703:
687:
662:
654:TKNN formula
641:
639:
612:
604:
593:
587:
581:
564:
558:
537:
533:
516:
508:
447:
439:Applications
421:
409:
403:
392:
224:
217:
210:
199:
189:
176:
172:Hall voltage
163:
160:
63:
52:temperatures
35:
31:
29:
7674:Spintronics
7659:Hall effect
7554:Soft matter
7454:Ferromagnet
7272:Drude model
7241:Berry phase
7221:Hall effect
6150:(6): 1839.
6015:: 583â596.
5467:Hall effect
4225:Fermi level
3757:. The sign
2866:-direction
690:Carl Frosch
576:.807 Ί
413:Fermi level
54:and strong
44:Hall effect
7653:Categories
7469:Spin glass
7464:Metamagnet
7444:Paramagnet
7331:Conduction
7307:BCS theory
7148:Superfluid
7143:Supersolid
6941:Postscript
6689:1909.04828
6622:1507.04541
6615:(2): 200.
6549:1511.07381
5988:(9): 547.
5820:2024-05-18
5816:. May 2024
5791:2018-11-19
5756:2024-05-18
5752:. May 2024
5574:1601.01697
5494:References
5472:Hall probe
5097:Bohr model
4411:one finds
2713:Lorentzian
2567:Dirac comb
629:... Ί
60:resistance
7527:Polariton
7434:Diamagnet
7382:Couplings
7358:Conductor
7353:Semimetal
7338:Insulator
7214:Phenomena
7138:Fermi gas
6969:(8): 38.
6897:0902.1356
6730:202558675
6714:0036-8075
6649:2334-2536
6574:0028-0836
6509:1749-4885
6462:119098087
6454:1050-2947
6391:cite book
5868:206528413
5720:1364-503X
5599:118614055
5301:ϵ
5297:π
5263:−
5235:−
5229:∞
5178:π
5167:ω
5136:ω
4978:ν
4903:ν
4897:σ
4894:⇒
4878:ν
4857:ρ
4818:ν
4741:ρ
4711:ν
4647:ρ
4617:ρ
4581:ν
4555:≠
4552:ρ
4505:ρ
4490:ρ
4486:−
4470:ρ
4466:−
4452:ρ
4437:ρ
4422:σ
4394:−
4390:ρ
4383:σ
4335:ν
4267:ν
4259:and give
4198:ν
4175:∝
4145:ν
4125:ν
4104:ν
4014:ν
3991:ν
3967:Γ
3953:ω
3949:ℏ
3929:Γ
3915:ω
3911:ℏ
3891:Γ
3871:δ
3843:ω
3839:ℏ
3809:Δ
3785:−
3733:μ
3655:μ
3638:±
3629:Δ
3599:±
3555:ω
3551:ℏ
3545:ε
3542:Δ
3509:ℏ
3505:π
3499:∗
3475:ω
3471:ℏ
3424:ℏ
3421:π
3405:ω
3399:∗
3349:ℏ
3346:π
3327:ω
3321:∗
3283:ω
3271:ℏ
3268:π
3196:π
3168:Δ
3130:ϕ
3126:ϕ
2886:π
2824:×
2715:profile.
2689:τ
2660:τ
2656:ℏ
2648:Γ
2620:ω
2616:ℏ
2601:ε
2597:Δ
2577:δ
2540:ℏ
2536:π
2530:∗
2405:, not on
2207:ω
2203:ℏ
2184:ε
2180:≡
2168:ε
2134:ℏ
2090:∗
2061:ω
2016:ε
1957:−
1935:ω
1929:∗
1895:∂
1885:∂
1874:∗
1856:ℏ
1850:−
1801:ℏ
1745:^
1651:ψ
1490:∗
1471:ℏ
1461:π
1430:ε
1359:ε
1307:∂
1297:∂
1286:∗
1268:ℏ
1262:−
1225:ε
1212:ε
1205:ε
1181:ε
1161:∇
1158:ℏ
1152:−
1086:ψ
1083:ε
1056:ψ
1014:−
995:∗
951:∗
900:ϕ
771:magnesium
734:In 1980,
688:In 1957,
650:Anomalies
140:ν
40:quantized
7601:Category
7582:Colloids
6922:18432240
6783:14161004
6722:31780626
6582:27281214
6517:33927607
6358:10674643
6350:17255474
6301:46256393
6293:17303717
5860:20689008
5728:16147506
5648:10038961
5543:32728252
5477:Graphene
5430:See also
3580:fermions
2709:Gaussian
1642:, i.e.,
767:graphene
526:standard
7613:Commons
7577:Polymer
7544:Polaron
7522:Plasmon
7502:Exciton
7036:Bibcode
6971:Bibcode
6930:4402113
6902:Bibcode
6863:Bibcode
6836:9265777
6816:Bibcode
6763:Bibcode
6694:Bibcode
6676:Science
6657:1645962
6627:Bibcode
6590:4468395
6554:Bibcode
6489:Bibcode
6434:Bibcode
6328:Bibcode
6320:Science
6273:Bibcode
6255:Science
6228:Bibcode
6191:Bibcode
6152:Bibcode
6117:Bibcode
6081:Physics
5911:Bibcode
5840:Science
5700:Bibcode
5626:Bibcode
5579:Bibcode
5521:Bibcode
5004:Photons
816:states.
748:exactly
740:silicon
684:History
617:
601:
521:
505:
385:
373:
369:
357:
353:
341:
337:
325:
321:
309:
305:
293:
289:
277:
273:
261:
257:
245:
241:
229:
203:is the
193:is the
185:current
180:channel
170:is the
114:channel
38:) is a
7512:Phonon
7507:Magnon
7265:Theory
7123:Plasma
7113:Liquid
7019:
7001:
6928:
6920:
6884:Nature
6834:
6781:
6728:
6720:
6712:
6655:
6647:
6609:Optica
6588:
6580:
6572:
6536:Nature
6515:
6507:
6460:
6452:
6379:
6356:
6348:
6299:
6291:
6088:: 46.
6052:
6027:
5944:
5866:
5858:
5726:
5718:
5669:
5646:
5597:
5541:
5513:Nature
3678:being
2914:being
2680:being
2052:where
1122:where
779:ZnOâMg
695:MOSFET
161:where
7487:Anyon
7108:Solid
6926:S2CID
6892:arXiv
6832:S2CID
6806:arXiv
6779:S2CID
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