237:
172:
3070:
31:
4015:
2234:
2237:
Hofstadter's butterfly phase diagram at zero temperature. The horizontal axis indicates electron density, starting with no electrons from the left. The vertical axis indicates the strength of the magnetic flux, starting from zero at the bottom, the pattern repeats periodically for higher fields. The
3660:
Roushan, P.; Neill, C.; Tangpanitanon, J.; Bastidas, V. M.; Megrant, A.; Barends, R.; Chen, Y.; Chen, Z.; Chiaro, B.; Dunsworth, A.; Fowler, A.; Foxen, B.; Giustina, M.; Jeffrey, E.; Kelly, J.; Lucero, E.; Mutus, J.; Neeley, M.; Quintana, C.; Sank, D.; Vainsencher, A.; Wenner, J.; White, T.; Neven,
3524:
Ponomarenko, L. A.; Gorbachev, R. V.; Yu, G. L.; Elias, D. C.; Jalil, R.; Patel, A. A.; Mishchenko, A.; Mayorov, A. S.; Woods, C. R.; Wallbank, J. R.; Mucha-Kruczynski, M.; Piot, B. A.; Potemski, M.; Grigorieva, I. V.; Novoselov, K. S.; Guinea, F.; Fal’ko, V. I.; Geim, A. K. (30 May 2013). "Cloning
179:
In 1997 the
Hofstadter butterfly was reproduced in experiments with a microwave guide equipped with an array of scatterers. The similarity between the mathematical description of the microwave guide with scatterers and Bloch's waves in the magnetic field allowed the reproduction of the Hofstadter
97:
in perpendicular magnetic fields. It gives a graphical representation of the spectrum of Harper's equation at different frequencies. One key aspect of the mathematical structure of this spectrum – the splitting of energy bands for a specific value of the magnetic field, along a single dimension
2736:
3460:
Dean, C. R.; Wang, L.; Maher, P.; Forsythe, C.; Ghahari, F.; Gao, Y.; Katoch, J.; Ishigami, M.; Moon, P.; Koshino, M.; Taniguchi, T.; Watanabe, K.; Shepard, K. L.; Hone, J.; Kim, P. (30 May 2013). "Hofstadter's butterfly and the fractal quantum Hall effect in moiré superlattices".
547:
2238:
colors represent the Chern numbers of the gaps in the spectrum, also known as the TKNN (Thouless, Kohmoto, Nightingale and Nijs) integers. Blueish cold colors indicate negative Chern numbers, warm red colors indicate positive Chern numbers, white indicates zero.
3596:
Hunt, B.; Sanchez-Yamagishi, J. D.; Young, A. F.; Yankowitz, M.; LeRoy, B. J.; Watanabe, K.; Taniguchi, T.; Moon, P.; Koshino, M.; Jarillo-Herrero, P.; Ashoori, R. C. (2013). "Massive Dirac fermions and
Hofstadter butterfly in a van der Waals heterostructure".
3661:
H.; Angelakis, D. G.; Martinis, J. (2017-12-01) . "Spectroscopic signatures of localization with interacting photons in superconducting qubits" [Spectral signatures of many-body localization with interacting photons].
1785:
2615:
3732:
Lu, Xiaobo; Lian, Biao; Chaudhary, Gaurav; Piot, Benjamin A.; Romagnoli, Giulio; Watanabe, Kenji; Taniguchi, Takashi; Poggio, Martino; MacDonald, Allan H.; Bernevig, B. Andrei; Efetov, Dmitri K. (2021-07-27).
62:
and is one of the early examples of modern scientific data visualization. The name reflects the fact that, as
Hofstadter wrote, "the large gaps form a very striking pattern somewhat resembling a butterfly."
3108:
and temperature, has infinitely many phases. Thouless and coworkers showed that each phase is characterized by an integral Hall conductance, where all integer values are allowed. These integers are known as
777:
328:
703:
2126:
1474:
1089:
1179:
2604:
1225:
2048:
1882:
1615:
831:
636:
1974:
957:
3015:
2985:
2317:
2203:
2156:
1832:
2808:
3215:
1927:
582:
2908:
725:
284:
2934:
2440:
2343:
1000:
1665:
2363:
2223:
1521:
258:
113:, his paper was influential in directing further research. It predicted on theoretical grounds that the allowed energy level values of an electron in a two-dimensional
3035:
2954:
2828:
2386:
2280:
2260:
2176:
1998:
1127:
106:
values of the magnetic field against all energy values, creating the two-dimensional plot that first revealed the spectrum's uniquely recursive geometric properties.
2534:
3308:
2879:
667:
1635:
884:
1547:
3055:
2848:
2779:
2759:
2508:
2484:
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2406:
904:
855:
308:
3914:
290:
In his original paper, Hofstadter considers the following derivation: a charged quantum particle in a two-dimensional square lattice, with a lattice spacing
2731:{\displaystyle n=\int _{-4}^{\epsilon _{\rm {F}}}\rho (\epsilon )\mathrm {d} \epsilon \;;\;n_{0}=\int _{-4}^{4}\rho (\epsilon )\mathrm {d} \epsilon }
1673:
1003:
3057:
or for lattices of finite size. The scale at which the butterfly can be resolved in a real experiment depends on the system's specific conditions.
3970:
3931:
82:
The first mathematical description of electrons on a 2D lattice, acted on by a perpendicular homogeneous magnetic field, was studied by
217:, published results from a simulation of 2D electrons in a perpendicular magnetic field using interacting photons in 9 superconducting
738:
206:
substrates. In this instance the butterfly spectrum results from the interplay between the applied magnetic field and the large-scale
4160:
542:{\displaystyle W(\mathbf {k} )=E_{0}(\cos k_{x}a+\cos k_{y}a)={\frac {E_{0}}{2}}(e^{ik_{x}a}+e^{-ik_{x}a}+e^{ik_{y}a}+e^{-ik_{y}a})}
3907:
1228:
214:
4110:
672:
1237:
314:, under a perpendicular static homogeneous magnetic field restricted to a single Bloch band. For a 2D square lattice, the
236:
2053:
1009:
192:
143:'s newly coined word "fractal" was introduced in an English text. Hofstadter also discusses the figure in his 1979 book
4165:
3900:
3735:"Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle"
198:
In 2013, three separate groups of researchers independently reported evidence of the
Hofstadter butterfly spectrum in
1132:
2560:
1188:
71:
2936:). This equation characterizes all the solutions of Harper's equation. Most importantly, one can derive that when
2003:
728:
3122:
1837:
1556:
782:
587:
39:
1932:
3370:
1638:
311:
4000:
913:
3940:
145:
3168:(1976). "Energy levels and wavefunctions of Bloch electrons in rational and irrational magnetic fields".
2993:
2963:
2181:
2134:
117:, as a function of a magnetic field applied perpendicularly to the system, formed what is now known as a
3165:
2289:
1977:
1793:
210:
that develops when the graphene lattice is oriented with near zero-angle mismatch to the boron nitride.
4014:
2784:
46:
is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular
4170:
4105:
3980:
3862:
3823:
3756:
3682:
3618:
3546:
3482:
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3179:
2551:
1887:
732:
556:
110:
102:
in 1964 (in a paper cited by
Hofstadter), but Hofstadter greatly expanded upon that work by plotting
4116:
4053:
3990:
3104:
of electrons in a two-dimensional square lattice, as a function of a perpendicular magnetic field,
2283:
319:
171:
160:
130:
121:. That is, the distribution of energy levels for small-scale changes in the applied magnetic field
67:
2884:
708:
263:
4043:
3960:
3923:
3746:
3714:
3672:
3642:
3608:
3578:
3536:
3506:
3472:
3405:
Albrecht, C.; Smet, J. H.; von
Klitzing, K.; Weiss, D.; Umansky, V.; Schweizer, H. (2001-01-01).
3368:
Kuhl, U.; Stöckmann, H.-J. (13 April 1998). "Microwave realization of the
Hofstadter butterfly".
3256:
3105:
2913:
2411:
2322:
965:
213:
In
September 2017, John Martinis's group at Google, in collaboration with the Angelakis group at
184:
59:
1644:
129:
seen in the large-scale structure. "Gplot", as
Hofstadter called the figure, was described as a
2348:
2208:
1482:
243:
4155:
4131:
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3774:
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2547:
2368:
Harper's equation has the particular property that the solutions depend on the rationality of
907:
834:
152:
140:
135:
51:
3037:. This discontinuity is nonphysical, and continuity is recovered for a finite uncertainty in
3020:
3017:
forms a self-similar fractal that is discontinuous between rational and irrational values of
2939:
2813:
2371:
2265:
2245:
2161:
1983:
1094:
4083:
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3426:
3387:
3348:
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3235:
3187:
2513:
639:
240:
Hofstadter butterfly is the graphical solution to Harper's equation, where the energy ratio
225:
207:
3303:
2857:
2262:
with period 1 (it repeats for each quantum flux per unit cell). The graph in the region of
645:
4073:
4038:
2543:
2443:
1620:
860:
55:
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1526:
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114:
94:
83:
47:
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2487:
1182:
315:
203:
3646:
3407:"Evidence of Hofstadter's Fractal Energy Spectrum in the Quantized Hall Conductance"
17:
4078:
3851:"Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems"
3582:
3110:
2851:
1479:
Considering that the particle can only hop between points in the lattice, we write
90:
30:
3850:
175:
A simulation of electrons via superconducting qubits yields
Hofstadter's butterfly
3391:
156:
122:
118:
3812:"A Result Not Dependent on Rationality for Bloch Electrons in a Magnetic Field"
3811:
3430:
3353:
3328:
1780:{\displaystyle g_{n+1}+g_{n-1}+2\cos(2\pi n\alpha -\nu )g_{n}=\epsilon g_{n},}
99:
66:
The Hofstadter butterfly plays an important role in the theory of the integer
3882:
3778:
3702:
3438:
3288:
3191:
4063:
3835:
3769:
3694:
3630:
3796:
3710:
3638:
3574:
3502:
3446:
2233:
1637:
depends on the energy, in order to obtain Harper's equation (also known as
27:
Fractal describing the theorised behaviour of electrons in a magnetic field
4033:
199:
3558:
3494:
3257:"Single Band Motion of Conduction Electrons in a Uniform Magnetic Field"
155:
and his team discovered that the butterfly's wings are characterized by
98:(energy) – had been previously mentioned in passing by Soviet physicist
3849:
Analytis, James G.; Blundell, Stephen J.; Ardavan, Arzhang (May 2004).
3566:
126:
3874:
3240:
2536:, the energy bands converge to thin energy bands corresponding to the
89:
Hofstadter first described the structure in 1976 in an article on the
3892:
1550:
3329:"Quantized Hall conductance in a two-dimensional periodic potential"
2242:
Due to the cosine function's properties, the pattern is periodic on
149:. The structure became generally known as "Hofstadter's butterfly".
3751:
3677:
3613:
3541:
3477:
2232:
218:
170:
29:
1929:
is proportional to the magnetic flux through a lattice cell and
772:{\displaystyle \hbar \mathbf {k} \to \mathbf {p} -q\mathbf {A} }
221:. The simulation recovered Hofstadter's butterfly, as expected.
187:, and coworkers realized an experimental setup to test Thouless
58:
nature of the spectrum was discovered in the 1976 Ph.D. work of
3896:
1231:
to obtain the following time-independent Schrödinger equation:
3304:"Energy Spectrum of a Conduction Electron in a Magnetic Field"
3064:
2881:
corresponds to the levels of the completely filled band (from
735:, replacing the crystal momentum with the canonical momentum
3327:
Thouless D., Kohmoto M, Nightngale and M. den-Nijs (1982).
3081:
698:{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }
2292:
2056:
2006:
2000:
can also be expressed in terms of the magnetic length
3043:
3023:
2996:
2966:
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2268:
2248:
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1986:
1935:
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892:
863:
843:
785:
741:
711:
675:
648:
590:
559:
331:
296:
266:
246:
4136:
By Hofstadter and the Fluid Analogies Research Group
2229:
Solutions to Harper's equation and Wannier treatment
2121:{\textstyle \alpha =(2\pi )^{-1}(a/l_{\rm {m}})^{2}}
1469:{\displaystyle E\psi (x,y)={\frac {E_{0}}{2}}\left.}
180:
butterfly for periodic sequences of the scatterers.
4092:
4022:
3930:
191:'s predictions about Hofstadter's butterfly with a
3049:
3029:
3009:
2979:
2948:
2928:
2902:
2873:
2842:
2822:
2802:
2773:
2753:
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1992:
1968:
1921:
1876:
1826:
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1659:
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1609:
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1515:
1468:
1219:
1173:
1121:
1084:{\displaystyle e^{ip_{j}a}\psi (x,y)=\psi (x+a,y)}
1083:
994:
951:
898:
878:
849:
825:
771:
719:
697:
661:
630:
576:
541:
302:
278:
252:
2131:Hofstadter's butterfly is the resulting plot of
3739:Proceedings of the National Academy of Sciences
3309:Journal of Experimental and Theoretical Physics
3216:"A topological look at the quantum Hall effect"
3160:
3158:
3525:of Dirac fermions in graphene superlattices".
3261:Proceedings of the Physical Society. Section A
3156:
3154:
3152:
3150:
3148:
3146:
3144:
3142:
3140:
3138:
2228:
1174:{\displaystyle \psi (\mathbf {r} )=\psi (x,y)}
669:is an empirical parameter. The magnetic field
224:In 2021 the butterfly was observed in twisted
3908:
2599:{\displaystyle {\frac {n}{n_{0}}}=S+T\alpha }
1549:are integers. Hofstadter makes the following
1220:{\displaystyle W(\mathbf {p} -q\mathbf {A} )}
8:
2043:{\textstyle l_{\rm {m}}={\sqrt {\hbar /eB}}}
3214:Avron J, Osadchy D., and Seiler R. (2003).
86:and his student R. G. Harper in the 1950s.
3915:
3901:
3893:
2676:
2672:
2225:that are a solution to Harper's equation.
1877:{\displaystyle \alpha =\phi (B)/\phi _{0}}
1610:{\displaystyle \psi (x,y)=g_{n}e^{i\nu m}}
826:{\displaystyle \mathbf {p} =(p_{x},p_{y})}
631:{\displaystyle \mathbf {k} =(k_{x},k_{y})}
260:is plotted as a function of the flux ratio
3786:
3768:
3750:
3676:
3612:
3540:
3476:
3352:
3239:
3042:
3022:
3001:
2995:
2971:
2965:
2960:, there are infinitely many solution for
2941:
2915:
2886:
2865:
2859:
2835:
2815:
2786:
2766:
2746:
2720:
2702:
2694:
2681:
2664:
2643:
2642:
2637:
2629:
2617:
2573:
2564:
2562:
2515:
2495:
2471:
2451:
2424:
2413:
2393:
2373:
2365:is necessarily bounded between -4 and 4.
2350:
2324:
2299:
2291:
2267:
2247:
2210:
2189:
2183:
2163:
2142:
2136:
2112:
2101:
2100:
2091:
2076:
2055:
2027:
2022:
2012:
2011:
2005:
1985:
1958:
1940:
1934:
1913:
1889:
1868:
1859:
1839:
1818:
1809:
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1752:
1700:
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1622:
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1528:
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1429:
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1369:
1271:
1265:
1239:
1209:
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1142:
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1025:
1017:
1011:
981:
973:
967:
917:
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862:
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764:
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712:
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409:
394:
372:
353:
338:
330:
295:
265:
245:
235:
34:Rendering of the butterfly by Hofstadter
3134:
3061:Phase diagram, conductance and topology
2546:showed that by taking into account the
2024:
1969:{\displaystyle \phi _{0}=2\pi \hbar /q}
1955:
1453:
1393:
910:). For convenience we choose the gauge
742:
159:, which provide a way to calculate the
2850:counts the number of states up to the
3971:Fluid Concepts and Creative Analogies
952:{\displaystyle \mathbf {A} =(0,Bx,0)}
731:, can be taken into account by using
7:
3209:
3207:
3205:
3203:
3201:
2810:is the density of states at a given
109:Written while Hofstadter was at the
3010:{\displaystyle \epsilon _{\alpha }}
2980:{\displaystyle \epsilon _{\alpha }}
2312:{\textstyle \alpha ={\frac {1}{2}}}
2198:{\displaystyle \epsilon _{\alpha }}
2151:{\displaystyle \epsilon _{\alpha }}
2721:
2665:
2644:
2102:
2013:
1827:{\displaystyle \epsilon =2E/E_{0}}
1181:is the particle's two-dimensional
684:
25:
4013:
3068:
2803:{\displaystyle \rho (\epsilon )}
2158:as a function of the flux ratio
1210:
1199:
1143:
918:
787:
765:
754:
746:
713:
691:
677:
592:
567:
339:
202:devices fabricated on hexagonal
2388:. By imposing periodicity over
1922:{\displaystyle \phi (B)=Ba^{2}}
857:is the charge of the particle (
577:{\displaystyle W(\mathbf {k} )}
4111:Indiana University Bloomington
2797:
2791:
2717:
2711:
2661:
2655:
2554:that describes the system, as
2109:
2085:
2073:
2063:
1900:
1894:
1856:
1850:
1745:
1724:
1575:
1563:
1422:
1404:
1362:
1344:
1335:
1317:
1308:
1290:
1259:
1247:
1214:
1195:
1168:
1156:
1147:
1139:
1078:
1060:
1051:
1039:
946:
925:
820:
794:
750:
625:
599:
571:
563:
536:
426:
403:
359:
343:
335:
1:
3810:Wannier, G. H. (1978-08-01).
310:, is described by a periodic
195:in a superlattice potential.
183:In 2001, Christian Albrecht,
2903:{\displaystyle \epsilon =-4}
720:{\displaystyle \mathbf {A} }
279:{\displaystyle 2\pi \alpha }
193:two-dimensional electron gas
3855:American Journal of Physics
3392:10.1103/PhysRevLett.80.3232
3281:10.1088/0370-1298/68/10/304
2929:{\displaystyle \epsilon =4}
2435:{\displaystyle \alpha =P/Q}
2338:{\displaystyle \epsilon =0}
2205:is the set of all possible
995:{\displaystyle e^{ip_{j}a}}
228:at the second magic angle.
72:topological quantum numbers
4187:
3431:10.1103/PhysRevLett.86.147
3354:10.1103/PhysRevLett.49.405
3255:Harper, P G (1955-10-01).
1660:{\displaystyle \lambda =1}
4130:Edited by Hofstadter and
4126:
4011:
3302:Azbel', Mark Ya. (1964).
2358:{\displaystyle \epsilon }
2218:{\displaystyle \epsilon }
1516:{\displaystyle x=na,y=ma}
729:magnetic vector potential
253:{\displaystyle \epsilon }
4161:Condensed matter physics
3192:10.1103/PhysRevB.14.2239
2510:energy bands. For large
584:is the energy function,
40:condensed matter physics
3836:10.1002/pssb.2220880243
3816:Physica Status Solidi B
3770:10.1073/pnas.2100006118
3695:10.1126/science.aao1401
3631:10.1126/science.1237240
3411:Physical Review Letters
3371:Physical Review Letters
3333:Physical Review Letters
3030:{\displaystyle \alpha }
2949:{\displaystyle \alpha }
2823:{\displaystyle \alpha }
2408:, one can show that if
2381:{\displaystyle \alpha }
2275:{\displaystyle \alpha }
2255:{\displaystyle \alpha }
2171:{\displaystyle \alpha }
1993:{\displaystyle \alpha }
1639:almost Mathieu operator
1122:{\displaystyle j=x,y,z}
163:in Hofstadter's model.
133:in his 1976 article in
4049:Hofstadter's butterfly
3166:Hofstadter, Douglas R.
3051:
3031:
3011:
2981:
2950:
2930:
2904:
2875:
2844:
2824:
2804:
2775:
2755:
2732:
2600:
2530:
2529:{\displaystyle Q\gg P}
2504:
2480:
2460:
2436:
2402:
2382:
2359:
2339:
2313:
2276:
2256:
2239:
2219:
2199:
2172:
2152:
2122:
2044:
1994:
1970:
1923:
1878:
1828:
1781:
1661:
1631:
1611:
1543:
1517:
1470:
1221:
1175:
1123:
1085:
996:
953:
900:
880:
851:
827:
773:
721:
699:
663:
632:
578:
543:
304:
287:
280:
254:
176:
44:Hofstadter's butterfly
35:
4001:Surfaces and Essences
3052:
3032:
3012:
2982:
2951:
2931:
2905:
2876:
2874:{\displaystyle n_{0}}
2845:
2825:
2805:
2776:
2756:
2733:
2601:
2531:
2505:
2481:
2461:
2437:
2403:
2383:
2360:
2340:
2314:
2277:
2257:
2236:
2220:
2200:
2173:
2153:
2123:
2045:
1995:
1978:magnetic flux quantum
1971:
1924:
1879:
1829:
1782:
1662:
1632:
1612:
1544:
1518:
1471:
1222:
1176:
1124:
1086:
997:
954:
901:
881:
852:
828:
774:
722:
700:
664:
662:{\displaystyle E_{0}}
633:
579:
544:
305:
281:
255:
239:
174:
33:
4106:Egbert B. Gebstadter
3981:Le Ton beau de Marot
3041:
3021:
2994:
2964:
2940:
2914:
2885:
2858:
2834:
2814:
2785:
2765:
2745:
2616:
2561:
2552:Diophantine equation
2514:
2494:
2490:, there are exactly
2470:
2450:
2412:
2392:
2372:
2349:
2323:
2290:
2282:between 0 and 1 has
2266:
2246:
2209:
2182:
2162:
2135:
2054:
2004:
1984:
1933:
1888:
1838:
1794:
1674:
1645:
1630:{\displaystyle \nu }
1621:
1557:
1527:
1483:
1238:
1189:
1133:
1095:
1010:
1004:translation operator
966:
914:
890:
879:{\displaystyle q=-e}
861:
841:
783:
739:
733:Peierls substitution
709:
673:
646:
588:
557:
329:
312:Schrödinger equation
294:
264:
244:
111:University of Oregon
18:Hofstadter butterfly
4117:Victim of the Brain
3991:I Am a Strange Loop
3941:Gödel, Escher, Bach
3867:2004AmJPh..72..613A
3828:1978PSSBR..88..757W
3761:2021PNAS..11800006L
3745:(30): e2100006118.
3687:2017Sci...358.1175R
3671:(6367): 1175–1179.
3623:2013Sci...340.1427H
3607:(6139): 1427–1430.
3559:10.1038/nature12187
3551:2013Natur.497..594P
3495:10.1038/nature12186
3487:2013Natur.497..598D
3423:2001PhRvL..86..147A
3384:1998PhRvL..80.3232K
3345:1982PhRvL..49..405T
3273:1955PPSA...68..874H
3232:2003PhT....56h..38A
3184:1976PhRvB..14.2239H
2707:
2651:
2550:, one can obtain a
2284:reflection symmetry
1542:{\displaystyle n,m}
320:dispersion relation
146:Gödel, Escher, Bach
131:recursive structure
68:quantum Hall effect
4166:1976 introductions
3961:Metamagical Themas
3924:Douglas Hofstadter
3106:chemical potential
3080:. You can help by
3047:
3027:
3007:
2977:
2946:
2926:
2900:
2871:
2840:
2820:
2800:
2781:are integers, and
2771:
2751:
2728:
2690:
2625:
2596:
2526:
2500:
2476:
2456:
2432:
2398:
2378:
2355:
2335:
2309:
2272:
2252:
2240:
2215:
2195:
2168:
2148:
2118:
2040:
1990:
1966:
1919:
1874:
1824:
1777:
1657:
1627:
1607:
1539:
1513:
1466:
1217:
1171:
1119:
1081:
992:
949:
896:
886:for the electron,
876:
847:
823:
769:
717:
695:
659:
628:
574:
539:
300:
288:
276:
250:
185:Klaus von Klitzing
177:
70:and the theory of
60:Douglas Hofstadter
36:
4143:
4142:
4132:Daniel C. Dennett
4100:Robert Hofstadter
4059:Hofstadter points
3875:10.1119/1.1615568
3535:(7451): 594–597.
3471:(7451): 598–602.
3378:(15): 3232–3235.
3241:10.1063/1.1611351
3171:Physical Review B
3123:Aubry–André model
3098:
3097:
3050:{\displaystyle B}
2990:The union of all
2958:irrational number
2843:{\displaystyle n}
2774:{\displaystyle T}
2754:{\displaystyle S}
2579:
2548:density of states
2503:{\displaystyle Q}
2479:{\displaystyle Q}
2459:{\displaystyle P}
2401:{\displaystyle n}
2307:
2038:
1980:. The flux ratio
1280:
908:elementary charge
899:{\displaystyle e}
850:{\displaystyle q}
835:momentum operator
424:
303:{\displaystyle a}
232:Theoretical model
153:David J. Thouless
141:Benoit Mandelbrot
139:, written before
136:Physical Review B
16:(Redirected from
4178:
4084:Superrationality
4069:Platonia dilemma
4054:Hofstadter's law
4017:
4006:
3996:
3986:
3976:
3966:
3956:
3946:
3917:
3910:
3903:
3894:
3887:
3886:
3846:
3840:
3839:
3807:
3801:
3800:
3790:
3772:
3754:
3729:
3723:
3722:
3680:
3657:
3651:
3650:
3616:
3593:
3587:
3586:
3544:
3521:
3515:
3514:
3480:
3457:
3451:
3450:
3402:
3396:
3395:
3365:
3359:
3358:
3356:
3324:
3318:
3317:
3299:
3293:
3292:
3252:
3246:
3245:
3243:
3211:
3196:
3195:
3178:(6): 2239–2249.
3162:
3093:
3090:
3072:
3065:
3056:
3054:
3053:
3048:
3036:
3034:
3033:
3028:
3016:
3014:
3013:
3008:
3006:
3005:
2986:
2984:
2983:
2978:
2976:
2975:
2955:
2953:
2952:
2947:
2935:
2933:
2932:
2927:
2909:
2907:
2906:
2901:
2880:
2878:
2877:
2872:
2870:
2869:
2849:
2847:
2846:
2841:
2829:
2827:
2826:
2821:
2809:
2807:
2806:
2801:
2780:
2778:
2777:
2772:
2760:
2758:
2757:
2752:
2737:
2735:
2734:
2729:
2724:
2706:
2701:
2686:
2685:
2668:
2650:
2649:
2648:
2647:
2636:
2605:
2603:
2602:
2597:
2580:
2578:
2577:
2565:
2535:
2533:
2532:
2527:
2509:
2507:
2506:
2501:
2485:
2483:
2482:
2477:
2465:
2463:
2462:
2457:
2441:
2439:
2438:
2433:
2428:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2364:
2362:
2361:
2356:
2344:
2342:
2341:
2336:
2318:
2316:
2315:
2310:
2308:
2300:
2281:
2279:
2278:
2273:
2261:
2259:
2258:
2253:
2224:
2222:
2221:
2216:
2204:
2202:
2201:
2196:
2194:
2193:
2177:
2175:
2174:
2169:
2157:
2155:
2154:
2149:
2147:
2146:
2127:
2125:
2124:
2119:
2117:
2116:
2107:
2106:
2105:
2095:
2084:
2083:
2049:
2047:
2046:
2041:
2039:
2031:
2023:
2018:
2017:
2016:
1999:
1997:
1996:
1991:
1975:
1973:
1972:
1967:
1962:
1945:
1944:
1928:
1926:
1925:
1920:
1918:
1917:
1883:
1881:
1880:
1875:
1873:
1872:
1863:
1833:
1831:
1830:
1825:
1823:
1822:
1813:
1786:
1784:
1783:
1778:
1773:
1772:
1757:
1756:
1711:
1710:
1692:
1691:
1666:
1664:
1663:
1658:
1636:
1634:
1633:
1628:
1616:
1614:
1613:
1608:
1606:
1605:
1590:
1589:
1548:
1546:
1545:
1540:
1522:
1520:
1519:
1514:
1475:
1473:
1472:
1467:
1462:
1458:
1457:
1456:
1452:
1397:
1396:
1392:
1281:
1276:
1275:
1266:
1227:as an effective
1226:
1224:
1223:
1218:
1213:
1202:
1180:
1178:
1177:
1172:
1146:
1128:
1126:
1125:
1120:
1090:
1088:
1087:
1082:
1035:
1034:
1030:
1029:
1001:
999:
998:
993:
991:
990:
986:
985:
958:
956:
955:
950:
921:
905:
903:
902:
897:
885:
883:
882:
877:
856:
854:
853:
848:
833:is the particle
832:
830:
829:
824:
819:
818:
806:
805:
790:
778:
776:
775:
770:
768:
757:
749:
726:
724:
723:
718:
716:
704:
702:
701:
696:
694:
680:
668:
666:
665:
660:
658:
657:
640:crystal momentum
637:
635:
634:
629:
624:
623:
611:
610:
595:
583:
581:
580:
575:
570:
548:
546:
545:
540:
535:
534:
530:
529:
506:
505:
501:
500:
480:
479:
475:
474:
451:
450:
446:
445:
425:
420:
419:
410:
399:
398:
377:
376:
358:
357:
342:
309:
307:
306:
301:
285:
283:
282:
277:
259:
257:
256:
251:
226:bilayer graphene
161:Hall conductance
21:
4186:
4185:
4181:
4180:
4179:
4177:
4176:
4175:
4146:
4145:
4144:
4139:
4122:
4088:
4074:Six nines in pi
4039:BlooP and FlooP
4027:
4025:
4018:
4009:
4004:
3994:
3984:
3974:
3964:
3954:
3944:
3926:
3921:
3891:
3890:
3848:
3847:
3843:
3809:
3808:
3804:
3731:
3730:
3726:
3659:
3658:
3654:
3595:
3594:
3590:
3523:
3522:
3518:
3459:
3458:
3454:
3404:
3403:
3399:
3367:
3366:
3362:
3326:
3325:
3321:
3301:
3300:
3296:
3267:(10): 874–878.
3254:
3253:
3249:
3213:
3212:
3199:
3164:
3163:
3136:
3131:
3119:
3094:
3088:
3085:
3078:needs expansion
3063:
3039:
3038:
3019:
3018:
2997:
2992:
2991:
2967:
2962:
2961:
2938:
2937:
2912:
2911:
2883:
2882:
2861:
2856:
2855:
2832:
2831:
2812:
2811:
2783:
2782:
2763:
2762:
2743:
2742:
2677:
2638:
2614:
2613:
2569:
2559:
2558:
2544:Gregory Wannier
2512:
2511:
2492:
2491:
2468:
2467:
2448:
2447:
2444:rational number
2410:
2409:
2390:
2389:
2370:
2369:
2347:
2346:
2321:
2320:
2288:
2287:
2264:
2263:
2244:
2243:
2231:
2207:
2206:
2185:
2180:
2179:
2160:
2159:
2138:
2133:
2132:
2108:
2096:
2072:
2052:
2051:
2007:
2002:
2001:
1982:
1981:
1936:
1931:
1930:
1909:
1886:
1885:
1864:
1836:
1835:
1814:
1792:
1791:
1764:
1748:
1696:
1677:
1672:
1671:
1643:
1642:
1619:
1618:
1591:
1581:
1555:
1554:
1525:
1524:
1481:
1480:
1425:
1365:
1286:
1282:
1267:
1236:
1235:
1187:
1186:
1131:
1130:
1093:
1092:
1021:
1013:
1008:
1007:
977:
969:
964:
963:
912:
911:
888:
887:
859:
858:
839:
838:
810:
797:
781:
780:
737:
736:
707:
706:
671:
670:
649:
644:
643:
615:
602:
586:
585:
555:
554:
521:
510:
492:
484:
466:
455:
437:
429:
411:
390:
368:
349:
327:
326:
292:
291:
262:
261:
242:
241:
234:
169:
95:Bloch electrons
80:
54:. The fractal,
28:
23:
22:
15:
12:
11:
5:
4184:
4182:
4174:
4173:
4168:
4163:
4158:
4148:
4147:
4141:
4140:
4138:
4137:
4134:
4127:
4124:
4123:
4121:
4120:
4113:
4108:
4103:
4096:
4094:
4090:
4089:
4087:
4086:
4081:
4076:
4071:
4066:
4061:
4056:
4051:
4046:
4041:
4036:
4030:
4028:
4023:
4020:
4019:
4012:
4010:
4008:
4007:
3997:
3987:
3977:
3967:
3957:
3947:
3936:
3934:
3928:
3927:
3922:
3920:
3919:
3912:
3905:
3897:
3889:
3888:
3861:(5): 613–618.
3841:
3822:(2): 757–765.
3802:
3724:
3652:
3588:
3516:
3452:
3417:(1): 147–150.
3397:
3360:
3339:(6): 405–408.
3319:
3294:
3247:
3197:
3133:
3132:
3130:
3127:
3126:
3125:
3118:
3115:
3096:
3095:
3089:September 2020
3075:
3073:
3062:
3059:
3046:
3026:
3004:
3000:
2974:
2970:
2945:
2925:
2922:
2919:
2899:
2896:
2893:
2890:
2868:
2864:
2839:
2819:
2799:
2796:
2793:
2790:
2770:
2750:
2739:
2738:
2727:
2723:
2719:
2716:
2713:
2710:
2705:
2700:
2697:
2693:
2689:
2684:
2680:
2675:
2671:
2667:
2663:
2660:
2657:
2654:
2646:
2641:
2635:
2632:
2628:
2624:
2621:
2607:
2606:
2595:
2592:
2589:
2586:
2583:
2576:
2572:
2568:
2525:
2522:
2519:
2499:
2475:
2455:
2431:
2427:
2423:
2420:
2417:
2397:
2377:
2354:
2334:
2331:
2328:
2306:
2303:
2298:
2295:
2271:
2251:
2230:
2227:
2214:
2192:
2188:
2167:
2145:
2141:
2115:
2111:
2104:
2099:
2094:
2090:
2087:
2082:
2079:
2075:
2071:
2068:
2065:
2062:
2059:
2037:
2034:
2030:
2026:
2021:
2015:
2010:
1989:
1965:
1961:
1957:
1954:
1951:
1948:
1943:
1939:
1916:
1912:
1908:
1905:
1902:
1899:
1896:
1893:
1871:
1867:
1862:
1858:
1855:
1852:
1849:
1846:
1843:
1821:
1817:
1812:
1808:
1805:
1802:
1799:
1788:
1787:
1776:
1771:
1767:
1763:
1760:
1755:
1751:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1709:
1706:
1703:
1699:
1695:
1690:
1687:
1684:
1680:
1656:
1653:
1650:
1626:
1604:
1601:
1598:
1594:
1588:
1584:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1538:
1535:
1532:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1477:
1476:
1465:
1461:
1455:
1451:
1447:
1444:
1441:
1438:
1435:
1432:
1428:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1395:
1391:
1387:
1384:
1381:
1378:
1375:
1372:
1368:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1285:
1279:
1274:
1270:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1216:
1212:
1208:
1205:
1201:
1197:
1194:
1185:. One can use
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1145:
1141:
1138:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1033:
1028:
1024:
1020:
1016:
989:
984:
980:
976:
972:
948:
945:
942:
939:
936:
933:
930:
927:
924:
920:
895:
875:
872:
869:
866:
846:
822:
817:
813:
809:
804:
800:
796:
793:
789:
767:
763:
760:
756:
752:
748:
744:
715:
693:
689:
686:
683:
679:
656:
652:
627:
622:
618:
614:
609:
605:
601:
598:
594:
573:
569:
565:
562:
551:
550:
538:
533:
528:
524:
520:
517:
513:
509:
504:
499:
495:
491:
487:
483:
478:
473:
469:
465:
462:
458:
454:
449:
444:
440:
436:
432:
428:
423:
418:
414:
408:
405:
402:
397:
393:
389:
386:
383:
380:
375:
371:
367:
364:
361:
356:
352:
348:
345:
341:
337:
334:
299:
275:
272:
269:
249:
233:
230:
168:
165:
157:Chern integers
115:square lattice
84:Rudolf Peierls
79:
76:
48:magnetic field
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4183:
4172:
4169:
4167:
4164:
4162:
4159:
4157:
4154:
4153:
4151:
4135:
4133:
4129:
4128:
4125:
4119:
4118:
4114:
4112:
4109:
4107:
4104:
4101:
4098:
4097:
4095:
4091:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4031:
4029:
4021:
4016:
4003:
4002:
3998:
3993:
3992:
3988:
3983:
3982:
3978:
3973:
3972:
3968:
3963:
3962:
3958:
3953:
3952:
3948:
3943:
3942:
3938:
3937:
3935:
3933:
3929:
3925:
3918:
3913:
3911:
3906:
3904:
3899:
3898:
3895:
3884:
3880:
3876:
3872:
3868:
3864:
3860:
3856:
3852:
3845:
3842:
3837:
3833:
3829:
3825:
3821:
3817:
3813:
3806:
3803:
3798:
3794:
3789:
3784:
3780:
3776:
3771:
3766:
3762:
3758:
3753:
3748:
3744:
3740:
3736:
3728:
3725:
3720:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3688:
3684:
3679:
3674:
3670:
3666:
3665:
3656:
3653:
3648:
3644:
3640:
3636:
3632:
3628:
3624:
3620:
3615:
3610:
3606:
3602:
3601:
3592:
3589:
3584:
3580:
3576:
3572:
3568:
3564:
3560:
3556:
3552:
3548:
3543:
3538:
3534:
3530:
3529:
3520:
3517:
3512:
3508:
3504:
3500:
3496:
3492:
3488:
3484:
3479:
3474:
3470:
3466:
3465:
3456:
3453:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3416:
3412:
3408:
3401:
3398:
3393:
3389:
3385:
3381:
3377:
3373:
3372:
3364:
3361:
3355:
3350:
3346:
3342:
3338:
3334:
3330:
3323:
3320:
3316:(3): 634–645.
3315:
3311:
3310:
3305:
3298:
3295:
3290:
3286:
3282:
3278:
3274:
3270:
3266:
3262:
3258:
3251:
3248:
3242:
3237:
3233:
3229:
3225:
3221:
3220:Physics Today
3217:
3210:
3208:
3206:
3204:
3202:
3198:
3193:
3189:
3185:
3181:
3177:
3173:
3172:
3167:
3161:
3159:
3157:
3155:
3153:
3151:
3149:
3147:
3145:
3143:
3141:
3139:
3135:
3128:
3124:
3121:
3120:
3116:
3114:
3112:
3111:Chern numbers
3107:
3103:
3102:phase diagram
3092:
3083:
3079:
3076:This section
3074:
3071:
3067:
3066:
3060:
3058:
3044:
3024:
3002:
2998:
2988:
2972:
2968:
2959:
2943:
2923:
2920:
2917:
2897:
2894:
2891:
2888:
2866:
2862:
2853:
2837:
2817:
2794:
2788:
2768:
2748:
2725:
2714:
2708:
2703:
2698:
2695:
2691:
2687:
2682:
2678:
2673:
2669:
2658:
2652:
2639:
2633:
2630:
2626:
2622:
2619:
2612:
2611:
2610:
2593:
2590:
2587:
2584:
2581:
2574:
2570:
2566:
2557:
2556:
2555:
2553:
2549:
2545:
2541:
2539:
2538:Landau levels
2523:
2520:
2517:
2497:
2489:
2488:prime numbers
2486:are distinct
2473:
2453:
2445:
2429:
2425:
2421:
2418:
2415:
2395:
2375:
2366:
2352:
2332:
2329:
2326:
2304:
2301:
2296:
2293:
2286:in the lines
2285:
2269:
2249:
2235:
2226:
2212:
2190:
2186:
2165:
2143:
2139:
2129:
2113:
2097:
2092:
2088:
2080:
2077:
2069:
2066:
2060:
2057:
2035:
2032:
2028:
2019:
2008:
1987:
1979:
1963:
1959:
1952:
1949:
1946:
1941:
1937:
1914:
1910:
1906:
1903:
1897:
1891:
1869:
1865:
1860:
1853:
1847:
1844:
1841:
1819:
1815:
1810:
1806:
1803:
1800:
1797:
1774:
1769:
1765:
1761:
1758:
1753:
1749:
1742:
1739:
1736:
1733:
1730:
1727:
1721:
1718:
1715:
1712:
1707:
1704:
1701:
1697:
1693:
1688:
1685:
1682:
1678:
1670:
1669:
1668:
1654:
1651:
1648:
1640:
1624:
1602:
1599:
1596:
1592:
1586:
1582:
1578:
1572:
1569:
1566:
1560:
1552:
1536:
1533:
1530:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1463:
1459:
1449:
1445:
1442:
1439:
1436:
1433:
1430:
1426:
1419:
1416:
1413:
1410:
1407:
1401:
1398:
1389:
1385:
1382:
1379:
1376:
1373:
1370:
1366:
1359:
1356:
1353:
1350:
1347:
1341:
1338:
1332:
1329:
1326:
1323:
1320:
1314:
1311:
1305:
1302:
1299:
1296:
1293:
1287:
1283:
1277:
1272:
1268:
1262:
1256:
1253:
1250:
1244:
1241:
1234:
1233:
1232:
1230:
1206:
1203:
1192:
1184:
1183:wave function
1165:
1162:
1159:
1153:
1150:
1136:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1075:
1072:
1069:
1066:
1063:
1057:
1054:
1048:
1045:
1042:
1036:
1031:
1026:
1022:
1018:
1014:
1005:
987:
982:
978:
974:
970:
960:
943:
940:
937:
934:
931:
928:
922:
909:
893:
873:
870:
867:
864:
844:
836:
815:
811:
807:
802:
798:
791:
761:
758:
734:
730:
687:
681:
654:
650:
641:
620:
616:
612:
607:
603:
596:
560:
531:
526:
522:
518:
515:
511:
507:
502:
497:
493:
489:
485:
481:
476:
471:
467:
463:
460:
456:
452:
447:
442:
438:
434:
430:
421:
416:
412:
406:
400:
395:
391:
387:
384:
381:
378:
373:
369:
365:
362:
354:
350:
346:
332:
325:
324:
323:
321:
317:
316:tight binding
313:
297:
273:
270:
267:
247:
238:
231:
229:
227:
222:
220:
216:
215:CQT Singapore
211:
209:
208:moiré pattern
205:
204:boron nitride
201:
196:
194:
190:
186:
181:
173:
166:
164:
162:
158:
154:
150:
148:
147:
142:
138:
137:
132:
128:
124:
120:
116:
112:
107:
105:
101:
96:
92:
91:energy levels
87:
85:
77:
75:
73:
69:
64:
61:
57:
53:
49:
45:
41:
32:
19:
4115:
4079:Strange loop
4048:
4024:Concepts and
3999:
3989:
3979:
3969:
3959:
3951:The Mind's I
3949:
3939:
3858:
3854:
3844:
3819:
3815:
3805:
3742:
3738:
3727:
3668:
3662:
3655:
3604:
3598:
3591:
3532:
3526:
3519:
3468:
3462:
3455:
3414:
3410:
3400:
3375:
3369:
3363:
3336:
3332:
3322:
3313:
3307:
3297:
3264:
3260:
3250:
3226:(8): 38–42.
3223:
3219:
3175:
3169:
3099:
3086:
3082:adding to it
3077:
2989:
2852:Fermi energy
2740:
2608:
2542:
2367:
2345:. Note that
2241:
2130:
2050:, such that
1789:
1478:
961:
552:
289:
223:
212:
197:
188:
182:
178:
167:Confirmation
151:
144:
134:
108:
103:
88:
81:
65:
56:self-similar
43:
37:
4171:Hall effect
3567:10261/93894
1229:Hamiltonian
962:Using that
123:recursively
119:fractal set
4150:Categories
3752:2006.13963
3678:1709.07108
3129:References
1006:, so that
100:Mark Azbel
4064:MU puzzle
3883:0002-9505
3779:0027-8424
3719:206662292
3703:0036-8075
3614:1303.6942
3542:1212.5012
3511:119210000
3478:1212.4783
3439:0031-9007
3289:0370-1298
3025:α
3003:α
2999:ϵ
2973:α
2969:ϵ
2944:α
2918:ϵ
2895:−
2889:ϵ
2818:α
2795:ϵ
2789:ρ
2726:ϵ
2715:ϵ
2709:ρ
2696:−
2692:∫
2670:ϵ
2659:ϵ
2653:ρ
2640:ϵ
2631:−
2627:∫
2594:α
2521:≫
2446:), where
2416:α
2376:α
2353:ϵ
2327:ϵ
2294:α
2270:α
2250:α
2213:ϵ
2191:α
2187:ϵ
2166:α
2144:α
2140:ϵ
2078:−
2070:π
2058:α
2025:ℏ
1988:α
1956:ℏ
1953:π
1938:ϕ
1892:ϕ
1866:ϕ
1848:ϕ
1842:α
1798:ϵ
1762:ϵ
1743:ν
1740:−
1737:α
1731:π
1722:
1705:−
1649:λ
1625:ν
1600:ν
1561:ψ
1454:ℏ
1417:−
1402:ψ
1394:ℏ
1371:−
1342:ψ
1324:−
1315:ψ
1288:ψ
1245:ψ
1204:−
1154:ψ
1137:ψ
1058:ψ
1037:ψ
871:−
759:−
751:→
743:ℏ
688:×
685:∇
516:−
461:−
388:
366:
274:α
271:π
248:ϵ
4156:Fractals
4102:(father)
4034:Ambigram
4026:projects
3797:34301893
3711:29191906
3647:37694594
3639:23686343
3575:23676678
3503:23676673
3447:11136115
3117:See also
2178:, where
1617:, where
1523:, where
1091:, where
779:, where
705:, where
200:graphene
127:patterns
125:repeats
4093:Related
4044:Copycat
3863:Bibcode
3824:Bibcode
3788:8325360
3757:Bibcode
3683:Bibcode
3664:Science
3619:Bibcode
3600:Science
3583:4431176
3547:Bibcode
3483:Bibcode
3419:Bibcode
3380:Bibcode
3341:Bibcode
3269:Bibcode
3228:Bibcode
3180:Bibcode
2830:. Here
1976:is the
1002:is the
906:is the
638:is the
318:energy
78:History
52:lattice
4005:(2013)
3995:(2007)
3985:(1997)
3975:(1995)
3965:(1985)
3955:(1981)
3945:(1979)
3881:
3795:
3785:
3777:
3717:
3709:
3701:
3645:
3637:
3581:
3573:
3528:Nature
3509:
3501:
3464:Nature
3445:
3437:
3287:
2956:is an
2854:, and
2741:where
2609:where
1790:where
1551:ansatz
642:, and
553:where
219:qubits
189:et al.
3932:Books
3747:arXiv
3715:S2CID
3673:arXiv
3643:S2CID
3609:arXiv
3579:S2CID
3537:arXiv
3507:S2CID
3473:arXiv
2319:and
50:in a
3879:ISSN
3793:PMID
3775:ISSN
3707:PMID
3699:ISSN
3635:PMID
3571:PMID
3499:PMID
3443:PMID
3435:ISSN
3285:ISSN
3100:The
2761:and
2466:and
1834:and
1641:for
1129:and
837:and
727:the
3871:doi
3832:doi
3783:PMC
3765:doi
3743:118
3691:doi
3669:358
3627:doi
3605:340
3563:hdl
3555:doi
3533:497
3491:doi
3469:497
3427:doi
3388:doi
3349:doi
3277:doi
3236:doi
3188:doi
3084:.
2910:to
2442:(a
1719:cos
1667:):
385:cos
363:cos
322:is
104:all
93:of
38:In
4152::
3877:.
3869:.
3859:72
3857:.
3853:.
3830:.
3820:88
3818:.
3814:.
3791:.
3781:.
3773:.
3763:.
3755:.
3741:.
3737:.
3713:.
3705:.
3697:.
3689:.
3681:.
3667:.
3641:.
3633:.
3625:.
3617:.
3603:.
3577:.
3569:.
3561:.
3553:.
3545:.
3531:.
3505:.
3497:.
3489:.
3481:.
3467:.
3441:.
3433:.
3425:.
3415:86
3413:.
3409:.
3386:.
3376:80
3374:.
3347:.
3337:49
3335:.
3331:.
3314:19
3312:.
3306:.
3283:.
3275:.
3265:68
3263:.
3259:.
3234:.
3224:53
3222:.
3218:.
3200:^
3186:.
3176:14
3174:.
3137:^
3113:.
2987:.
2540:.
2128:.
1884:,
1553::
959:.
74:.
42:,
3916:e
3909:t
3902:v
3885:.
3873::
3865::
3838:.
3834::
3826::
3799:.
3767::
3759::
3749::
3721:.
3693::
3685::
3675::
3649:.
3629::
3621::
3611::
3585:.
3565::
3557::
3549::
3539::
3513:.
3493::
3485::
3475::
3449:.
3429::
3421::
3394:.
3390::
3382::
3357:.
3351::
3343::
3291:.
3279::
3271::
3244:.
3238::
3230::
3194:.
3190::
3182::
3091:)
3087:(
3045:B
2924:4
2921:=
2898:4
2892:=
2867:0
2863:n
2838:n
2798:)
2792:(
2769:T
2749:S
2722:d
2718:)
2712:(
2704:4
2699:4
2688:=
2683:0
2679:n
2674:;
2666:d
2662:)
2656:(
2645:F
2634:4
2623:=
2620:n
2591:T
2588:+
2585:S
2582:=
2575:0
2571:n
2567:n
2524:P
2518:Q
2498:Q
2474:Q
2454:P
2430:Q
2426:/
2422:P
2419:=
2396:n
2333:0
2330:=
2305:2
2302:1
2297:=
2114:2
2110:)
2103:m
2098:l
2093:/
2089:a
2086:(
2081:1
2074:)
2067:2
2064:(
2061:=
2036:B
2033:e
2029:/
2020:=
2014:m
2009:l
1964:q
1960:/
1950:2
1947:=
1942:0
1915:2
1911:a
1907:B
1904:=
1901:)
1898:B
1895:(
1870:0
1861:/
1857:)
1854:B
1851:(
1845:=
1820:0
1816:E
1811:/
1807:E
1804:2
1801:=
1775:,
1770:n
1766:g
1759:=
1754:n
1750:g
1746:)
1734:n
1728:2
1725:(
1716:2
1713:+
1708:1
1702:n
1698:g
1694:+
1689:1
1686:+
1683:n
1679:g
1655:1
1652:=
1603:m
1597:i
1593:e
1587:n
1583:g
1579:=
1576:)
1573:y
1570:,
1567:x
1564:(
1537:m
1534:,
1531:n
1511:a
1508:m
1505:=
1502:y
1499:,
1496:a
1493:n
1490:=
1487:x
1464:.
1460:]
1450:/
1446:a
1443:x
1440:B
1437:q
1434:i
1431:+
1427:e
1423:)
1420:a
1414:y
1411:,
1408:x
1405:(
1399:+
1390:/
1386:a
1383:x
1380:B
1377:q
1374:i
1367:e
1363:)
1360:a
1357:+
1354:y
1351:,
1348:x
1345:(
1339:+
1336:)
1333:y
1330:,
1327:a
1321:x
1318:(
1312:+
1309:)
1306:y
1303:,
1300:a
1297:+
1294:x
1291:(
1284:[
1278:2
1273:0
1269:E
1263:=
1260:)
1257:y
1254:,
1251:x
1248:(
1242:E
1215:)
1211:A
1207:q
1200:p
1196:(
1193:W
1169:)
1166:y
1163:,
1160:x
1157:(
1151:=
1148:)
1144:r
1140:(
1117:z
1114:,
1111:y
1108:,
1105:x
1102:=
1099:j
1079:)
1076:y
1073:,
1070:a
1067:+
1064:x
1061:(
1055:=
1052:)
1049:y
1046:,
1043:x
1040:(
1032:a
1027:j
1023:p
1019:i
1015:e
988:a
983:j
979:p
975:i
971:e
947:)
944:0
941:,
938:x
935:B
932:,
929:0
926:(
923:=
919:A
894:e
874:e
868:=
865:q
845:q
821:)
816:y
812:p
808:,
803:x
799:p
795:(
792:=
788:p
766:A
762:q
755:p
747:k
714:A
692:A
682:=
678:B
655:0
651:E
626:)
621:y
617:k
613:,
608:x
604:k
600:(
597:=
593:k
572:)
568:k
564:(
561:W
549:,
537:)
532:a
527:y
523:k
519:i
512:e
508:+
503:a
498:y
494:k
490:i
486:e
482:+
477:a
472:x
468:k
464:i
457:e
453:+
448:a
443:x
439:k
435:i
431:e
427:(
422:2
417:0
413:E
407:=
404:)
401:a
396:y
392:k
382:+
379:a
374:x
370:k
360:(
355:0
351:E
347:=
344:)
340:k
336:(
333:W
298:a
286:.
268:2
20:)
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