86:). So unless there are cancellations arising from some symmetry of the system, the quantum-mechanical sum over all multi-particle states involves an integral over a function that is highly oscillatory, hence hard to evaluate numerically, particularly in high dimension. Since the dimension of the integral is given by the number of particles, the sign problem becomes severe in the
1317:
73:
In physics the sign problem is typically (but not exclusively) encountered in calculations of the properties of a quantum mechanical system with large number of strongly interacting fermions, or in field theories involving a non-zero density of strongly interacting fermions. Because the particles are
1395:
is a highly oscillatory function in the configuration space, so if one uses Monte Carlo methods to evaluate the numerator and denominator, each of them will evaluate to a very small number, whose exact value is swamped by the noise inherent in the Monte Carlo sampling process. The "badness" of the
945:
A field theory with a non-positive weight can be transformed to one with a positive weight by incorporating the non-positive part (sign or complex phase) of the weight into the observable. For example, one could decompose the weighting function into its modulus and phase:
1062:
1885:
In systems with a moderate sign problem, such as field theories at a sufficiently high temperature or in a sufficiently small volume, the sign problem is not too severe and useful results can be obtained by various methods, such as more carefully tuned reweighting,
711:
1881:
solution to the sign problem. This leaves open the possibility that there may be solutions that work in specific cases, where the oscillations of the integrand have a structure that can be exploited to reduce the numerical errors.
1873:, implying that a full and generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time. If (as is generally suspected) there are no polynomial-time solutions to NP problems (see
1856:
1615:
The decomposition of the weighting function into modulus and phase is just one example (although it has been advocated as the optimal choice since it minimizes the variance of the denominator). In general one could write
1546:
1611:
is an energy density. The number of Monte Carlo sampling points needed to obtain an accurate result therefore rises exponentially as the volume of the system becomes large, and as the temperature goes to zero.
2058:
These achieve an exponential speed-up by decomposing the fermion world lines into clusters that contribute independently. Cluster algorithms have been developed for certain theories, but not for the
1312:{\displaystyle \langle A\rangle _{\rho }={\frac {\int D\sigma A\exp(i\theta )\,p}{\int D\sigma \exp(i\theta )\,p}}={\frac {\langle A\exp(i\theta )\rangle _{p}}{\langle \exp(i\theta )\rangle _{p}}}.}
488:
1451:
2108:
One fixes the location of nodes (zeros) of the multiparticle wavefunction, and uses Monte Carlo methods to obtain an estimate of the energy of the ground state, subject to that constraint.
2119:
Stochastically and strategically sampling
Feynman diagrams can also render the sign problem more tractable for a Monte Carlo approach which would otherwise be computationally unworkable.
1696:
776:
1022:
341:
902:
546:
1393:
931:
2172:
Loh, E. Y.; Gubernatis, J. E.; Scalettar, R. T.; White, S. R.; Scalapino, D. J.; Sugar, R. L. (1990). "Sign problem in the numerical simulation of many-electron systems".
858:
396:
612:
367:
238:
286:
1754:
2045:
1998:
812:
743:
604:
2787:
Li, Zi-Xiang; Jiang, Yi-Fan; Yao, Hong (2016). "Majorana-Time-Reversal
Symmetries: A Fundamental Principle for Sign-Problem-Free Quantum Monte Carlo Simulations".
1952:
1928:
1908:
832:
180:
2288:
Anagnostopoulos, K. N.; Nishimura, J. (2002). "New approach to the complex-action problem and its application to a nonperturbative study of superstring theory".
2018:
1728:
1609:
1589:
1569:
1349:
1054:
575:
511:
258:
203:
1322:
Note that the desired expectation value is now a ratio where the numerator and denominator are expectation values that both use a positive weighting function
1762:
183:
289:
2734:
Li, Zi-Xiang; Jiang, Yi-Fan; Yao, Hong (2015). "Solving the fermion sign problem in quantum Monte Carlo simulations by
Majorana representation".
2516:
Troyer, Matthias; Wiese, Uwe-Jens (2005). "Computational
Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations".
2999:
1459:
42:
fail because of the near-cancellation of the positive and negative contributions to the integral. Each has to be integrated to very high
2097:
2084:. So far, the algorithm has been found to evade the sign problem in test models that have a sign problem but do not involve fermions.
2673:
Aarts, Gert (2009). "Can
Stochastic Quantization Evade the Sign Problem? The Relativistic Bose Gas at Finite Chemical Potential".
2912:
Houcke, Kris Van; Kozik, Evgeny; Prokof'ev, Nikolay V.; Svistunov, Boris
Vladimirovich (2010-01-01). "Diagrammatic Monte Carlo".
93:
The sign problem is one of the major unsolved problems in the physics of many-particle systems, impeding progress in many areas:
2152:
Sources for this section include
Chandrasekharan & Wiese (1999) and Kieu & Griffin (1994), in addition to those cited.
2989:
404:
2463:
Barbour, I. M.; Morrison, S. E.; Klepfish, E. G.; Kogut, J. B.; Lombardo, M.-P. (1998). "Results on Finite
Density QCD".
2994:
2849:
1399:
2853:
264:
of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the
2636:
Alexandru, Andrei; Basar, Gokce; Bedaque, Paulo; Warrington, Neill (2022). "Complex paths around the sign problem".
159:
In a field-theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion
2129:
78:
is inapplicable, and one is forced to use brute-force numerical methods. Because the particles are fermions, their
100:— It prevents the numerical solution of systems with a high density of strongly correlated electrons, such as the
778:
can be calculated by performing the sum over field configurations numerically, using standard techniques such as
1622:
2114:
748:
97:
952:
548:
is a matrix that encodes how the fermions were coupled to the bosons. The expectation value of an observable
298:
2075:
779:
2402:
Kieu, T. D.; Griffin, C. J. (1994). "Monte Carlo simulations with indefinite and complex-valued measures".
863:
2063:
47:
43:
35:
2080:
The sum over configurations is obtained as the equilibrium distribution of states explored by a complex
1887:
82:
changes sign when any two fermions are interchanged (due to the anti-symmetry of the wave function, see
2346:
Chandrasekharan, Shailesh; Wiese, Uwe-Jens (1999). "Meron-Cluster
Solution of Fermion Sign Problems".
2931:
2868:
2806:
2753:
2692:
2600:
2535:
2482:
2421:
2365:
2307:
2234:
2181:
2134:
516:
143:
131:
54:
1354:
907:
1970:
1874:
814:
is non-positive. This typically occurs in theories of fermions when the fermion chemical potential
87:
75:
20:
706:{\displaystyle \langle A\rangle _{\rho }={\frac {\int D\sigma \,A\,\rho }{\int D\sigma \,\rho }}.}
2965:
2921:
2830:
2796:
2769:
2743:
2716:
2682:
2645:
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2590:
2559:
2525:
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2445:
2411:
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2355:
2323:
2297:
2224:
160:
58:
837:
372:
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2894:
2822:
2708:
2551:
2437:
2197:
2081:
349:
208:
39:
271:
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2939:
2884:
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2700:
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2608:
2543:
2490:
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2315:
2268:
2189:
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1733:
261:
2023:
1976:
788:
719:
580:
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1937:
1913:
1893:
817:
165:
107:
83:
2935:
2872:
2810:
2757:
2696:
2604:
2539:
2486:
2425:
2369:
2311:
2238:
2185:
2003:
1704:
1594:
1574:
1554:
1325:
1030:
934:
551:
496:
243:
188:
121:
117:
2494:
65:, or in field theories involving a non-zero density of strongly interacting fermions.
2983:
2385:
2327:
2059:
1931:
1851:{\displaystyle \left\langle {\frac {\rho }{p}}\right\rangle _{p}\propto \exp(-fV/T),}
101:
2969:
2834:
2773:
2720:
2622:
2563:
2502:
2449:
2100:
can help to solve the fermion sign problem in a class of fermionic many-body models.
2818:
2704:
1730:
can be any positive weighting function (for example, the weighting function of the
1453:: if it is much less than 1, then the sign problem is severe. It can be shown that
139:
125:
79:
2547:
2659:
2943:
2880:
2377:
135:
32:
2765:
2319:
288:(which may have been originally part of the theory, or have been produced by a
937:, so Monte Carlo importance sampling cannot be used to evaluate the integral.
2961:
2433:
2193:
2257:"Lattice calculations at non-zero chemical potential: The QCD phase diagram"
1962:
There are various proposals for solving systems with a severe sign problem:
834:
is nonzero, i.e. when there is a nonzero background density of fermions. If
112:
90:. The field-theoretic manifestation of the sign problem is discussed below.
2952:
2898:
2826:
2712:
2555:
2441:
2201:
2530:
2360:
53:
The sign problem is one of the major unsolved problems in the physics of
28:
2613:
2595:
2578:
2477:
2416:
2273:
2256:
2302:
1870:
62:
1541:{\displaystyle \langle \exp(i\theta )\rangle _{p}\propto \exp(-fV/T),}
745:
is positive, then it can be interpreted as a probability measure, and
2889:
2801:
2650:
2926:
2748:
2687:
2229:
2215:
de
Forcrand, Philippe (2010). "Simulating QCD at finite density".
1861:
which again goes to zero exponentially in the large-volume limit.
265:
205:
by summing over all classical field configurations, weighted by
1756:
theory). The badness of the sign problem is then measured by
1396:
sign problem is measured by the smallness of the denominator
2854:"Fixed-Node Quantum Monte Carlo Method for Lattice Fermions"
2848:
Van Bemmel, H. J. M.; Ten Haaf, D. F. B.; Van Saarloos, W.;
577:
is therefore an average over all configurations weighted by
369:
represents the measure for the sum over all configurations
57:. It often arises in calculations of the properties of a
46:
in order for their difference to be obtained with useful
2026:
2006:
1979:
1940:
1916:
1896:
1765:
1736:
1707:
1625:
1597:
1577:
1557:
1462:
1402:
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1065:
1033:
955:
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820:
791:
751:
722:
615:
583:
554:
519:
499:
407:
375:
352:
301:
274:
246:
211:
191:
168:
483:{\displaystyle \rho =\det(M(\mu ,\sigma ))\exp(-S),}
2039:
2012:
1992:
1946:
1922:
1902:
1850:
1748:
1722:
1690:
1603:
1583:
1563:
1540:
1446:{\displaystyle \langle \exp(i\theta )\rangle _{p}}
1445:
1387:
1343:
1311:
1048:
1016:
925:
896:
860:, there is no particle–antiparticle symmetry, and
852:
826:
806:
770:
737:
705:
598:
569:
540:
505:
482:
390:
361:
335:
280:
252:
232:
197:
174:
61:system with large number of strongly interacting
867:
423:
2341:
2339:
2337:
513:is now the action of the bosonic fields, and
27:is the problem of numerically evaluating the
8:
1494:
1463:
1434:
1403:
1294:
1263:
1252:
1209:
1073:
1066:
759:
752:
623:
616:
2465:Nuclear Physics B - Proceedings Supplements
2250:
2248:
2020:-dimensional manifold embedded in complex
1691:{\displaystyle \rho =p{\frac {\rho }{p}},}
2951:
2925:
2888:
2800:
2747:
2686:
2649:
2612:
2594:
2529:
2476:
2415:
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2005:
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1978:
1939:
1915:
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1807:
1771:
1764:
1735:
1706:
1653:
1624:
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1576:
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1437:
1401:
1356:
1327:
1297:
1255:
1206:
1187:
1136:
1085:
1076:
1064:
1032:
983:
954:
909:
865:
839:
819:
790:
771:{\displaystyle \langle A\rangle _{\rho }}
762:
750:
721:
684:
660:
647:
635:
626:
614:
582:
553:
518:
498:
406:
374:
351:
317:
300:
273:
245:
210:
190:
167:
1969:The field space is complexified and the
1017:{\displaystyle \rho =p\,\exp(i\theta ),}
138:to predict the phases and properties of
2164:
2145:
336:{\displaystyle Z=\int D\sigma \,\rho ,}
2397:
2395:
292:to make the fermion action quadratic)
120:and hence limits our understanding of
1865:Methods for reducing the sign problem
897:{\displaystyle \det(M(\mu ,\sigma ))}
7:
2098:Hubbard-Stratonovich transformations
398:of the bosonic fields, weighted by
290:Hubbard–Stratonovich transformation
146:, the problem is also known as the
14:
155:The sign problem in field theory
38:of a large number of variables.
2579:"Lattice QCD at Finite Density"
780:Monte Carlo importance sampling
541:{\displaystyle M(\mu ,\sigma )}
2819:10.1103/PhysRevLett.117.267002
2705:10.1103/PhysRevLett.102.131601
1842:
1822:
1797:
1791:
1783:
1777:
1717:
1711:
1679:
1673:
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1644:
1635:
1629:
1532:
1512:
1490:
1487:
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1472:
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1427:
1421:
1412:
1388:{\displaystyle \exp(i\theta )}
1382:
1379:
1373:
1364:
1338:
1332:
1290:
1287:
1281:
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1245:
1239:
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1115:
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1100:
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1005:
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990:
980:
974:
965:
959:
926:{\displaystyle \rho (\sigma )}
920:
914:
891:
888:
876:
870:
801:
795:
732:
726:
694:
688:
670:
664:
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523:
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327:
321:
227:
218:
16:Problem in applied mathematics
1:
2548:10.1103/PhysRevLett.94.170201
2495:10.1016/S0920-5632(97)00484-2
1571:is the volume of the system,
785:The sign problem arises when
116:calculation of properties of
3000:Unsolved problems in physics
2660:10.1103/RevModPhys.94.015006
2944:10.1016/j.phpro.2010.09.034
2881:10.1103/PhysRevLett.72.2442
2577:Schmidt, Christian (2006).
2378:10.1103/PhysRevLett.83.3116
3016:
2766:10.1103/PhysRevB.91.241117
2320:10.1103/PhysRevD.66.106008
2130:Method of stationary phase
2096:representation to perform
853:{\displaystyle \mu \neq 0}
391:{\displaystyle \sigma (x)}
2638:Reviews of Modern Physics
1056:is real and positive, so
134:— It prevents the use of
2434:10.1103/PhysRevE.49.3855
2194:10.1103/PhysRevB.41.9301
2115:Diagrammatic Monte Carlo
1958:List: Current Approaches
1591:is the temperature, and
362:{\displaystyle D\sigma }
233:{\displaystyle \exp(-S)}
98:Condensed matter physics
2861:Physical Review Letters
2789:Physical Review Letters
2675:Physical Review Letters
2518:Physical Review Letters
2348:Physical Review Letters
2106:Fixed-node Monte Carlo:
2076:Stochastic quantization
904:, and hence the weight
281:{\displaystyle \sigma }
2261:Proceedings of Science
2255:Philipsen, O. (2008).
2062:of electrons, nor for
2041:
2014:
1994:
1948:
1924:
1904:
1852:
1750:
1749:{\displaystyle \mu =0}
1724:
1692:
1605:
1585:
1565:
1542:
1447:
1389:
1345:
1313:
1050:
1018:
927:
898:
854:
828:
808:
772:
739:
707:
600:
571:
542:
507:
484:
392:
363:
337:
282:
254:
234:
199:
176:
148:complex action problem
74:strongly interacting,
25:numerical sign problem
2990:Statistical mechanics
2850:Van Leeuwen, J. M. J.
2069:the theory of quarks.
2042:
2040:{\displaystyle C^{N}}
2015:
1995:
1993:{\displaystyle R^{N}}
1971:path integral contour
1949:
1925:
1905:
1888:analytic continuation
1853:
1751:
1725:
1693:
1606:
1586:
1566:
1543:
1448:
1390:
1351:. However, the phase
1346:
1314:
1051:
1019:
941:Reweighting procedure
928:
899:
855:
829:
809:
807:{\displaystyle \rho }
773:
740:
738:{\displaystyle \rho }
708:
601:
599:{\displaystyle \rho }
572:
543:
508:
485:
393:
364:
338:
283:
255:
235:
200:
177:
55:many-particle systems
2135:Oscillatory integral
2090:Majorana algorithms:
2056:-cluster algorithms:
2024:
2004:
1977:
1967:Contour deformation:
1947:{\displaystyle \mu }
1938:
1923:{\displaystyle \mu }
1914:
1903:{\displaystyle \mu }
1894:
1877:), then there is no
1869:The sign problem is
1763:
1734:
1705:
1623:
1595:
1575:
1555:
1460:
1400:
1355:
1326:
1063:
1031:
953:
908:
864:
838:
827:{\displaystyle \mu }
818:
789:
749:
720:
613:
581:
552:
517:
497:
405:
373:
350:
299:
272:
244:
209:
189:
182:. One evaluates the
175:{\displaystyle \mu }
166:
144:lattice field theory
132:Quantum field theory
2995:Numerical artifacts
2936:2010PhPro...6...95V
2873:1994PhRvL..72.2442V
2811:2016PhRvL.117z7002L
2758:2015PhRvB..91x1117L
2697:2009PhRvL.102m1601A
2614:10.22323/1.032.0021
2605:2006slft.confE..21S
2540:2005PhRvL..94q0201T
2487:1998NuPhS..60..220B
2426:1994PhRvE..49.3855K
2370:1999PhRvL..83.3116C
2312:2002PhRvD..66j6008A
2274:10.22323/1.077.0011
2239:2010arXiv1005.0539D
2186:1990PhRvB..41.9301L
1875:P versus NP problem
88:thermodynamic limit
76:perturbation theory
21:applied mathematics
2037:
2010:
1990:
1944:
1920:
1900:
1848:
1746:
1720:
1688:
1601:
1581:
1561:
1538:
1443:
1385:
1341:
1309:
1046:
1014:
933:, is in general a
923:
894:
850:
824:
804:
768:
735:
703:
596:
567:
538:
503:
480:
388:
359:
333:
278:
250:
230:
195:
184:partition function
172:
161:chemical potential
110:— It prevents the
59:quantum mechanical
2867:(15): 2442–2445.
2852:; An, G. (1994).
2736:Physical Review B
2471:(1998): 220–233.
2404:Physical Review E
2354:(16): 3116–3119.
2290:Physical Review D
2180:(13): 9301–9307.
2174:Physical Review B
2082:Langevin equation
2013:{\displaystyle N}
1973:is deformed from
1801:
1723:{\displaystyle p}
1683:
1604:{\displaystyle f}
1584:{\displaystyle T}
1564:{\displaystyle V}
1344:{\displaystyle p}
1304:
1201:
1049:{\displaystyle p}
698:
570:{\displaystyle A}
506:{\displaystyle S}
253:{\displaystyle S}
198:{\displaystyle Z}
40:Numerical methods
3007:
2974:
2973:
2955:
2929:
2914:Physics Procedia
2909:
2903:
2902:
2892:
2858:
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2839:
2838:
2804:
2784:
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2777:
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2664:
2663:
2653:
2633:
2627:
2626:
2616:
2598:
2574:
2568:
2567:
2533:
2531:cond-mat/0408370
2513:
2507:
2506:
2480:
2460:
2454:
2453:
2419:
2410:(5): 3855–3859.
2399:
2390:
2389:
2363:
2361:cond-mat/9902128
2343:
2332:
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2305:
2285:
2279:
2278:
2276:
2252:
2243:
2242:
2232:
2212:
2206:
2205:
2169:
2153:
2150:
2094:Majorana fermion
2046:
2044:
2043:
2038:
2036:
2035:
2019:
2017:
2016:
2011:
1999:
1997:
1996:
1991:
1989:
1988:
1953:
1951:
1950:
1945:
1932:Taylor expansion
1929:
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1909:
1907:
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833:
831:
830:
825:
813:
811:
810:
805:
777:
775:
774:
769:
767:
766:
744:
742:
741:
736:
712:
710:
709:
704:
699:
697:
673:
636:
631:
630:
605:
603:
602:
597:
576:
574:
573:
568:
547:
545:
544:
539:
512:
510:
509:
504:
489:
487:
486:
481:
397:
395:
394:
389:
368:
366:
365:
360:
342:
340:
339:
334:
287:
285:
284:
279:
259:
257:
256:
251:
239:
237:
236:
231:
204:
202:
201:
196:
181:
179:
178:
173:
3015:
3014:
3010:
3009:
3008:
3006:
3005:
3004:
2980:
2979:
2978:
2977:
2953:1854/LU-3234513
2911:
2910:
2906:
2856:
2847:
2846:
2842:
2786:
2785:
2781:
2733:
2732:
2728:
2672:
2671:
2667:
2635:
2634:
2630:
2596:hep-lat/0610116
2576:
2575:
2571:
2515:
2514:
2510:
2478:hep-lat/9705042
2462:
2461:
2457:
2417:hep-lat/9311072
2401:
2400:
2393:
2345:
2344:
2335:
2287:
2286:
2282:
2254:
2253:
2246:
2214:
2213:
2209:
2171:
2170:
2166:
2161:
2156:
2151:
2147:
2143:
2126:
2027:
2022:
2021:
2002:
2001:
1980:
1975:
1974:
1960:
1936:
1935:
1912:
1911:
1892:
1891:
1890:from imaginary
1867:
1787:
1773:
1767:
1766:
1761:
1760:
1732:
1731:
1703:
1702:
1669:
1655:
1621:
1620:
1593:
1592:
1573:
1572:
1553:
1552:
1493:
1458:
1457:
1433:
1398:
1397:
1353:
1352:
1324:
1323:
1293:
1262:
1251:
1208:
1150:
1087:
1072:
1061:
1060:
1029:
1028:
951:
950:
943:
906:
905:
862:
861:
836:
835:
816:
815:
787:
786:
758:
747:
746:
718:
717:
674:
637:
622:
611:
610:
579:
578:
550:
549:
515:
514:
495:
494:
403:
402:
371:
370:
348:
347:
297:
296:
270:
269:
242:
241:
207:
206:
187:
186:
164:
163:
157:
108:Nuclear physics
84:Pauli principle
71:
17:
12:
11:
5:
3013:
3011:
3003:
3002:
2997:
2992:
2982:
2981:
2976:
2975:
2904:
2840:
2795:(26): 267002.
2779:
2742:(24): 241117.
2726:
2681:(13): 131601.
2665:
2628:
2569:
2524:(17): 170201.
2508:
2455:
2391:
2333:
2303:hep-th/0108041
2296:(10): 106008.
2280:
2244:
2207:
2163:
2162:
2160:
2157:
2155:
2154:
2144:
2142:
2139:
2138:
2137:
2132:
2125:
2122:
2121:
2120:
2110:
2109:
2102:
2101:
2086:
2085:
2071:
2070:
2049:
2048:
2034:
2030:
2009:
1987:
1983:
1959:
1956:
1943:
1919:
1899:
1866:
1863:
1859:
1858:
1847:
1844:
1841:
1837:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1810:
1805:
1799:
1796:
1793:
1790:
1785:
1782:
1779:
1776:
1770:
1745:
1742:
1739:
1719:
1716:
1713:
1710:
1699:
1698:
1687:
1681:
1678:
1675:
1672:
1667:
1664:
1661:
1658:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1600:
1580:
1560:
1549:
1548:
1537:
1534:
1531:
1527:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1500:
1496:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1440:
1436:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1340:
1337:
1334:
1331:
1320:
1319:
1308:
1300:
1296:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1258:
1254:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1205:
1199:
1196:
1193:
1190:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1148:
1145:
1142:
1139:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1084:
1079:
1075:
1071:
1068:
1045:
1042:
1039:
1036:
1025:
1024:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
982:
979:
976:
973:
970:
967:
964:
961:
958:
942:
939:
935:complex number
922:
919:
916:
913:
893:
890:
887:
884:
881:
878:
875:
872:
869:
849:
846:
843:
823:
803:
800:
797:
794:
765:
761:
757:
754:
734:
731:
728:
725:
714:
713:
702:
696:
693:
690:
687:
683:
680:
677:
672:
669:
666:
663:
659:
656:
653:
650:
646:
643:
640:
634:
629:
625:
621:
618:
595:
592:
589:
586:
566:
563:
560:
557:
537:
534:
531:
528:
525:
522:
502:
491:
490:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
387:
384:
381:
378:
358:
355:
344:
343:
332:
329:
326:
323:
320:
316:
313:
310:
307:
304:
277:
249:
229:
226:
223:
220:
217:
214:
194:
171:
156:
153:
152:
151:
129:
118:nuclear matter
105:
70:
67:
15:
13:
10:
9:
6:
4:
3:
2:
3012:
3001:
2998:
2996:
2993:
2991:
2988:
2987:
2985:
2971:
2967:
2963:
2959:
2954:
2949:
2945:
2941:
2937:
2933:
2928:
2923:
2919:
2915:
2908:
2905:
2900:
2896:
2891:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2855:
2851:
2844:
2841:
2836:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2803:
2798:
2794:
2790:
2783:
2780:
2775:
2771:
2767:
2763:
2759:
2755:
2750:
2745:
2741:
2737:
2730:
2727:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2689:
2684:
2680:
2676:
2669:
2666:
2661:
2657:
2652:
2647:
2643:
2639:
2632:
2629:
2624:
2620:
2615:
2610:
2606:
2602:
2597:
2592:
2588:
2584:
2580:
2573:
2570:
2565:
2561:
2557:
2553:
2549:
2545:
2541:
2537:
2532:
2527:
2523:
2519:
2512:
2509:
2504:
2500:
2496:
2492:
2488:
2484:
2479:
2474:
2470:
2466:
2459:
2456:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2423:
2418:
2413:
2409:
2405:
2398:
2396:
2392:
2387:
2383:
2379:
2375:
2371:
2367:
2362:
2357:
2353:
2349:
2342:
2340:
2338:
2334:
2329:
2325:
2321:
2317:
2313:
2309:
2304:
2299:
2295:
2291:
2284:
2281:
2275:
2270:
2266:
2262:
2258:
2251:
2249:
2245:
2240:
2236:
2231:
2226:
2222:
2218:
2211:
2208:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2168:
2165:
2158:
2149:
2146:
2140:
2136:
2133:
2131:
2128:
2127:
2123:
2118:
2116:
2112:
2111:
2107:
2104:
2103:
2099:
2095:
2091:
2088:
2087:
2083:
2079:
2077:
2073:
2072:
2068:
2065:
2061:
2060:Hubbard model
2057:
2055:
2051:
2050:
2032:
2028:
2007:
1985:
1981:
1972:
1968:
1965:
1964:
1963:
1957:
1955:
1941:
1934:in powers of
1933:
1917:
1897:
1889:
1883:
1880:
1876:
1872:
1864:
1862:
1845:
1839:
1835:
1831:
1828:
1825:
1819:
1816:
1813:
1808:
1803:
1794:
1788:
1780:
1774:
1768:
1759:
1758:
1757:
1743:
1740:
1737:
1714:
1708:
1685:
1676:
1670:
1662:
1656:
1647:
1641:
1638:
1632:
1626:
1619:
1618:
1617:
1613:
1598:
1578:
1558:
1535:
1529:
1525:
1521:
1518:
1515:
1509:
1506:
1503:
1498:
1484:
1478:
1475:
1469:
1466:
1456:
1455:
1454:
1438:
1424:
1418:
1415:
1409:
1406:
1376:
1370:
1367:
1361:
1358:
1335:
1329:
1306:
1298:
1284:
1278:
1275:
1269:
1266:
1256:
1242:
1236:
1233:
1227:
1224:
1218:
1212:
1203:
1194:
1188:
1178:
1172:
1169:
1163:
1160:
1157:
1154:
1151:
1143:
1137:
1127:
1121:
1118:
1112:
1109:
1103:
1097:
1094:
1091:
1088:
1082:
1077:
1069:
1059:
1058:
1057:
1040:
1034:
1011:
1002:
996:
993:
987:
984:
977:
971:
968:
962:
956:
949:
948:
947:
940:
938:
936:
917:
911:
885:
882:
879:
873:
847:
844:
841:
821:
798:
792:
783:
781:
763:
755:
729:
723:
700:
691:
685:
681:
678:
675:
667:
661:
654:
648:
644:
641:
638:
632:
627:
619:
609:
608:
607:
590:
584:
561:
555:
532:
529:
526:
520:
500:
477:
468:
462:
459:
453:
450:
441:
438:
435:
429:
420:
414:
408:
401:
400:
399:
382:
376:
356:
353:
330:
324:
318:
314:
311:
308:
305:
302:
295:
294:
293:
291:
275:
267:
263:
247:
224:
221:
215:
212:
192:
185:
169:
162:
154:
149:
145:
141:
137:
133:
130:
127:
126:neutron stars
123:
119:
115:
114:
109:
106:
103:
102:Hubbard model
99:
96:
95:
94:
91:
89:
85:
81:
77:
68:
66:
64:
60:
56:
51:
49:
45:
41:
37:
34:
30:
26:
22:
2917:
2913:
2907:
2864:
2860:
2843:
2792:
2788:
2782:
2739:
2735:
2729:
2678:
2674:
2668:
2641:
2637:
2631:
2586:
2582:
2572:
2521:
2517:
2511:
2468:
2464:
2458:
2407:
2403:
2351:
2347:
2293:
2289:
2283:
2264:
2260:
2220:
2216:
2210:
2177:
2173:
2167:
2148:
2113:
2105:
2089:
2074:
2066:
2052:
1966:
1961:
1884:
1878:
1868:
1860:
1700:
1614:
1550:
1321:
1026:
944:
784:
715:
492:
345:
158:
147:
140:quark matter
111:
92:
80:wavefunction
72:
52:
31:of a highly
24:
18:
2000:to another
136:lattice QCD
33:oscillatory
2984:Categories
2920:: 95–105.
2802:1601.05780
2651:2007.05436
2644:: 015006.
2159:References
2962:1875-3892
2927:0802.2923
2890:1887/5478
2749:1408.2269
2688:0810.2089
2386:119061060
2328:119384615
2230:1005.0539
2141:Footnotes
1942:μ
1918:μ
1898:μ
1826:−
1820:
1814:∝
1795:σ
1781:σ
1775:ρ
1738:μ
1715:σ
1677:σ
1663:σ
1657:ρ
1648:σ
1633:σ
1627:ρ
1516:−
1510:
1504:∝
1495:⟩
1485:σ
1479:θ
1470:
1464:⟨
1435:⟩
1425:σ
1419:θ
1410:
1404:⟨
1377:σ
1371:θ
1362:
1336:σ
1295:⟩
1285:σ
1279:θ
1270:
1264:⟨
1253:⟩
1243:σ
1237:θ
1228:
1219:σ
1210:⟨
1195:σ
1179:σ
1173:θ
1164:
1158:σ
1152:∫
1144:σ
1128:σ
1122:θ
1113:
1104:σ
1095:σ
1089:∫
1078:ρ
1074:⟩
1067:⟨
1041:σ
1003:σ
997:θ
988:
978:σ
963:σ
957:ρ
918:σ
912:ρ
886:σ
880:μ
845:≠
842:μ
822:μ
799:σ
793:ρ
764:ρ
760:⟩
753:⟨
730:σ
724:ρ
692:σ
686:ρ
682:σ
676:∫
668:σ
662:ρ
655:σ
645:σ
639:∫
628:ρ
624:⟩
617:⟨
591:σ
585:ρ
562:σ
533:σ
527:μ
469:σ
460:−
454:
442:σ
436:μ
415:σ
409:ρ
377:σ
357:σ
325:σ
319:ρ
315:σ
309:∫
276:σ
222:−
216:
170:μ
113:ab initio
44:precision
2970:16490610
2899:10055881
2835:24661656
2827:28059531
2774:86865851
2721:12719451
2713:19392346
2623:14890549
2589:: 21.1.
2564:11394699
2556:15904269
2503:16172956
2450:46652412
2124:See also
1910:to real
1804:⟩
1769:⟨
240:, where
69:Overview
63:fermions
48:accuracy
36:function
29:integral
2932:Bibcode
2869:Bibcode
2807:Bibcode
2754:Bibcode
2693:Bibcode
2601:Bibcode
2583:Pos Lat
2536:Bibcode
2483:Bibcode
2442:9961673
2422:Bibcode
2366:Bibcode
2308:Bibcode
2267:: 011.
2235:Bibcode
2223:: 010.
2217:Pos Lat
2202:9993272
2182:Bibcode
1879:generic
1871:NP-hard
268:fields
266:bosonic
260:is the
2968:
2960:
2897:
2833:
2825:
2772:
2719:
2711:
2621:
2562:
2554:
2501:
2448:
2440:
2384:
2326:
2200:
2092:Using
2047:space.
1701:where
1551:where
1027:where
493:where
346:where
262:action
142:. (In
122:nuclei
23:, the
2966:S2CID
2922:arXiv
2857:(PDF)
2831:S2CID
2797:arXiv
2770:S2CID
2744:arXiv
2717:S2CID
2683:arXiv
2646:arXiv
2619:S2CID
2591:arXiv
2560:S2CID
2526:arXiv
2499:S2CID
2473:arXiv
2446:S2CID
2412:arXiv
2382:S2CID
2356:arXiv
2324:S2CID
2298:arXiv
2225:arXiv
2054:Meron
1930:, or
2958:ISSN
2895:PMID
2823:PMID
2709:PMID
2552:PMID
2438:PMID
2198:PMID
2067:i.e.
124:and
2948:hdl
2940:doi
2885:hdl
2877:doi
2815:doi
2793:117
2762:doi
2701:doi
2679:102
2656:doi
2609:doi
2587:021
2544:doi
2491:doi
2430:doi
2374:doi
2316:doi
2269:doi
2221:010
2190:doi
2064:QCD
1817:exp
1507:exp
1467:exp
1407:exp
1359:exp
1267:exp
1225:exp
1161:exp
1110:exp
985:exp
868:det
716:If
451:exp
424:det
213:exp
19:In
2986::
2964:.
2956:.
2946:.
2938:.
2930:.
2916:.
2893:.
2883:.
2875:.
2865:72
2863:.
2859:.
2829:.
2821:.
2813:.
2805:.
2791:.
2768:.
2760:.
2752:.
2740:91
2738:.
2715:.
2707:.
2699:.
2691:.
2677:.
2654:.
2642:94
2640:.
2617:.
2607:.
2599:.
2585:.
2581:.
2558:.
2550:.
2542:.
2534:.
2522:94
2520:.
2497:.
2489:.
2481:.
2469:60
2467:.
2444:.
2436:.
2428:.
2420:.
2408:49
2406:.
2394:^
2380:.
2372:.
2364:.
2352:83
2350:.
2336:^
2322:.
2314:.
2306:.
2294:66
2292:.
2265:77
2263:.
2259:.
2247:^
2233:.
2219:.
2196:.
2188:.
2178:41
2176:.
1954:.
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