1362:
1014:
907:
1357:{\displaystyle G_{ijkl}^{\mathrm {conn} }=-\langle c_{i}{\bar {c}}_{j}c_{k}{\bar {c}}_{l}\rangle +\langle c_{i}{\bar {c}}_{j}\rangle \langle c_{k}{\bar {c}}_{l}\rangle -\langle c_{i}{\bar {c}}_{l}\rangle \langle c_{k}{\bar {c}}_{j}\rangle =\left.{\frac {\delta ^{2}W}{\delta J_{ji}\delta J_{lk}}}\right|_{J=0}}
602:
1841:
The diagrams are closed as they do not have any external legs, i.e., no particles going in or out of the diagram. They are “bold” because they are formulated in terms of the interacting or bold propagator rather than the non-interacting one. They are two-particle irreducible since they do not
2379:
492:
1791:
51:, the Luttinger–Ward functional is the sum of all closed, bold, two-particle irreducible diagrams, i.e., all diagrams without particles going in or out that do not fall apart if one removes two propagator lines. It is usually written as
902:{\displaystyle G_{i_{1}j_{1}\ldots i_{N}j_{N}}=-\langle c_{i_{1}}{\bar {c}}_{j_{1}}\cdots c_{i_{N}}{\bar {c}}_{j_{N}}\rangle ={\frac {-1}{Z}}\left.{\frac {\delta ^{N}Z}{\delta J_{j_{1}i_{1}}\cdots \delta J_{j_{N}i_{N}}}}\right|_{J=0}}
2225:
responding to one-body external fields apparently satisfies the space- and time- translational symmetries as well as the abelian gauge symmetry (phase symmetry), as long as the equation of motion is given with the derivative of
2162:
1958:
1556:
2475:
2070:
1631:
2686:
3084:
Rentrop, J. F.; Meden, V.; Jakobs, S. G. (2016). "Renormalization group flow of the
Luttinger–Ward functional: Conserving approximations and application to the Anderson impurity model".
1835:
2552:
2260:
311:
3137:
Kozik, E.; Ferrero, M.; Georges, A. (2015). "Nonexistence of the
Luttinger-Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models".
3199:
Schaefer, T.; Rohringer, G.; Gunnarsson, O.; Ciuchi, S.; Sangiovanni, G.; Toschi, A. (2013). "Divergent
Precursors of the Mott-Hubbard Transition at the Two-Particle Level".
2816:
3021:
Kotliar, G.; Savrasov, S. Y.; Haule, K.; Oudovenko, V. S.; Parcollet, O.; Marianetti, C. A. (2006). "Electronic structure calculations with dynamical mean-field theory".
2586:
2410:
2255:
2203:
1002:
949:
177:
in 1961. Some authors use the terms interchangeably; if a distinction is made, then the Baym–Kadanoff functional is identical to the two-particle irreducible effective
2854:
1866:
1815:
1655:
327:
207:
2171:. In particular, the irreducible vertex functions show a set of divergencies, which causes the self-energy to bifurcate into a physical and an unphysical solution.
1663:
1431:
2753:
2618:
1980:
113:
78:
561:
2223:
2000:
1821:. Along the lines of the proof of the linked-cluster theorem, one can show that this is the generating functional for the two-particle irreducible propagators.
1398:
590:
516:
259:
153:
133:
1829:
Diagrammatically, the
Luttinger–Ward functional is the sum of all closed, bold, two-particle irreducible Feynman diagrams (also known as “skeleton” diagrams):
2075:
1874:
314:
44:
1834:
1637:
Unlike the connected case, one more step is required to obtain a generating functional from the two-particle irreducible effective action
1439:
2415:
2012:
3265:
1561:
2381:
is needed to satisfy the conservation law. This is nothing but the completely-integrable condition, implying the existence of
2691:
2623:
1657:
because of the presence of a non-interacting part. By subtracting it, one obtains the
Luttinger–Ward functional:
318:
33:
25:
1982:
is a generating functional for irreducible vertex quantities: the first functional derivative with respect to
2374:{\displaystyle {\frac {\delta \Sigma (1,\left)}{\delta G(2)}}={\frac {\delta \Sigma (2,\left)}{\delta G(1)}}}
1368:
3048:
2866:
2480:
913:
267:
593:
3218:
3157:
3103:
3040:
2992:
2954:
2909:
3053:
2757:
17:
3242:
3208:
3181:
3147:
3119:
3093:
3066:
3030:
2925:
2899:
2560:
2384:
2229:
2177:
487:{\displaystyle Z=\int \mathrm {D} \exp \!{\Big (}-S+\sum _{ij}{\bar {c}}_{i}J_{ij}c_{j}{\Big )}}
2205:, thanks to the Noether's theorem. This is followed by the fact that the equation of motion of
1786:{\displaystyle \Phi =\Gamma -\Gamma _{0}=\Gamma -\mathrm {tr} \log(-G)-\mathrm {tr} (\Sigma G)}
954:
919:
3270:
3234:
3173:
2006:, while the second derivative gives the partially two-particle irreducible four-point vertex:
2821:
2257:. Note that reverse is also true. Based on the diagramatic analysis, what Baym found is that
1851:
1800:
1640:
183:
3226:
3165:
3139:
3111:
3058:
3000:
2962:
2917:
2597:
1406:
262:
178:
159:
37:
2890:
Potthoff, M. (2003). "Self-energy-functional approach to systems of correlated electrons".
2697:
2603:
1965:
83:
54:
2945:
2943:
Luttinger, J. M.; Ward, J. C. (1960). "Ground-State Energy of a Many-Fermion System. II".
1846:
1401:
158:
The
Luttinger–Ward functional has no direct physical meaning, but it is useful in proving
48:
41:
29:
525:
3222:
3161:
3107:
3044:
2996:
2958:
2913:
2208:
1985:
1374:
566:
501:
220:
138:
118:
1433:
as the source and obtains the 2PI functional, also known as Baym–Kadanoff functional:
3259:
3123:
3070:
2871:
2174:
Baym and
Kadanoff showed that we can satisfy the conservation law for any functional
2168:
2157:{\displaystyle \Gamma _{ijkl}={\frac {\delta ^{2}\Phi }{\delta G_{ij}\delta G_{kl}}}}
3246:
3185:
2929:
1008:
Green's function. As an example, the two particle connected Green's function reads:
3230:
3169:
519:
174:
2983:
Baym, G.; Kadanoff, L. P. (1961). "Conservation Laws and
Correlation Functions".
2921:
2688:(A ring diagram consists of polarisation bubbles connected by interaction lines).
2167:
While the
Luttinger–Ward functional exists, it can be shown to be not unique for
2003:
1953:{\displaystyle \Omega =\mathrm {tr} \log(-G)+\mathrm {tr} (\Sigma G)+\Phi \left}
1818:
170:
3115:
1367:
To pass to the two-particle irreducible (2PI) effective action, one performs a
3062:
2966:
3004:
3238:
3177:
3035:
2904:
1400:
to a new binary source field. One chooses an, at this point arbitrary,
209:, which differs from the Luttinger–Ward functional by a trivial term.
563:
is the sum of all (possibly disconnected), closed
Feynman diagrams.
3098:
1551:{\displaystyle \Gamma ={\Big -\sum _{ij}J_{ij}G_{ij}{\Big ]}_{J=J}}
3213:
3152:
2470:{\displaystyle \Sigma \left={\frac {\delta \Phi \left}{\delta G}}}
2065:{\displaystyle \Sigma _{ij}={\frac {\delta \Phi }{\delta G_{ij}}}}
2694:
is equivalent to taking only purely local diagrams into account:
1842:
become disconnected if we sever up to two fermionic lines.
1274:
788:
1626:{\displaystyle G_{ij}=-{\frac {\delta W}{\delta J_{ij}}}}
2824:
2760:
2700:
2626:
2606:
2563:
2483:
2418:
2387:
2263:
2232:
2211:
2180:
2078:
2015:
1988:
1968:
1877:
1854:
1803:
1666:
1643:
1564:
1442:
1409:
1377:
1017:
957:
951:
is the sum of all closed, connected, bare diagrams.
922:
605:
569:
528:
504:
330:
270:
223:
186:
141:
121:
86:
57:
2848:
2810:
2747:
2680:
2612:
2580:
2546:
2469:
2404:
2373:
2249:
2217:
2197:
2156:
2064:
1994:
1974:
1952:
1860:
1809:
1785:
1649:
1625:
1550:
1425:
1392:
1356:
996:
943:
901:
584:
555:
510:
486:
305:
253:
201:
147:
127:
107:
72:
2588:approximately. Such approximations are called as
1522:
1460:
479:
384:
381:
2477:(recall the completely-integrable condition for
1845:The Luttinger–Ward functional is related to the
2681:{\displaystyle \Phi \approx GUG+GUGGUG+\ldots }
2557:Thus the remainig problem is how to determine
518:is a binary source field. By expansion in the
1004:in turn is the generating functional for the
8:
1266:
1234:
1231:
1199:
1193:
1161:
1158:
1126:
1120:
1059:
754:
662:
3016:
3014:
217:Given a system characterized by the action
3212:
3151:
3097:
3052:
3034:
2903:
2823:
2790:
2774:
2759:
2727:
2711:
2699:
2625:
2605:
2562:
2482:
2436:
2417:
2386:
2319:
2264:
2262:
2231:
2210:
2179:
2142:
2126:
2108:
2101:
2083:
2077:
2050:
2032:
2020:
2014:
1987:
1967:
1913:
1884:
1876:
1853:
1802:
1763:
1734:
1701:
1665:
1642:
1611:
1584:
1569:
1563:
1527:
1521:
1520:
1510:
1497:
1484:
1459:
1458:
1441:
1414:
1408:
1376:
1342:
1326:
1310:
1283:
1276:
1260:
1249:
1248:
1241:
1225:
1214:
1213:
1206:
1187:
1176:
1175:
1168:
1152:
1141:
1140:
1133:
1114:
1103:
1102:
1095:
1085:
1074:
1073:
1066:
1037:
1036:
1022:
1016:
956:
921:
887:
872:
862:
857:
839:
829:
824:
797:
790:
760:
746:
741:
730:
729:
720:
715:
700:
695:
684:
683:
674:
669:
648:
638:
625:
615:
610:
604:
568:
527:
503:
478:
477:
471:
458:
448:
437:
436:
426:
405:
404:
383:
382:
364:
363:
349:
329:
297:
286:
285:
275:
269:
237:
236:
222:
185:
165:The functional is closely related to the
140:
135:is the one-particle Green's function and
120:
85:
56:
2978:
2976:
2882:
7:
2547:{\displaystyle df=A(x,y)dx+B(x,y)dy}
596:of the N-particle Green's function:
306:{\displaystyle c_{i},{\bar {c}}_{i}}
2764:
2701:
2627:
2607:
2564:
2442:
2419:
2388:
2325:
2270:
2233:
2181:
2114:
2080:
2038:
2017:
1969:
1936:
1924:
1917:
1914:
1888:
1885:
1878:
1855:
1804:
1774:
1767:
1764:
1738:
1735:
1719:
1698:
1682:
1667:
1644:
1443:
1047:
1044:
1041:
1038:
916:asserts that the effective action
350:
187:
87:
58:
38:bare electron-electron interaction
14:
1833:
3231:10.1103/PhysRevLett.110.246405
3170:10.1103/PhysRevLett.114.156402
2805:
2767:
2742:
2704:
2636:
2630:
2535:
2523:
2508:
2496:
2365:
2359:
2348:
2328:
2310:
2304:
2293:
2273:
1930:
1921:
1907:
1898:
1780:
1771:
1757:
1748:
1728:
1722:
1713:
1707:
1691:
1685:
1676:
1670:
1599:
1593:
1543:
1537:
1474:
1468:
1452:
1446:
1387:
1381:
1298:
1292:
1254:
1219:
1181:
1146:
1108:
1079:
991:
985:
967:
961:
812:
806:
780:
774:
735:
689:
579:
573:
550:
538:
442:
416:
410:
395:
375:
369:
354:
340:
334:
291:
248:
242:
227:
196:
190:
102:
90:
67:
61:
1:
2811:{\displaystyle \approx \Phi }
1558: with
169:constructed independently by
2620:to so-called ring diagrams:
2600:is equivalent to truncating
2596:The (fully self-consistent)
2892:European Physical Journal B
2692:Dynamical mean field theory
3287:
3116:10.1103/PhysRevB.93.195160
2922:10.1140/epjb/e2003-00121-8
2581:{\displaystyle \Phi \left}
2405:{\displaystyle \Phi \left}
2250:{\displaystyle \Phi \left}
2198:{\displaystyle \Phi \left}
3063:10.1103/RevModPhys.78.865
2856:are lattice site indices.
997:{\displaystyle W=-\log Z}
944:{\displaystyle W=-\log Z}
155:is the bare interaction.
22:Luttinger–Ward functional
3266:Condensed matter physics
2967:10.1103/PhysRev.118.1417
2590:conserving approximation
317:can be expressed as the
167:Baym–Kadanoff functional
26:Joaquin Mazdak Luttinger
3005:10.1103/PhysRev.124.287
2849:{\displaystyle i,j,k,l}
1861:{\displaystyle \Omega }
1810:{\displaystyle \Sigma }
1650:{\displaystyle \Gamma }
202:{\displaystyle \Gamma }
45:one-particle propagator
2850:
2812:
2749:
2682:
2614:
2582:
2548:
2471:
2406:
2375:
2251:
2219:
2199:
2158:
2066:
1996:
1976:
1954:
1862:
1811:
1787:
1651:
1627:
1552:
1427:
1426:{\displaystyle G_{ij}}
1394:
1358:
998:
945:
914:linked-cluster theorem
903:
586:
557:
512:
488:
307:
255:
203:
149:
129:
109:
74:
2851:
2813:
2750:
2748:{\displaystyle \Phi }
2683:
2615:
2613:{\displaystyle \Phi }
2583:
2549:
2472:
2407:
2376:
2252:
2220:
2200:
2159:
2067:
1997:
1977:
1975:{\displaystyle \Phi }
1955:
1863:
1812:
1788:
1652:
1628:
1553:
1428:
1395:
1359:
999:
946:
904:
594:generating functional
587:
558:
513:
489:
308:
256:
204:
150:
130:
110:
108:{\displaystyle \Phi }
75:
73:{\displaystyle \Phi }
32:in 1960, is a scalar
2822:
2758:
2698:
2624:
2604:
2561:
2481:
2416:
2385:
2261:
2230:
2209:
2178:
2076:
2013:
1986:
1966:
1875:
1852:
1801:
1664:
1641:
1562:
1440:
1407:
1375:
1015:
955:
920:
603:
567:
526:
502:
328:
268:
221:
184:
139:
119:
84:
55:
3223:2013PhRvL.110x6405S
3162:2015PhRvL.114o6402K
3108:2016PhRvB..93s5160R
3045:2006RvMP...78..865K
2997:1961PhRv..124..287B
2959:1960PhRv..118.1417L
2914:2003EPJB...32..429P
2867:Luttinger's theorem
2169:Hubbard-like models
1052:
556:{\displaystyle Z=Z}
18:solid state physics
2846:
2808:
2745:
2678:
2610:
2578:
2544:
2467:
2402:
2371:
2247:
2215:
2195:
2154:
2062:
1992:
1972:
1950:
1858:
1807:
1783:
1647:
1623:
1548:
1492:
1423:
1390:
1369:Legendre transform
1354:
1018:
994:
941:
899:
582:
553:
508:
484:
434:
315:partition function
303:
251:
199:
145:
125:
105:
70:
2592:. Some examples:
2465:
2369:
2314:
2218:{\displaystyle G}
2152:
2060:
1995:{\displaystyle G}
1621:
1480:
1393:{\displaystyle W}
1336:
1257:
1222:
1184:
1149:
1111:
1082:
881:
784:
738:
692:
585:{\displaystyle Z}
522:, one finds that
511:{\displaystyle J}
445:
422:
413:
372:
294:
254:{\displaystyle S}
245:
160:conservation laws
148:{\displaystyle U}
128:{\displaystyle G}
3278:
3251:
3250:
3216:
3196:
3190:
3189:
3155:
3140:Phys. Rev. Lett.
3134:
3128:
3127:
3101:
3081:
3075:
3074:
3056:
3038:
3036:cond-mat/0511085
3018:
3009:
3008:
2980:
2971:
2970:
2953:(5): 1417–1427.
2940:
2934:
2933:
2907:
2905:cond-mat/0301137
2887:
2855:
2853:
2852:
2847:
2817:
2815:
2814:
2809:
2804:
2803:
2782:
2781:
2754:
2752:
2751:
2746:
2741:
2740:
2719:
2718:
2687:
2685:
2684:
2679:
2619:
2617:
2616:
2611:
2598:GW approximation
2587:
2585:
2584:
2579:
2577:
2553:
2551:
2550:
2545:
2476:
2474:
2473:
2468:
2466:
2464:
2456:
2455:
2437:
2432:
2411:
2409:
2408:
2403:
2401:
2380:
2378:
2377:
2372:
2370:
2368:
2351:
2347:
2320:
2315:
2313:
2296:
2292:
2265:
2256:
2254:
2253:
2248:
2246:
2224:
2222:
2221:
2216:
2204:
2202:
2201:
2196:
2194:
2163:
2161:
2160:
2155:
2153:
2151:
2150:
2149:
2134:
2133:
2117:
2113:
2112:
2102:
2097:
2096:
2071:
2069:
2068:
2063:
2061:
2059:
2058:
2057:
2041:
2033:
2028:
2027:
2001:
1999:
1998:
1993:
1981:
1979:
1978:
1973:
1959:
1957:
1956:
1951:
1949:
1920:
1891:
1867:
1865:
1864:
1859:
1837:
1816:
1814:
1813:
1808:
1792:
1790:
1789:
1784:
1770:
1741:
1706:
1705:
1656:
1654:
1653:
1648:
1632:
1630:
1629:
1624:
1622:
1620:
1619:
1618:
1602:
1585:
1577:
1576:
1557:
1555:
1554:
1549:
1547:
1546:
1526:
1525:
1518:
1517:
1505:
1504:
1491:
1464:
1463:
1432:
1430:
1429:
1424:
1422:
1421:
1399:
1397:
1396:
1391:
1363:
1361:
1360:
1355:
1353:
1352:
1341:
1337:
1335:
1334:
1333:
1318:
1317:
1301:
1288:
1287:
1277:
1265:
1264:
1259:
1258:
1250:
1246:
1245:
1230:
1229:
1224:
1223:
1215:
1211:
1210:
1192:
1191:
1186:
1185:
1177:
1173:
1172:
1157:
1156:
1151:
1150:
1142:
1138:
1137:
1119:
1118:
1113:
1112:
1104:
1100:
1099:
1090:
1089:
1084:
1083:
1075:
1071:
1070:
1051:
1050:
1035:
1003:
1001:
1000:
995:
950:
948:
947:
942:
908:
906:
905:
900:
898:
897:
886:
882:
880:
879:
878:
877:
876:
867:
866:
846:
845:
844:
843:
834:
833:
815:
802:
801:
791:
785:
783:
769:
761:
753:
752:
751:
750:
740:
739:
731:
727:
726:
725:
724:
707:
706:
705:
704:
694:
693:
685:
681:
680:
679:
678:
655:
654:
653:
652:
643:
642:
630:
629:
620:
619:
591:
589:
588:
583:
562:
560:
559:
554:
517:
515:
514:
509:
493:
491:
490:
485:
483:
482:
476:
475:
466:
465:
453:
452:
447:
446:
438:
433:
415:
414:
406:
388:
387:
374:
373:
365:
353:
312:
310:
309:
304:
302:
301:
296:
295:
287:
280:
279:
263:Grassmann fields
260:
258:
257:
252:
247:
246:
238:
208:
206:
205:
200:
154:
152:
151:
146:
134:
132:
131:
126:
114:
112:
111:
106:
79:
77:
76:
71:
49:Feynman diagrams
3286:
3285:
3281:
3280:
3279:
3277:
3276:
3275:
3256:
3255:
3254:
3201:Phys. Rev. Lett
3198:
3197:
3193:
3136:
3135:
3131:
3083:
3082:
3078:
3054:10.1.1.475.7032
3020:
3019:
3012:
2985:Physical Review
2982:
2981:
2974:
2946:Physical Review
2942:
2941:
2937:
2889:
2888:
2884:
2880:
2863:
2820:
2819:
2786:
2770:
2756:
2755:
2723:
2707:
2696:
2695:
2622:
2621:
2602:
2601:
2567:
2559:
2558:
2479:
2478:
2457:
2445:
2438:
2422:
2414:
2413:
2391:
2383:
2382:
2352:
2337:
2321:
2297:
2282:
2266:
2259:
2258:
2236:
2228:
2227:
2207:
2206:
2184:
2176:
2175:
2138:
2122:
2118:
2104:
2103:
2079:
2074:
2073:
2046:
2042:
2034:
2016:
2011:
2010:
1984:
1983:
1964:
1963:
1939:
1873:
1872:
1850:
1849:
1847:grand potential
1827:
1799:
1798:
1697:
1662:
1661:
1639:
1638:
1607:
1603:
1586:
1565:
1560:
1559:
1519:
1506:
1493:
1438:
1437:
1410:
1405:
1404:
1373:
1372:
1322:
1306:
1302:
1279:
1278:
1273:
1272:
1247:
1237:
1212:
1202:
1174:
1164:
1139:
1129:
1101:
1091:
1072:
1062:
1013:
1012:
953:
952:
918:
917:
868:
858:
853:
835:
825:
820:
816:
793:
792:
787:
786:
770:
762:
742:
728:
716:
711:
696:
682:
670:
665:
644:
634:
621:
611:
606:
601:
600:
592:in turn is the
565:
564:
524:
523:
500:
499:
467:
454:
435:
326:
325:
284:
271:
266:
265:
219:
218:
215:
182:
181:
137:
136:
117:
116:
82:
81:
53:
52:
47:. In terms of
30:John Clive Ward
12:
11:
5:
3284:
3282:
3274:
3273:
3268:
3258:
3257:
3253:
3252:
3207:(24): 246405.
3191:
3146:(15): 156402.
3129:
3092:(19): 195160.
3076:
3029:(3): 865–951.
3023:Rev. Mod. Phys
3010:
2991:(2): 287–299.
2972:
2935:
2898:(4): 429–436.
2881:
2879:
2876:
2875:
2874:
2869:
2862:
2859:
2858:
2857:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2807:
2802:
2799:
2796:
2793:
2789:
2785:
2780:
2777:
2773:
2769:
2766:
2763:
2744:
2739:
2736:
2733:
2730:
2726:
2722:
2717:
2714:
2710:
2706:
2703:
2689:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2609:
2576:
2573:
2570:
2566:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2463:
2460:
2454:
2451:
2448:
2444:
2441:
2435:
2431:
2428:
2425:
2421:
2400:
2397:
2394:
2390:
2367:
2364:
2361:
2358:
2355:
2350:
2346:
2343:
2340:
2336:
2333:
2330:
2327:
2324:
2318:
2312:
2309:
2306:
2303:
2300:
2295:
2291:
2288:
2285:
2281:
2278:
2275:
2272:
2269:
2245:
2242:
2239:
2235:
2214:
2193:
2190:
2187:
2183:
2165:
2164:
2148:
2145:
2141:
2137:
2132:
2129:
2125:
2121:
2116:
2111:
2107:
2100:
2095:
2092:
2089:
2086:
2082:
2056:
2053:
2049:
2045:
2040:
2037:
2031:
2026:
2023:
2019:
1991:
1971:
1961:
1960:
1948:
1945:
1942:
1938:
1935:
1932:
1929:
1926:
1923:
1919:
1916:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1890:
1887:
1883:
1880:
1857:
1839:
1838:
1826:
1823:
1806:
1795:
1794:
1782:
1779:
1776:
1773:
1769:
1766:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1740:
1737:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1704:
1700:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1646:
1635:
1634:
1617:
1614:
1610:
1606:
1601:
1598:
1595:
1592:
1589:
1583:
1580:
1575:
1572:
1568:
1545:
1542:
1539:
1536:
1533:
1530:
1524:
1516:
1513:
1509:
1503:
1500:
1496:
1490:
1487:
1483:
1479:
1476:
1473:
1470:
1467:
1462:
1457:
1454:
1451:
1448:
1445:
1420:
1417:
1413:
1389:
1386:
1383:
1380:
1365:
1364:
1351:
1348:
1345:
1340:
1332:
1329:
1325:
1321:
1316:
1313:
1309:
1305:
1300:
1297:
1294:
1291:
1286:
1282:
1275:
1271:
1268:
1263:
1256:
1253:
1244:
1240:
1236:
1233:
1228:
1221:
1218:
1209:
1205:
1201:
1198:
1195:
1190:
1183:
1180:
1171:
1167:
1163:
1160:
1155:
1148:
1145:
1136:
1132:
1128:
1125:
1122:
1117:
1110:
1107:
1098:
1094:
1088:
1081:
1078:
1069:
1065:
1061:
1058:
1055:
1049:
1046:
1043:
1040:
1034:
1031:
1028:
1025:
1021:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
940:
937:
934:
931:
928:
925:
910:
909:
896:
893:
890:
885:
875:
871:
865:
861:
856:
852:
849:
842:
838:
832:
828:
823:
819:
814:
811:
808:
805:
800:
796:
789:
782:
779:
776:
773:
768:
765:
759:
756:
749:
745:
737:
734:
723:
719:
714:
710:
703:
699:
691:
688:
677:
673:
668:
664:
661:
658:
651:
647:
641:
637:
633:
628:
624:
618:
614:
609:
581:
578:
575:
572:
552:
549:
546:
543:
540:
537:
534:
531:
507:
496:
495:
481:
474:
470:
464:
461:
457:
451:
444:
441:
432:
429:
425:
421:
418:
412:
409:
403:
400:
397:
394:
391:
386:
380:
377:
371:
368:
362:
359:
356:
352:
348:
345:
342:
339:
336:
333:
300:
293:
290:
283:
278:
274:
250:
244:
241:
235:
232:
229:
226:
214:
211:
198:
195:
192:
189:
144:
124:
104:
101:
98:
95:
92:
89:
69:
66:
63:
60:
24:, proposed by
13:
10:
9:
6:
4:
3:
2:
3283:
3272:
3269:
3267:
3264:
3263:
3261:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3215:
3210:
3206:
3202:
3195:
3192:
3187:
3183:
3179:
3175:
3171:
3167:
3163:
3159:
3154:
3149:
3145:
3142:
3141:
3133:
3130:
3125:
3121:
3117:
3113:
3109:
3105:
3100:
3095:
3091:
3087:
3080:
3077:
3072:
3068:
3064:
3060:
3055:
3050:
3046:
3042:
3037:
3032:
3028:
3024:
3017:
3015:
3011:
3006:
3002:
2998:
2994:
2990:
2986:
2979:
2977:
2973:
2968:
2964:
2960:
2956:
2952:
2948:
2947:
2939:
2936:
2931:
2927:
2923:
2919:
2915:
2911:
2906:
2901:
2897:
2893:
2886:
2883:
2877:
2873:
2872:Ward identity
2870:
2868:
2865:
2864:
2860:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2800:
2797:
2794:
2791:
2787:
2783:
2778:
2775:
2771:
2761:
2737:
2734:
2731:
2728:
2724:
2720:
2715:
2712:
2708:
2693:
2690:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2633:
2599:
2595:
2594:
2593:
2591:
2574:
2571:
2568:
2555:
2541:
2538:
2532:
2529:
2526:
2520:
2517:
2514:
2511:
2505:
2502:
2499:
2493:
2490:
2487:
2484:
2461:
2458:
2452:
2449:
2446:
2439:
2433:
2429:
2426:
2423:
2398:
2395:
2392:
2362:
2356:
2353:
2344:
2341:
2338:
2334:
2331:
2322:
2316:
2307:
2301:
2298:
2289:
2286:
2283:
2279:
2276:
2267:
2243:
2240:
2237:
2212:
2191:
2188:
2185:
2172:
2170:
2146:
2143:
2139:
2135:
2130:
2127:
2123:
2119:
2109:
2105:
2098:
2093:
2090:
2087:
2084:
2054:
2051:
2047:
2043:
2035:
2029:
2024:
2021:
2009:
2008:
2007:
2005:
1989:
1946:
1943:
1940:
1933:
1927:
1910:
1904:
1901:
1895:
1892:
1881:
1871:
1870:
1869:
1868:of a system:
1848:
1843:
1836:
1832:
1831:
1830:
1824:
1822:
1820:
1777:
1760:
1754:
1751:
1745:
1742:
1731:
1725:
1716:
1710:
1702:
1694:
1688:
1679:
1673:
1660:
1659:
1658:
1615:
1612:
1608:
1604:
1596:
1590:
1587:
1581:
1578:
1573:
1570:
1566:
1540:
1534:
1531:
1528:
1514:
1511:
1507:
1501:
1498:
1494:
1488:
1485:
1481:
1477:
1471:
1465:
1455:
1449:
1436:
1435:
1434:
1418:
1415:
1411:
1403:
1384:
1378:
1370:
1349:
1346:
1343:
1338:
1330:
1327:
1323:
1319:
1314:
1311:
1307:
1303:
1295:
1289:
1284:
1280:
1269:
1261:
1251:
1242:
1238:
1226:
1216:
1207:
1203:
1196:
1188:
1178:
1169:
1165:
1153:
1143:
1134:
1130:
1123:
1115:
1105:
1096:
1092:
1086:
1076:
1067:
1063:
1056:
1053:
1032:
1029:
1026:
1023:
1019:
1011:
1010:
1009:
1007:
988:
982:
979:
976:
973:
970:
964:
958:
938:
935:
932:
929:
926:
923:
915:
894:
891:
888:
883:
873:
869:
863:
859:
854:
850:
847:
840:
836:
830:
826:
821:
817:
809:
803:
798:
794:
777:
771:
766:
763:
757:
747:
743:
732:
721:
717:
712:
708:
701:
697:
686:
675:
671:
666:
659:
656:
649:
645:
639:
635:
631:
626:
622:
616:
612:
607:
599:
598:
597:
595:
576:
570:
547:
544:
541:
535:
532:
529:
521:
505:
472:
468:
462:
459:
455:
449:
439:
430:
427:
423:
419:
407:
401:
398:
392:
389:
378:
366:
360:
357:
346:
343:
337:
331:
324:
323:
322:
320:
319:path integral
316:
298:
288:
281:
276:
272:
264:
239:
233:
230:
224:
212:
210:
193:
180:
176:
172:
168:
163:
161:
156:
142:
122:
99:
96:
93:
64:
50:
46:
43:
39:
35:
31:
27:
23:
19:
3204:
3200:
3194:
3143:
3138:
3132:
3089:
3086:Phys. Rev. B
3085:
3079:
3026:
3022:
2988:
2984:
2950:
2944:
2938:
2895:
2891:
2885:
2589:
2556:
2173:
2166:
1962:
1844:
1840:
1828:
1796:
1636:
1366:
1005:
911:
520:Dyson series
497:
261:in terms of
216:
213:Construction
175:Leo Kadanoff
166:
164:
157:
42:renormalized
21:
15:
2004:self-energy
1819:self-energy
171:Gordon Baym
3260:Categories
3099:1602.06120
2878:References
2412:such that
2002:gives the
1825:Properties
34:functional
3214:1303.0246
3153:1407.5687
3124:119212288
3071:119099745
3049:CiteSeerX
2765:Φ
2762:≈
2702:Φ
2676:…
2640:≈
2628:Φ
2608:Φ
2565:Φ
2459:δ
2443:Φ
2440:δ
2420:Σ
2389:Φ
2354:δ
2326:Σ
2323:δ
2299:δ
2271:Σ
2268:δ
2234:Φ
2182:Φ
2136:δ
2120:δ
2115:Φ
2106:δ
2081:Γ
2044:δ
2039:Φ
2036:δ
2018:Σ
1970:Φ
1937:Φ
1925:Σ
1902:−
1896:
1879:Ω
1856:Ω
1805:Σ
1775:Σ
1761:−
1752:−
1746:
1732:−
1720:Γ
1699:Γ
1695:−
1683:Γ
1668:Φ
1645:Γ
1605:δ
1588:δ
1582:−
1482:∑
1478:−
1444:Γ
1320:δ
1304:δ
1281:δ
1267:⟩
1255:¯
1235:⟨
1232:⟩
1220:¯
1200:⟨
1197:−
1194:⟩
1182:¯
1162:⟨
1159:⟩
1147:¯
1127:⟨
1121:⟩
1109:¯
1080:¯
1060:⟨
1057:−
1006:connected
980:
974:−
936:
930:−
851:δ
848:⋯
818:δ
795:δ
764:−
755:⟩
736:¯
709:⋯
690:¯
663:⟨
660:−
632:…
443:¯
424:∑
411:¯
390:−
370:¯
347:∫
292:¯
243:¯
188:Γ
88:Φ
59:Φ
3271:Fermions
3247:14280120
3239:25165946
3186:23241294
3178:25933324
2930:55745257
2861:See also
2818:, where
2072:;
115:, where
40:and the
3219:Bibcode
3158:Bibcode
3104:Bibcode
3041:Bibcode
2993:Bibcode
2955:Bibcode
2910:Bibcode
1817:is the
36:of the
3245:
3237:
3184:
3176:
3122:
3069:
3051:
2928:
1797:where
1402:convex
498:where
313:, the
179:action
20:, the
3243:S2CID
3209:arXiv
3182:S2CID
3148:arXiv
3120:S2CID
3094:arXiv
3067:S2CID
3031:arXiv
2926:S2CID
2900:arXiv
3235:PMID
3174:PMID
912:The
173:and
28:and
3227:doi
3205:110
3166:doi
3144:114
3112:doi
3059:doi
3001:doi
2989:124
2963:doi
2951:118
2918:doi
2554:).
1893:log
1743:log
1371:of
977:log
933:log
379:exp
80:or
16:In
3262::
3241:.
3233:.
3225:.
3217:.
3203:.
3180:.
3172:.
3164:.
3156:.
3118:.
3110:.
3102:.
3090:93
3088:.
3065:.
3057:.
3047:.
3039:.
3027:78
3025:.
3013:^
2999:.
2987:.
2975:^
2961:.
2949:.
2924:.
2916:.
2908:.
2896:32
2894:.
321::
162:.
3249:.
3229::
3221::
3211::
3188:.
3168::
3160::
3150::
3126:.
3114::
3106::
3096::
3073:.
3061::
3043::
3033::
3007:.
3003::
2995::
2969:.
2965::
2957::
2932:.
2920::
2912::
2902::
2844:l
2841:,
2838:k
2835:,
2832:j
2829:,
2826:i
2806:]
2801:i
2798:i
2795:i
2792:i
2788:U
2784:,
2779:i
2776:i
2772:G
2768:[
2743:]
2738:l
2735:k
2732:j
2729:i
2725:U
2721:,
2716:j
2713:i
2709:G
2705:[
2673:+
2670:G
2667:U
2664:G
2661:G
2658:U
2655:G
2652:+
2649:G
2646:U
2643:G
2637:]
2634:G
2631:[
2575:]
2572:G
2569:[
2542:y
2539:d
2536:)
2533:y
2530:,
2527:x
2524:(
2521:B
2518:+
2515:x
2512:d
2509:)
2506:y
2503:,
2500:x
2497:(
2494:A
2491:=
2488:f
2485:d
2462:G
2453:]
2450:G
2447:[
2434:=
2430:]
2427:G
2424:[
2399:]
2396:G
2393:[
2366:)
2363:1
2360:(
2357:G
2349:)
2345:]
2342:G
2339:[
2335:,
2332:2
2329:(
2317:=
2311:)
2308:2
2305:(
2302:G
2294:)
2290:]
2287:G
2284:[
2280:,
2277:1
2274:(
2244:]
2241:G
2238:[
2213:G
2192:]
2189:G
2186:[
2147:l
2144:k
2140:G
2131:j
2128:i
2124:G
2110:2
2099:=
2094:l
2091:k
2088:j
2085:i
2055:j
2052:i
2048:G
2030:=
2025:j
2022:i
1990:G
1947:]
1944:G
1941:[
1934:+
1931:)
1928:G
1922:(
1918:r
1915:t
1911:+
1908:)
1905:G
1899:(
1889:r
1886:t
1882:=
1793:,
1781:)
1778:G
1772:(
1768:r
1765:t
1758:)
1755:G
1749:(
1739:r
1736:t
1729:]
1726:G
1723:[
1717:=
1714:]
1711:G
1708:[
1703:0
1692:]
1689:G
1686:[
1680:=
1677:]
1674:G
1671:[
1633:.
1616:j
1613:i
1609:J
1600:]
1597:J
1594:[
1591:W
1579:=
1574:j
1571:i
1567:G
1544:]
1541:G
1538:[
1535:J
1532:=
1529:J
1523:]
1515:j
1512:i
1508:G
1502:j
1499:i
1495:J
1489:j
1486:i
1475:]
1472:J
1469:[
1466:W
1461:[
1456:=
1453:]
1450:G
1447:[
1419:j
1416:i
1412:G
1388:]
1385:J
1382:[
1379:W
1350:0
1347:=
1344:J
1339:|
1331:k
1328:l
1324:J
1315:i
1312:j
1308:J
1299:]
1296:J
1293:[
1290:W
1285:2
1270:=
1262:j
1252:c
1243:k
1239:c
1227:l
1217:c
1208:i
1204:c
1189:l
1179:c
1170:k
1166:c
1154:j
1144:c
1135:i
1131:c
1124:+
1116:l
1106:c
1097:k
1093:c
1087:j
1077:c
1068:i
1064:c
1054:=
1048:n
1045:n
1042:o
1039:c
1033:l
1030:k
1027:j
1024:i
1020:G
992:]
989:J
986:[
983:Z
971:=
968:]
965:J
962:[
959:W
939:Z
927:=
924:W
895:0
892:=
889:J
884:|
874:N
870:i
864:N
860:j
855:J
841:1
837:i
831:1
827:j
822:J
813:]
810:J
807:[
804:Z
799:N
781:]
778:0
775:[
772:Z
767:1
758:=
748:N
744:j
733:c
722:N
718:i
713:c
702:1
698:j
687:c
676:1
672:i
667:c
657:=
650:N
646:j
640:N
636:i
627:1
623:j
617:1
613:i
608:G
580:]
577:J
574:[
571:Z
551:]
548:0
545:=
542:J
539:[
536:Z
533:=
530:Z
506:J
494:,
480:)
473:j
469:c
463:j
460:i
456:J
450:i
440:c
431:j
428:i
420:+
417:]
408:c
402:,
399:c
396:[
393:S
385:(
376:]
367:c
361:,
358:c
355:[
351:D
344:=
341:]
338:J
335:[
332:Z
299:i
289:c
282:,
277:i
273:c
249:]
240:c
234:,
231:c
228:[
225:S
197:]
194:G
191:[
143:U
123:G
103:]
100:U
97:,
94:G
91:[
68:]
65:G
62:[
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