112:. In the Ising model, the lattice problem is mapped onto an effective single site problem, whose magnetization is to reproduce the lattice magnetization through an effective "mean-field". This condition is called the self-consistency condition. It stipulates that the single-site observables should reproduce the lattice "local" observables by means of an effective field. While the N-site Ising Hamiltonian is hard to solve analytically (to date, analytical solutions exist only for the 1D and 2D case), the single-site problem is easily solved.
944:
656:
939:{\displaystyle H_{\text{AIM}}=\underbrace {\sum _{p}\epsilon _{p}a_{p}^{\dagger }a_{p}} _{H_{\text{bath}}}+\underbrace {\sum _{p\sigma }\left(V_{p}^{\sigma }c_{\sigma }^{\dagger }a_{p\sigma }+h.c.\right)} _{H_{\text{mix}}}+\underbrace {Un_{\uparrow }n_{\downarrow }-\mu \left(n_{\uparrow }+n_{\downarrow }\right)} _{H_{\text{loc}}}}
1390:
This hybridization function describes the dynamics of electrons hopping in and out of the bath. It should reproduce the lattice dynamics such that the impurity Green's function is the same as the local lattice Green's function. It is related to the non-interacting Green's function by the relation:
185:
is time-dependent, or dynamical. This also points to the major difference between the Ising MFT and DMFT: Ising MFT maps the N-spin problem into a single-site, single-spin problem. DMFT maps the lattice problem onto a single-site problem, but the latter fundamentally remains a N-body problem which
2144:
In order to find the local lattice Green's function, one has to determine the hybridization function such that the corresponding impurity Green's function will coincide with the sought-after local lattice Green's function. The most widespread way of solving this problem is by using a forward
2120:
This approximation becomes exact in the limit of lattices with infinite coordination, that is when the number of neighbors of each site is infinite. Indeed, one can show that in the diagrammatic expansion of the lattice self-energy, only local diagrams survive when one goes into the infinite
3131:
In order to improve on the DMFT approximation, the
Hubbard model can be mapped on a multi-site impurity (cluster) problem, which allows one to add some spatial dependence to the impurity self-energy. Clusters contain 4 to 8 sites at low temperature and up to 100 sites at high temperature.
68:
problem, called an impurity model. While the lattice problem is in general intractable, the impurity model is usually solvable through various schemes. The mapping in itself does not constitute an approximation. The only approximation made in ordinary DMFT schemes is to assume the lattice
1969:
353:
2809:
2591:
1349:
2124:
Thus, as in classical mean-field theories, DMFT is supposed to get more accurate as the dimensionality (and thus the number of neighbors) increases. Put differently, for low dimensions, spatial fluctuations will render the DMFT approximation less reliable.
1505:
3148:, can be obtained also through diagrammatic extensions of DMFT using a combination of analytical and numerical techniques. The starting point of the dynamical vertex approximation and of the dual fermion approach is the local
2391:
3135:
The
Typical Medium Dynamical Cluster Approximation (TMDCA) is a non-perturbative approach for obtaining the electronic ground state of strongly correlated many-body systems, built on the dynamical cluster approximation (DCA).
1790:
1198:
3611:
G. Rohringer; H. Hafermann; A. Toschi; A. Katanin; A. E. Antipov; M. I. Katsnelson; A. I. Lichtenstein; A. N. Rubtsov; K. Held (2018). "Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory".
2115:
1798:
230:
127:. Thus, the self-consistency condition for DMFT is for the impurity Green's function to reproduce the lattice local Green's function through an effective mean-field which, in DMFT, is the hybridization function
630:) through a hybridization function. The Anderson model corresponding to our single-site model is a single-orbital Anderson impurity model, whose hamiltonian formulation, on suppressing some spin 1/2 indices
2883:
2652:
2454:
3065:
2647:
2447:
1229:
496:
2303:
1693:
1385:
2252:
2017:
628:
421:
1587:
2027:
The only DMFT approximations (apart from the approximation that can be made in order to solve the
Anderson model) consists in neglecting the spatial fluctuations of the lattice
1629:
1548:
1113:
977:
1081:
1033:
183:
154:
1004:
1634:
593:
3085:
648:
376:
3105:
1613:
1397:
1224:
3318:
G. Kotliar; S. Y. Savrasov; K. Haule; V. S. Oudovenko; O. Parcollet; C. A. Marianetti (2006). "Electronic structure calculations with dynamical mean-field theory".
3184:
A. Georges; G. Kotliar; W. Krauth; M. Rozenberg (1996). "Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions".
2932:
2183:
2999:
2952:
2912:
2203:
2163:
1053:
565:(AIM). This model describes the interaction of one site (the impurity) with a "bath" of electronic levels (described by the annihilation and creation operators
550:
527:
441:
222:
2311:
49:
calculations, breaks down. Dynamical mean-field theory, a non-perturbative treatment of local interactions between electrons, bridges the gap between the
2894:
The local lattice Green's function and other impurity observables can be used to calculate a number of physical quantities as a function of correlations
1698:
1121:
3119:
Extended DMFT yields a local impurity self energy for non-local interactions and hence allows us to apply DMFT for more general models such as the
1964:{\displaystyle G_{\mathrm {imp} }(i\omega _{n})=G_{ii}(i\omega _{n})=\sum _{k}{\frac {1}{i\omega _{n}+\mu -\epsilon (k)-\Sigma (k,i\omega _{n})}}}
348:{\displaystyle H_{\text{Hubbard}}=t\sum _{\langle ij\rangle \sigma }c_{i\sigma }^{\dagger }c_{j\sigma }+U\sum _{i}n_{i\uparrow }n_{i\downarrow }}
2037:
3009:
DMFT has several extensions, extending the above formalism to multi-orbital, multi-site problems, long-range correlations and non-equilibrium.
561:
The
Hubbard model is in general intractable under usual perturbation expansion techniques. DMFT maps this lattice model onto the so-called
17:
3789:"Numerical implementation of dynamical mean field theory for disordered systems: application to the Lotka–Volterra model of ecosystems"
2804:{\displaystyle \Sigma _{\mathrm {imp} }(i\omega _{n})=({\mathcal {G}}_{0})^{-1}(i\omega _{n})-(G_{\mathrm {imp} })^{-1}(i\omega _{n})}
2816:
3420:. Lectures on the Physics of Highly Correlated Electron Systems VIII. Vol. 715. American Institute of Physics. pp. 3–74.
2586:{\displaystyle \Delta (i\omega )=i\omega _{n}+\mu -G_{\mathrm {loc} }^{-1}(i\omega _{n})-\Sigma _{\mathrm {imp} }(i\omega _{n})}
506:
3160:
DMFT has been employed to study non-equilibrium transport and optical excitations. Here, the reliable calculation of the AIM's
124:
3929:
3020:
3164:
remains a big challenge. DMFT has also been applied to ecological models in order to describe the mean-field dynamics of a
73:
to be a momentum-independent (local) quantity. This approximation becomes exact in the limit of lattices with an infinite
1639:
3416:
Antoine
Georges (2004). "Strongly Correlated Electron Materials: Dynamical Mean-Field Theory and Electronic Structure".
3451:
3017:
DMFT can be extended to
Hubbard models with multiple orbitals, namely with electron-electron interactions of the form
2598:
2398:
3667:
A. Toschi; A. Katanin; K. Held (2007). "Dynamical vertex approximation: A step beyond dynamical mean-field theory".
1344:{\displaystyle \Delta _{\sigma }(i\omega _{n})=\sum _{p}{\frac {|V_{p}^{\sigma }|^{2}}{i\omega _{n}-\epsilon _{p}}}}
505:
only one orbital contributes to the electronic properties (as might be the case of copper atoms in superconducting
1619:
446:
2257:
1654:
1511:
Solving the
Anderson impurity model consists in computing observables such as the interacting Green's function
1354:
204:
The
Hubbard model describes the onsite interaction between electrons of opposite spin by a single parameter,
3914:
2212:
1977:
598:
38:
3144:
Spatial dependencies of the self energy beyond DMFT, including long-range correlations in the vicinity of a
381:
3846:
3564:"Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions"
1553:
3722:
Aoki, Hideo; Tsuji, Naoto; Eckstein, Martin; Kollar, Marcus; Oka, Takashi; Werner, Philipp (2014-06-24).
1615:. It is a difficult but not intractable problem. There exists a number of ways to solve the AIM, such as
93:
1083:
describes the hybridization (or coupling) between the impurity and the bath through hybridization terms
3924:
3919:
50:
3858:
3896:
DMFT for two-site
Hubbard dimer: in Solving the strong-correlation problem in materials, Eva Pavarini
3614:
3320:
3186:
3108:
1514:
1086:
955:
42:
1059:
1011:
159:
130:
3891:
3563:
982:
54:
568:
3272:
562:
2136:, the pronounced changes before the phase transition are not reflected in the DMFT self-energy.
1500:{\displaystyle ({\mathcal {G}}_{0})^{-1}(i\omega _{n})=i\omega _{n}+\mu -\Delta (i\omega _{n})}
89:
3788:
3549:"Embedded Dynamical Mean Field Theory, an electronic structure package implementing DFT+DMFT"
3070:
1624:
633:
423:
denote the creation and annihilation operators of an electron on a localized orbital on site
361:
3267:
3090:
1592:
1203:
3888:
DMFT for two-site
Hubbard dimer: in Dynamical Mean-Field Theory for Materials, Eva Pavarini
3810:
3745:
3688:
3633:
3585:
3514:
3460:
3388:
3339:
3281:
3232:
3195:
2917:
2168:
92:
is increased. It has been successfully applied to real materials, in combination with the
8:
3165:
74:
3814:
3749:
3692:
3637:
3589:
3518:
3464:
3392:
3343:
3285:
3236:
3199:
3800:
3769:
3735:
3704:
3678:
3649:
3623:
3575:
3530:
3504:
3494:
K. Held (2007). "Electronic Structure Calculations using Dynamical Mean Field Theory".
3476:
3421:
3379:
3355:
3329:
2984:
2937:
2897:
2188:
2148:
2133:
1038:
535:
512:
426:
207:
3882:
3876:
3870:
3826:
3773:
3761:
3723:
3708:
3669:
3653:
3562:
Ekuma, C.E.; Terletska, H.; Tam, K.-M.; Meng, Z.-Y.; Moreno, J.; Jarrell, M. (2014).
3359:
3297:
3248:
3223:
3161:
2959:
2386:{\displaystyle \Sigma (k,i\omega _{n})\approx \Sigma _{\mathrm {imp} }(i\omega _{n})}
2129:
105:
61:
3864:
3534:
3480:
3818:
3753:
3696:
3641:
3593:
3522:
3468:
3431:
3396:
3347:
3289:
3240:
3203:
3145:
81:
41:. In such materials, the approximation of independent electrons, which is used in
3645:
3149:
2974:
3111:(DFT+DMFT) then allows for a realistic calculation of correlated materials.
3822:
3700:
3597:
3317:
3293:
2966:
1785:{\displaystyle G_{ii}(\tau )=-\langle Tc_{i}(\tau )c_{i}^{\dagger }(0)\rangle }
1193:{\displaystyle G_{\text{imp}}(\tau )=-\langle Tc(\tau )c^{\dagger }(0)\rangle }
1035:
describes the impurity, where two electrons interact with the energetical cost
85:
46:
3757:
3526:
3351:
3908:
3899:
3830:
3765:
3244:
3207:
120:
27:
Method to determine the electronic structure of strongly correlated materials
3885:
Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
3879:
Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
3873:
Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
3867:
Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
3610:
3472:
3401:
3374:
3301:
3252:
2110:{\displaystyle \Sigma (k,i\omega _{n})\approx \Sigma _{imp}(i\omega _{n})}
2022:
186:
captures the temporal fluctuations due to electron-electron correlations.
156:
of the impurity model. DMFT owes its name to the fact that the mean-field
3683:
3548:
3509:
3426:
3334:
3221:
A. Georges and G.Kotliar (1992). "Hubbard model in infinite dimensions".
3183:
2028:
123:) onto a single-site problem. In DMFT, the local observable is the local
109:
70:
3859:
Strongly Correlated Materials: Insights From Dynamical Mean-Field Theory
3865:
Lecture notes on the LDA+DMFT approach to strongly correlated materials
3496:
3435:
1651:
The self-consistency condition requires the impurity Green's function
3449:
John Hubbard (1963). "Electron Correlations in Narrow Energy Bands".
3120:
556:
2132:. Here, DMFT and classical mean-field theories result in mean-field
3805:
3628:
3740:
3580:
3883:
Lecture notes Dynamical Mean-Field Theory of Correlated Electrons
3724:"Nonequilibrium dynamical mean-field theory and its applications"
532:
the orbitals are so localized that only nearest-neighbor hopping
3892:
https://www.cond-mat.de/events/correl21/manuscripts/pavarini.pdf
3877:
Lecture notes DMFT – From Infinite Dimensions to Real Materials
2878:{\displaystyle G_{\mathrm {imp} }^{n}=G_{\mathrm {imp} }^{n+1}}
104:
The DMFT treatment of lattice quantum models is similar to the
2128:
Spatial fluctuations also become relevant in the vicinity of
189:
3666:
1118:
The Matsubara Green's function of this model, defined by
3787:
Roy, F; Biroli, G; Bunin, G; Cammarota, C (2019-11-29).
3220:
3786:
3721:
2023:
DMFT approximation: locality of the lattice self-energy
224:. The Hubbard Hamiltonian may take the following form:
37:) is a method to determine the electronic structure of
3375:"Dynamical mean-field theory for correlated electrons"
3265:
3060:{\displaystyle U_{\alpha \beta }n_{\alpha }n_{\beta }}
2977:(compressibility, optical conductivity, specific heat)
3561:
3093:
3073:
3023:
2987:
2940:
2920:
2900:
2819:
2655:
2601:
2457:
2401:
2314:
2260:
2215:
2191:
2171:
2151:
2040:
1980:
1801:
1701:
1657:
1595:
1556:
1517:
1400:
1357:
1232:
1206:
1124:
1089:
1062:
1041:
1014:
985:
958:
659:
636:
601:
571:
538:
515:
449:
429:
384:
364:
233:
210:
162:
133:
1695:
to coincide with the local lattice Green's function
2981:In particular, the drop of the double-occupancy as
1351:, which is the imaginary-time Fourier-transform of
3793:Journal of Physics A: Mathematical and Theoretical
3099:
3079:
3059:
2993:
2946:
2926:
2906:
2877:
2803:
2641:
2595:Solve the AIM for a new impurity Green's function
2585:
2441:
2385:
2297:
2246:
2197:
2177:
2157:
2109:
2011:
1963:
1784:
1687:
1607:
1581:
1542:
1499:
1379:
1343:
1218:
1192:
1107:
1075:
1047:
1027:
998:
971:
938:
642:
622:
587:
557:The auxiliary problem: the Anderson impurity model
544:
521:
490:
435:
415:
370:
347:
216:
177:
148:
3268:"Correlated Lattice Fermions in d = ∞ Dimensions"
3001:increases is a signature of the Mott transition.
2813:Go back to step 2 until convergence, namely when
3906:
3107:denote different orbitals. The combination with
2642:{\displaystyle G_{\mathrm {imp} }(i\omega _{n})}
2442:{\displaystyle G_{\mathrm {loc} }(i\omega _{n})}
108:(MFT) treatment of classical models such as the
99:
80:One of DMFT's main successes is to describe the
3415:
979:describes the non-correlated electronic levels
199:
3214:
2031:, by equating it to the impurity self-energy:
3871:Lecture notes DMFT at 25: Infinite Dimensions
1646:
3448:
3372:
1779:
1730:
1187:
1150:
264:
255:
1200:, is entirely determined by the parameters
491:{\displaystyle n_{i}=c_{i}^{\dagger }c_{i}}
358:where, on suppressing the spin 1/2 indices
3900:https://doi.org/10.1007/s40766-021-00025-8
3313:
3311:
3139:
3012:
501:The following assumptions have been made:
3804:
3739:
3682:
3627:
3604:
3579:
3508:
3425:
3400:
3333:
2914:, bandwidth, filling (chemical potential
2298:{\displaystyle \Sigma (k,i\omega _{n})=0}
1688:{\displaystyle G_{\mathrm {imp} }(\tau )}
1226:and the so-called hybridization function
190:Description of DMFT for the Hubbard model
3409:
3366:
3168:with a thermodynamic number of species.
1380:{\displaystyle \Delta _{\sigma }(\tau )}
3493:
3308:
3171:
2247:{\displaystyle \Sigma (k,i\omega _{n})}
2012:{\displaystyle \Sigma (k,i\omega _{n})}
623:{\displaystyle a_{p\sigma }^{\dagger }}
115:Likewise, DMFT maps a lattice problem (
14:
3907:
3660:
3442:
3259:
3177:
2145:recursion method, namely, for a given
416:{\displaystyle c_{i}^{\dagger },c_{i}}
3004:
1582:{\displaystyle \Delta (i\omega _{n})}
3487:
2395:Compute the local Green's function
1640:Continuous-time quantum Monte Carlo
1550:for a given hybridization function
194:
24:
3452:Proceedings of the Royal Society A
3155:
2858:
2855:
2852:
2832:
2829:
2826:
2763:
2760:
2757:
2702:
2668:
2665:
2662:
2657:
2614:
2611:
2608:
2558:
2555:
2552:
2547:
2510:
2507:
2504:
2458:
2414:
2411:
2408:
2358:
2355:
2352:
2347:
2315:
2261:
2216:
2073:
2041:
1981:
1930:
1814:
1811:
1808:
1670:
1667:
1664:
1557:
1475:
1407:
1359:
1234:
163:
134:
53:gas limit and the atomic limit of
25:
3941:
3852:
3266:W. Metzner; D. Vollhardt (1989).
3162:Green function out of equilibrium
2451:Compute the dynamical mean field
2019:denotes the lattice self-energy.
3114:
2962:(which gives the band structure)
2139:
3780:
3715:
3555:
3126:
2889:
1620:Numerical renormalization group
1543:{\displaystyle G(i\omega _{n})}
1108:{\displaystyle V_{p}^{\sigma }}
972:{\displaystyle H_{\text{bath}}}
64:lattice problem to a many-body
3541:
2971:the double occupancy of a site
2798:
2782:
2770:
2748:
2742:
2726:
2714:
2696:
2690:
2674:
2636:
2620:
2580:
2564:
2540:
2524:
2470:
2461:
2436:
2420:
2380:
2364:
2340:
2318:
2308:Make the DMFT approximation:
2286:
2264:
2241:
2219:
2104:
2088:
2066:
2044:
2006:
1984:
1955:
1933:
1924:
1918:
1871:
1855:
1836:
1820:
1776:
1770:
1752:
1746:
1721:
1715:
1682:
1676:
1576:
1560:
1537:
1521:
1494:
1478:
1447:
1431:
1419:
1401:
1374:
1368:
1300:
1279:
1259:
1243:
1184:
1178:
1165:
1159:
1141:
1135:
1076:{\displaystyle H_{\text{mix}}}
1028:{\displaystyle H_{\text{loc}}}
907:
894:
873:
863:
340:
327:
178:{\displaystyle \Delta (\tau )}
172:
166:
149:{\displaystyle \Delta (\tau )}
143:
137:
96:of density functional theory.
13:
1:
1630:Iterative perturbation theory
999:{\displaystyle \epsilon _{p}}
100:Relation to mean-field theory
39:strongly correlated materials
3930:Electronic structure methods
3847:Strongly correlated material
3646:10.1103/RevModPhys.90.025003
588:{\displaystyle a_{p\sigma }}
200:Single-orbital Hubbard model
7:
3861:G. Kotliar and D. Vollhardt
3840:
2649:, extract its self-energy:
529:-bands are non-degenerate),
94:local density approximation
60:DMFT consists in mapping a
31:Dynamical mean-field theory
18:Dynamical mean field theory
10:
3946:
3701:10.1103/PhysRevB.75.045118
3598:10.1103/PhysRevB.89.081107
3418:AIP Conference Proceedings
3294:10.1103/PhysRevLett.62.324
1647:Self-consistency equations
1635:Non-crossing approximation
3758:10.1103/RevModPhys.86.779
3728:Reviews of Modern Physics
3615:Reviews of Modern Physics
3527:10.1080/00018730701619647
3352:10.1103/RevModPhys.78.865
3321:Reviews of Modern Physics
3187:Reviews of Modern Physics
3109:density functional theory
43:density functional theory
3823:10.1088/1751-8121/ab1f32
3245:10.1103/PhysRevB.45.6479
3208:10.1103/RevModPhys.68.13
55:condensed-matter physics
3273:Physical Review Letters
3140:Diagrammatic extensions
3080:{\displaystyle \alpha }
3013:Multi-orbital extension
2209:Start with a guess for
643:{\displaystyle \sigma }
563:Anderson impurity model
371:{\displaystyle \sigma }
90:electronic correlations
3473:10.1098/rspa.1963.0204
3402:10.1002/andp.201100250
3101:
3100:{\displaystyle \beta }
3081:
3061:
2995:
2948:
2928:
2908:
2879:
2805:
2643:
2587:
2443:
2387:
2299:
2248:
2199:
2179:
2159:
2111:
2013:
1965:
1786:
1689:
1609:
1608:{\displaystyle U,\mu }
1583:
1544:
1501:
1381:
1345:
1220:
1219:{\displaystyle U,\mu }
1194:
1109:
1077:
1049:
1029:
1000:
973:
940:
644:
624:
589:
546:
523:
492:
437:
417:
372:
349:
218:
179:
150:
84:between a metal and a
3373:D. Vollhardt (2012).
3102:
3082:
3062:
2996:
2949:
2929:
2909:
2880:
2806:
2644:
2588:
2444:
2388:
2300:
2249:
2200:
2180:
2160:
2112:
2014:
1966:
1787:
1690:
1625:Exact diagonalization
1610:
1584:
1545:
1502:
1382:
1346:
1221:
1195:
1110:
1078:
1050:
1030:
1001:
974:
941:
645:
625:
590:
552:is taken into account
547:
524:
493:
438:
418:
373:
350:
219:
180:
151:
88:when the strength of
3915:Correlated electrons
3172:References and notes
3091:
3071:
3021:
2985:
2938:
2927:{\displaystyle \mu }
2918:
2898:
2817:
2653:
2599:
2455:
2399:
2312:
2258:
2213:
2189:
2178:{\displaystyle \mu }
2169:
2149:
2121:coordination limit.
2038:
1978:
1799:
1699:
1655:
1593:
1554:
1515:
1398:
1355:
1230:
1204:
1122:
1087:
1060:
1039:
1012:
983:
956:
657:
634:
599:
569:
536:
513:
447:
427:
382:
362:
231:
208:
160:
131:
51:nearly free electron
3815:2019JPhA...52V4001R
3750:2014RvMP...86..779A
3693:2007PhRvB..75d5118T
3638:2018RvMP...90b5003R
3590:2014PhRvB..89h1107E
3519:2007AdPhy..56..829H
3465:1963RSPSA.276..238H
3393:2012AnP...524....1V
3344:2006RvMP...78..865K
3286:1989PhRvL..62..324M
3237:1992PhRvB..45.6479G
3200:1996RvMP...68...13G
3150:two-particle vertex
2934:), and temperature
2874:
2842:
2523:
1769:
1297:
1104:
795:
780:
711:
619:
477:
399:
289:
3380:Annalen der Physik
3097:
3077:
3057:
3005:Extensions of DMFT
2991:
2975:response functions
2944:
2924:
2904:
2875:
2846:
2820:
2801:
2639:
2583:
2498:
2439:
2383:
2295:
2244:
2195:
2175:
2155:
2134:critical exponents
2107:
2009:
1961:
1886:
1782:
1755:
1685:
1605:
1579:
1540:
1497:
1377:
1341:
1283:
1274:
1216:
1190:
1105:
1090:
1073:
1045:
1025:
996:
969:
936:
935:
921:
847:
833:
781:
766:
760:
740:
726:
697:
686:
640:
620:
602:
585:
542:
519:
488:
463:
433:
413:
385:
368:
345:
318:
272:
271:
214:
175:
146:
3925:Quantum mechanics
3920:Materials science
3670:Physical Review B
3568:Physical Review B
3459:(1365): 238–257.
3436:10.1063/1.1800733
3231:(12): 6479–6483.
3224:Physical Review B
2994:{\displaystyle U}
2960:spectral function
2947:{\displaystyle T}
2907:{\displaystyle U}
2198:{\displaystyle T}
2158:{\displaystyle U}
2130:phase transitions
1959:
1877:
1339:
1265:
1132:
1070:
1048:{\displaystyle U}
1022:
966:
931:
853:
851:
843:
748:
746:
744:
736:
677:
675:
673:
667:
545:{\displaystyle t}
522:{\displaystyle d}
436:{\displaystyle i}
309:
250:
241:
217:{\displaystyle U}
106:mean-field theory
16:(Redirected from
3937:
3835:
3834:
3808:
3784:
3778:
3777:
3743:
3719:
3713:
3712:
3686:
3684:cond-mat/0603100
3664:
3658:
3657:
3631:
3608:
3602:
3601:
3583:
3574:(8): 081107(R).
3559:
3553:
3552:
3545:
3539:
3538:
3512:
3510:cond-mat/0511293
3491:
3485:
3484:
3446:
3440:
3439:
3429:
3427:cond-mat/0403123
3413:
3407:
3406:
3404:
3370:
3364:
3363:
3337:
3335:cond-mat/0511085
3315:
3306:
3305:
3263:
3257:
3256:
3218:
3212:
3211:
3181:
3146:phase transition
3106:
3104:
3103:
3098:
3086:
3084:
3083:
3078:
3066:
3064:
3063:
3058:
3056:
3055:
3046:
3045:
3036:
3035:
3000:
2998:
2997:
2992:
2953:
2951:
2950:
2945:
2933:
2931:
2930:
2925:
2913:
2911:
2910:
2905:
2884:
2882:
2881:
2876:
2873:
2862:
2861:
2841:
2836:
2835:
2810:
2808:
2807:
2802:
2797:
2796:
2781:
2780:
2768:
2767:
2766:
2741:
2740:
2725:
2724:
2712:
2711:
2706:
2705:
2689:
2688:
2673:
2672:
2671:
2648:
2646:
2645:
2640:
2635:
2634:
2619:
2618:
2617:
2592:
2590:
2589:
2584:
2579:
2578:
2563:
2562:
2561:
2539:
2538:
2522:
2514:
2513:
2488:
2487:
2448:
2446:
2445:
2440:
2435:
2434:
2419:
2418:
2417:
2392:
2390:
2389:
2384:
2379:
2378:
2363:
2362:
2361:
2339:
2338:
2304:
2302:
2301:
2296:
2285:
2284:
2253:
2251:
2250:
2245:
2240:
2239:
2204:
2202:
2201:
2196:
2185:and temperature
2184:
2182:
2181:
2176:
2164:
2162:
2161:
2156:
2116:
2114:
2113:
2108:
2103:
2102:
2087:
2086:
2065:
2064:
2018:
2016:
2015:
2010:
2005:
2004:
1970:
1968:
1967:
1962:
1960:
1958:
1954:
1953:
1905:
1904:
1888:
1885:
1870:
1869:
1854:
1853:
1835:
1834:
1819:
1818:
1817:
1791:
1789:
1788:
1783:
1768:
1763:
1745:
1744:
1714:
1713:
1694:
1692:
1691:
1686:
1675:
1674:
1673:
1614:
1612:
1611:
1606:
1588:
1586:
1585:
1580:
1575:
1574:
1549:
1547:
1546:
1541:
1536:
1535:
1506:
1504:
1503:
1498:
1493:
1492:
1465:
1464:
1446:
1445:
1430:
1429:
1417:
1416:
1411:
1410:
1386:
1384:
1383:
1378:
1367:
1366:
1350:
1348:
1347:
1342:
1340:
1338:
1337:
1336:
1324:
1323:
1310:
1309:
1308:
1303:
1296:
1291:
1282:
1276:
1273:
1258:
1257:
1242:
1241:
1225:
1223:
1222:
1217:
1199:
1197:
1196:
1191:
1177:
1176:
1134:
1133:
1130:
1114:
1112:
1111:
1106:
1103:
1098:
1082:
1080:
1079:
1074:
1072:
1071:
1068:
1054:
1052:
1051:
1046:
1034:
1032:
1031:
1026:
1024:
1023:
1020:
1005:
1003:
1002:
997:
995:
994:
978:
976:
975:
970:
968:
967:
964:
945:
943:
942:
937:
934:
933:
932:
929:
922:
917:
916:
912:
911:
910:
898:
897:
877:
876:
867:
866:
846:
845:
844:
841:
834:
829:
828:
824:
808:
807:
794:
789:
779:
774:
759:
739:
738:
737:
734:
727:
722:
721:
720:
710:
705:
696:
695:
685:
669:
668:
665:
649:
647:
646:
641:
629:
627:
626:
621:
618:
613:
594:
592:
591:
586:
584:
583:
551:
549:
548:
543:
528:
526:
525:
520:
497:
495:
494:
489:
487:
486:
476:
471:
459:
458:
442:
440:
439:
434:
422:
420:
419:
414:
412:
411:
398:
393:
377:
375:
374:
369:
354:
352:
351:
346:
344:
343:
331:
330:
317:
302:
301:
288:
283:
270:
243:
242:
239:
223:
221:
220:
215:
195:The DMFT mapping
184:
182:
181:
176:
155:
153:
152:
147:
125:Green's function
82:phase transition
21:
3945:
3944:
3940:
3939:
3938:
3936:
3935:
3934:
3905:
3904:
3855:
3843:
3838:
3785:
3781:
3720:
3716:
3665:
3661:
3609:
3605:
3560:
3556:
3547:
3546:
3542:
3492:
3488:
3447:
3443:
3414:
3410:
3371:
3367:
3316:
3309:
3264:
3260:
3219:
3215:
3182:
3178:
3174:
3158:
3156:Non-equilibrium
3142:
3129:
3117:
3092:
3089:
3088:
3072:
3069:
3068:
3051:
3047:
3041:
3037:
3028:
3024:
3022:
3019:
3018:
3015:
3007:
2986:
2983:
2982:
2939:
2936:
2935:
2919:
2916:
2915:
2899:
2896:
2895:
2892:
2863:
2851:
2850:
2837:
2825:
2824:
2818:
2815:
2814:
2792:
2788:
2773:
2769:
2756:
2755:
2751:
2736:
2732:
2717:
2713:
2707:
2701:
2700:
2699:
2684:
2680:
2661:
2660:
2656:
2654:
2651:
2650:
2630:
2626:
2607:
2606:
2602:
2600:
2597:
2596:
2574:
2570:
2551:
2550:
2546:
2534:
2530:
2515:
2503:
2502:
2483:
2479:
2456:
2453:
2452:
2430:
2426:
2407:
2406:
2402:
2400:
2397:
2396:
2374:
2370:
2351:
2350:
2346:
2334:
2330:
2313:
2310:
2309:
2280:
2276:
2259:
2256:
2255:
2235:
2231:
2214:
2211:
2210:
2190:
2187:
2186:
2170:
2167:
2166:
2150:
2147:
2146:
2142:
2098:
2094:
2076:
2072:
2060:
2056:
2039:
2036:
2035:
2025:
2000:
1996:
1979:
1976:
1975:
1949:
1945:
1900:
1896:
1892:
1887:
1881:
1865:
1861:
1846:
1842:
1830:
1826:
1807:
1806:
1802:
1800:
1797:
1796:
1764:
1759:
1740:
1736:
1706:
1702:
1700:
1697:
1696:
1663:
1662:
1658:
1656:
1653:
1652:
1649:
1594:
1591:
1590:
1570:
1566:
1555:
1552:
1551:
1531:
1527:
1516:
1513:
1512:
1488:
1484:
1460:
1456:
1441:
1437:
1422:
1418:
1412:
1406:
1405:
1404:
1399:
1396:
1395:
1362:
1358:
1356:
1353:
1352:
1332:
1328:
1319:
1315:
1311:
1304:
1299:
1298:
1292:
1287:
1278:
1277:
1275:
1269:
1253:
1249:
1237:
1233:
1231:
1228:
1227:
1205:
1202:
1201:
1172:
1168:
1129:
1125:
1123:
1120:
1119:
1099:
1094:
1088:
1085:
1084:
1067:
1063:
1061:
1058:
1057:
1040:
1037:
1036:
1019:
1015:
1013:
1010:
1009:
990:
986:
984:
981:
980:
963:
959:
957:
954:
953:
928:
924:
923:
906:
902:
893:
889:
888:
884:
872:
868:
862:
858:
854:
852:
840:
836:
835:
800:
796:
790:
785:
775:
770:
765:
761:
752:
747:
745:
733:
729:
728:
716:
712:
706:
701:
691:
687:
681:
676:
674:
664:
660:
658:
655:
654:
635:
632:
631:
614:
606:
600:
597:
596:
576:
572:
570:
567:
566:
559:
537:
534:
533:
514:
511:
510:
482:
478:
472:
467:
454:
450:
448:
445:
444:
428:
425:
424:
407:
403:
394:
389:
383:
380:
379:
363:
360:
359:
336:
332:
323:
319:
313:
294:
290:
284:
276:
254:
238:
234:
232:
229:
228:
209:
206:
205:
202:
197:
192:
161:
158:
157:
132:
129:
128:
102:
28:
23:
22:
15:
12:
11:
5:
3943:
3933:
3932:
3927:
3922:
3917:
3903:
3902:
3897:
3894:
3889:
3886:
3880:
3874:
3868:
3862:
3854:
3853:External links
3851:
3850:
3849:
3842:
3839:
3837:
3836:
3799:(48): 484001.
3779:
3734:(2): 779–837.
3714:
3659:
3603:
3554:
3540:
3503:(6): 829–926.
3486:
3441:
3408:
3365:
3307:
3280:(3): 324–327.
3258:
3213:
3175:
3173:
3170:
3157:
3154:
3141:
3138:
3128:
3125:
3116:
3113:
3096:
3076:
3054:
3050:
3044:
3040:
3034:
3031:
3027:
3014:
3011:
3006:
3003:
2990:
2979:
2978:
2972:
2969:
2967:kinetic energy
2963:
2943:
2923:
2903:
2891:
2888:
2887:
2886:
2872:
2869:
2866:
2860:
2857:
2854:
2849:
2845:
2840:
2834:
2831:
2828:
2823:
2811:
2800:
2795:
2791:
2787:
2784:
2779:
2776:
2772:
2765:
2762:
2759:
2754:
2750:
2747:
2744:
2739:
2735:
2731:
2728:
2723:
2720:
2716:
2710:
2704:
2698:
2695:
2692:
2687:
2683:
2679:
2676:
2670:
2667:
2664:
2659:
2638:
2633:
2629:
2625:
2622:
2616:
2613:
2610:
2605:
2593:
2582:
2577:
2573:
2569:
2566:
2560:
2557:
2554:
2549:
2545:
2542:
2537:
2533:
2529:
2526:
2521:
2518:
2512:
2509:
2506:
2501:
2497:
2494:
2491:
2486:
2482:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2449:
2438:
2433:
2429:
2425:
2422:
2416:
2413:
2410:
2405:
2393:
2382:
2377:
2373:
2369:
2366:
2360:
2357:
2354:
2349:
2345:
2342:
2337:
2333:
2329:
2326:
2323:
2320:
2317:
2306:
2294:
2291:
2288:
2283:
2279:
2275:
2272:
2269:
2266:
2263:
2243:
2238:
2234:
2230:
2227:
2224:
2221:
2218:
2194:
2174:
2154:
2141:
2138:
2118:
2117:
2106:
2101:
2097:
2093:
2090:
2085:
2082:
2079:
2075:
2071:
2068:
2063:
2059:
2055:
2052:
2049:
2046:
2043:
2024:
2021:
2008:
2003:
1999:
1995:
1992:
1989:
1986:
1983:
1972:
1971:
1957:
1952:
1948:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1903:
1899:
1895:
1891:
1884:
1880:
1876:
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1868:
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1860:
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1805:
1781:
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1762:
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1726:
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1717:
1712:
1709:
1705:
1684:
1681:
1678:
1672:
1669:
1666:
1661:
1648:
1645:
1644:
1643:
1637:
1632:
1627:
1622:
1604:
1601:
1598:
1578:
1573:
1569:
1565:
1562:
1559:
1539:
1534:
1530:
1526:
1523:
1520:
1509:
1508:
1496:
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1487:
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1480:
1477:
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1471:
1468:
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1440:
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1370:
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1361:
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1327:
1322:
1318:
1314:
1307:
1302:
1295:
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1268:
1264:
1261:
1256:
1252:
1248:
1245:
1240:
1236:
1215:
1212:
1209:
1189:
1186:
1183:
1180:
1175:
1171:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1128:
1116:
1115:
1102:
1097:
1093:
1066:
1055:
1044:
1018:
1007:
993:
989:
962:
947:
946:
927:
920:
915:
909:
905:
901:
896:
892:
887:
883:
880:
875:
871:
865:
861:
857:
850:
839:
832:
827:
823:
820:
817:
814:
811:
806:
803:
799:
793:
788:
784:
778:
773:
769:
764:
758:
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751:
743:
732:
725:
719:
715:
709:
704:
700:
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690:
684:
680:
672:
663:
639:
617:
612:
609:
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582:
579:
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558:
555:
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553:
541:
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518:
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98:
86:Mott insulator
47:band structure
26:
9:
6:
4:
3:
2:
3942:
3931:
3928:
3926:
3923:
3921:
3918:
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3848:
3845:
3844:
3832:
3828:
3824:
3820:
3816:
3812:
3807:
3802:
3798:
3794:
3790:
3783:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3742:
3737:
3733:
3729:
3725:
3718:
3710:
3706:
3702:
3698:
3694:
3690:
3685:
3680:
3677:(4): 045118.
3676:
3672:
3671:
3663:
3655:
3651:
3647:
3643:
3639:
3635:
3630:
3625:
3622:(4): 025003.
3621:
3617:
3616:
3607:
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3587:
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3506:
3502:
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3470:
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3462:
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3437:
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3423:
3419:
3412:
3403:
3398:
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3390:
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3323:
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3314:
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3299:
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3201:
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3180:
3176:
3169:
3167:
3163:
3153:
3151:
3147:
3137:
3133:
3124:
3122:
3115:Extended DMFT
3112:
3110:
3094:
3074:
3052:
3048:
3042:
3038:
3032:
3029:
3025:
3010:
3002:
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2527:
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2375:
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2331:
2327:
2324:
2321:
2307:
2292:
2289:
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2277:
2273:
2270:
2267:
2236:
2232:
2228:
2225:
2222:
2208:
2207:
2206:
2192:
2172:
2152:
2140:The DMFT loop
2137:
2135:
2131:
2126:
2122:
2099:
2095:
2091:
2083:
2080:
2077:
2069:
2061:
2057:
2053:
2050:
2047:
2034:
2033:
2032:
2030:
2020:
2001:
1997:
1993:
1990:
1987:
1950:
1946:
1942:
1939:
1936:
1927:
1921:
1915:
1912:
1909:
1906:
1901:
1897:
1893:
1889:
1882:
1878:
1874:
1866:
1862:
1858:
1850:
1847:
1843:
1839:
1831:
1827:
1823:
1803:
1795:
1794:
1793:
1773:
1765:
1760:
1756:
1749:
1741:
1737:
1733:
1727:
1724:
1718:
1710:
1707:
1703:
1679:
1659:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1617:
1616:
1602:
1599:
1596:
1571:
1567:
1563:
1532:
1528:
1524:
1518:
1489:
1485:
1481:
1472:
1469:
1466:
1461:
1457:
1453:
1450:
1442:
1438:
1434:
1426:
1423:
1413:
1394:
1393:
1392:
1388:
1371:
1363:
1333:
1329:
1325:
1320:
1316:
1312:
1305:
1293:
1288:
1284:
1270:
1266:
1262:
1254:
1250:
1246:
1238:
1213:
1210:
1207:
1181:
1173:
1169:
1162:
1156:
1153:
1147:
1144:
1138:
1126:
1100:
1095:
1091:
1064:
1056:
1042:
1016:
1008:
991:
987:
960:
952:
951:
950:
925:
918:
913:
903:
899:
890:
885:
881:
878:
869:
859:
855:
848:
837:
830:
825:
821:
818:
815:
812:
809:
804:
801:
797:
791:
786:
782:
776:
771:
767:
762:
756:
753:
749:
741:
730:
723:
717:
713:
707:
702:
698:
692:
688:
682:
678:
670:
661:
653:
652:
651:
637:
615:
610:
607:
603:
580:
577:
573:
564:
539:
531:
516:
508:
504:
503:
502:
499:
483:
479:
473:
468:
464:
460:
455:
451:
430:
408:
404:
400:
395:
390:
386:
365:
337:
333:
324:
320:
314:
310:
306:
303:
298:
295:
291:
285:
280:
277:
273:
267:
261:
258:
251:
247:
244:
235:
227:
226:
225:
211:
187:
169:
140:
126:
122:
121:Hubbard model
118:
113:
111:
107:
97:
95:
91:
87:
83:
78:
76:
72:
67:
63:
58:
56:
52:
48:
44:
40:
36:
32:
19:
3796:
3792:
3782:
3731:
3727:
3717:
3674:
3668:
3662:
3619:
3613:
3606:
3571:
3567:
3557:
3543:
3500:
3495:
3489:
3456:
3450:
3444:
3417:
3411:
3384:
3378:
3368:
3325:
3319:
3277:
3271:
3261:
3228:
3222:
3216:
3191:
3185:
3179:
3159:
3143:
3134:
3130:
3127:Cluster DMFT
3118:
3016:
3008:
2980:
2893:
2890:Applications
2254:(typically,
2143:
2127:
2123:
2119:
2026:
1973:
1650:
1510:
1389:
1117:
948:
560:
500:
357:
203:
116:
114:
103:
79:
75:coordination
65:
59:
34:
30:
29:
3387:(1): 1–19.
2029:self-energy
1006:of the bath
110:Ising model
71:self-energy
3909:Categories
3806:1901.10036
3629:1705.00024
3497:Adv. Phys.
3328:(3): 865.
1642:algorithms
45:and usual
3831:1751-8113
3774:119213862
3766:0034-6861
3741:1310.5329
3709:119538856
3654:119186041
3581:1402.4190
3360:119099745
3194:(1): 13.
3166:community
3121:t-J model
3095:β
3075:α
3053:β
3043:α
3033:β
3030:α
2922:μ
2790:ω
2775:−
2746:−
2734:ω
2719:−
2682:ω
2658:Σ
2628:ω
2572:ω
2548:Σ
2544:−
2532:ω
2517:−
2496:−
2493:μ
2481:ω
2468:ω
2459:Δ
2428:ω
2372:ω
2348:Σ
2344:≈
2332:ω
2316:Σ
2278:ω
2262:Σ
2233:ω
2217:Σ
2173:μ
2096:ω
2074:Σ
2070:≈
2058:ω
2042:Σ
1998:ω
1982:Σ
1947:ω
1931:Σ
1928:−
1916:ϵ
1913:−
1910:μ
1898:ω
1879:∑
1863:ω
1828:ω
1780:⟩
1766:†
1750:τ
1731:⟨
1728:−
1719:τ
1680:τ
1603:μ
1568:ω
1558:Δ
1529:ω
1486:ω
1476:Δ
1473:−
1470:μ
1458:ω
1439:ω
1424:−
1372:τ
1364:σ
1360:Δ
1330:ϵ
1326:−
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1294:σ
1267:∑
1251:ω
1239:σ
1235:Δ
1214:μ
1188:⟩
1174:†
1163:τ
1151:⟨
1148:−
1139:τ
1101:σ
988:ϵ
919:⏟
908:↓
895:↑
882:μ
879:−
874:↓
864:↑
831:⏟
805:σ
792:†
787:σ
777:σ
757:σ
750:∑
724:⏟
708:†
689:ϵ
679:∑
638:σ
616:†
611:σ
581:σ
474:†
396:†
366:σ
341:↓
328:↑
311:∑
299:σ
286:†
281:σ
268:σ
265:⟩
256:⟨
252:∑
170:τ
164:Δ
141:τ
135:Δ
62:many-body
3841:See also
3535:15466043
3481:35439962
3302:10040203
3253:10000408
509:, whose
507:cuprates
3811:Bibcode
3746:Bibcode
3689:Bibcode
3634:Bibcode
3586:Bibcode
3515:Bibcode
3461:Bibcode
3389:Bibcode
3340:Bibcode
3282:Bibcode
3233:Bibcode
3196:Bibcode
240:Hubbard
3829:
3772:
3764:
3707:
3652:
3533:
3479:
3358:
3300:
3251:
3067:where
1974:where
949:where
650:, is:
443:, and
3801:arXiv
3770:S2CID
3736:arXiv
3705:S2CID
3679:arXiv
3650:S2CID
3624:arXiv
3576:arXiv
3531:S2CID
3505:arXiv
3477:S2CID
3422:arXiv
3356:S2CID
3330:arXiv
66:local
3827:ISSN
3762:ISSN
3298:PMID
3249:PMID
3087:and
2965:the
2958:the
1589:and
965:bath
735:bath
595:and
119:the
117:e.g.
35:DMFT
3819:doi
3754:doi
3697:doi
3642:doi
3594:doi
3523:doi
3469:doi
3457:276
3432:doi
3397:doi
3385:524
3348:doi
3290:doi
3241:doi
3204:doi
1507:(1)
1131:imp
1069:mix
1021:loc
930:loc
842:mix
666:AIM
3911::
3825:.
3817:.
3809:.
3797:52
3795:.
3791:.
3768:.
3760:.
3752:.
3744:.
3732:86
3730:.
3726:.
3703:.
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3648:.
3640:.
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3620:90
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3570:.
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3501:56
3475:.
3467:.
3455:.
3430:.
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3383:.
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3354:.
3346:.
3338:.
3326:78
3324:.
3310:^
3296:.
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3278:62
3276:.
3270:.
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3229:45
3227:.
3202:.
3192:68
3190:.
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3123:.
2954::
2205::
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1792::
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3803::
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3049:n
3039:n
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2885:.
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2045:(
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1994:i
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1985:(
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1934:(
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