533:
1926:
22:
165:
155:
On the other hand, the dynamics of the models solvable by the Bethe ansatz is two-body reducible: the many-body scattering matrix is a product of two-body scattering matrices. Many-body collisions happen as a sequence of two-body collisions and the many-body wave function can be represented in a form
891:
solutions (by N. Andrei and C. Destri and by C.J. Bolech and N. Andrei). Recently several models solvable by Bethe ansatz were realized experimentally in solid states and optical lattices. An important role in the theoretical description of these experiments was played by Jean-SĂ©bastien Caux and
890:
model (by P.B. Wiegmann in 1980 and independently by N. Andrei, also in 1980) and the
Anderson model (by P.B. Wiegmann in 1981, and by N. Kawakami and A. Okiji in 1981) are also both based on the Bethe ansatz. There exist multi-channel generalizations of these two models also amenable to exact
528:{\displaystyle \Psi _{M}(j_{1},\cdots ,j_{M})=\prod _{M\geq a>b\geq 1}{\text{sgn}}(j_{a}-j_{b})\sum _{P\in {\mathfrak {S}}_{M}}(-1)^{}\exp \left(i\sum _{a=1}^{M}k_{P_{a}}j_{a}+{\frac {i}{2}}\sum _{M\geq a>b\geq 1}\mathrm {sgn} (j_{a}-j_{b})\phi (k_{P_{a}},k_{P_{b}})\right)}
1704:
1556:
1073:
1199:
2018:
1962: J. des
Cloizeaux and J. J. Pearson obtain the correct spectrum of the Heisenberg antiferromagnet (spinon dispersion relation), showing that it differs from Andersonâs spin-wave theory predictions (the constant prefactor is
1262:
1567:
1352:
1315:
958:
2980:
Yang, C. N.; Yang, C. P. (7 October 1966). "One-Dimensional Chain of
Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System".
635:
604:
2945:
Yang, C. N.; Yang, C. P. (7 October 1966). "One-Dimensional Chain of
Anisotropic Spin-Spin Interactions. I. Proof of Bethe's Hypothesis for Ground State in a Finite System".
2059:
generalizes Lieb and
Liniger's solution of the ÎŽ-function interacting Bose gas to arbitrary permutation symmetry of the wavefunction, giving birth to the nested Bethe ansatz.
824:
1081:
667:
1341:
709:
1813:
1734:
796:
756:
148:) is the anti-symmetrized (symmetrized) product of one-body wave functions. Models solvable by the Bethe ansatz are not free: the two-body sector has a non-trivial
1839:
1786:
1760:
1885:
776:
729:
556:
1998:
156:
which contains only elements from two-body wave functions. The many-body scattering matrix is equal to the product of pairwise scattering matrices.
3089:
Lieb, Elliott H.; Wu, F. Y. (17 June 1968). "Absence of Mott
Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension".
2027:
40:
1204:
2840:
Lieb, Elliott H.; Liniger, Werner (15 May 1963). "Exact
Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State".
2052:
rigorously prove that the ground-state of the
Heisenberg chain is given by the Bethe ansatz. They study properties and applications in and.
2078:
2049:
3124:
Yang, C. N.; Yang, C. P. (July 1969). "Thermodynamics of a OneâDimensional System of Bosons with
Repulsive DeltaâFunction Interaction".
2067:
2910:
Griffiths, Robert B. (3 February 1964). "Magnetization Curve at Zero
Temperature for the Antiferromagnetic Heisenberg Linear Chain".
1699:{\displaystyle \theta _{1}(\lambda _{a})-{\frac {1}{N}}\sum _{b=1}^{M}\theta _{2}(\lambda _{a}-\lambda _{b})=2\pi {\frac {I_{a}}{N}}}
860:
lead to the Bethe ansatz equations or simply Bethe equations. In logarithmic form the Bethe ansatz equations can be generated by the
2251:
2462:
Bolech, C. J.; Andrei, N. (2002). "Solution of the Two-Channel Anderson Impurity Model: Implications for the Heavy Fermion UBe13".
1551:{\displaystyle \left^{N}=\prod _{b\neq a}^{M}{\frac {\lambda _{a}-\lambda _{b}+i}{\lambda _{a}-\lambda _{b}-i}},\qquad a=1,...,M}
1987:
proposes an oversimplified ansatz which miscounts the number of solutions to the Schrödinger equation for the Heisenberg chain.
3015:
Yang, C. N.; Yang, C. P. (4 November 1966). "One-Dimensional Chain of Anisotropic Spin-Spin Interactions. III. Applications".
830:. This form is universal (at least for non-nested systems), with the momentum and scattering functions being model-dependent.
2614:
2581:
1068:{\displaystyle H=J\sum _{j=1}^{N}\mathbf {S} _{j}\cdot \mathbf {S} _{j+1},\qquad \mathbf {S} _{j+N}\equiv \mathbf {S} _{j}.}
2221:
1267:
144:
models. One can say that the dynamics of a free model is one-body reducible: the many-body wave function for fermions (
2770:
des Cloizeaux, Jacques; Pearson, J. J. (1 December 1962). "Spin-Wave Spectrum of the Antiferromagnetic Linear Chain".
3185:
2376:
Kawakami, Norio; Okiji, Ayao (1981). "Exact expression of the ground-state energy for the symmetric anderson model".
2231:
908:
876:
58:
3190:
2081:
obtain the thermodynamics of the Lieb-Liniger model, providing the basis of the thermodynamic Bethe ansatz (TBA).
2102:
Bethe, H. (March 1931). "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette".
952:
The Heisenberg antiferromagnetic chain is defined by the Hamiltonian (assuming periodic boundary conditions)
857:
99:
2875:
Lieb, Elliott H. (15 May 1963). "Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum".
3052:"Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction"
609:
561:
107:
2805:
Anderson, P. W. (1 June 1952). "An Approximate Quantum Theory of the Antiferromagnetic Ground State".
2137:
1194:{\displaystyle \phi (k_{a}(\lambda _{a}),k_{b}(\lambda _{b}))=\theta _{2}(\lambda _{a}-\lambda _{b})}
838:
1994:
proposes the correct ansatz and carefully shows that it yields the correct number of eigenfunctions.
801:
1977:
1078:
This model is solvable using the (coordinate) Bethe ansatz. The scattering phase shift function is
834:
1853:
640:
118:
1899:
1323:
1864:
672:
911:
is the method of solution by algebraic Bethe ansatz, and the two are practically synonymous.
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2342:
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2178:
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1712:
781:
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Orbach, R. (15 October 1958). "Linear Antiferromagnetic Chain with Anisotropic Coupling".
8:
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2003:
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provide the exact solution of the 1d ÎŽ-function interacting Bose gas (now known as the
2012:
1908:
1870:
761:
714:
541:
103:
1947:
124:"Bethe ansatz problems" were one of the topics featuring in the "To learn" section of
36:
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2015:
uses the Bethe ansatz to solve the Heisenberg model with anisotropic interactions.
159:
The generic form of the (coordinate) Bethe ansatz for a many-body wavefunction is
2531:
Faddeev, Ludwig (1992). "How Algebraic Bethe Ansatz works for integrable model".
2063:
2023:
841:
is valid for models solvable by the Bethe ansatz, even for models of interacting
125:
3110:
3075:
2606:
2440:
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that "has allowed a wide class of nonlinear evolution equations to be solved."
869:
861:
3174:
2931:
2896:
2861:
2791:
2573:
2501:
2448:
2419:
Andrei, N.; Destri, C. (1984). "Solution of the Multichannel Kondo Problem".
2405:
2362:
2319:
2247:
2198:
1889:
853:
827:
3036:
3001:
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obtains the magnetization curve of the Heisenberg model at zero temperature.
2826:
2509:
1951:
1858:
1257:{\displaystyle \theta _{n}(\lambda )\equiv 2\arctan {\frac {2\lambda }{n}}}
849:
84:
2333:
Wiegmann, P.B. (1980). "Towards an exact solution of the Anderson model".
2476:
1984:
1894:
903:
There are many similar methods which come under the name of Bethe ansatz
865:
94:, most commonly for one-dimensional lattice models. It was first used by
2034:). Lieb studies the spectrum and defines two basic types of excitations.
3051:
2694:
2651:
2537:
2190:
2115:
1991:
934:
Thermodynamic Bethe ansatz (C.N. Yang & C.P. Yang
918:
114:
95:
3145:
2138:"Richard Feynman's blackboard at time of his death | Caltech Archives"
879:, or algebraic Bethe ansatz, which gives an ansatz for the underlying
2716:
HulthĂ©n, Lamek (1938). "Ăber das Austauschproblem eines Kristalles".
2630:
Heisenberg, W. (September 1928). "Zur Theorie des Ferromagnetismus".
1940:
1936:
149:
140:, models solvable by the Bethe ansatz can be contrasted with free
141:
72:
2220:
Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1997-03-06).
864:. The square of the norm of Bethe wave function is equal to the
2290:
Andrei, N. (1980). "Diagonalization of the Kondo Hamiltonian".
842:
80:
2008:
obtains the exact ground-state energy of the Heisenberg model.
1264:
in which the momentum has been conveniently reparametrized as
145:
2673:
Bloch, F. (March 1930). "Zur Theorie des Ferromagnetismus".
2223:
Quantum Inverse Scattering Method and Correlation Functions
1849:
The following systems can be solved using the Bethe ansatz
2597:
Sklyanin, E.K. (October 1990). "Functional Bethe Ansatz".
2554:
Non-Linear Equations in Classical and Quantum Field Theory
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The (here, periodic) boundary conditions impose the
1310:{\displaystyle k(\lambda )=\pi -2\arctan 2\lambda }
31:
may be too technical for most readers to understand
2552:Sklyanin, E. K. (1985). "The quantum Toda chain".
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2130:
113:Since then the method has been extended to other
3172:
2455:
2369:
837:guarantees consistency of the construction. The
2412:
2252:"Exact solution of s-d exchange model at T = 0"
2092:
731:taking values either positive or negative one,
637:is the set of all permutations of the integers
2167:"Calculation of norms of Bethe wave functions"
2375:
2461:
2418:
2839:
2326:
798:is the scattering phase shift function and
152:, which in general depends on the momenta.
2629:
2909:
2536:
2475:
2283:
1561:or more conveniently in logarithmic form
59:Learn how and when to remove this message
43:, without removing the technical details.
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3014:
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128:'s blackboard at the time of his death.
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2599:Integrable and Superintegrable Systems
2289:
2171:Communications in Mathematical Physics
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2101:
922:
886:The exact solutions of the so-called
41:make it understandable to non-experts
3049:
2874:
1919:
875:A substantial generalization is the
15:
1736:are distinct half-odd integers for
630:{\displaystyle {\mathfrak {S}}_{M}}
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293:
13:
948:Heisenberg antiferromagnetic chain
812:
809:
806:
599:{\displaystyle j_{a},a=1,\cdots M}
441:
438:
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170:
14:
3202:
3156:
909:quantum inverse scattering method
877:quantum inverse scattering method
711:is the parity of the permutation
3163:Introduction to the Bethe Ansatz
1924:
1844:
1052:
1031:
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994:
20:
3126:Journal of Mathematical Physics
3117:
3082:
3050:Yang, C. N. (4 December 1967).
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1911:(through Heisenberg spin chain)
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1028:
758:is the (quasi-)momentum of the
2240:
2226:. Cambridge University Press.
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1939:format but may read better as
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819:{\displaystyle \mathrm {sgn} }
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136:In the framework of many-body
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2494:10.1103/PhysRevLett.88.237206
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1915:
131:
2556:. Lecture Notes in Physics.
2398:10.1016/0375-9601(81)90663-0
2355:10.1016/0375-9601(80)90212-1
2142:digital.archives.caltech.edu
907:Algebraic Bethe ansatz. The
858:Periodic boundary conditions
558:is the number of particles,
100:eigenvalues and eigenvectors
7:
3111:10.1103/PhysRevLett.20.1445
3076:10.1103/PhysRevLett.19.1312
2070:solve the 1d Hubbard model.
942:
662:{\displaystyle 1,\cdots ,M}
10:
3207:
2607:10.1142/9789812797179_0002
2441:10.1103/PhysRevLett.52.364
2312:10.1103/PhysRevLett.45.379
1709:where the quantum numbers
98:in 1931 to find the exact
1336:{\displaystyle \lambda .}
917:Coordinate Bethe ansatz (
839:Pauli exclusion principle
3186:Condensed matter physics
2932:10.1103/PhysRev.133.A768
2897:10.1103/PhysRev.130.1616
2862:10.1103/PhysRev.130.1605
2792:10.1103/PhysRev.128.2131
2718:Arkiv Mat. Astron. Fysik
2574:10.1007/3-540-15213-X_80
2265:(7): 364. Archived from
928:Functional Bethe ansatz
3191:Exactly solvable models
3091:Physical Review Letters
3056:Physical Review Letters
3037:10.1103/PhysRev.151.258
3002:10.1103/PhysRev.150.327
2967:10.1103/PhysRev.150.321
2757:10.1103/PhysRev.112.309
2464:Physical Review Letters
2421:Physical Review Letters
2292:Physical Review Letters
1948:converting this section
1854:Anderson impurity model
704:{\displaystyle (-1)^{}}
102:of the one-dimensional
2827:10.1103/PhysRev.86.694
2675:Zeitschrift fĂŒr Physik
2632:Zeitschrift fĂŒr Physik
2104:Zeitschrift fĂŒr Physik
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365:
83:for finding the exact
1882:
1865:Heisenberg spin chain
1836:
1810:
1808:{\displaystyle I_{j}}
1783:
1757:
1731:
1729:{\displaystyle I_{j}}
1701:
1610:
1553:
1430:
1338:
1312:
1259:
1196:
1070:
971:
914:Analytic Bethe ansatz
872:of the Yang action.
821:
793:
791:{\displaystyle \phi }
773:
753:
751:{\displaystyle k_{a}}
726:
706:
664:
632:
601:
553:
530:
345:
2163:Korepin, Vladimir E.
1871:
1819:
1792:
1766:
1740:
1713:
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959:
835:YangâBaxter equation
802:
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673:
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542:
166:
3138:1969JMP....10.1115Y
3103:1968PhRvL..20.1445L
3068:1967PhRvL..19.1312Y
3029:1966PhRv..151..258Y
2994:1966PhRv..150..327Y
2959:1966PhRv..150..321Y
2924:1964PhRv..133..768G
2889:1963PhRv..130.1616L
2854:1963PhRv..130.1605L
2819:1952PhRv...86..694A
2784:1962PhRv..128.2131D
2749:1958PhRv..112..309O
2687:1930ZPhy...61..206B
2644:1928ZPhy...49..619H
2566:1985LNP...226..196S
2486:2002PhRvL..88w7206B
2433:1984PhRvL..52..364A
2390:1981PhLA...86..483K
2347:1980PhLA...80..163W
2304:1980PhRvL..45..379A
2183:1982CMaPh..86..391K
2039:Robert B. Griffiths
1867:for arbitrary spin
1834:{\displaystyle (N)}
1781:{\displaystyle N-M}
1762:even, integers for
1755:{\displaystyle N-M}
931:Nested Bethe ansatz
2695:10.1007/BF01339661
2652:10.1007/BF01328601
2191:10.1007/BF01212176
2116:10.1007/BF01341708
2032:Lieb-Liniger model
2013:Raymond Lee Orbach
1950:, if appropriate.
1909:Eight-vertex model
1900:LiebâLiniger model
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3146:10.1063/1.1664947
3097:(25): 1445â1448.
3062:(23): 1312â1315.
2918:(3A): A768âA775.
2638:(9â10): 619â636.
2616:978-981-02-0316-0
2583:978-3-540-15213-2
2378:Physics Letters A
2335:Physics Letters A
1974:Werner Heisenberg
1969:
1968:
1880:{\displaystyle s}
1694:
1608:
1515:
1415:
1252:
771:{\displaystyle a}
724:{\displaystyle P}
551:{\displaystyle M}
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150:scattering matrix
138:quantum mechanics
104:antiferromagnetic
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3132:(7): 1115â1122.
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2883:(4): 1616â1624.
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2778:(5): 2131â2135.
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2341:(2â3): 163â167.
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2110:(3â4): 205â226.
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1954:is available.
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1890:Hubbard model
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85:wavefunctions
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49:February 2020
42:
38:
32:
29:This article
27:
18:
17:
3167:
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2328:
2295:
2291:
2285:
2274:. Retrieved
2267:the original
2262:
2259:JETP Letters
2258:
2242:
2222:
2215:
2174:
2170:
2157:
2145:. Retrieved
2141:
2132:
2107:
2103:
1957:
1952:Editing help
1934:
1863:XXX and XXZ
1859:Gaudin model
1848:
1708:
1560:
1344:
1318:
1077:
951:
902:
887:
885:
874:
854:Fermi sphere
850:ground state
847:
832:
537:
158:
154:
135:
123:
112:
77:Bethe ansatz
76:
70:
55:
46:
30:
2560:: 196â233.
2019:different).
2002: [
1985:Felix Bloch
1895:Kondo model
1815:defined mod
899:Terminology
866:determinant
862:Yang action
115:spin chains
87:of certain
3175:Categories
2276:2019-05-17
2086:References
1992:Hans Bethe
1976:publishes
1916:Chronology
1788:odd (with
132:Discussion
96:Hans Bethe
3181:Magnetism
2703:120459635
2660:122524239
2502:0031-9007
2449:0031-9007
2406:0375-9601
2363:0375-9601
2320:0031-9007
2207:122250890
2199:0010-3616
2124:124225487
2079:C.P. Yang
2075:C.N. Yang
2057:C.N. Yang
2050:C.P. Yang
2046:C.N. Yang
1978:his model
1773:−
1747:−
1677:π
1659:λ
1655:−
1646:λ
1633:θ
1612:∑
1598:−
1586:λ
1573:θ
1509:−
1500:λ
1496:−
1487:λ
1469:λ
1465:−
1456:λ
1439:≠
1432:∏
1401:−
1392:λ
1366:λ
1328:λ
1305:λ
1299:
1290:−
1287:π
1278:λ
1246:λ
1237:
1228:≡
1222:λ
1210:θ
1180:λ
1176:−
1167:λ
1154:θ
1135:λ
1106:λ
1086:ϕ
1048:≡
1005:⋅
973:∑
786:ϕ
680:−
651:⋯
591:⋯
538:in which
475:ϕ
459:−
427:≥
415:≥
408:∑
347:∑
335:
310:−
288:∈
281:∑
264:−
238:≥
226:≥
219:∏
196:⋯
171:Ψ
2601:: 8â33.
2518:15180985
2510:12059396
2250:(1980).
2165:(1982).
2068:F. Y. Wu
1319:rapidity
943:Examples
3134:Bibcode
3099:Bibcode
3064:Bibcode
3025:Bibcode
2990:Bibcode
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