Integrability of the planar N = 4 gauge theory. and the Hubbard model. A. Rej, M. Staudacher AEI Golm D. Serban SPhT Saclay

Transcription

1 Integrability of the planar N = 4 gauge theory and the Hubbard model A. Rej, M. Staudacher AEI Golm D. Serban SPhT Saclay Banff, February 16, 2006

2 Overview Integrability of the All loop integrability and the BDS ansatz The Hubbard model N = 4 Conclusions and perspectives SYM: the three loop result

3 Integrability in AdS/CFT: sigma models vs. spin chains String side: non-linear sigma model on the coset space P SL(2, 2 4)/SO(4, 1) SO(5) non-local conserved charges [Bena, Polchinski, Roiban 03 ] rotating string solutions at large J classical integrable model (Neumann system) [Frolov, Tseytlin, Arutyunov, Russo ] [Metsaev, Tseytlin 98] E(λ) J = ε 0 ( λ J 2 ) + 1 J ε 1 ( λ J 2 ) + 1 ( ) λ J ε J 2 - full solution for the classical sigma model algebraic curve [Kazakov, Marshakov, Minahan, Zarembo; Kazakov, Zarembo; Schäfer-Namecki; Kazakov, Beisert, Sakai, 04] - quantizing the string sigma model [Kazakov et al.; Zarembo, Klose 06]

4 Integrability in AdS/CFT: sigma models vs. spin chains perturbative N = 4 so(6) sector at one loop is integrable [Minahan, Zarembo 02] psl(2,2 4) sector at one loop is integrable [Beisert, Staudacher 03] su(2 3) sector is integrable up to three loop order [Beisert, Kristjansen, Staudacher; Beisert 03] e.g. su(2) sector: Φ = Φ 3 + iφ 4, Z = Φ 1 + iφ 2, D = L + λ L L 2(1 P i,i+1 ) + λ 2 (8P i,i+1 2P i,i+2 6) +... i=1 i=1 Bethe Ansatz

5 Integrability: sigma models vs. spin chains Integrability in AdS/CFT: sigma models vs. spin chains comparison of the Bethe Ansatz and string results: E(λ) J L L = λ ( a L a ) ( ) 1 λ 2 ( L b L b ) 1 L λ 1 ( ) λ + 1 ( ) λ J 2 J ε λ/j 2 1, J J 2 = ε 0 ( λ J 2 ) + 1 J ε 1 Discrepancy at three loop order! [Callan et al. 03] BMN scaling ) ( ) ( Bethe Ansatz solution to three loop order [Serban, Staudacher 04] solution of the classical sigma model [KMMZ 04] ) ( order of limits? non-analytic corrections to strings [Beisert, Tseytlin 05] non-perturbative mixing of the sectors [Minahan; Alday, Arutyunov, Frolov 05] ( λ/j 2) 5/2

6 All loop integrability: the BDS conjecture [Beisert, Dippel, Staudacher 04] There is a unique spin chain obeying: - diagrammatic constraint - integrability up to five loops - BMN scaling candidate for the dilatation operator all loop Bethe ansatz for L infinite : e ip kl = M j k u k u j + i u k u j i, k = 1,..., M, u(p) = 1 2 cot p g 2 sin 2 p 2, E(g) = M g g 2 M k= g 2 sin 2 p k 2. = L + g 2 E(g), g 2 λ 8π 2 all loop psl(2,2 4): [Beisert, Staudacher; Beisert 05]

7 ( )( ) ( { } { } ) H 8 = {} + ( 1043 ) ( ) 4 12α + 4β 1 {1} α 2β1 4β 2 {1, 3} + ( ) 5 + 2α + 4β 2 + 4β 3 {1, 4} {1, 5} + ( )( ) 11α 4β 1 + 2β 3 {1, 2} + {2, 1} 1 4 {1, 3, 5} + ( α + 2β )( ) 1 2β 3 {1, 3, 2} + {2, 1, 3} + ( )( ) 3 α 2β 3 {1, 2, 4} + {1, 3, 4} + {1, 4, 3} + {2, 1, 4} ( ) {1, 2, 5} + {1, 4, 5} + {1, 5, 4} + {2, 1, 5} ( α + 2β )( ) ( 1 4β 3 {1, 2, 3} + {3, 2, 1} α 2β 1) {2, 1, 3, 2} + ( β )( ) 2 {1, 3, 2, 5} + {1, 3, 5, 4} + {1, 4, 3, 5} + {2, 1, 3, 5} + ( α + 2β )( ) 1 2β 2 {1, 3, 2, 4} + {2, 1, 4, 3} + ( 3 4 2β )( ) ( )( ) 2 {1, 2, 5, 4} + {2, 1, 4, 5} β2 {1, 2, 4, 5} + {2, 1, 5, 4} + ( α β )( ) 1 + β 2 + 3β 3 {1, 2, 4, 3} + {1, 4, 3, 2} + {2, 1, 3, 4} + {3, 2, 1, 4} ( ) β 2 {1, 2, 3, 5} + {1, 3, 4, 5} + {1, 5, 4, 3} + {3, 2, 1, 5} + ( α + 2β )( ) 3 {1, 2, 3, 4} + {4, 3, 2, 1} + ( 7 8 α + 2β )( ) 3 {1, 4, 3, 2, 5} + {2, 1, 3, 5, 4} + ( α)( {1, 3, 2, 5, 4} + {2, 1, 4, 3, 5} ) + ( α β )( ) 3 {1, 3, 2, 4, 3} + {2, 1, 3, 2, 4} + {2, 1, 4, 3, 2} + {3, 2, 1, 4, 3} + ( 1 4 2β )( ) 3 {1, 2, 5, 4, 3} + {3, 2, 1, 4, 5} + ( α + β )( ) 3 {1, 2, 4, 3, 5} + {1, 3, 2, 4, 5} + {2, 1, 5, 4, 3} + {3, 2, 1, 5, 4} + ( 1 2 α β )( ) 3 {1, 2, 3, 5, 4} + {1, 5, 4, 3, 2} + {2, 1, 3, 4, 5} + {4, 3, 2, 1, 5} ( ) {1, 2, 3, 4, 5} + {5, 4, 3, 2, 1} 7 8 {m, n, p} i P i+m,i+m+1 P i+n,i+n+1 P i+p,i+p+1

8 BDS ansatz from the Hubbard model at half filling [Rej, Serban, Staudacher 05] energy of the AF state [RSS; Zarembo 05] : E AF (g) = 4L dt J 0 ( 2gt) J 1 ( 2gt) 2g 0 t 1 + e t Lieb, Wu 1968! ( ) 1-d Hubbard model: itinerant fermions with onsite repulsion ( H = 1 L ( ) e iφ c i,σ c i+1,σ + e iφ c i+1,σ c i,σ 1 L c 2 g g 2 i, c i, c i, c i,, i=1 σ=, i=1 - solved by (nested) Bethe Ansatz t = 1 U = 2/g Heisenberg model at half filling and g = 0 ground state: ferromagnetic state... >

9 Hubbard model at half filling E projection to a spin Hamiltonian ( strong coupling or g 0 ) at [Klein, Seitz 73; Takahashi 77] g = 0 the onsite part dominates states:... > 1/g 2 singly-occupied states fluctuations g 2 spin permutation g 4 (unwanted) four spin term [Takahashi 77] twisted boundary conditions for odd chains: t 1L = t 1L Aharonov-Bohm flux Φ = π(l + 1) 2

10 ( ) Projection of the Hubbard model on the spin space: h = L (h 2 + g 2 h 4 + g 4 h ), i=1 h 2 = 1 2 (1 σ iσ i+1 ), h 4 = (1 σ i σ i+1 ) (1 σ iσ i+2 ), h 6 = 15 4 (1 σ i σ i+1 ) 3 2 (1 σ i σ i+2 ) (1 σ i σ i+3 ) 1 8 (1 σ i σ i+3 )(1 σ i+1 σ i+2 ) (1 σ i σ i+2 )(1 σ i+1 σ i+3 ). dilatation operator in the su(2) sector!

11 BDS ansatz from Lieb-Wu equations Lieb-Wu equations (half filling): e i q nl = L n=1 M j=1 u j 2g sin( q n + φ) i/2 u j 2g sin( q n + φ) + i/2, u k 2g sin( q n + φ) + i/2 u k 2g sin( q n + φ) i/2 = M j=1 j k n = 1,..., L u k u j + i u k u j i, k = 1,..., M E = 2 g L cos( q n + φ) n=1 φ = π(l + 1) 2L g 0 Heisenberg Bethe ansatz L fermions, L large integral equations [Lieb, Wu 68]

12 ) BDS ansatz from Lieb-Wu equations Shiba (particle/hole) transformation: c i, = c i, c i, = c i, H(g; φ, φ) H( g; π φ, φ) M g 2 attractive interaction Dual Lieb-Wu equations e iq nl = 2M n=1 M j=1 u j 2g sin(q n φ) i/2 u j 2g sin(q n φ) + i/2, u k 2g sin(q n φ) + i/2 u k 2g sin(q n φ) i/2 = M j=1 j k n = 1,..., 2M u k u j + i u k u j i, k = 1,..., M 2M fermions, M magnons E = M g 2 2 g 2M n=1 cos(q n φ).

13 BDS ansatz from Lieb-Wu equations Magnons as bound states of fermions: Lieb -Wu equations have bound-states solutions (strings) [Takahashi 72] q-u strings: 2 fermions ( q 1 and q 2 ) and one rapidity u 1st LW ± equation sinh β = q 1 φ = π 2 + p 2 + i β, q 2 φ = π 2 + p 2 i β ( 2 u ± i/2 = ( p ) 2 g cos 2 iβ L g sin p u(p) = 1 2 cot p 1 + 8g 2 2 sin 2 p 2, 2 E(p) = 1 ( g g 2 sin 2 p ) 2 1 M magnons: 1st LW equation: e ip kl = M j k u k u j + i u k u j i, k = 1,..., M, BDS equation

14 ( ) Finite size corrections: O(e βl ) O g 1 e βl g 2L as expected No order-of-limits problem! - Hubbard model has a space of states much larger than the su(2) sector of the dilatation operator! Solutions with real q? important at finite g - comparison with the strings solutions around the AF state qualitatively correct [Roiban, Tîrziu, Tseytlin 06]

15 Conclusions Extension of the Hubbard model to psl(2,2 4)? Four loop computation in the gauge theory/ direct derivation from the gauge theory? Comparison with the Bethe ansatz solution for the string sigma model? [Kazakov et al. 06] [Zarembo, Klose 06]