Thirring Model with Jump Defect

Thirring Model with J ump Defect A.R. Agu irre ∗ Instituto de Física T eórica, UNESP , São P au lo, Brazil. E-mail: aleroagu@ift .unesp.br J.F . Gomes Instituto de Física T eórica, UNESP , São P aulo , Brazil. E-mail: jfg@ift.unes p.br L.H. Ymai Instituto de Física T eórica, UNESP , São P aulo , Brazil. E-mail: leandroy@ift .unesp.br A.H. Zimerman Instituto de Física T eórica, UNESP , São P aulo , Brazil. E-mail: zimerman@ift .unesp.br The purpo se of our work is to extend the formulation of classical af fine T oda Mod els in the presence of jump defects to pur e fermion ic Thirring mo del. As a first attempt we constru ct the Lagrang ian of the Grassmanian Th irring model with jump defect (of Backlu nd type) and present its conser ved modified momen tum and energy expressions giving a first indication of its integra- bility . 5th Internation al Sc hool on F ield Theory and Gr avitation , April 20 - 24 2009 Cuiabá city , Brazil ∗ Poster section. c  Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it / Thirring Model with J ump Defect A.R. Aguirre 1. Intr oduction Recently , there has been some interes t in th e questio n of w hether relati vistic inte grable field theori es admit certain disconti nuities (or jump defects) and, if so, what kind of properties they might ha ve [ 1 , 2, 3, 4 , 5]. In particul ar , in [1, 2, 3 , 4, 5] using the L agrang ian formulation, it was noted that s ome bosoni c field theorie s could allo w disco ntinuiti es like ju mp defec ts and yet remain classic ally inte grable. The best unders tood of these m odels is the sinh-Gord on in w hich diffe rent solito n soluti ons are connected in such way that the integ rability is preserve d. The defect condit ions relating fields e v aluated as limits from both side s of the defect turn out to be descri bed by Bäcklund t ransformat ion locate d at the defect. This fac t sh eds an inte resting ne w light on the role which the Bäcklund transfor mations hav e played in the dev elopment of soliton theory . More recen tly , the exten sion to classical integr able N = 1 and N = 2 super sinh-Gordo n models in v ol ving bosons an d fermio ns ad mitting jump defects hav e bee n stud ied [ 6 , 7] usin g bot h the Lagrangi an appro ach and the zero cur va ture formali sm. Here we stud y the Thirrin g model with jump defects as an example of pure fermion ic c ase. In this work we exp licitly construct its modified conser ved momen tum an d e ner gy which giv es a v ery strong s uggestio n of its class ical in tegra bility . 2. Lagrangian Ap pr oach The Lagrangian density for Grassmanian T hirring model w ith jump de fect can be written as follo ws, L = θ ( − x ) L 1 + θ ( x ) L 2 + δ ( x ) L D , (2.1) where L p = i 2 ψ ( p ) 1 ( ∂ t − ∂ x ) ψ † ( p ) 1 + i 2 ψ † ( p ) 1 ( ∂ t − ∂ x ) ψ ( p ) 1 + i 2 ψ ( p ) 2 ( ∂ t + ∂ x ) ψ † ( p ) 2 + i 2 ψ † ( p ) 2 ( ∂ t + ∂ x ) ψ ( p ) 2 + m  ψ ( p ) 1 ψ † ( p ) 2 + ψ ( p ) 2 ψ † ( p ) 1  − g  ψ † ( p ) 1 ψ † ( p ) 2 ψ ( p ) 2 ψ ( p ) 1  , (2.2) is the lagrang ian density for Thirring model [ 8 ] describing massi ve two-co mponent Dirac fi elds ( ψ ( p ) 1 , ψ ( p ) 2 ) with p = 1 correspond ing to x < 0, p = 2 correspondi ng to x > 0, and g is cou pling consta nt. The defect contrib ution located at x = 0 is described by L D = 1 2 " 2 ia m X † ∂ t X + i ψ ( 1 ) 1 ψ † ( 2 ) 1 + i ψ ( 1 ) 2 ψ † ( 2 ) 2 − i ψ ( 2 ) 1 ψ † ( 1 ) 1 − i ψ ( 2 ) 2 ψ † ( 1 ) 2 + +  i ( ψ ( 2 ) 1 − ψ ( 1 ) 1 ) + a ( ψ ( 2 ) 2 + ψ ( 1 ) 2 )  X † +  i ( ψ † ( 2 ) 1 − ψ † ( 1 ) 1 ) − a ( ψ † ( 2 ) 2 + ψ † ( 1 ) 2 )  X − iag 2 m  i ( ψ † ( 2 ) 1 − ψ † ( 1 ) 1 ) ψ ( 2 ) 1 ψ ( 1 ) 1 + a − 1 ( ψ † ( 2 ) 2 + ψ † ( 1 ) 2 ) ψ ( 2 ) 2 ψ ( 1 ) 2  X † − iag 2 m  − i ψ † ( 1 ) 1 ψ † ( 2 ) 1 ( ψ ( 2 ) 1 − ψ ( 1 ) 1 ) + a − 1 ψ † ( 1 ) 2 ψ † ( 2 ) 2 ( ψ ( 2 ) 2 + ψ ( 1 ) 2 )  X + ag m ψ † ( 1 ) 1 ψ † ( 2 ) 1 ψ ( 2 ) 1 ψ ( 1 ) 1 + g am ψ † ( 1 ) 2 ψ † ( 2 ) 2 ψ ( 2 ) 2 ψ ( 1 ) 2 # (2.3) Here, we introdu ce auxiliary fi elds X and X † . 2 Thirring Model with J ump Defect A.R. Aguirre The field equa tions for x 6 = 0 are obtained as i ( ∂ t − ∂ x ) ψ ( p ) 1 = m ψ ( p ) 2 + g ψ † ( p ) 2 ψ ( p ) 2 ψ ( p ) 1 , (2.4) i ( ∂ t + ∂ x ) ψ ( p ) 2 = m ψ ( p ) 1 + g ψ † ( p ) 1 ψ ( p ) 1 ψ ( p ) 2 , (2.5) i ( ∂ t − ∂ x ) ψ † ( p ) 1 = − m ψ † ( p ) 2 − g ψ † ( p ) 1 ψ † ( p ) 2 ψ ( p ) 2 , (2.6) i ( ∂ t + ∂ x ) ψ † ( p ) 2 = − m ψ † ( p ) 1 − g ψ † ( p ) 2 ψ † ( p ) 1 ψ ( p ) 1 , (2.7) which are the equatio ns of motion for the Thirring model. For x = 0, the equations correspo nding to defect condit ions are X = ( ψ ( 2 ) 1 + ψ ( 1 ) 1 ) + iag 2 m ( ψ † ( 2 ) 1 + ψ † ( 1 ) 1 ) ψ ( 2 ) 1 ψ ( 1 ) 1 = ia − 1 ( ψ ( 2 ) 2 − ψ ( 1 ) 2 ) − g 2 a 2 m ( ψ † ( 2 ) 2 − ψ † ( 1 ) 2 ) ψ ( 2 ) 2 ψ ( 1 ) 2 (2.8) and its hermitian conjug ated X † = ( ψ † ( 2 ) 1 + ψ † ( 1 ) 1 ) − iag 2 m ( ψ ( 2 ) 1 + ψ ( 1 ) 1 ) ψ † ( 1 ) 1 ψ † ( 2 ) 1 = − ia − 1 ( ψ † ( 2 ) 2 − ψ † ( 1 ) 2 ) − g 2 a 2 m ( ψ ( 2 ) 2 − ψ ( 1 ) 2 ) ψ † ( 1 ) 2 ψ † ( 2 ) 2 (2.9) togeth er w ith ∂ t X = ma − 1 2 ( ψ ( 2 ) 1 − ψ ( 1 ) 1 ) − im 2 ( ψ ( 2 ) 2 + ψ ( 1 ) 2 ) − g 4  i ( ψ † ( 2 ) 1 − ψ † ( 1 ) 1 ) ψ ( 2 ) 1 ψ ( 1 ) 1 + a − 1 ( ψ † ( 2 ) 2 + ψ † ( 1 ) 2 ) ψ ( 2 ) 2 ψ ( 1 ) 2  , (2.10) ∂ t X † = ma − 1 2 ( ψ † ( 2 ) 1 − ψ † ( 1 ) 1 ) + im 2 ( ψ † ( 2 ) 2 + ψ † ( 1 ) 2 ) + g 4  i ψ † ( 1 ) 1 ψ † ( 2 ) 1 ( ψ ( 2 ) 1 − ψ ( 1 ) 1 ) − a − 1 ψ † ( 1 ) 2 ψ † ( 2 ) 2 ( ψ ( 2 ) 2 + ψ ( 1 ) 2 )  (2.11) Notice that we can also write X = ( ψ ( 2 ) 1 + ψ ( 1 ) 1 ) + iag 2 m ψ ( 2 ) 1 ψ ( 1 ) 1 X † = ( ψ ( 2 ) 1 + ψ ( 1 ) 1 ) + iag 2 m ψ ( 1 ) 1 X † X = ( ψ ( 2 ) 1 + ψ ( 1 ) 1 ) − iag 2 m ψ ( 2 ) 1 X † X (2.12) and simila rly for X † . A s a conseq uence of (2.10) , (2.11) an d the T hirring e quation s of motio n (2.4) - (2.7), we obtain  ∂ t + ∂ x  X = m a − 1  ψ ( 2 ) 1 − ψ ( 1 ) 1  − ig 2  ψ † ( 2 ) 1 − ψ † ( 1 ) 1  ψ ( 2 ) 1 ψ ( 1 ) 1 , (2.13)  ∂ t − ∂ x  X = − im  ψ ( 2 ) 2 + ψ ( 1 ) 2  − g 2 a  ψ † ( 2 ) 2 + ψ † ( 1 ) 2  ψ ( 2 ) 2 ψ ( 1 ) 2 . (2.14) which can be re-written as  ∂ t + ∂ x  X = m a − 1 X − 2 ma − 1 ψ ( 1 ) 1 − ig ψ † ( 1 ) 1 ψ ( 1 ) 1 X − igX † X ψ ( 1 ) 1 , (2.15)  ∂ t − ∂ x  X = − maX − 2 mi ψ ( 1 ) 2 − ig ψ † ( 1 ) 2 ψ ( 1 ) 2 X − gaX † X ψ ( 1 ) 2 . (2.16) 3 Thirring Model with J ump Defect A.R. Aguirre The equations (2.8,2 .9 , 2.12,2.13 , 2.14) are the Bäck lund tran sformation fo r the classic al antico m- muting Thirring model [9]. In other words, in this approach the B äcklun d transformat ion at x = 0 repres ents the bound ary conditions between two regi ons. 3. Conserv ed Momentum and Energy The canon ical momentum (which is not expect ed to be preserve d) is giv en by P = 0 Z − ∞ d x  i 2  ψ ( 1 ) 1 ∂ x ψ † ( 1 ) 1 + ψ † ( 1 ) 1 ∂ x ψ ( 1 ) 1 + ψ ( 1 ) 2 ∂ x ψ † ( 1 ) 2 + ψ † ( 1 ) 2 ∂ x ψ ( 1 ) 2   + ∞ Z 0 d x  i 2  ψ ( 2 ) 1 ∂ x ψ † ( 2 ) 1 + ψ † ( 2 ) 1 ∂ x ψ ( 2 ) 1 + ψ ( 2 ) 2 ∂ x ψ † ( 2 ) 2 + ψ † ( 2 ) 2 ∂ x ψ ( 2 ) 2   . (3.1) Using the field equation s (2.4,2.5,2.6 , 2.7 ) we obtain d P d t =  m ( ψ ( 1 ) 1 ψ † ( 1 ) 2 + ψ ( 1 ) 2 ψ † ( 1 ) 1 ) − g ψ † ( 1 ) 1 ψ † ( 1 ) 2 ψ ( 1 ) 2 ψ ( 1 ) 1 + i 2 ( ψ † ( 1 ) 1 ∂ t ψ ( 1 ) 1 + ψ ( 1 ) 1 ∂ t ψ † ( 1 ) 1 + ψ † ( 1 ) 2 ∂ t ψ ( 1 ) 2 + ψ ( 1 ) 2 ∂ t ψ † ( 1 ) 2 ) i x = 0 − h m ( ψ ( 2 ) 1 ψ † ( 2 ) 2 + ψ ( 2 ) 2 ψ † ( 2 ) 1 ) − g ψ † ( 2 ) 1 ψ † ( 2 ) 2 ψ ( 2 ) 2 ψ ( 2 ) 1 + i 2 ( ψ † ( 2 ) 1 ∂ t ψ ( 2 ) 1 + ψ ( 2 ) 1 ∂ t ψ † ( 2 ) 1 + ψ † ( 2 ) 2 ∂ t ψ ( 2 ) 2 + ψ ( 2 ) 2 ∂ t ψ † ( 2 ) 2 )  x = 0 (3.2) Consider ing the boundary cond itions (2.8 , 2.9,2.13 , 2.14 ) the right hand side be comes a total time deri v ati ve . T hus, we found a functi onal P D gi ven by P D = ia m  X ∂ t X † − ( ∂ t X ) X †  − i 2  ψ ( 1 ) 1 ψ † ( 2 ) 1 + ψ † ( 1 ) 1 ψ ( 2 ) 1 + 3 ψ ( 1 ) 2 ψ † ( 2 ) 2 + 3 ψ † ( 1 ) 2 ψ ( 2 ) 2  − ga 2 m ψ † ( 1 ) 1 ψ † ( 2 ) 1 ψ ( 2 ) 1 ψ ( 1 ) 1 + g 2 ma ψ † ( 1 ) 2 ψ † ( 2 ) 2 ψ ( 2 ) 2 ψ ( 1 ) 2 , (3.3) so that P = P + P D is conserve d. This ‘modified’ moment um P appears to be a ‘total’ momen tum which is preserv ed containing bulk and defect contr ibu tions. In the case of the ener gy E = 0 Z − ∞ d x  i 2  ψ ( 1 ) 1 ∂ x ψ † ( 1 ) 1 + ψ † ( 1 ) 1 ∂ x ψ ( 1 ) 1 − ψ ( 1 ) 2 ∂ x ψ † ( 1 ) 2 − ψ † ( 1 ) 2 ∂ x ψ ( 1 ) 2  − m  ψ ( 1 ) 1 ψ † ( 1 ) 2 + ψ ( 1 ) 2 ψ † ( 1 ) 1  + g ψ † ( 1 ) 1 ψ † ( 1 ) 2 ψ ( 1 ) 2 ψ ( 1 ) 1 i + ∞ Z 0 d x  ( 1 ) ↔ ( 2 )  , (3.4) the ener gy-li ke conserve d quantity is E = E + E D , where E D = ia m  X ∂ t X † − ( ∂ t X ) X †  − i 2  ψ ( 1 ) 1 ψ † ( 2 ) 1 + ψ † ( 1 ) 1 ψ ( 2 ) 1 + ψ ( 1 ) 2 ψ † ( 2 ) 2 + ψ † ( 1 ) 2 ψ ( 2 ) 2  − ga 2 m ψ † ( 1 ) 1 ψ † ( 2 ) 1 ψ ( 2 ) 1 ψ ( 1 ) 1 − g 2 ma ψ † ( 1 ) 2 ψ † ( 2 ) 2 ψ ( 2 ) 2 ψ ( 1 ) 2 . (3.5) 4 Thirring Model with J ump Defect A.R. Aguirre W e notice that this fun ctional E D appear s on-she ll to be the def ect la grangian L D . This property alread y was found in th e sinh-Gordon mod el with defect [2]. Then, in spite of the loss of translatio n in v ar iance, the fields can exchang e both ener gy and momentum with the defect. In this work we hav e studied the classic al inte grab ility of the Grassmanian Thirring model with jump defect by con structin g the lowes t conserv ed quantitie s, namely , the modified momentu m and ener gy . The inte grabilit y of the mode l in v olve s also higher conserv ation la ws w hich are encoded within the Lax pair formalism. This wor k is in progress . Acknowledgmen ts W e would like to th ank th e organ isers of the FIFT H INTERNA TION AL SC HOOL ON FIEL D THEOR Y A ND GRA VIT A T ION for the opportunity to present these ideas. ARA and LH Y thank F AP ESP , AHZ and JFG CNPq for financial suppor t. Refer ences [1] S. Ghosal and A. Zamolodchikov , Int. J . Mod. Physics A 9 (1994 ). 3841, [ hep-th/9 3060 02 ]. [2] P . Bo wcock, E. Corrig an and C. Zambon, Int. J . Mod . Physics A 19 (Supplemen t) ( 2004) 82-91, [ hep-th /03050 22 ]. [3] P . Bo wcock, E. Corrig an and C. Zambon, J . of Hig h Ener g y Physics JHEP (2004) 056, [ hep-th /04010 20 ]. [4] E. Corrigan and C . 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