Bhattacharyya, Bibhas

Roy, G K

IACS Institutional Repository

Indian J. Phys. 69 A (1), 85 -  90 (1995)U P  A— an international journalPenson-K olb-H ubbard M odel: A R enorm alisation  G roup  StudyBibhas Bhattacharyya and G. K. RoySSM P Division, S aha  In s titu te  of Nuclear Physics,l/A F -B id h an n ag a r, C a lcu tta  -  700 064, India.A b s t r a c t :  We have stud ied  the  Penson-K olb-H ubbard (P K H ) m odel in one dim ension ( Id )  by m eans of a  real space renorm alisation  group (RG) m ethod for th e  half-filled band . Different phases are identified by study ing  the  RG-flow p a tte rn , the  energy gap and different correlation functions. T he phase d iagram  consists of four phases: a  spin density  wave (SD W ), a  strong  coupling superconducting  phase (SSC), a  weak coupling super­conducting phase (W SC ) and a nearly m etallic phase. For the  negative value of the  pa ir hopping am plitude in troduced in th is m odel we find th a t the  pa ir-pair correlation indicates a  superconducting  phase  for which the ccntre-of m ass of the  pairs move w ith a  m om entum  7T.K e y w o rd s :  Penson-K olb-H ubbard m odel, real space R G , repairing  P A C S  n u m b e r s :  64.60.Ak, 71.27.+ a , 74.25.Dw1 . I n t r o d u c t io nT he a ttrac tiv e  (negative U-t) H ubbard  m odel has been vigorously pursued[1] to s tu d y  th e  crossover from  the  weak (BCS) to  strong  (Bose) coupling regime of high-Te m aterials which, given th e ir very sho rt coherence lengths and  unusual norm al s ta te  p ro perties , are likely to  be in th e  in te rm m edia te  region. However, such a  m odel w ith  a  zero-range instan teneous in te rac tio n  m ay give rise to some unw anted  features. In fact th e  fall of Tc in th is m odel shows a  wrong dependence [2 ] on U/t  in  th e  com posite boson regim e (\U\/t >  1 ) where a pure H ubbard-like m odel goes classical; its applicability  in th is lim it thus becomes questionable. T h e  sim plest way ou t of th is im passe is to  use a  non-© 1995IACSlocal pairing in teraction  — the so-called Pens on-Kolb m odel [3]. A relatively less stud ied  m odel em bracing the basic features of these  system s corresponds to  th e  H ubbard  generalisation [4] of the  Penson-K olb ham ilto n ian ,H  = t 53  °J«r0>  + ^ $ 3 n,'Tn,'l “ V  53  “ /* $ 3 n»>» (1)<0>,<r • <0> •>where the  singlet pair operator d, =  o , | 0 ,-j and r is the  chem ical po ten tia l; < ij  >  represents nearest neighbours. T he precise m icroscopic ro u te  tp  th e  pair hopping term  is not specified in this phenom enological m odel. Independent of the pairing origin, this phenomenological m odelling of th e  p a irin g  process has the m erit of adm itting  different g round-states so th a t  one can eventually com pare w ith experim ents to  identify the  ap p ro p ria te  p a ram ete r s^ace. A continuation of such phenomenology seems necessary before one can hope for a  complete theory of high-Tc superconductivity .T he ground s ta te  (determ ined by U/t,V/t)  p ro p erties  of th e  half-filled (/x = U/ 2  by particle-hole sym m etry) 1 -d Penson-K olb-H ubb#ard  (P K H ) model have been studied for U > 0 and for bo th  V > 0  and V <  0 . In  th is  work we shall be chiefly concerned w ith the results in the  V < 0 sector of the  p aram eter space. However, we shall quote some results for the  V >  0 case for elucidating the qualitative difference in results for these two cases.2. RG MethodologyWe use a  simple real space RG technique [5] to  s tu d y  th e  g round  s ta te  of the PK H  m odel on a  linear chain for half-filling. T h e  ham ilto n ian  has several conserved quantities like particle  num ber N t to ta l sp in  S and  spin  com ponent St etc. T he la ttice  is divided into 3-sitc blocks. T h e  b lock-ham ilton ian  is diagonalised exactly. Only four low lying energy s ta te s  a re  re ta in ed  w ith in  a block. T he  param eters in the  ham ilton ian  ( 1 ) are th en  renorm alised  within this tru n ca ted  basis set [5]. This m ethod leads to  th e  RG equationsU' =  U +  2 ( £ 2  -  £ 3), V' =  (<1 2 / 2  +-  [{(a a^i +  a i ^ )  +  \ /3 a 263 +  C1362 -1- ^^20463) /2> /2]2t ,y! =  E2 - E 3 + r = U'/2 tH6  Bibhas Bhatlacharyya and G K Roy(2)87where Ej  an d  E3 axe th e  lowest eigenvalues of the  sym m etric m atrices M  and  N respectively,Penson-Kolb-Hubbard model: a renormalisation xronp study0 s/2t 0 0 \s/21 0 s/i1 2t0 s/it U -sp2V0 21 -y /iV  UN =U spit 0 \spit U - V  spit I 0 s/6t 0 /and ( 0 1 , 0 3 , 0 3 , 0 4 )  and  ( 6 1 , 6 2 , 6 3 )  are the  corresponding eigenvectors. T hese axe first-order RG equations, rela ting  the  renorm alised p aram ete rs  ( [ / ',  V " ,i ')  at the n - th  stag e  w ith  th e  previously ob tained  ite ra ted  values.We have com puted  th e  g-transform s of the  spin-spin, density-density  and pair-pair correlation  functions to s tu d y  the  SDW , CD W  (charge density wave) and SC (superconducting) orderings. Such correlation functions are defined asC(q) = N~ 1 J 2 <  AiAj > exp[iq(Ri -  Rj)),•jwhere, Ai =  ( n , |  -  n ^ ) , ( l  -  n ;j -  n jj)  or d; for SDW , CD W  or SC correlations respectively. T h is p a rticu la r  m ethod  has succeeded in  deriving qualita tive  features qu ite  reliably w hen aplied to  itin e ran t electron m odels [5] though  th e  quantitave accuracy is no t always unquestionable [6 ].3. The Phase DiagramT h e  phase  d iagram  of F ig .l is ob tained  from  the  tra jec to ries  of th e  coupling param eters (U'ft*, \V'\ft') under th e  RG ite ra tions. In itia l po in ts in the  two regions SD W  an d  SC D W  (identified la te r  w ith sh o rt range charge density  wave order) flow in to  th e  stab le  fixed points ( 0 0 , 0 ) and ( - 0 0 , 0 ) respectively; the  points in  th e  W SC  and  SSC phases go to  ( - 0 0 , 0 0 ). T hus the  flow -pattern  is unable to  discern precisely th e  W S C /S S C  tran sitio n  from  a  weak coupling SC state to  a  com posite  boson phase  ( it is ob ta ined  la te r  from  th e  single partic le  excitation gap). T h e  SDW  boundary  tu rns ou t to  be th e  line of zeroes of th e  energy gap Uoo~ th e  lim iting  value U' reaches after infin ite ite ra tio n s while t' and V7  separa te ly  go to  zero. T h e  SD W  region (U^ o > 0 ) is characterised  by88 Bibhas Bhattacharyya and G K Roya  gap in the charge sector, while below this boundary a  gap in th e  spin sector emerges ([/«, <  0). An unstable Axed point (F P ) appears a t  (3 .145,3.617) on this boundary; this separates two critical lines on th is boundary .T he m etallic critical line extends from the unstab le  F P  to  the  free fermionic stable FP  (0,0); all points on it flow to (0,0) under RG ite ra tio n s. T h e  other p a rt of the SDW line beyond the unstable FP  extends to  th e  F P  (oo ,oo) with ^71^1 — 4>/2/5 (<  4 /x , the exact value). In the bounded weah coupling region below the SDW boundary the gap H 7 J  > 0  (especially in  the  SC D W  phase). As the W SC region is approached across the S C D W /W S C  boun d ary  th e  pair hopping mechanism begins to dom inate over th e  single partic le  hopping. This is reflected in the fact th a t  from the W SC region onw ards \V\jt -* oo under RG flow (F ig .l) . T here is no qualitative change in the  flow p a tte rn  across the W SC /SSC  line, as m entioned earlier. Dut the gap rises a b ru p tly  in the SSC region which implies th a t the electrons get tigh tly  paired  in th is  region (a89composite boson phase). T h e  knees of the  curves {/<*, vs. \V\fi (F ig .2 ) trace  out the W S C /S S C  boundary. T h e  phase boundaries m eet a t  a  tc trac ritica l point — (3.0, 3.51) very close to the  unstab le  FP. I t  is notew orthy th a t  all th e  param eters have th e  sam e order of m agnitude a t  th is point as expected .T h a t the  region Uqo >  0 corresponds to  an SDW  phase is su p ported  from the p lot of SDW  correlation C sd w (<7) vs- U/t (Fig. 3). C sd w (<z) takes large value for q =  jr in  th is region im plying an anti ferrom agnetic order. S im ilarly the SC phase is identified by th e  behaviour of the  SC correlation C sc (? )  for an app ropria te  q. For V >  0  it  is found th a t C sc(tf) blows up in th e  SC phase for q =  0 which m eans th a t pairs w ith ccntrc-of-m ass a t  rest arc  form ed. In  sharp con trast to  this kind of fam iliar pairing, we find th a t for V <  0 C sc (? ) rises up for q = v  (F ig.4) as we approach the  SSC phase th rough  the W SC  region.Penson-Kolb-Hubbard model: a renomudisation group studyPairing w ith  such non-zero ccntrc-of-m ass m om entum  is generally referred to as >7-pairing  [7]. In  the  SCDW  phase there is no long range order because none of the C(q)’s take  large values com pared to  their free ferm ionic values a t  (0 ,0 ). This phase  is nearly  m etallic  since | (7cx>| — 0  here. However, the  q = ir CDW  correlation dom inates over all o ther correlations in th is phase. T h is  shows that a  6 h o rt range CDW  order is p resent in this phase. T h is  is in m arked contrast to  th e  case of V >  0. T h ere  is a  sim ilar nearly  m etallic  phase  in th e  weak coupling region. B u t th a t  phase is dom inated by a  q =  0 SC correlation69A(1)-13ra th e r  th an  a  CDW  one. T he PK H  m odel flows in to  th e  a ttra c tiv e  H ubbard  m odel under successive RG ite ra tions in th is weak coupling region below  the SDW  boundary. For the  a ttrac tiv e  H ubbard  m odel the  SC and  th e  CDW  channels are degenerate for half-filling. Dut we find th a t  th is  never happens for the  PK H  m odel. This is due to the  short range pa iring  flu c tu a tio n  which com petes w ith the single particle  kinetic energy t.4 . C o n c lu s io nWe have studied the Pcnson-K olb-lIubbard  m odel in 1-d and  for hhlf-filling by m eans of a  real space RG technique for bo th  V  >  0 and  V  <  0. F o n th e  caseV <  0, which we are chiefly concerned w ith in th is  work, we find four ^ phases: SDW , SCDW , W SC and SSC. T he superconducting  phases are  found  tb  show 77-pairing in contrary to  the  fam iliar q =  0  pairing  in the* SC p h ase  of theV > 0 case. Also th e  short range CDW order found in  the  near-m eta lic  weak coupling regime is an intrinsic feature of th e  V <  0 case -  th e  corresponding region for V > 0  having a  short range SC order. T h is work m ay b e  ex tended  to improve the  quan tita tive  accuracy by su itab le  im provem ent o f th e  present RG scheme [6 ], This m odel also requires fu rth e r s tu d y  in h igher dim ensions for comparison w ith  experim ents.R e fe re n c e s  :[1] J . Luo and N. E. Bickers, Phys. Rev. B 4 8 , 15983 (1993); M . R anderia  et 0 /., Phys. Rev. L ett. 69 , 2001 (1992); L. B elkhir an d  M . R an d eria , Phys. Rev. B 4 5 , 5087 (1992); R . T . S calettar et al.} P hys. R ev . L e tt. 6 2 , 1407(1989).[2 ] P. Nozieres and S. Schm itt-R ink, J . Low. Tem p. P hys. 5 9 , 195 (1985).[3] K. A. Penson and M. Kolb, Phys. Rev. B 3 3 ,1 6 6 3  (1986); J . S ta t .  Phys. 44 , 129 (1986).[4] Achille Hui and S. Doniach, Phys. Rev. B 48 , 2063 (1993).[5] Jorge E. Hirsch, Phys. Rev. B 2 2 , 5259 (1980); C. D asg u p ta  and  P. Pfeuty, J . Phys. C: Solid S ta te  Phys. 14 , 717 (1981).[6 ] S. R. W hite  and R . M. Noack, Phys. Rev. L e tt. 6 8 , 3487 (1992).[7] C. N. Yang, Phys. R ev. L ett. 63, 2144 (1989).90 Bibhas Bhattacharyya and G K Roy

Penson-Kolb-Hubbard Model : A Renormalisation Group Study

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Penson-Kolb-Hubbard Model : A Renormalisation Group Study

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