PHYSICAL REVIEW B 71, 172509 ͑2005͒ Spinon-holon binding in t-J model Tai-Kai Ng Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay Road, Hong Kong ͑Received 21 December 2004; published 27 May 2005͒ Using a phenomenological model, we discuss the consequences of spinon-holon binding in the U͑1͒ slave- boson approach to t-J model in the weak-coupling limit. We find that spinon-holon binding produces a pseudogap normal state with a segmented Fermi surface and the superconducting state is formed by opening an “additional” d-wave gap on the segmented Fermi surface. The d-wave gap merges with the pseudogap smoothly as temperature T!0. The quasiparticles in the superconducting state are coupled to external elec- tromagnetic field with a coupling constant of order x␥/2 ͑x=hole concentration͒, where 0ഛ␥ഛ1. DOI: 10.1103/PhysRevB.71.172509 PACS number͑s͒: 74.20.Mn, 74.25.Jb ͑ ͒ ͑ ͒ ␮͑␮ ͒ ␥͑ជ͒ ͓ ͑ ͒ ͑ ͔͒ The U 1 slave-boson mean-field theory SBMFT of the where j=i+ ˆ =x,y , and k =2 cos kx +cos ky . t-J model has been used by many authors as a starting point ⌬͑kជ͒=͑3J/4͒⌬¯ (cos͑k ͒−cos͑k ͒) is a spinon pairing field for the theory of high-T superconductors.1–4 With suitable x y c ⌬¯ ͗ ͘ refinements, the theory can explain a lot of the qualitative where = fi"f j# − fi#f j" . The mean-field dispersion for the 2–5 ͑ជ͒ ͱ␰2 ͉⌬͑ជ͉͒2 1,5 features of the cuprates. However, the theory does not spinon is Ef k =± kជ + k . produce a correct description of the low-energy quasiparticle The boson ͑holon͒ mean-field Hamiltonian is properties in the underdoped regime. It predicts a very strong renormalization of quasiparticle charge in the superconduct- U 5,6 7,8 h ⑀͑ជ͒ + h ⌬ ͑ + ͒͑ + ͒ ͑ ͒ ing state which is not observed experimentally. It has HMF = ͚ q bqជ bqជ + ͚ h bqជ + b−qជ bqជ + b−qជ , 2 been suggested3,9 that the failure of SBMFT in describing qជ 2 qជ quasiparticles is due to the lack of consideration of confine- ment between low-energy spinons and holons coming from where we have introduced a short-ranged hole-hole repulsion ϳ strong gauge field fluctuations. This scenario has been stud- term Uh t which is treated by usual Bogoliubov ied phenomenologically in the SU͑2͒ formulation of the t-J approximation10 and model, where it was found that spinon-holon binding leads to formation of half-pocket ͑segmented͒ Fermi surfaces 1 3,9 ⌬ 2 ͗ ͘ ͑ ͒ in the normal state, and a rather normal d-wave supercon- h = ¯x + ͚ bqជb−qជ , 3 V ductor state. In this paper, we study spinon-holon binding qជ in the U͑1͒ slave-boson formulation of the t-J model, ¯ assuming that an effective spinon-holon interaction Uo where x is the hole Bose-condensation amplitude. Note which is constant at distance range dϽlϳ␲−1x−1/2 that ¯xϽx in the presence of hole-hole and holon-spinon ⑀͑ជ͒ ␹␥͑ជ͒ ␮ ͑x=hole concentration͒ exists. Our goal is to understand how interactions. q =−t¯ q + b. The existence of Ͼ ␮ ␹␥͑ ͒ the quasiparticle properties in SBMFT are modified in the Bose-condensation ¯x 0 implies b =t¯ 0,0 . The presence of this phenomenological interaction in the under- mean-field dispersion for the holon excitation is ͑ជ͒ ͱ⑀͑ជ͒2 ⑀͑ជ͒ ⌬ 5 doped ͑small x͒ regime. We find that there exist two regimes Eh q = q +2 q Uh h. ͑ in the x-Uo phase diagram. For small Uo weak-coupling We assume an effective spinon-holon interaction of form regime͒ Bose condensation of holes exists and the properties of the system can be studied in a small-x expansion, whereas + + H = U ͑x͒ ͚ ͑f ជ bជ͒͑b f ជ ជ ␴͒, ͑4͒ Bose condensation of holes vanishes for large U ͑strong- c o k+qជ␴ q qជЈ k+qЈ o ͉͑ជ ជ͉ ͉ជ͉ ͉ជЈ͉Ͻ⌳͒␴ coupling regime͒ and a new state which cannot be described k − k F, q , q by small-x expansion is formed. We shall concentrate on the quasiparticle properties in the weak-coupling regime in this where we shall consider x-dependent spinon-holon interac- ͑ ͒ϳ ␩ paper. tions of form Uo x t/x for reasons which will become −1 ͱ We consider a model Hamiltonian on a two-dimensional clear later. ⌳ ϳ␲ x. We assume that the binding potential f b is effective only at a small range of momentum ഛ⌳ around square lattice, H=HMF+HMF+Hc, where qជ =0 and around the spinon Fermi surface. Gaussian fluctua- f + * tions above SBMFT result in an effective spinon-holon inter- ͚ ␰ជ ជ ជ ͚ ͓⌬¯ ͑ជ͒͑ ជ ជ ជ ជ ͒ ͔ HMF = kfk␴fk␴ + k fk"f−k# − fk#f−k" + H.C. , ϳ 5 action with Uo t. We shall examine the effect of spinon- kជ␴ kជ holon binding for various values of ␩ in the following. ͑1͒ The electron Green’s function is computed in a general- ized self-consistent Born approximation that involves self- is the fermion ͑spinon͒ mean-field Hamiltonian consistent evaluation of the electron and boson Green’s func- ␰ជ ͓ ¯2 ͑ ͒␹͔␥͑ជ͒ ␮ ␹ ͚͗ + ͘ in SBMFT. k =− tb + 3J/8 ¯ k − f, ¯ = ␴fi␴f j␴ tions, 1098-0121/2005/71͑17͒/172509͑4͒/$23.00172509-1 ©2005 The American Physical Society BRIEF REPORTS PHYSICAL REVIEW B 71, 172509 ͑2005͒ G ͑k͒ F ͑k͒ G ͑q͒ F ͑q͒ evaluated self-consistently here for simplicity. ¯ ͑ ͒ ͩ c c ͪ ¯ ͑ ͒ ͩ b b ͪ Gc k = , Gb q = , Equations ͑4͒–͑10͒ can be solved in a small-x expansion F*͑k͒ − G ͑− k͒ F*͑q͒ G ͑− q͒ c c b b when ␩Ͻ1/2. In this limit, we may perform a small qជ ,⍀ ͑5͒ expansion of the holon Green’s function self-energies to ob- tain where k=͑kជ ,i␻͒. The self-consistent equation is ͑ ͒ ⍀¯ + ¯⑀ ជ G¯ ͑k͒ = G¯ 0 ͑k͒ + ͓U G¯ ͑k͔͒, ͑6a͒ ͑ ͒ −q c c o c Gb q = , 2 ͑⍀¯ − ¯⑀ជ͒͑⍀¯ + ¯⑀ ជ͒ + ͉⌬ ͑0͉͒ where q −q b ͑ ͒ ͑ ͒ * ͑0͒ gc k fc k − ⌬ ͑0͒ G¯ ͑k͒ = ͩ ͪ, ͑6b͒ ͑ ͒ b ͑ ͒ c *͑ ͒ ͑ ͒ Fb q = , 8 fc k gc − k ͑⍀¯ ⑀ ͒͑⍀¯ ⑀ ͒ ͉⌬ ͑ ͉͒2 − ¯qជ + ¯−qជ + b 0 and ⍀¯ −1␻ ⌬ ͑ ͒ ⌬ ͑ជ ⍀ ͒ where =Zb , b 0 = b q=0, =0 , −1 ͚ ͑ជ ␻ץ ␻͒ ¯⑀ជ ⑀ជץ ͚ ͑ជ ␻͒ץ͑ 1 g ͑k͒ = xZ ͑T͒g ͑k͒ + ͚ g ͑k + q͒G ͑q͒, Zb =1− b q=0, / ␻=0, and q = q +( b q, ជ͒2 ជ2 ϳ ជ2 ␮͑ץ ͒ c g f ␤ f b V q =0 / q )qជ=0q teffq − h. A self-consistent evaluation of ជ͒2 ϳ͑ץ ͚ץ ͒ ␻ϳ͑ 2ץ ͚ץ the self-energies gives b / U0 /J ¯x, b / q U0 ⌬ ͑ ͒ϳ ⌬ ͑ 2 ͒ 3/2 ϳ 1 and b 0 Uh h − Uo /J ¯x . Therefore, teff t+Uo and f ͑k͒ = xZ ͑T͒f ͑k͒ − ͚ f ͑k + q͒F ͑q͒, ͑6c͒ ϳ ␩Ͻ c f f ␤ f b Zb 1 for 1/2 at small x. At zero temperature, the boson V q occupation number is a sum of two terms, x=¯x+xnbc, where ͒ ͑ ␻ ␰ ␻2 ͑ជ͒2 ͑ ͒ ⌬ ␻2 ϳͱ⌬ ͑ ͒ where gf(k = i + kជ)/(i −Ef k ) and f f k =− kជ /(i xnbc b 0 ¯x/teff is the density of uncondensed bosons −E ͑kជ͒2) are the mean-field normal and anomalous Green’s arising from interactions. With Eqs. ͑3͒ and ͑8͒, we obtain a f ⌬ ͑ ͒ ͱ ͑ ͒ self-consistent equation for b 0 of form functions of the spinons, respectively. xZg͑f͒ T are effec- tive Bose-condensation amplitudes in the normal and anoma- ⌬ ͑0͒ϳax¯͑1/2͒͑1+␩͒ͱ⌬ ͑0͒ + U ¯x − bx¯3/2−2␩, lous Green’s functions and b b h ϳ ϳ 2 where a Uh, b t /J are numerical factors. It is easy to see U ⌬ ͑ ͒ϳ ␩Ͻ ͓U G ͑k͔͒ = ¯␪͑kជ͒ o ͚ g ͑k + q͒G ͑q͒, that b 0 ¯x for 1/4 when holon-holon repulsion domi- o c ␤ f b ⌬ ͑ ͒ϳ 3/2−2␩ ␩Ͼ V ͉qជ͉Ͻ⌳ nates, and b 0 ¯x for 1/4 where spinon-holon ϳ 1+␩/2 interaction dominates. Correspondingly, xnbc ¯x for ␩Ͻ ϳ 1+͑1/2͒͑1/2−␩͒ ␩Ͼ U 1/4 and xncb ¯x for 1/4. In particular, ͓ ͑ ͔͒ ¯␪͑ជ͒ o ͑ ͒ ͑ ͒ ͑ ͒ UoFc k = k ͚ f f k + q Fb q , 6d x Ͼ¯x for ␩Ͼ1/2 where Bose-condensation vanishes and V␤ ncb ͉qជ͉Ͻ⌳ our small-x expansion breaks down. In this case, a new state ¯␪͑ជ͒ ␪͑⌳ ͉ជ ជ ͉͒ that cannot be described by SBMFT as a starting point is where k = − k−kF . The boson Green’s functions are formed. The corresponding x−Uo diagram has a phase given by ϳ 1/2 Ͻ boundary at Uc 1/x . For Uo Uc, the system is in the ⍀ ⑀ ⌺ ͑ ͒ ⌬ ͑ ͒ i − qជ − b q b q weak-coupling regime with nonzero Bose-condensation am- G¯ −1͑q͒ = ͩ ͪ, ¯ Þ b ⌬*͑ ͒ ⍀ ⑀ ⌺ ͑ ͒ plitude x 0, whereas Bose condensation vanishes for b q − i − −qជ − b − q Ͼ Uo Uc. We shall examine the quasiparticle properties in the ͑7a͒ weak-coupling regime in the following.