Spinon-Holon Binding in T-J Model

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 electromagnetic 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,

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 obtain 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 effective 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. First, we consider zero temperature. In this case, where ͑ ͒ ͑ ͒ ϳ xZg 0 =xZf 0 =¯x x and the electron Green’s functions can U2 be written as ⌺ ͑q͒ = U ⌬ + o ͚ G ͑k͒g ͑k + q͒, b h h ␤ c f V ͉ជ͉ Ͻ⌳ ¯ k −kf ¯x͑␻ + ␰ជ͒ ͑ ͒ϳ ͑ ͒ k ͑ ␥/2͒ Gc k Ginc k + + O x , 2 2 2 ␻ ¯ U − Ekជ ⌬ ͑q͒ = U ⌬ − o ͚ F*͑k͒f ͑k + q͒. ͑7b͒ b h h ␤ c f V ͉ជ͉ Ͻ⌳ k −kf ¯x⌬ជ ͑ ͒ϳ k ͑ ␥/2͒ ͑ ͒ Fc k − + O x , 9 The system is in a superconducting state if both the electron 2 2 ␻ − ¯Eជ normal and anomalous Green’s functions are nonzero, and is k in the normal state if only the normal Green’s function is where ␥=1−␩ for ␩Ͻ1/4 and ␥=3͑1/2−␩͒ for ␩Ͼ1/4, nonzero. Self-consistent determination of the Green’s func- 2 2 2 ¯␰ជ =␰ជ +U ¯x and ¯Eជ =¯␰ជ +⌬ជ. G is a smooth background tions is not performed in the previous SU͑2͒ theory.3,9 We k k o k k k inc ﬁnd that self-consistent determination of electron and boson coming from convolution of the spinon and holon Green’s Green’s function is important for identifying the strong and functions. The quasiparticle behavior is determined by the ϳ ͉␻͉ϳ¯ Շ weak coupling regimes whereas self-consistency for the terms with weight ¯x. For Ekជ Uo¯x, the quasiparticle ␥ spinon Green’s function does not affect the qualitative prop- term is of order O͑1͒ and is larger than the O͑x /2͒ term for erties of the system. The spinon Green’s function is not ␩Ͻ1/2. This is also the energy range where the spinon-

172509-2 BRIEF REPORTS PHYSICAL REVIEW B 71, 172509 ͑2005͒ holon bound states are stable. The bound states become un- stable and decay into separate spinons and holons at ¯ ജ Ekជ Uo¯x. Ͼ ϳ At temperatures T Tc TBE where Bose-condensation vanishes, the anomalous Green’s function is zero. For T just above TBE, we may consider the bosons to be “almost 3 ͑ ജ ͒ϳ ͑ ജ ͒ Bose-condensed” where Zg T TBE 1 and Zf T TBE ϳ0. We obtain with this approximation

x͑␻¯ + ¯␰Јជ͒ ͑ ͒ϳ ͑ ͒ k ͑ ␥/2͒ ͑ ͒ Gc k Ginc k + + O x , 10 ␻2 ¯Ј2 ¯ − E kជ

2 2 2 ¯␰Јជ ␰ជ ␻ ␻ ¯Ј ¯␰Ј ⌬ where k = k +Uox/2, ¯ = −Uox/2, and E kជ = kជ + kជ. No- tice that for each momentum kជ, there exist two branches of quasiparticles with energies ±¯E, both in the superconducting and the normal states. The transition from superconducting to normal state as the temperature rises from 0 to Tc can also be studied in the almost Bose-condensed approximation where we approxi- ͑ ͒ϳ Ͼ ͑ ͒Ͼ FIG. 1. Tunneling density of states for x=0,0.05,0.1. mate Zg T 1 and 1 Zf T 0. With this, we obtain after some algebra, pinned at zero energy, whereas it is pinned at an energy ជ ϳ x u ͑k͒ v ͑k͒ E Uox/2 in the normal state. G ͑k͒ϳG ͑k͒ + ͚ ͩ i + i ͪ + O͑x␥/2͒ c inc ␻ ͑ជ͒ ␻ ͑ជ͒ Correspondingly, the position of the Fermi surface in the 4 i=1,2 − Ei k + Ei k normal state deﬁned by ¯EЈជ =−U x/2 is downshifted from ͑ ͒ k o 11 ¯ the nodal point EЈkជ =0, resulting in formation of segmented 2 ͑ជ͒ ⌬2 ͑ c ͑ ͒͑ ͒͒2 c Fermi surface below the nodal points as in SU͑2͒ theory.3,9 where E1͑2͒ k = Zf kជ + Ekជ + − Uox/2 , Ekជ 2 2 ជ c We show in Fig. 2 the electron normal state occupation num- = ͱ¯␰Јជ +͑1−Z ͒⌬ជ, and u͑v͒ ͑k͒ = ͑1+¯␰Јជ /Eជ͕͒1+͑−͓͒͑¯␰Јជ k f k 1 k k k ber computed at x=0.1. The electron occupation number ex- ͒ ͑ជ͔͒ ͑ ͒͑ ͓͒⌬2 c ͑ជ͒ ͔͖ ͑ ͒ ͑ជ͒ +Uox/2 /E1 k + − 1−Zf kជ /(EkជE1 k ) , u v 2 k hibits “pocket” structures although the discontinuity across ͑ ¯␰Ј c͕͒ ͑ ͓͒͑¯␰Ј ͒ ͑ជ͔͒ ͑ ͒͑ ͒ the Fermi surface occurs only on the inner side of the pocket = 1− kជ / Ekជ 1+ − kជ + Uox/2 / E2 k − + 1−Zf 2 c ជ ͑Fermi arc͒, in agreement with photoemission experiments.13 ϫ͓⌬ជ / (EជE ͑k͒)͔͖. A corresponding expression also exists k k 2 The superconducting transition at T is driven by opening for F ͑k͒. c c an “additional” superconducting gap ͑ϳZ1/2⌬ជ͒ at the Fermi We observe that four branches of quasiparticles exist at f k arc. The opening of the superconducting gap doubles the intermediate temperature 0ϽTϽT . However u =v =0 at c 2 2 branches of quasiparticles at an intermediate temperature be- T=0 and v =u =0 at TജT , showing that two of the four 1 2 c low T . However the superconducting order parameter branches of quasiparticles vanish at these temperatures. c merges with the pseudogap smoothly as temperature T→0, These results together suggest a rather unconventional pic- where two of the four branches of quasiparticles vanish. In ture for the low-temperature behavior of high-T cuprates in c Fig. 3, we show the electron DOS computed in our theory at our theory. The normal state is a pseudogap state with two x=0.1 for three different values of Z =0.0 ͑normal state͒, branches of quasiparticles where the tunneling density of f states ͑DOS͒ has a d-wave like gap in the spectrum. Notice that unlike the d-density wave state,11 the present pseudogap state does not break translational symmetry. The normal density of states computed by Eq. ͑10͒ for three different dop- ings x=0,0.05,0.1 with J=1, t=3, and Uh =Uo =4 and energy resolution ⌬␻ϳ0.15J is shown in Fig. 1. The cutoff in momentum space is implemented by introducing a cutoff factor 2 2 e−q /⌳ in the numerical integrations. We note that the energy resolution in our calculation is not high enough to provide quantitative information of the system. Nevertheless, the shifting of DOS into quasiparticle spectral weight as x in- creases with a d-wave gap structure is clear from the ﬁgure. This global feature is in agreement with tunneling measurements.12 The DOS in the superconducting state is FIG. 2. Electron occupation in the normal state for x=0.1 at the ഛ ഛ␲ similar to the normal state except for the sharper d-wave gap momentum region 0 kx,ky . The lines are equally spaced con- ϳ ϳ structures and that the superconducting DOS minimum is stant contours with nmax 0.52 and nmin 0.31.

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the t-J model. The paramagnetic coupling to electromagnetic ﬁeld is in linear response,

+ + ␦ ͑ ͒ ជ ␹ ជ ͑ជ͒ HP =−2it͚ sin k␮ ͩ¯x͚ fkជ+qជ␴fk␴ + ¯bkជ+qជbkͪA␮ q , kជ.qជ ␴

where the first term represents coupling through spinons and the second term represents coupling through holons. The ជ paramagnetic coupling of A to quasiparticles ͑ϭspinon- -holon bound state in our theory͒ can be evaluated by exam- ining the first-order change in electron Green’s functions by ␦ HP. After some lengthy algebra we find that at zero temperature, the nodal quasiparticles in the superconducting state are minimally coupled to external magnetic field as in usual d-wave superconductors with coupling constant ϳ͑ ͒͑ 1/2͒ϳ ␥/2 ␥Ͼ qeff Uo /J xncb/¯x x . We note that 0 in the ͑ Ͼ␩Ͼ ͒ weak-coupling regime 1/2 0 and qeff vanishes as x →0, although the vanishing rate is slower than predicted in pure SBMFT. The quasiparticle charge becomes of order O͑1͒ only in the strong-coupling regime where Bose conden- FIG. 3. Electron density of states at x=0 for three different sation vanishes. values of z=0.0,0.5,0.1. Summarizing, we have examined the effect of spinon- ͑ Ͼ ͒ holon binding on the quasiparticle properties in the U͑1͒ Zf =0.5 T 0 superconducting state , and Zf =1.0 ͑T=0 superconducting state͒. The gradual opening of gap at slave-boson approach to the t-J model. In the weak-coupling ͑ ͒ regime, we find that spinon-holon binding produces a seg- the Fermi arc Zf =0.5 and merging of superconducting gap with pseudogap at T=0͑Z =1͒ is clear from the figure. The mented Fermi surface at normal state and a rather normal f d-wave superconductor at T=0, in agreement with photo- smooth merging of the superconducting gap with the emission experiments.13 The theory predicts a rather non- pseudogap reflects the common origin of the two gaps in our trivial crossover behavior between T=T to T=0 where four theory and would be absent in theories where the two gaps c branches of quasiparticles exist in the intermediate state. are of different origins. This prediction is yet to be tested in photoemission or tun- A problem associated with the SBMFT approach to high- neling experiments. The problem of quasiparticle charge is T superconductors is that the quasiparticle charge q in- c eff not completely resolved in our theory. Our analysis suggests ferred from the temperature-dependent London penetration that quasiparticle charge would be of order one, only in the depth is of order O͑1͒ experimentally,7 but is of order x in strong-coupling regime when Bose-condensation amplitude SBMFT.5,6 A recent experiment also indicates that the simple vanishes. A fully self-consistent treatment in this regime is quasiparticle picture for temperature-dependent London pen- missing and will be the subject of future investigations. etration depth14 may be violated at the extremely low-doping regime.8 In the following, we study how quasiparticles in our The author thanks P. A. Lee for interesting him in this theory couple to an external electromagnetic field. The elec- investigation and for many useful comments. This work is ជ tromagnetic field A couples to electrons through the t term in supported by HKRGC through Grant No. 602803.

1 G. Kotliar and J. Liu, Phys. Rev. B 38, 5142 ͑1988͒; H. Fuku- 8 M. R. Trunin, Y. A. Nefyodov, and A. F. Shevchun, cond-mat/ yama, Prog. Theor. Phys. Suppl. 108, 287 ͑1992͒; F. C. Zhang 0312566. and T. M. Rice, Phys. Rev. B 37, R3759 ͑1988͒. 9 X.-.G. Wen and P. A. Lee, Phys. Rev. Lett. 76, 503 ͑1996͒. 2 ͑ ͒ P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621 1992 . 10 See, for example, G. D. Mahan, Many Particle Physics ͑Plenum, 3 P. A. Lee, N. Nagaosa, T. K. Ng, and X. G. Wen, Phys. Rev. B New York, 1990͒. 57, 6003 ͑1998͒. 11 S. Chakravarty et al., Phys. Rev. B 63, 094503 ͑2001͒. 4 D.-H. Lee, Phys. Rev. Lett. 84, 2694 ͑2000͒. 12 See, for example, K. McElroy et al., cond-mat/0406491; Y. 5 T. K. Ng, Phys. Rev. B 69, 125112 ͑2004͒. 6 L. B. Ioffe and A. J. Millis, J. Phys. Chem. Solids 63, 2259 Kohsaka et al., cond-mat/0406089. 13 ͑2002͒. A. Damascelli, Z. Hussain, and Z.-.X. Shen, Rev. Mod. Phys. 75, 7 See, for example, T. Schneider, cond-mat/0308595 and references 473 ͑2003͒; see also H. Matsui et al., cond-mat/0304505. therein. 14 A. C. Durst and P. A. Lee, Phys. Rev. B 62, 1270 ͑2000͒.

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