PHYSICAL REVIEW B VOLUME 61, NUMBER 18 1 MAY 2000-II
Nature of spin-charge separation in the t-J model
Z. Y. Weng, D. N. Sheng, and C. S. Ting Texas Center for Superconductivity and Department of Physics, University of Houston, Houston, Texas 77204-5506 ͑Received 10 November 1999͒ Quasiparticle properties are explored in an effective theory of the t-J model which includes two important components: spin-charge separation and unrenormalizable phase shift. We show that the phase shift effect indeed causes the system to be a non-Fermi liquid as conjectured by Anderson on general grounds. But this phase shift also drastically changes a conventional perception of quasiparticles in a spin-charge separation state: an injected hole will remain stable due to the confinement of spinon and holon by the phase shift field despite the background is a spinon-holon sea. True deconfinement only happens in the zero-doping limit where a bare hole will lose its integrity and decay into holon and spinon elementary excitations. The Fermi-surface structure is completely different in these two cases, from a large band-structure-like one to four Fermi points in one-hole case, and we argue that the so-called underdoped regime actually corresponds to a situation in between, where the ‘‘gaplike’’ effect is amplified further by a microscopic phase separation at low temperature. Unique properties of the single-electron propagator in both normal and superconducting states are studied by using the equation of motion method. We also comment on some influential ideas proposed in the literature related to the Mott-Hubbard insulator and offer a unified view based on the present consistent theory.
I. INTRODUCTION later referred to as the deconfinement, in order to distinguish it from the narrow meaning of the spin-charge separation High-Tc cuprates are regarded by many as essentially a about elementary excitations. We will see later that these two doped Mott-Hubbard insulator.1 At half-filling such an insu- are generally not the same thing. lator is a pure antiferromagnet with only the spin degrees of The second idea, the so-called ‘‘unrenormalizable phase freedom not being frozen at low energy. And a metallic shift,’’3,4 may be described as follows. In the presence of an phase with gapless charge degrees of freedom emerge after upper Hubbard band, adding a hole to the lower Hubbard holes are added to the filled lower Hubbard band. To char- band could change the whole Hilbert space due to the on-site acterize the doped Mott-Hubbard insulator in the metallic Coulomb interaction: The entire spectrum of momentum k’s regime, two important ideas were originally introduced by may be shifted through the phase shift effect. It leads to the Anderson: spin-charge separation2 and the unrenormalizable orthogonality of a bare doped hole state with its true ground phase shift effect.3,4 The first one is about elementary exci- state such that the quasiparticle weight Zϵ0, the key crite- tations of such a system and the second one is responsible for rion for a non-Fermi liquid. In general, it implies its non-Fermi-liquid behavior. ⌰ ϭ i i ͑ ͒ The spin-charge separation idea may be generally stated ci eie , 1.2 as the existence of two independent elementary excitations, where ei is related to elementary excitation fields, e.g., charge-neutral spinon and spinless holon, which carry spin † h f in a spin-charge separation framework. Such an ex- 1/2 and charge ϩe, respectively. It can be easily visualized i i pression means that in order for a bare hole created by c to ͑ ͒ 5 i in a short-range resonating-valence-bond RVB state and become low-lying elementary excitations, a many-body has become a widely used terminology in literature, often ⌰ phase shift i must take place in the background. In momen- with an additional meaning attached to it. For example, a tum space, it is easy to see how such a phase shift changes spin-charge separation may be mathematically realized in the 6 the Hilbert space by shifting k values. Note that ek so-called slave-particle representation of the t-J model, ϭ͚ † Ј kЈhkЈf kϩkЈ where k and k belong to the same set of ϭ † ͑ ͒ ci hi f i , 1.1 where the no-double-occupancy constraint, reflecting the Hubbard gap in its extreme limit, is handled by an equality † ϩ͚ † ϭ hi hi f i f i 1 which commutes with the Hamiltonian. Here one sees the close relation of the spin-charge separation and the constraint condition through the counting of the quantum numbers. But the spin-charge separation also ac- † quires a new meaning here: If those holon (hi ) and spinon ( f i) fields indeed describe elementary excitations, the hole ͑electron͒ is no longer a stable object and must decay into a holon-spinon pair once being injected into the system as FIG. 1. Schematical illustration of the quasiparticle deconfine- shown by Fig. 1͑a͒. This instability of a quasiparticle will be ment ͑a͒ and confinement ͑b͒ due to the phase shift field.
0163-1829/2000/61͑18͒/12328͑14͒/$15.00PRB 61 12 328 ©2000 The American Physical Society PRB 61 NATURE OF SPIN-CHARGE SEPARATION IN THE t-J MODEL 12 329 quantized values ͓for example, in a two-dimensional ͑2D͒ The phase shift field in 2D is related to a nonlocal vortexlike ϫ i⌰ˆ i i⌰string i square sample with size L L, the momentum is quantized at operator by e iϵ(Ϫ) e i ͓the sign (Ϫ) keeps ϭ k␣ (2 /L)n under the periodic boundary condition with track of the Marshall sign just for convenience͔ with the ␣ϭ ϭ ͔ ⌰ 15 x, y and n integer . But because of a nontrivial i , vorticity given by ck and ek generally may no longer be described by the same set of k’s or in the same Hilbert space, which thus ,Ͷ dr ٌ⌰stringϭϮ ͚ ͚ͫ ␣nb Ϫ1Ϫnhͬ l␣ l constitutes an essential basis for a possible non-Fermi liquid. ⌫ • l⌫ ␣ The 1D Hubbard model serves as a marvelous example in ͑1.4͒ favor of the decomposition ͑1.2͒ over Eq. ͑1.1͒. The quanti- tative value of the phase shift was actually determined by where ⌫ is an arbitrary closed loop without passing any lat- Anderson and Ren2,7 using the Bethe-ansatz solution8 in the tice site except the site i and the summation on the right-hand large U limit, and Z was shown to decay at large sample size side ͑RHS͒ of Eq. ͑1.4͒ runs over lattice sites l within the ⌫ b h with a finite exponent. An independent path-integral loop . Here nl␣ and nl are spinon and holon number op- approach9 without using the Bethe ansatz also reaches the erators, respectively. same conclusion which supports Eq. ͑1.2͒ as the correct de- Such a vortexlike phase shift originates from the fact that composition of true holon and spinon. a doped hole moving on an AF spin background will always Another important property of the Mott-Hubbard insula- pick up sequential ϩ and Ϫ signs, (ϩ1)ϫ(Ϫ1)ϫ(Ϫ1)ϫ tor, which is well known but has not been fully appreciated, •••, first identified in Ref. 18. These signs come from the is the bosonization of the electrons at half-filling. Namely, Marshall signs hidden in the AF background which are the fermionic nature of the electrons completely disappears scrambled by the hopping of the doped hole on its path, and is replaced by a bosonic one. This is one of the most determined by simply counting the index of each spin peculiar features of the Mott-Hubbard insulator due to the exchanged with the hole during its hopping.18 The signifi- strong on-site Coulomb interaction. In fact, under the no- cance of such a phase string is that it represents the sole double-occupancy constraint, the t-J model reduces to the source to generate phase frustrations in the t-J model ͑at Heisenberg model in this limit. Its ground state for any finite finite doping, the only additional signs coming from the fer- bipartite lattice is singlet according to Marshall10 and the mionic statistics of doped holes in the original slave-fermion wave function is real and satisfies a trivial Marshall sign rule representation are also counted in͒. Namely, the ground-state as opposed to a much complicated ‘‘sign problem’’ associ- wave function would become real and there should be no ated with the fermionic statistics in a conventional fermionic ‘‘sign problem’’ only if such a phase string effect is absent system. This bosonization is the reason behind a very suc- ͑like in the zero-doping case͒. The phase shift field in Eq. cessful bosonic RVB description of the antiferromagnet: The ͑1.3͒ precisely keeps track of such a phase string effect15 and variational bosonic RVB wave functions can produce strik- therefore can be considered to be a general consequence of ingly accurate ground-state energy11,12 as well as an elemen- the t-J model. tary excitation spectrum over the whole Brillouin zone.12 A The decomposition ͑1.3͒ defines a unique spin-charge mean-field bosonic RVB approach,13 known as Schwinger- separation theory where the relation between the physical boson mean-field theory ͑SBMFT͒, provides a fairly accurate electron operator and the internal elementary excitations, ho- and mathematically useful framework for both zero- and lon and spinon, is explicitly given. The thermodynamic prop- finite-temperature spin-spin correlations. erties will be obtained in terms of the energy spectra of holon Starting from either the slave-boson14 or slave-fermion15 and spinon fields, while the physically observable quantities representation, a 2D version of the decomposition ͑1.2͒ has will be determined based on Eq. ͑1.3͒ where the singular been previously constructed such that the electron bosoniza- phase shift field with vorticities is to play a very essential tion can be naturally realized at half-filling to restore the role in contrast to the conventional spin-charge separation correct antiferromagnetic ͑AF͒ correlations. In the 1D case, theories in the slave-particle decompositions of Eq. ͑1.1͒. this decomposition also recovers the aforementioned spinon- Note that the total vorticity of Eq. ͑1.4͒ is always equal to holon decoupling and reproduces the correct Luttinger-liquid 2ϫinteger due to the no-double-occupancy constraint such 15 ⌰ˆ behavior. Even in the two-leg ladder system where holons that the phase shift factor ei i be single valued. Such a and spinons are recombined together to form quasiparticles phase shift field will play different roles in different chan- in the strong rung case,16 a many-body phase shift field in ϩ nels. For example, in the spin channel one has Si this decomposition still exists at finite doping, playing a non- † i Ϫi[⌰stringϪ⌰string] ϭb b ↓(Ϫ1) e i↑ i↓ where the total vorticity trivial role. Such a decomposition form can be generally i↑ i written as ͒ ͑ Ϫ Ͷ ٌ͑⌰stringϪ⌰string͒ϭϮ h dr ↑ ↓ 2 ͚ nl . 1.5 ⌰ˆ ⌫ • l⌫ ϭ † i i ͑ ͒ ci hi bie . 1.3 It obviously vanishes at ␦→0(␦ is the doping concentra- It may be properly called a bosonization formulation as the ͒ † tion so that the aforementioned bosonization is naturally holon operator hi and spinon operator bi are both bosonic realized. And at finite doping, the vorticity shown in Eq. fields here. They still satisfy the no-double-occupancy con- ͑1.5͒ reflects the recovered fermionic effect and is respon- † ϩ͚ † ϭ straint hi hi bibi 1. The fermionic nature of ci is to sible for a doping-dependent incommensurate momentum ⌰ˆ 19 be represented through the phase shift field i which re- structure in the dynamic spin susceptibility function which places the description of Fermi-surface patches and Fermi- provides a unique reconciliation of neutron scattering and surface fluctuations in the usual bosonization language.17,4 NMR measurements in the cuprates. On the other hand, in 12 330 Z. Y. WENG, D. N. SHENG, AND C. S. TING PRB 61 the singlet pairing channel, the phase shift field appearing in non-Fermi-liquid state. In Sec. II C, we demonstrate how the the local pairing operator will contribute to a vorticity phase shift field causes the confinement of the holon and spinon within a quasiparticle except in the zero-doping limit. In Sec. II D, we study the single-electron propagator in both ,Ͷ dr ٌ͑⌰stringϩ⌰string͒ϭϮ2 ͚ ͚ͫ ␣nb Ϫ1ͬ ↑ ↓ l␣ normal and superconducting states based on an equation-of- ⌫ • l⌫ ␣ motion approach. We then discuss an underdoped case as a ͑1.6͒ crossover regime from a Fermi-point structure in the one- which decides the phase coherence of Cooper pairs. Accord- hole case with the holon-spinon deconfinement to a large ing to Eq. ͑1.6͒, besides a trivial 2 flux quantum per site, Fermi surface in the confinement case. Finally, Sec. III is each spinon also carries a fictitious (Ϯ)2 flux tube. To devoted to discussing some of the most influential ideas pro- achieve the phase coherence or superconducting condensa- posed in the literature related to the Mott-Hubbard insulator tion, ↑ and ↓ spinons have to be paired off to remove the and high-Tc cuprates and offers a unified view based on the vorticities associated with individual spinons in Eq. ͑1.6͒, present consistent theory. 20,21 which then connects Tc to a characteristic spinon energy scale, in consistency with the experimental result of cuprate II. PROPERTIES OF A QUASIPARTICLE IN THE superconductors and resolving the issue why Tc is too high SPIN-CHARGE SEPARATION STATE in usual RVB theories. The purpose of the present work is to explore the conse- A. Effective spin-charge separation theory quences of the bosonization decomposition ͑1.3͒ in 2D qua- The decomposition ͑1.3͒ determines an effective spin- siparticle channel. First of all, we show that the phase shift charge separation theory of the t-J model in which spinon field indeed causes the quasiparticle weight Z to vanish. and holon fields constitute the elementary particles. Before Namely, this spin-charge separation state is a 2D non-Fermi proceeding to the discussion of the quasiparticle properties in liquid, a fact almost trivial in such a particular formulation. next subsections, we first briefly review some basic features A surprising ‘‘by-product’’ of this phase shift field is that it of this theory based on Refs. 15 and 20. ⌰string also plays a role of confinement force to ‘‘glue’’ spinon and In the operator formalism, the phase shift field i holon constituents together inside a quasiparticle, as illus- satisfying Eq. ͑1.4͒ can be explicitly written down in a spe- trated in Fig. 1͑b͒. In other words, a hole injected into this cific gauge as follows:15 system generally does not break up into spinon-holon el- ementary particles, even though the background is a spinon- i holon sea. Such a quasiparticle may be regarded as a spinon- ⌰stringϵ ͓⌽bϪ⌽h͔, ͑2.1͒ i 2 i i holon bound state or more properly a collective mode but will generally remain incoherent due to the same phase shift where field. Due to the confinement, an equation-of-motion descrip- tion of the quasiparticle excitation is constructed, in which ⌽bϭ͚ ͑l͒ͩ ͚ ␣nb Ϫ1ͪ ͑2.2͒ i i l␣ the dominant ‘‘scattering’’ process is described as the ‘‘vir- lÞi ␣ tual’’ decay of the quasiparticle into holon-spinon compos- ite. In the superconducting phase, the composite nature of the and quasiparticle predicts a unique non-BCS structure for the single-electron Green’s function which is consistent with the ⌽hϭ ͑ ͒ h ͑ ͒ i ͚ i l nl . 2.3 experimental measurements. In particular, we find the resto- lÞi ration of the quasiparticle coherence with regard to the inco- herence in the normal state. Here i(l) is defined as an angle A true deconfinement or instability of the quasiparticle only happens in the zero-doping limit where an injected hole ͑l͒ϭIm ln͑z Ϫz ͒, ͑2.4͒ indeed can decay into a holon and spinon pair, which i i l 22 provides a consistent explanation of angle-resolved photo- ϭ ϩ ͑ ͒ 23 with zi xi iyi representing the complex coordinate of a emission spectroscopy ARPES measurements. The con- lattice site i. trast of a large band-structure-like Fermi surface in the con- In 2D, an effective Hamiltonian based on the decomposi- finement phase to the four Fermi points in the deconfinement tion ͑1.3͒ after a generalized mean-field decoupling20 in the phase at the zero-doping limit may provide a unique expla- t-J model can be written down: nation for the ARPES experimental measurements in cuprate superconductors. In the weakly doped regime, a ‘‘partial’’ ϭ ϩ ͑ ͒ deconfinement of the quasiparticle between full-blown de- Heff Hh Hs , 2.5 confinement and confinement will be reflected in the single- where the holon Hamiltonian electron Green’s function which may explain the ‘‘spin-gap’’ phenomenon.24 f † The remainder of the paper is organized as follows. In ϭϪ ͑ iAij͒ ϩ ͑ ͒ Hh th͚ e hi h j H.c. 2.6 Sec. II, we first briefly review the effective spin-charge sepa- ͗ij͘ ration theory based on the decomposition ͑1.3͒. Then in Sec. II B, we show that the phase shift field leads to Zϭ0, i.e., a and the spinon Hamiltonian PRB 61 NATURE OF SPIN-CHARGE SEPARATION IN THE t-J MODEL 12 331
h † † ϭϪ ͓͑ i Aij͒ ϩ ͔ Hs Js ͚ e bib jϪ H.c. ͗ij͘
s h † Ϫ ͑ i Aij͒ ͑ ͒ ͚ Jij e bib j , 2.7 ij ϳ ϳ s ϳ␦ Þ with th t, Js J. In the second term of Hs , Jij t 0 only for i and j on the same sublattice sites, which originates from 20 Ht where a phase shift occurs in the spinon mean-field wave function and results in the same-sublattice hopping of f h spinons. The lattice gauge fields Aij and Aij in the specific gauge choice of Eqs. ͑2.2͒ and ͑2.3͒ are given by
1 A f ϭ ͚ ͓ ͑l͒Ϫ ͑l͔͒ͩ ͚ nb Ϫ1ͪ ϵAs Ϫ0 ij i j l ij ij 2 lÞi, j ͑2.8͒ and FIG. 2. Dynamical spin susceptibility at the AF vector (,)in 1 ␦ϭ h ϭ ͓ ͑ ͒Ϫ ͑ ͔͒ h ͑ ͒ the uniform phase for 0 and 0.143, respectively. The inset: the Aij ͚ i l j l nl . 2.9 2 lÞi, j doping dependence of the characteristic spin ‘‘resonancelike’’ en- ϭ ͒ ergy Eg 2Es (Es denotes the corresponding spinon energy . s h In general, Aij and Aij can be regarded as ‘‘mutual’’ Chern- h Simons lattice gauge fields as, for example, Aij is determined susceptibility function at the AF vector ( , ) is illustrated by the density distribution of holons but only seen by in Fig. 2. Note that Eg in Fig. 2 is twice as large as the spinons. corresponding spinon energy Es . The doping dependence of The above effective theory is based on a RVB pairing Eg is shown in the inset. Finally, the superconducting order 20 ⌬SCϭ͚͗ ͘ ͑ order parameter parameter ij cic jϪ has a finite value for nearest-neighboring i and j) in the ground state,20 Ϫ h ⌬sϭ ͗ i Aij ͘ ͑ ͒ ͚ e bib jϪ , 2.10 SC † ⌽bϩ⌽b † ⌬ ϭ⌬s͑Ϫ ͒i͗ i( i j )/2 ͘Þ ͑ ͒ ij 1 hi e h j 0, 2.11 which in the zero-doping limit ␦→0 reduces to the well- due to the Bose condensation of holons as well as the pairing 13 h ϭ ⌬sÞ known bosonic RVB order parameter as Aij 0. And Heff of spinons which leads to 0 and the vortex-antivortex 13 i(⌽bϩ⌽b)/2 recovers the Schwinger-boson mean-field Hamiltonian of confinement in e i j . Thus the ground state is always the Heisenberg model. So this theory can well describe AF superconducting condensed with a pairing symmetry of d- correlations at half-filling. At finite doping, the ‘‘mutual’’ wave-like.20 f h Chern-Simons gauge fields Aij and Aij will play important Besides the above uniform ground state, possible nonuni- roles in shaping superconductivity, magnetic, and transport form solutions characterized by the coexistence of the Bose properties, and some very interesting similarities with cu- condensations of holons and spinons have been also dis- prate superconductors have been discussed based on this cussed in Ref. 20 where the spinon spectrum has only a model.20 In contrast to the slave-fermion approach,25 ⌬s re- pseudogap. In any case, the ground state can be regarded as mains the only order parameter at finite doping, controlling a spinon-holon sea, and low-lying elementary excitations are ͗ ͘ϭ the short-range spin-spin correlations as Si•Sj described in terms of spinons and holons. What we are Ϫ1/2͉⌬s͉2 for nearest-neighboring i and j. It is noted that mainly interested in this paper is to answer the question how due to the presence of the RVB pairing ͑2.10͒, the conven- a hole ͑electron͒ as a composite of spinon and holon behaves tional gauge fluctuations26,27 are suppressed as the gauge in- in this spin-charge separation state. This is one of the most Ϫ † ϭ͓ † i i͔͓ i i͔ variance hi bi hi e bie is apparently broken by fundamental questions not only because it can be directly ⌬s. Here spinons no longer contribute to transport and are tested in an ARPES measurement, but also because it will really charge-neutral particles. make a crucial distinction between a conventional Fermi liq- In the ground state of the uniform-phase solution ͑Ref. uid and a non-Fermi liquid. Let us begin with the question: if ͒ f h f 20 , Aij and Aij both become simplified: Aij simply de- this spin-charge separation state is a non-Fermi liquid. ͚ f ϷϪ͚ 0 ϭϪ scribes a flux per plaquette, ᮀAij ᮀ ij since s B. Non-Fermi liquid with ZÄ0 Aij is suppressed due to the spinon pairing in the ground h ͚ ¯ f ϭ␦ state; Aij describes a uniform flux ᮀAij due to the The definition of the quasiparticle weight Zk at momen- ͓ ϭ͉ ⌿ Ϫ ͉ ͉⌿ ͉2 Bose condensation of holons. Self-consistently, Hh Eq. tum k is given by Zk ͗ G(Ne 1) ck G(Ne)͘ , and it ͑2.6͔͒ determines a Bose-condensed ground state of holons measures the overlap of a bare hole state at momentum k, f where the flux produced by Aij merely enlarges the effec- created by ck in the ground state of Ne electrons, with the ͱ ¯ h ground state of N Ϫ1 electrons. For a Fermi-liquid state, tive mass near the band edge by 2. On the other hand, Aij e ͓ ͑ ͔͒ one always has Z Þ0 at the Fermi momentum k .IfZ in Hs Eq. 2.7 leads to a ‘‘resonancelike’’ energy structure k f f k in the spinon spectrum and the corresponding dynamic spin ϭ0 for any k, then the system is a non-Fermi liquid by 12 332 Z. Y. WENG, D. N. SHENG, AND C. S. TING PRB 61
definition. In the following we will show that the bare hole ͉⌿ state ck G(Ne)͘ acquires an ‘‘angular’’ momentum due to ⌰ˆ the vorticities in e i in the decomposition ͑1.3͒. Due to such a distinct symmetry, it is always orthogonal to the ground ͉⌿ Ϫ ϵ state G(Ne 1)͘, leading to Z 0. One can construct a ‘‘rotational’’ operation by making a transformation
͒→ ͒ϩ ͑ ͒ i͑l i͑l . 2.12
It corresponds to a simple change of reference axis for the FIG. 3. Schematical illustration of the case when a quasiparticle ͑ ͒ angle function i(l) defined in Eq. 2.4 . It is easy to see that decays into spinon and holon constituents. the Hamiltonians ͑2.6͒ and ͑2.7͒ are invariant since the gauge fields, in which i(l) appears, are obviously not changed: C. Quasiparticle: Spinon-holon confinement h, f → h, f ͉⌿ ͘ Aij Aij . Both the ground state G as well as single- † The difference in symmetry between a quasiparticle and a valued h and b fields are apparently independent of . i i holon-spinon pair implies that the former cannot simply de- But a bare hole state will change under the transformation cay into the latter even though they share the same quantum ͑2.12͒ as follows: numbers of charge and spin. In this section, we demonstrate that generally the holon and spinon constituents will be con- ͉⌿ ͘→ i Pi ϫ ͉⌿ ͘ ͑ ͒ ci G e ci G 2.13 fined by the phase shift field within a quasiparticle although ⌰ˆ the background is a spinon-holon sea. i i ϭ zϪ h due to the phase shift factor e with Pi S N /2 Intuitively such a confinement is easy to understand: If Ϫ Ϫ Ϫ (N 1 )/2 which remains an integer for a bipartite lat- the holon and spinon constituents inside a quasiparticle could z h tice (S and N denote total spin and hole numbers, respec- move away independently by themselves, as schematically tively, and N is the lattice size͒. This implies that c ͉⌿ ͘ ⌰ˆ i G shown in Fig. 3, the vortex phase shift field ei i left behind indeed has a nontrivial ‘‘angular’’ momentum in contrast to ͉⌿ would cost a logarithmically divergent energy as to be shown G͘ which carries none. below. But a quasiparticle state c ͉⌿ ͘ as a local excitation It is probably more transparent to see the origin of the i G should only cost a finite energy relative to the ground-state angular momentum if one rewrites, for example, energy. Such a discrepancy can be reconciled only if the holon and spinon constituents no longer behave as free el- ⌰ˆ i i↓ϭ ͑ Ϫ ͒1/2 ͑ Ϫ ͒1/2 ementary excitations: They have to absorb the effect of the e ͟ zi zl↑ ͟ zi* zl*↓ lÞi lÞi vortexlike phase shift and by doing so make themselves bound together. ⌰ˆ ϫ ͑ Ϫ ͒1/2 ͑ Ϫ ͒1/2ϫ ͑ ͒ ͉⌿Ј ϵ i i͉⌿ ͟ zi zlh ͟ zi* zl* Fi , 2.14 Let us consider ͘ e G͘ and compute the energy lÞi lÞi cost for the vortexlike phase shift: ↑ where zl↑ , zl↓ , and zlh denote the complex coordinates of , ↓ ϭ͟ ͉ Ϫ ͉ ͗⌿Ј͉H ͉⌿Ј͘Ϫ͗⌿ ͉H ͉⌿ ͘. ͑2.17͒ spinons, and holons, respectively. And Fi lÞi zi zl is eff G eff G a constant ͑which is obtained by using the no-double- occupancy constraint͒. It is important to note that despite the We first focus on the contribution from the holon part Hh ͓ ͑ ͔͒ h ϭ͗⌿ ͉ ͉⌿ ͘ fractional ͑‘‘semion’’͒ exponents of 1/2 in Eq. ͑2.14͒,itcan Eq. 2.6 . Define EG G Hh G . One has ˆ † f i⌰ ↓ Ϫ ͗⌿ ͉ iAlm͉⌿ ͘ϭ h be directly verified that the phase shift field e i remains th G hl hme G E /4N for any nearest-neighbor single valued under the no-double-occupancy constraint. link ͑lm͒ due to the translational symmetry. Then a straight- Generally the vortex field ͑2.14͒ introduces an extra angular forward manipulation leads to momentum which can be easily identified as Eh h ͗⌿Ј͉ ͉⌿Ј͘Ϫ h ϭϪ G ͕ Ϫ ͓ ͑ ͒Ϫ ͑ ͔͒ ͖ N Hh EG ͚ 1 cos i l i m /2 . lϭSzϩ . ͑2.15͒ 2N ͗lm͘ 2 ͑2.18͒ Here l is always an integer. Then one has Notice that if the (lm) link ͑say, along the xˆ direction͒ is far ⌿ Ϫ ͉͒ ͉⌿ ͒ ϭ ͑ ͒ away from the site i, then one has ͉ (l)Ϫ (m)͉ ͗ G͑Ne 1 ci G͑Ne ͘ 0 2.16 i i →a͉sin ͉/r where r denotes the distance between the center due to the orthogonal condition28 as lÞ0 ͓SzϭO(1), Nh of the link and the site i and is the azimuth angle. Then it ϭ ͔ ͉⌿ O(N) at finite doping for ci G͘. By extending the is easy to see that the summation over those links on the same argument, one can quickly see that the bare hole state RHS of Eq. ͑2.18͒ will contribute as ͐rdrd sin2/r2ϰln R ͉⌿ ͉⌿ Ϫ ci G(Ne)͘ has no overlap not only with G(Ne 1)͘ but (R denotes the sample size͒. Namely, the vortexlike phase also with all the elementary excitations composed of simple shift will cost a logarithmically diverged energy if it is left ϭ holons and spinons with l 0. So ci is more like a creation alone. It should be noted that the same conclusion still holds operator of a ‘‘collective’’ mode whose quantum number l is if one replaces Heff by the exact t-J model in the represen- different from a simple spinon-holon pair. tation of Eq. ͑1.3͒͑Ref. 15͒. PRB 61 NATURE OF SPIN-CHARGE SEPARATION IN THE t-J MODEL 12 333
Hence the vortexlike phase shift field has to be absorbed be bound to the total phase shift fields together to eliminate by the holon and spinon fields in order to keep the quasipar- the divergent energy while maintaining single-valuedness. There is another way to see this. Note that (m ) ticle energy finite. In the following, let us illustrate how this msϪ1 s 1 ⌽b ͑ ͒ will happen. We first use the vortex phase 2 in Eq. 2.1 Ϫ (m ) describes the angle between the nearest- i msϩ1 s as an example. Let us write down the following identity: neighboring links (msϪ1 ,ms) and (msϩ1 ,ms); it can have an uncertainty by Ϯ2ϫinteger, and it is easy to see that the 1 1 iKb(c ) ϪiKh(c ) expͩ i ⌽b ͪ ϭͫ expͩ i͚ A f ͪ exp͓iKb͑c ͔͒ͬexpͩ i ⌽b ͪ , phase string factors e h and e s in Eqs. ͑2.19͒ and i h j 2 ch 2 ͑2.21͒ are not well defined by themselves as they are multi- ͑2.19͒ valued except at ␦ϭ0. On the other hand, if one chooses ϭ ϭ in which ch cs c in Fig. 3, the mathematical ambiguity is elimi- nated in the total phase string field, ϵ b Ϫ h ͚ f ϵ͚ f ͑ ͒ K͑c͒ K ͑c͒ K ͑c͒ A Am m ϩ , 2.20 c s s s 1 h kc ϭ ϵ ϭ ͚ ͓ Ϫ ͑m ͒Ϫ ϩ ͑m ͔͒ where m0 i, m1 ,...,mk j are sequential lattice sites on ms 1 s ms 1 s ch sϭ1 b an arbitrary path ch connecting i and j. And K (ch) ϵ 1 ͚ ͓ Ϫ ͔ ͚ ␣ b Ϫ 1 2 (m ) (m ) ( ␣ n ␣ 1) which is a b h s msϪ1 s msϩ1 s ms ϫ ͩ ͚ ␣n Ϫ1Ϫn ͪ , ͑2.23͒ m ␣ m 2 ␣ s s stringlike field only involving spinons on the path ch .By contrast, the line summation ͚ A f is contributed by spinons ch since by using the no-double-occupancy constraint, one can 1 ͚ ␣ b Ϫ Ϫ h ϭϪ ϩ ϩ b from the whole system nonlocally. Note that, according to show that 2 ( ␣ n ␣ 1 n ) (1 )/2 n ms ms ms ͓ ͑ ͔͒ f iK(c) Hh Eq. 2.6 , holons see the gauge field A in the Hamil- which is an integer such that e remains single valued. tonian, and if a holon moves from site i to j via the same path Physically, it is because a fermionic quasiparticle may not c , it should acquire a phase factor exp(Ϫi͚ Af) which can decay into two bosonic holon and spinon elementary excita- h ch exactly compensate the similar phase in Eq. ͑2.19͒. In other tions in 2D. The only exception is in the zero-doping limit. i⌽b/2 We have pointed out in the Introduction that at half-filling words, if the vortex phase factor e i is bound to a holon to ⌽b the ‘‘fermionic’’ nature of the electrons essentially disap- ˜ †ϭ † i i /2 form a new composite object hi hi e , then there will be pears and is replaced by a ‘‘bosonic’’ one due to the no- no more vortex effect as it moves on the path ch shown in double-occupancy constraint. Then it is not surprising that in h Fig. 3, except for a phase string field K (ch) left on its path, the one-hole doped case which is adjacent to the half-filling, and the new object should cost only a finite energy. the deconfinement of holon-spinon can happen as a result of Ϫ ⌽h ͑ ͒ Similarly, for the vortex field ( /2) i in Eq. 2.1 one the electron ‘‘bosonization.’’ Indeed, in the zero-doping limit can rewrite ⌽h ͑ ͒ ⌽h i defined in the gauge 2.3 vanishes. Without i , the Ϫi⌽h/2 i⌽b /2 i⌰ˆ original reason for inseparable e i and e i in e i is h h h ⌰ˆ expͩ Ϫi ⌽ ͪ ϭͫ expͩ Ϫi͚ A ͪ exp͓ϪiK (c )͔ͬ i i i s no longer present: In this case, the phase shift field e 2 c s i⌽b/2 reduces to e i which itself becomes well defined, and can h solely accompany the holon during the propagation. As for ϫexpͩ Ϫi ⌽ ͪ , ͑2.21͒ ͑ ͒ 2 jЈ the spinon part, Hs in Eq. 2.7 reduces to the well-known SBMFT Hamiltonian with Ahϭ0 and the corresponding line in which ͚ h summation cA is also absent in the propagator. Without leading to the multivalue ambiguity, the quasiparticle will ͚ Ahϵ͚ Ah , ͑2.22͒ break into a spinon and a composite of holon-vortex phase msmsϩ1 cs s which can propagate independently. More discussions of the one-hole problem can be found in Sec. II D 3. where the line summation runs over a sequential lattice sites How a quasiparticle behaves in a spinon-holon sea as a on an arbitrary path c connecting i and jЈ shown in Fig. 3. s single entity at finite doping will be the subject of discussion And then we can similarly see that the spinon constituent of in the next subsection. In the following we will make several Ϫi⌽h/2 the quasiparticle also has to be bound to e in order to remarks on some implications of the confinement before compensate the logarithmically divergent energy cost by the concluding the present subsection. First of all, we note that a vortex structure, and the new composite will only leave a quasiparticle generally remains an incoherent excitation in phase string behind given by Kh(c )ϵ 1 ͚ ͓ (m ) s 2 s msϪ1 s contrast to the coherent spinons and holons and we assume Ϫ ͔ h m (ms) nm . that it will not contribute significantly to either thermody- sϩ1 s Ϫ However, there is one problem in the above argument namic and dynamic properties. In the equal-time limit tϭ0 , i⌽b/2 the single-electron propagator can be expressed as about the absorption of the vortex phase factors e i and i⌽h/2 e i by the holon and spinon constituents. Namely, these Ϫ iϪ j † h G ͑i, j;0 ͒ϭi͑Ϫ͒ ͫ b expͩ i ͚ A ͪ b ͬ ͳ e j i two phase factors are not single valued except in the zero- 2 c ⌰ˆ doping limit. In fact, only the total phase factor ei i is al- ways well defined and single valued as mentioned before. It ϫͫ h expͩ Ϫi͚ A f ͪ h†ͬeiK(c) ʹ . ͑2.24͒ j i thus means that both holon and spinon constituents have to c 12 334 Z. Y. WENG, D. N. SHENG, AND C. S. TING PRB 61
At finite t, temporal components have to be added to the line ergy if being left alone unscreened. Therefore, one has to ͚ h ͚ f summations, cA and cA , as well as in the phase string treat a quasiparticle as an independent collective excitation field K(c) above. Even though mathematically the path c in this spin-charge separation system. can be chosen arbitrarily in Eq. ͑2.24͒, a natural choice is for Involving infinite-body holons and spinons, a quasiparti- c to coincide with the real path of the quasiparticle such that cle cannot be simply described by the mean-field theory of the line summations can be precisely compensated by the individual holon and spinon. The previously discussed con- phases picked up by the holon and spinon constituents as finement is one example of the nonperturbative conse- mentioned above. In this case, all the singular phase effect quences caused by the infinite-body phase shift field. But will be tracked by eiK(c) which is nothing but the previously such a confinement of the holon and spinon inside a quasi- identified phase string effect,18 where it has been shown that particle will enable us to approach this problem from a dif- the phase string effect is nonrepairable and represents the ferent angle. dominant phase interference at low energy. Physically, it re- Here it may be instructive to recall how a low-lying col- flects the fermionic exchange relation between the quasipar- lective mode is determined in the BCS theory. In BCS mean- ticle under consideration and those electrons in the back- field theory, quasiparticle excitations are well defined with ground. Such a phase string field accompanying the an energy gap. But quasiparticle excitations do not exhaust propagation of the quasiparticle is a many-body operator in all the low-lying excitations, and there exists a collective terms of elementary holon and spinon fields. Even in the one mode in the absence of long-range Coulomb interaction, hole case, such a phase string effect results in the incoher- which may be also regarded as a ‘‘bound’’ state of a quasi- ency of the quasiparticle as has been discussed in Ref. 22. particle pair due to the residual attractive interaction. A cor- One may also see how a Fermi-surface structure is gener- rect way29 to handle this ‘‘bound’’ state is to use the full ⌰ˆ BCS Hamiltonian to first write down the equation of motion ated from the phase shift i in some limits. For example, in the 1D case ͑where Ah, f ϭ015͒, since one may always define for a quasiparticle pair and then apply the BCS mean-field ij treatment to linearize the equation to produce the gapless Ϫ (m )Ϫ ϩ (m )ϭϮ, the phase string factor ms 1 s ms 1 s iK(c) spectrum, which is equivalent to the random phase approxi- e in Eq. ͑2.24͒ can be written as 30 mation ͑RPA͒ scheme. Including the long-range Coulomb iϪ j ik (x Ϫx ) i␦k (x Ϫx ) (Ϫ) e f i j e f i j which produces the 1D Fermi interaction29 will turn this collective mode into the well- ϭϮ Ϫ␦ ͑ ␦ surface at k f (1 )/2 here k f denotes Fermi-surface known plasma mode. ͗␦ ͘ϭ fluctuations with k f 0 which is crucial to the Luttinger- Similarly we can establish an equation-of-motion descrip- 15͒ ⌰ˆ tion of the quasiparticle as a ‘‘collective mode,’’ which liquid behavior . In the 2D one-hole case, i also leads to a ‘‘remnant’’ Fermi-surface structure in the equal-time limit moves on the background of the mean-field spin-charge separation state. For this purpose, let us first write down the while it gives rise to four Fermi points k0 at low energy as discussed in Ref. 22. At finite doping, the doping-dependent full equation of motion of the hole operator in the Heisen- ͔ ϭ͓ ץ Ϫ incommensurate peaks in the dynamic spin susceptibility berg representation: i tci(t) HtϪJ ,ci(t) , based on function has been also related to such a phase shift field.19 In the exact t-J model, either in the decomposition ͑1.3͒ or general, the Luttinger-volume theorem may even be under- simply in the original c-operator representation, as follows: stood based on eiK(c) as it involves the counting of the t background electron numbers. Nevertheless, the precise ͓ ͔ϭ ͑ ϩ h͒ Ht ,ci 1 ni ͚ cl Fermi-surface topology will not be solely determined by the 2 lϭNN(i) phase shift field in 2D and one must take into account of the dynamic effect. ϩ ͑ zϩ Ϫ͒ ͑ ͒ t ͚ cl Si clϪSi 2.25 Finally, a stable but incoherent quasiparticle excitation in lϭNN(i) which a pair of holon and spinon are confined means that a photoemission experiment, in which such a quasiparticle ex- and citation can be created through ‘‘knocking out’’ an electron J by a photon, does not directly probe the intrinsic information ͓ ͔ϭ ͑ Ϫ h͒ HJ ,ci ci ͚ 1 nl of coherent elementary excitations anymore, and the energy- 4 lϭNN(i) momentum structure of the single-electron Green’s function J is no longer a basis as fundamental and useful as in the case Ϫ ͑ zϩ Ϫ͒ ͑ ͒ ͚ ci Sl ciϪSl . 2.26 of conventional Fermi-liquid metals to understand supercon- 2 lϭNN(i) ductivity, spin dynamics, and transport properties in other channels. Note that the above equations hold in the restricted Hilbert ͚ † space under the no-double-occupancy constraint: cici р1. D. Description of the quasiparticle: Equation-of-motion There are many papers in literature dealing with the t-J method model in the c-operator representation, in which the no ͚ † р Now imagine a bare hole is injected into the ground state double occupancy icici 1 is disregarded. As a conse- of Ne electrons. By symmetry, such a state should be or- quence, there is only a conventional scattering between the Ϫ thogonal to the ground state of Ne 1 electrons. Its dissolu- quasiparticle and spin fluctuations as suggested by Eqs. tion into a holon and a spinon is also prohibited by the sym- ͑2.25͒ and ͑2.26͒. This leads to a typical spin-fluctuation metry introduced by the phase shift field ͓Eq. ͑1.3͔͒ and the theory, which usually remains a Fermi-liquid theory with latter would otherwise cost a logarithmically divergent en- well-defined coherent quasiparticle excitations near the PRB 61 NATURE OF SPIN-CHARGE SEPARATION IN THE t-J MODEL 12 335
Fermi surface in contrast to the Zϭ0 conclusion obtained t t ϭ ͑1ϩ␦͒. ͑2.30͒ here. The problem with the spin-fluctuation theory is that the eff 2 crucial role of the no-double-occupancy constraint hidden in Eqs. ͑2.25͒ and ͑2.26͒ has been completely ignored which, in Generally the ‘‘decaying’’ terms do not contribute to a co- combination with the RVB spin pairing, is actually the key herent k-dependent correction due to the nature of the holon reason resulting in a spin-charge separation state in the and spinon excitations as well as the ‘‘smearing’’ caused by ˆ present effective theory of the t-J model. In such a backdrop ei⌰ in Eq. ͑2.27͒. But in the ground state, which is also of the holon-spinon sea, the scattering terms in Eqs. ͑2.25͒ superconducting, the ‘‘decaying’’ terms in Eq. ͑2.27͒ do pro- and ͑2.26͒ will actually produce a virtual ‘‘decaying’’ pro- duce a coherent contribution due to the composite nature of cess which is fundamentally different from the usual spin- the quasiparticle which will modify the solution of the fluctuation scattering in shaping the single-electron propaga- equation-of-motion. tor. By using the decomposition ͑1.3͒ and the mean-field or- 1. Ground state: A superconducting state der parameter ⌬s defined in Eq. ͑2.10͒, the high-order spin- ͑ ͒ In the mean-field ground state, the bosonic holons are fluctuation-scattering terms on the RHS of Eqs. 2.25 and ͗ †͘ϭ ϳͱ␦ Bose condensed with hi h0 and the superconduct- ͑2.26͒ can be ‘‘reduced’’ to the same order of linear c , and i ing order parameter ⌬SCÞ0 ͑see Sec. II A͒. The decomposi- we find tion ͑1.3͒ then implies that the electron c operator may be t rewritten in two parts: ͒␦Ϸ ϩ␦͒ ϩ Ϫ͒ ץ Ϫ i tci͑t ͑1 ͚ cl J͑1 ci 2 ϭ ϭ ϩ Ј ͑ ͒ l NN(i) ci h0ai ci , 2.31
i⌰ˆ † i⌰ˆ † 1 ⌰ˆ Ϫ h where a ϵb e i and cЈ ϭ(:h :)b e i with :h : Ϫ i l † i Ail i i i i i i tB0 ͚ e hl bie ϭ ϵ †Ϫ 4 l NN(i) hi h0. Correspondingly, a coherent term will emerge from the ‘‘decaying’’ terms in Eq. ͑2.27͒ which is linear in 3 ⌰ˆ h † ϩ ⌬s i i † † i Ailϩ a : J ͚ e hi blϪe , 8 lϭNN(i) ••• J-scattering term in Eq. ͑2.27͒ ͑2.27͒ ⌬SC ͑ ͒ 3 il where B0 is the modified but not an independent order → J ͚ ͩ ͪ h a† ϩhigh order. 0 lϪ 8 ϭ 2 parameter for the hopping term introduced in Ref. 20. In the l NN(i) h0 following we will discuss some unique quasiparticle proper- ͑ ͒ ties based on this equation. 2.32 So in the spin-charge separation ͑mean-field͒ background, In obtaining the RHS of the above expression, the supercon- the leading order effect of the ‘‘scattering’’ terms correspond ducting order parameter defined in Eq. ͑2.11͒ is used. to the decay of the quasiparticle: The terms in the second and Note that the t-scattering term in Eq. ͑2.27͒ gives rise to a ͑ ͒ ϰ third lines of Eq. 2.27 clearly indicate the tendency for the term h0ai which can be absorbed by the chemical poten- quasiparticle to break up into holon and spinon constituents. tial added to the equation. Then one finds This is in contrast to the conventional scatterings between ͑ ͒ Ӎ ϩ ץ the quasiparticle and spin fluctuations, as Eqs. 2.25 and Ϫ ͑ ͒ i tai teff ͚ al ai 2.26 would have suggested. Generally, the quasiparticle is lϭNN(i) expected to have an intrinsic broad spectral function ex- ⌬SC tended over the whole energy range 3 il ϩ J ͚ ͩ ͪ a† ϩhigh order, lϪ 8 ϭ 2 Ͼ ϩ ͑ ͒ l NN(i) h0 Equasiparticle Eholon Espinon 2.28 ͑2.33͒ because of the decomposition process. But the presence of ⌰ˆ Ј the phase factor ei in these ‘‘decaying’’ terms of Eq. ͑2.27͒ where the connection between a and c has been assumed to be in high order and thus is neglected in the leading order prevents a real decay of the quasiparticle since such a vortex † field would cost a logarithmically divergent energy as has approximation to get a closed form in linear a and a . Fi- been discussed before. Thus, even in the case of Eq. ͑2.28͒, nally, introducing the Bogoliubov transformation in the mo- the decaying of a quasiparticle remains only a virtual process mentum space which is an another way to understand the confinement dis- ϭ ␥ Ϫ ␥† ͑ ͒ cussed in Sec. II C. ak uk k vk ϪkϪ , 2.34 Without the ‘‘decaying’’ terms, the equation of motion we find that Eq. ͑2.33͒ can be reduced to ͑2.27͒ would become closed with an eigenspectrum in ͒ ͑ †␥ ϭ †␥ ץ Ϫ ͑ momentum-energy space besides a constant which can be i t k Ek k , 2.35 absorbed into the chemical potential͒: ␥† where k represents the creation operator of an eigenstate ⑀ ϭϪ ϩ ͒ ͑ ͒ of quasiparticle excitations with the energy spectrum k 2teff͑cos kx cos ky , 2.29 ϭͱ ⑀ Ϫ͒2ϩ͉⌬ ͉2 ͑ ͒ with Ek ͑ k k . 2.36 12 336 Z. Y. WENG, D. N. SHENG, AND C. S. TING PRB 61
͑see Fig. 4͒. In this sense, the quasiparticle partially restores its coherence in the superconducting state. Such a coherent part will disappear as a result of vanishing superfluid density. According to Eq. ͑2.31͒ one may rewrite the single-particle propagator as
Ӎ 2 ϩ Ј ͑ ͒ Ge h0Ga Ge , 2.40
where Ga denotes the propagator of a particles with omitting the crossing term between a and cЈ which is assumed negli- 2 FIG. 4. Low-lying excitations in the superconducting phase: The gible. Then h0Ga emerges as the ‘‘coherent’’ part of the ‘‘V’’ shape quasiparticle spectrum and the discrete spinon energy at Green’s function in superconducting state against the ‘‘nor- E . Ј s mal’’ part Ge :
⌬ 2 2 Here k is defined by u v 2 ͑ ͒ϳ 2ͩ k ϩ k ͪ ͑ ͒ h0Ga k, h0 Ϫ ϩ . 2.41 ⌬SC Ek Ek 3 k¿q ⌬ ϭ J͚ ⌫ ͩ ͪ , ͑2.37͒ k 4 q 2 q h0 Correspondingly, the total spectral function as the imaginary part of G in our theory can be written as ⌫ ϭ ϩ 2ϭ͓ ϩ ⑀ e with q cos qx cos qy . Like in BCS theory, uk 1 ( k Ϫ ͔ 2ϭ͓ Ϫ ⑀ Ϫ ͔ 2 )/Ek /2 and vk 1 ( k )/Ek /2. A ͑k,͒ϭh A ͑k,͒ϩAЈ͑k,͒. ͑2.42͒ ⑀ ϭ ⌬ e 0 a e The large ‘‘Fermi surface’’ is defined by k and k → ⌬ then represents the energy gap opened at the Fermi surface. So at h0 0, even though k does not scale with h0, the ⌬ 2 Note that k changes sign as superconducting coherent part h0Aa vanishes altogether, Ј Ͼ ⌬ ϭϪ⌬ ͑ ͒ with Ae reduced to the normal part Ae at T Tc . k¿Q k , 2.38 Ј Finally, in the present case, Ae (k, ) as a normal part has ϭ Ϯ Ϯ ⌫ ϭϪ⌫ ͑ ͒ with Q ( , ), by noting qϩQ q in Eq. 2.37 .It nothing to do with the procedure that leads to the spectrum ⌬ ϭ Ϯ ͑ ͒ means that k has opposite signs at k ( ,0) and (0, 2.36 with a d-wave gap, which is different from the slave- Ϯ) , indicating a d-wave symmetry near the Fermi surface. boson approach where the fermionic spinons are all paired ⌬SC In fact, since the pairing order parameter k is up such that even the part of the spectral function corre- 20 ⌬ Ј d-wave-like, k should be always d-wave-like with node sponding to Ae should also look like in a d-wave pairing ϭϮ ͑ ͒ ͐ e ϭ lines kx ky according to Eq. 2.37 . state. Due to the sum rule (d /2 )A (k, ) 1, the ‘‘nor- Ј Comparing to the conventional BCS theory with the mal’’ part Ae (k) is expected to be sort of suppressed by the 2 d-wave order parameter, there are several distinct features in emergence of the ‘‘coherent’’ h0Aa part, but since the latter the present case. First of all, besides the d-wave quasiparticle ␦ Ј is in order of , Ae should be still dominant away from the spectrum illustrated in Fig. 4 by the ‘‘V’’ shape lines along Fermi surface at small doping. It implies that even in super- the Fermi surface, there exists a discrete spinon excitation conducting state, a normal-state dispersion represented by ϳ␦ ͑ ͒ level at Es J horizontal line in Fig. 4 which leads to the peak of AЈ may still be present as a ‘‘hump’’ in the total ϭ ϳ ͑ ϳ ␦ e Eg 2Es 41 meV if J 100 meV) magnetic peak at spectral function A . Recent ARPES experiments have in- ϳ e 0.14 as reviewed in Sec. II A. This latter spin collective deed indicated33 the existence of a ‘‘hump’’ in the spectral mode is independent of the quasiparticle excitations at the function which clearly exhibits normal-state dispersion in the mean-field level. Fermi-surface portions near the areas of (Ϯ,0) and (0, Second, even though the superconducting order param- Ϯ) where the d-wave gap is maximum. ⌬SC ⌬ eter k and the energy gap k in the quasiparticle spectrum have the same symmetry, both are d-wave-like, and they can- 2. Normal state not be simply identified as the same quantity as in BCS ⌬SC In the normal state, without any coherent contribution, the theory. For example, while k apparently scales with the ͑ ͒ ␦ ϰͱ␦ ␦→ scattering terms in Eq. 2.27 only give rise to the virtual doping concentration (h0 ) and vanishes at 0, the ⌬ ͑ ͒ process for a quasiparticle to decay into the holon-spinon gap k defined in Eq. 2.37 is not, and can be extrapolated ͑ ͒ ϭ pairs. Based on Eq. 2.27 , the propagator can be determined to a finite value in the zero-doping limit where Tc 0. It according to the following standard equation of motion for means the single-particle Green’s function Ge(i, j;t): 2⌬ ͑Tϭ0͒ ϭ͑ ͓͒͗ ͑ ͔͒ † ͑ ͒͒͘ ͑ ץ k →ϱ͑͒ 2.39 tGe i, j;t t HtϪJ ,ci t c j 0 Tc Ϫ͑Ϫ ͒͗ † ͑ ͓͒ ͑ ͔͒͘Ϫ ␦͑ ͒␦ at ␦→0, whereas the BCS theory predicts a constant ϳ4.28 t c j 0 HtϪJ ,ci t i t i, j . 31͒ ͑ ͒ (d-wave case . The result 2.39 is consistent with the ͑2.43͒ ARPES measurements.32 Third, the quasiparticle gains a ‘‘coherent’’ part h0a If we simply neglect the scattering terms in zero-order ap- which should behave similarly to the conventional quasipar- proximation, a closed form for Ge can be obtained in Ͻ ticle in BCS theory as it does not further decay at Ek Es momentum-energy space: PRB 61 NATURE OF SPIN-CHARGE SEPARATION IN THE t-J MODEL 12 337
† ⌽b Ϫ ⌽b 1 ͑ ͒ϭϪ ͗ ͑ ͒͑ i i (t)/2 i j (0)/2͒ ͑ ͒͘ ͑ ͒ϳ ͑ ͒ G f i, j;t i Tthi t e e h j 0 Ge k, Ϫ ⑀ Ϫ͒ . 2.44 ͑ ͒ ͑ k 2.46 and ⑀ ͓ ͑ ͔͒ Here the quasiparticle spectrum k defined in Eq. 2.29 ͒ϭϪ Ϫ͒iϪ j ͒ is essentially the same as the original band-structure spec- Gb͑i, j;t i͑ ͗Ttbi͑t trum except for a factor of 2/(1ϩ␦)Ϸ2 enhancement in ef- Ϫ ⌽h ⌽h † ϫ͑ i i (t)/2 i j (0)/2͒ ͑ ͒͘ ͑ ͒ fective mass. It is noted that in the t-J model if the hopping e e b j 0 , 2.47 term described by the tight-binding model is replaced by a ⌽h ͑ ͒ more realistic band-structure model, like introducing the without the multivalue problem because i in Eq. 2.3 van- i⌽b/2 next-nearest-neighbor hopping terms, the above conclusion ishes and e i becomes well defined as discussed in Sec. ␦→ ͑ ͒ about the factor-of-2 enhancement of the effective mass still II C. At 0, Hs in Eq. 2.7 reduces to the SBMFT Hamil- 34 hϭ holds, in good agreement with ARPES experiments. Here tonian with A 0 and Gb becomes the conventional the reason for the mass enhancement is quite simple: At each Schwinger-boson propagator. Such a deconfinement can be step of hopping, the probability is roughly one-half for a hole also seen from the equation of motion ͑2.27͒ by noting that ⌰ˆ ⌽b not to change the surrounding singlet spin configuration, i i→ i i /2 † h ϭ e e can be absorbed by hi , while Ail 0, so that which in turn reduces the ‘‘bandwidth’’ of the quasiparticle the scattering term becomes a pure decaying process for the by a factor of 2. quasiparticle without any confining force. Due to such a true The expression ͑2.44͒ shows a ‘‘quasiparticle’’ peak at ͑ ͒ ⑀ Ϫ decaying, Eq. 2.27 actually describes in real time the first k and defines a large ‘‘Fermi surface’’ as an equal- ⑀ ϭ step towards dissolution for the quasiparticle. In particular, energy contour at k . Here is determined such that the large Fermi-surface structure originating from the bare Ϫ ͚ ϭϪ ϭ 35 i2 kGe(k,t 0) Ne . So the ‘‘Fermi-surface’’ struc- hopping term in Eq. ͑2.27͒ will no longer appear in the de- ture should look similar to that of a noninteracting band- composition form of the electron propagator ͑2.45͒, where structure fermion system as long as the virtual decaying pro- the residual Fermi surface ͑points͒ will solely come from the ͑ ͒ ͓ cess in Eq. 2.27 does not fundamentally alter it. As i⌽b/2 mentioned before, we do not expect such ‘‘decaying’’ terms oscillating part of the phase shift field e i in G f . ⑀ The single-electron propagator for the one-hole case has to significantly modify the k dependence of k since, for example, the spinon no longer has a well-defined spectrum in been discussed in detail in Ref. 22. Here the large Fermi ϭ Ϯ momentum space ͑see Sec. II A͒ and, in particular, the vortex surface is gone except for four Fermi points at k0 ( /2, ˆ Ϯ/2) with the remaining in part looking like they are all phase ei⌰ in Eq. ͑2.27͒ will further ‘‘smear out’’ the ‘‘gapped.’’ In fact, in the convolution form of Eq. ͑2.45͒ the k-dependent correction, if any, from Eq. ͑2.27͒.͔ ‘‘quasiparticle’’ peak ͑edge͒ is essentially determined by the So far we have not discussed the finite lifetime effect of a s ϭ ͱ Ϫ 2 ϭ ͑ ͒ spinon spectrum Ek 2.32J 1 sk with sk (sin kx quasiparticle due to the ‘‘decaying’’ terms in Eq. 2.27 . ϩ sin ky)/2 in SBMFT through the spinon propagator Gb , Even though the true breakup of a quasiparticle is prevented 22 by the phase shift field as discussed before, the virtual de- since the holon propagator G f is incoherent. The intrinsic broad feature of the spectral function found in Ref. 22 is due caying process should remain a very strong effect since the ͑ ͒ phase shift field only costs a logarithmically divergent en- to the convolution law of Eq. 2.45 and is a direct indication of the composite nature of the quasiparticle, which is also ergy at a large length scale. The corresponding confinement 23 force is rather weak and the virtual decaying process should consistent with the ARPES results as well as the earlier become predominant locally to cause an intrinsic broad fea- theoretical discussion in Ref. 36. ture in the spectral function at high energy. Such a broad Note that the Fermi points k0 coming from G f at low i⌽b/2 structure reflecting the decomposition in the one-hole case energy is due to the phase shift field e i appearing in it. In has been previously discussed in Ref. 22. At finite doping, Ref. 22, this is shown in the slave-fermion formulation how the ‘‘decaying’’ effect shapes the broadening of the which is related to the present formulation through a unitary † ⌽b quasiparticle peak will be a subject to be investigated else- 15 i i /2 transformation with hi e being replaced by a new ho- where. † lon operator f i . And the f holon will then pick up a phase ↓ N ↓ string factor (Ϫ1) c (N denotes the total number of ↓ spins 3. Destruction of Fermi surface: Deconfinement of spinon and c exchanged with the holon during its propagation along the holon path c connecting sites i and j) at low energy which can be The existence of a large Fermi surface, coinciding with written as the noninteracting band-structure one, can be attributed to ↓ ↓ ↓ N ϮiN ik (r Ϫr ) Ϯi␦N the integrity of the quasiparticle due to the confinement of ͑Ϫ1͒ c ϵe c ϭe 0• i j e c , ͑2.48͒ spinon and holon. But as pointed out in Sec. II C, such a ␦ ↓ϭ ↓Ϫ͗ ↓͘ ͗ b ͘ϭ confinement will disappear in the zero-doping limit. The where Nc Nc Nc , and nl↓ 1/2 is used. On the other Fermi surface structure will then be drastically changed. hand, in the equal-time (tϳ0Ϫ) limit, the singular oscillating In this limit, the single-electron propagator may be ex- part of eiK(c) in Eq. ͑2.24͒ will also contribute to a large pressed in the following decomposition form ‘‘remnant Fermi surface’’ in the momentum distribution function n(k) which can be regarded as a precursor of the Ϸ ͑ ͒ large Fermi surface in the confining phase at finite doping, Ge iGf •Gb , 2.45 and is also consistent with the ARPES experiment as dis- where cussed in Ref. 22. 12 338 Z. Y. WENG, D. N. SHENG, AND C. S. TING PRB 61
The above one-hole picture may have an important impli- found to be effectively confined which maintains the integ- cation for the so-called pseudo-gap phenomenon24 in the un- rity of a quasiparticle except for the case in the zero-doping derdoped region of the high-Tc cuprates. Even though the limit. In particular, the quasiparticle weight is zero since confinement will set in once the density of holes becomes there is no overlap between a doped hole state and the true finite, the ‘‘confining force’’ should remain weak at small ground state due to the symmetry difference introduced by doping, and one expects the virtual ‘‘decaying’’ process in the vortex phase shift. Such a quasiparticle is no longer a ͑ ͒ Eq. 2.27 to contribute significantly at weak doping to conventional Landau quasiparticle and is generally incoher- bridge the continuum evolution between the Fermi-point ent due to the virtual decaying process. Only in the super- structure in the zero-doping limit to a full large Fermi sur- conducting state can the coherence be partially regained by face at larger doping. Recall that in the one-hole case decay- the quasiparticle excitation. ing into spinon-holon composite happens around k0 at zero The physical origin of the ‘‘unrenormalizable phase energy transfer, while it costs higher energy near (,0) and shift’’ is based on the fact that a hole moving on an antifer- (0,), which should not be changed much at weak doping. romagnetic spin background will always pick up the phase In the confinement regime, the quasiparticle peak in the elec- string composed of a product of ϩ and Ϫ signs which de- tron spectral function defines a quasiparticle spectrum and a pend on the spins exchanged with the hole during its large Fermi surface as discussed before. Then due to the propagation.18 Such a phase string is nonrepairable at low virtual ‘‘decaying’’ process in the equation of motion ͑2.27͒ energy and is the only source to generate phase frustrations ͓as shown in Fig. 1͑b͔͒, the spectral function will become in the t-J model. The phase shift field in Eq. ͑1.3͒ precisely much broadened with its weight shifted toward higher en- keeps track of such a phase string effect15 and therefore ac- ergy like a gap opening near those portions of the Fermi curately describes the phase problem in the t-J model even at surface far away from k0, particularly around four corners the mean-field level discussed in Sec. II A. (Ϯ,0) and (0,Ϯ). With the increase of doping concen- Probably the best way to summarize the present work is to tration and reduction of the decaying effect, the suppressed compare the present self-consistent spin-charge separation quasiparticle peak can be gradually recovered starting from theory with some fundamental concepts and ideas proposed the inner parts of the Fermi surface towards four corners over years in the literature related to the doped Mott- (Ϯ,0) and (0,Ϯ). Eventually, a coherent Landau quasi- Hubbard insulator. particle may be even restored in the so-called overdoped re- RVB pairing. The present theory can be regarded as one gime, when the bosonic RVB ordering collapses such that of the RVB theories,1,37,4 where the spin RVB pairing is the the spin-charge separation disappears. driving force behind everything from spin-charge separation Furthermore, at small doping ͑underdoping͒, something to superconductivity. The key justification for this RVB more dramatic can happen in the model described by Eqs. theory is that it naturally recovers the bosonic RVB descrip- ͑2.6͒ and ͑2.7͒. In Ref. 20, a microscopic type of phase sepa- tion at half-filling, which represents11,12 the most accurate ration has been found in this regime which is characterized description of the antiferromagnet for both short-range and by the Bose condensation of bosonic spinon field. Since long-range AF correlations. In the metallic state at finite dop- 20 spinons are presumably condensed in hole-dilute regions, ing, the RVB order ⌬s defined in Eq. ͑2.10͒ reflects a partial the propagator will then exhibit features looking like in an ‘‘fermionization’’ from the original pure bosonic RVB pair- even weaker doping concentration or more ‘‘gap’’ like than h ing due to the gauge field Aij determined by doped holes. in a uniform case, below a characteristic temperature T* But it is still physically different from a full fermionic RVB which determines this microscopic phase separation. There- description.1,6,37 In contrast to the fermionic RVB order pa- fore, the ‘‘spin-gap’’ phenomenon related to the ARPES ⌬s 24 rameter, here serves as a ‘‘super’’ order parameter char- experiments in the underdoped cuprates may be understood acterizes a unified phase covering the antiferromagnetic in- as a ‘‘partial’’ deconfinement of holon and spinon whose sulating and metallic phases, and normal and effect is ‘‘amplified’’ through a microscopic phase separa- superconducting states altogether.20 tion in this weakly doped regime. As discussed in Ref. 20, Spin-charge separation. In our theory, elementary excita- T* also characterizes other ‘‘spin-gap’’ properties in mag- tions are described by charge-neutral spinon and spinless ho- netic and transport channels in this underdoping regime. lon fields, and the ground state may be viewed as a spinon- holon sea. Different from slave-particle decompositions, III. CONCLUSION AND DISCUSSION: A UNIFIED VIEW spinons and holons here are all bosonic in nature and the conventional gauge symmetry is broken by the RVB order- In this paper, we have studied the quasiparticle properties ing. But these spinons and holons in 2D still couple to each of doped holes based on an effective spin-charge separation other through the mutual Chern-Simons-like gauge interac- theory of the t-J model. The most unique result is that a tions which are crucial to Tc , anomalous transport and mag- quasiparticle remains stable as an independent excitation de- netic properties. The Bose condensation of holons corre- spite the existence of holon and spinon elementary excita- sponds to the superconducting state, while the Bose tions. The underlying physics is that in order for a doped condensation of spinons in the insulating phase gives rise to hole to evolve into elementary excitations described by a an AF long-range order. The spinon Bose condensation can holon and spinon, the whole system has to adjust itself glo- persist into the metallic regime, leading to a pseudogap bally which would take infinite time under a local perturba- phase with microscopic phase separation which can coexist tion. Such an adjustment is characterized by a vortexlike with superconductivity.20 phase shift as shown in Eq. ͑1.3͒. As a consequence of the Bosonization. The electron c operator expressed in terms phase shift effect, the holon and spinon constituents are of bosonic spinons and holons in Eq. ͑1.3͒ naturally realizes PRB 61 NATURE OF SPIN-CHARGE SEPARATION IN THE t-J MODEL 12 339 a special form of bosonization. A 2D bosonization descrip- tial point is that the holons and spinons are not confined in tion has been regarded by many4,17,38–40 as the long-sought the ground state but are only bound in quasiparticles as a technique to replace the perturbative many-body theory in kind of incoherent ͑many-body͒ excitations which have no dealing with a non-Fermi liquid. The 2D bosonization overlap with elementary holon and spinon excitations as scheme has been usually studied, as an analog to the success- guaranteed by symmetry. These incoherent quasiparticle ex- ful 1D version,41 in momentum space where Fermi surface citations should not have any significant contribution to the 4,17,38–40 ͑ patches have to be assumed first. In the present thermodynamic properties at least above Tc). At short dis- scheme, which is also applicable to 1D, the Fermi surface tance and high energy, the composite nature of a quasiparti- satisfying the Luttinger volume and the so-called Fermi- cle will become dominant which may well explain the broad surface fluctuations are presumably all generated by the intrinsic structure in the spectral function observed in the phase shift field in Eq. ͑1.3͒, which guarantees the fermionic ARPES experiments. The composite structure of the quasi- nature of the electron. Note that the vortex structure involved particle can even show up at low energy when one navigates in the phase shift field in the 2D case is the main distinction through different circumstances like the superconducting from the ⌰ function in conventional bosonization condensation, underdoping regime, etc., with some unique proposals.40 features different from the behavior expected from Landau Non-Fermi liquid. As a consequence of the phase shift quasiparticles. field, representing the Fermi surface ‘‘fluctuations,’’ the Fractional statistics. Laughlin44 made a compelling argu- ground state is a non-Fermi liquid with vanishing spectral ment right after the proposal of spin-charge separation that weight Z, consistent with the argument made by Anderson3,4 the holon and spinon should carry fractional statistics, by based on a ‘‘scattering’’ phase shift description. In 1D both making an analogy of the spin liquid state with a fractional methods are equivalent as the phase shift value in the latter quantum Hall state. Even though the absence of the time- can be determined quantitatively based on the exact Bethe- reversal symmetry-breaking evidence in experiments does string ansatz solution. But in 2D, the phase shift field ⌰ , not support the original version of fractional statistics i 44,45 which is obtained by keeping track of the nonrepairable ͑anyon͒ theories, the essential characterization of frac- phase string effect induced by the traveling holon, provides a tional statistics for the singlet spin liquid state is, surpris- unique many-body version with vorticities, and our model ingly, present in our theory in the form of the phase string shows how a concrete 2D non-Fermi liquid system can be effect as discussed in Ref. 15. But no explicit time-reversal 20 realized. symmetry is broken in this description. Quasiparticle: Spinon-holon confinement. In conventional A fractional statistics may sound strange as we have been spin-charge separation theories based on slave-particle talking about ‘‘bosonization’’ throughout the paper. But as ⌰string schemes, a quasiparticle does not exist at all: It always pointed out in Ref. 15, if the phase shift field i is to be breaks up and decays into spinon-holon elementary excita- ‘‘absorbed’’ by the bosonic spinon and holon fields, then the tions. This deconfinement has been widely perceived as a expression ͑1.3͒ can be regarded as a slave-semion decom- logical consequence of the spin-charge separation in the lit- position with the new ‘‘spinon’’ and ‘‘holon’’ fields being erature. But in the present paper, we have shown that the ‘‘bosonic’’ among themselves but satisfying a mutual frac- phase shift field actually confines the spinon-holon constitu- tional statistics between the spin and charge degrees of free- ents ͑at least at finite doping͒, which means that the integrity dom. The origin of mutual statistics can be traced back to the of a quasiparticle is still preserved even in a spin-charge nonrepairable phase string effect induced by a hole in the separation state. It may be considered as a U(1) version of antiferromagnetic spin background. At finite doping, the or- quark confinement, but with a twist: the stable quasiparticle der parameter ⌬s in Eq. ͑2.10͒ actually describes the RVB as a collective mode is generally not a coherent elementary pairing of spinons with mutual statistics which reduces to the excitation since during its propagation the phase shift field bosonic RVB only in the half-filling limit. Furthermore, in also induces a nonlocal phase string on its path. In other the bosonic representation of Eqs. ͑2.6͒ and ͑2.7͒ the lattice f h words, the quasiparticle here is not a Landau quasiparticle Chern-Simons fields Aij and Aij precisely keep track of mu- anymore. The confinement and nonrepairable phase string tual statistics,15 which are crucial to various peculiar proper- effect both reflect the fermionic nature of the quasiparticle; ties exhibited in the model. namely, the fermionic quasiparticle cannot simply decay into Gauge theory. Based on the slave-boson decomposition, it a bosonic spinon and holon, and the phase string effect has been shown26,27 that the gauge coupling is the most im- comes from sequential signs due to the exchange between portant low-energy interaction associated with spin-charge the propagating quasiparticle and the spinon-holon deconfinement there. The anomalous linear-temperature background—the latter after all is composed of fermionic resistivity27 has become the hallmark for anomalous trans- electrons in the original representation. port phenomenon based on the scattering between charge The issue of the possible confinement of spinon and holon carriers and gauge fluctuations. In the present theory, the was already raised by Laughlin42 along a different line of holons are also subject to strong random flux fluctuations, in reasoning. He has also discussed numerical and experimental terms of the effective holon Hamiltonian ͑2.7͒, in a uniform evidence that the spinons and holons may be seen, more normal state where it leads14 to the linear-T resistivity in sensibly, in high-energy spectroscopy like the way quarks consistency with the Monte Carlo numerical calculation.46 are seen in particle physics. In recent SU͑2͒ gauge theory,43 Other anomalous transport properties related to the cuprates an attraction between spinons and holons to form a bound may be also systematically explained in such a simple gauge state due to gauge fluctuations is also assumed in order to model based on some effective analytic treatment.14,47 In explain the ARPES data. But in the present work the essen- contrast to the slave-boson gauge theory, however, the 12 340 Z. Y. WENG, D. N. SHENG, AND C. S. TING PRB 61 spinon part does not participate in the transport phenomenon cles in the singlet channel can recover the coherency due to due to the RVB condensation (⌬sÞ0) persisting over the the cancellation of the frustration caused by the phase string normal state, and besides the Chern-Simons gauge fields effect. Thus, if one is to construct a phenomenological theory f ,h based on the electron representation, the superconductivity Aij , the conventional gauge interaction between spinons can be naturally viewed as due to an in-plane kinetic and holons is suppressed because of it. 54 Furthermore, the -flux phase48 and commensurate flux mechanism. phase49 in the mean-field slave-boson theory, and the recent Phase string and Z2 gauge theory. As pointed out at the SU͑2͒ gauge theory43 at small doping have a very close con- beginning of this section, our whole theory is built on the nection with the present approach: A fermionic spinon in the phase string effect identified in the t-J model. Namely, the main phase frustration induced by doping is characterized by presence of flux per plaquette is actually a precursor to a sequence of signs ͟ on a closed path c where ϭ become a bosonic one at half-filling under the lattice and c ij ij Ϯ1 denotes the index of spins exchanged with a hole at a no-double-occupancy constraint. In Ref. 14, how such a sta- link (ij) during its hopping. Thus, instead of working in the tistics transmutation occurs has been discussed, and in fact vortex representation of Eq. ͑1.3͒ where the singular phase the bosonization decomposition ͑1.3͒ was first obtained there string effect has been built into the wave functions with the based on the fermionic flux phase. In spite of the physical nonsingular part described by the Chern-Simons-like lattice proximity, however, the detailed mathematical structure of 15 ϩ 50,51 gauge fields, one may also directly construct a 2 1 Z2 the gauge description for the fermionic flux phase and 55 ͟ the present bosonic spinon description are obviously rather gauge theroy to deal with the singular phase string c ij . different. A discrete Z2 gauge theory here seems the most natural de- Superconducting mechanism. Anderson1 originally con- scription of the phase string effect as the sole phase frustra- tion in the t-J model, in contrast to the conventional con- jectured that the superconducting condensation may occur 26,27 once the RVB spin pairs in the insulating phase start to move tinuum gauge field description. like Cooper pairs in the doped case. The superconducting Finally, we emphasize the close connection between the condensation in the present theory indeed follows suit. But antiferromagnetism and superconductivity as both occur in a ⌬s there is an important subtlety here. Since the RVB paring unified RVB background controlled by . The relation be- order parameter ⌬s covers the normal state as well, there tween the AF insulating phase and the superconducting must be an another factor controlling the superconducting phase here is much more intrinsic than in conventional ap- transition: the phase coherence. Indeed, the vortex phase ⌽b proaches to the t-J model. Especially, the coexistence of appearing in the superconducting order parameter ͑2.11͒ is holon and spinon Bose condensations in the underdoped regime20 makes a group theory description of such a phase, the key to ensure the phase coherence at a relatively low ͑ ͒ 56 temperature compatible to a characteristic spin energy.20,21 In in the fashion of SO 5 theory, become possible but with other words, the phase coherence discussed by Emery and an important modification: Inhomogeneity must play a cru- Kivelson52 is realized by the phase shift field in the present cial role here in the Bose-condensed holon and spinon fields in order to describe this underdoped regime. A detailed in- theory which effectively resolves the issue why Tc is too high in previous RVB theories. vestigation along this line will be pursued elsewhere. Furthermore, the interlayer pair tunneling mechanism53 ACKNOWLEDGMENTS for superconductivity is also relevant to the present theory from a different angle. Recall that the quasiparticle does exist The present work was supported, in part, by Texas ARP in the present theory but is always incoherent just like the program No. 3652707 and a grant from the Robert A. Welch blocking of a coherent single-particle interlayer hopping con- foundation, and the State of Texas through the Texas Center jectured in Ref. 53. On the other hand, a pair of quasiparti- for Superconductivity at University of Houston.
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