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Pseudogap, competing order and coexistence of staggered flux and d-wave pairing in high-temperature superconductors

We study the t-J-V model of a doped Mott insulator in connection to high-T_c superconductors. The nearest neighbor Coulomb interaction (V) is treated quantum mechanically on equal footing as the antiferromagnetic exchange interaction (J). Motivated by the SU(2) symmetry at half-filling, we construct a large-N theory which allows a systematic study of the interplay between staggered flux order and superconductivity upon doping. We solve the model in the large-N limit and obtain the ground state properties and the phase diagram as a function of doping. We discuss the competition and the coexistence of the staggered flux and the d-wave superconductivity in the underdoped regime and the disappearance of superconductivity in the overdoped regime

The physics of doped Mott insulators has played a central role in the study of high-T c superconductors [1,2]. Traditionally, the Mott insulating state is derived from the strong onsite Coulomb repulsion and naturally accounts for the antiferromagnetic (AF) ordered state at half-filling [1]. There is increasing evidence that, due to large quantum fluctuations in spin-1/2 systems, several exotic quantum spin liquid states are likely hidden under the AF Mott insulating ground state. These states are competitive in energy and form a basis for the resonating valence bond route to superconductivity [1]. Several important, yet unresolved problems remain: Is the d-wave superconductor (dSC) the ground state when the longrange AF order is destroyed by the coherent motion of the doped holes? Does the competition between the dSC and other forms of order account for the pseudogap phenomenon in the underdoped regime? Does the physics of doped Mott insulator explain the disappearance of superconductivity in the overdoped regime?
In this paper we demonstrate that long-range Coulomb interactions, not crucial for discussing the Mott insulating state, play an important role in addressing these issues associated with doped Mott insulators. Because charge fluctuations are limited by the density of doped holes, long-range Coulomb interactions have been thought to be not important near the stoichiometric Mott limit. However, since the strength of the interaction is large compared to the Fermi energy, its effects become quickly significant with doping. We consider the microscopic t-J-V model where V stands for the nearestneighbor (NN) Coulomb repulsion. As in the t-Jmodel, the constraint of no double occupancy leads to an insulating state at half-filling. It is well known that the d-wave pairing state and the staggered flux phase (SFP) [3,4,5], identical at half-filling owing to an SU (2) gauge symmetry [6], are two competing phases at low doping. The SFP has a plaquette flux density wave (orbital current) order with d-wave symmetry. It is a metallic state with hole pockets and a pseudogap of the same symmetry as the dSC. Thus the SFP serves as a candidate pseudogap phase for underdoped cuprates, much as in the d-density wave scenario [7]. However, in the standard t-Jmodel, the ground state becomes the dSC upon doping [9]. The SFP was found to be unstable to superconductivity [10,11] and only exists at finite temperatures in a very narrow regime near half-filling [12].
We show that the NN Coulomb interaction introduces competing and coexisting order and changes significantly the structure of the phase diagram. It is crucial to treat the V -term quantum mechanically, on equal footing as the exchange J-term, since both pairing and screening originate at the same energy scale. Motivated by the SU (2) symmetry at half-filling, we construct an SU (N, q) (q =quaternion) generalization of the t-J-V model. While the SU (2N ) theory [13] favors the normal state and the SP (2N ) theory [14] favors the superconducting (SC) state in the large-N limit, the SU (N, q) theory accommodates both staggered flux (SF) and SC order, and allows a systematic study of their interplay. In the large-N limit, we find that the SF order coexists with the dSC below a critical doping x c where a quantum phase transition to a pure dSC phase takes place. The SC phase is destroyed in the overdoped regime beyond x p > x c . At finite temperatures, the onset of the SF order is at T SF > T c , giving rise to a pseudogap SFP above the SC transition temperature T c . We discuss the phase diagram, the signatures of the coexistence and the pseudogap phase.
We start with the large-N generalization of the t-J-V model on a square lattice, H = H tJ + H V , The operator c † iσ creates an electron with spin σ = 1,1; · · · ; N,N at site i under the constraint of no double occupancy for each spin flavor, n i = c † iσ c iσ ≤ N . The NN Coulomb repulsion between the electrons is given by V c , which is grouped together with −(J/4)n i n j from the exchange term and represented by the V -term in Eq. (2). The usual t-J-V model corresponds to N = 1. The projected Hilbert space can be treated in the slave boson formalism by writing c † iσ = f † iσ b i , where f † iσ is a spincarrying fermion and b i a spinless boson keeping track of empty sites, with the constraint Here 1 αβ is the identity matrix in the flavor space and (σ y ) τ τ ′ is the Pauli matrix acting on the spins. Under a unitary transfor- To construct the large-N theory, we form the flavor singlet by tracing over the flavor indices, which is a quaternion with elements: . They represent the bond (SF) and pairing degrees of freedom. In terms of u, the Hamiltonian can be written as, Here tr stands for the trace in spinor space, λ i is the Lagrange multiplier enforcing the local constraints, and µ f fixes the average fermion density. Note that for general N , H V cannot be written solely in terms of four fermions using the constraints. In the large-N limit, the bosons condense at low tem- x i is the local doping concentration [15]. Note that since charge fluctuations are suppressed, the V -term vanishes with vanishing boson amplitude on approaching half-filling where the SU (2) symmetry is restored. The interactions are decoupled by introducing the saddle point values χ ij = χ ij /N and where g χ = J/4 + x i x j V and g ∆ = J/4 − x i x j V are the fermion coupling constants to the bond and paring order parameters respectively. The effect of V is to reduce pairing while enhance the SF ordering. We focus on the uniform solutions with and ∆ s,d are real. The flux per plaquette is given by φ = 4 tan −1 (χ 2 /χ 1 ). ∆ s and ∆ d are the s-wave and d-wave pairing order parameters, respectively. The free energy per site is obtained analytically, where µ = µ f − λ. The k-sum is over the reduced zone due to the doubling of the unit cell, and E ks describe the two branches of the quasiparticle (QP) dispersion with γ ± = cos k x ± cos k y , α k = 2(xt + g χ χ 1 )γ + , and β k = 2g χ χ 2 γ − . Minimizing the free energy in Eq. (8), we obtain the self-consistent equations, In the absence of the NN Coulomb interaction, V = 0 and g χ = g ∆ . The equations for χ 2 and ∆ d imply that they cannot be simultaneously nonzero except at half-filling (µ = 0) [10]. The NN Coulomb interaction V > 0 leads to g χ > g ∆ away from half-filling, making it possible for the coexistence of the two types of order. We solve the self-consistent equations in (9) using J as the energy unit and t = 2J, V = 2J. The s-wave pairing ∆ s is found to be zero. In Fig. 1, ∆ d , χ 1 and χ 2 are plotted at T = 0 as a function of doping x. The results for V = 0 is also plotted for comparison. The NN Coulomb interaction introduces two important features. (i) The dSC order develops upon doping, but is destroyed beyond a critical doping x p ≃ 0.26, which is smaller than the naive estimate from the condition g ∆ = 0. The suppression of pairing by V in the overdoped regime is consistent with recent quantum Monte Carlo studies of the t-U -V model [16]. (ii) The SF order develops below a critical doping x c ≃ 0.1 and coexists with the dSC. There is a reduction of ∆ d concomitant with the development of the SF (χ 2 ) below x c , indicative of competing order. The values of x c and x p depend on the band structure. A next NN hopping t ′ < 0 moves both to higher values. Although the d-wave pairing amplitude ∆ d extrapolates to a nonzero value at half-filling, the two states are identical by the SU (2) gauge symmetry. In Fig. 2, the phase diagram of the t-J-V model is shown in the plane of temperature T versus doping x. The phase boundaries are the temperatures T SF and T c at which the SF and dSC order parameters vanish, respectively. The topology of the phase diagram resembles the one proposed for the cuprate high-T c superconductors by invoking competing order and quantum criticality [7]. The dSC sets in below x p . The transition temperature T c rises with decreasing doping and reaches its optimal value somewhat below the quantum critical point (QCP) at x c where coexisting SF order emerges. The T SF line shows a bulge when crossing the T c line at x ′ c ≃ 0.12, making it possible for a thermal transition from the dSC to the coexistence phase for x c < x < x ′ c . The phase boundary rises almost vertically at low temperatures, and the transition is weakly first order as is evidenced by the small discontinuity in the order parameters (Fig. 1). For x < x ′ c , T c < T SF , giving rise to a pseudogap SFP with orbital currents for T c < T < T SF in the underdoped regime. The phase diagram and the nature of the pseudogap phase are different from those in the SU (2) formulation of the t-Jmodel [8]. In that theory, the SU (2) symmetry at half-filling is extended to finite doping via the introduction of a slave boson doublet. The pseudogap phase is a fluctuating state between gauge equivalent SFP and the dSC. The static SF order develops only inside vortices where SC order is suppressed by the external magnetic field [17]. The present theory explores the consequences of symmetry breaking by doping, through the NN Coulomb interaction in particular, which stabilizes the static order of the SF in the pseudogap phase.
To illustrate the electronic structures, we calculate the tunneling density of states (LDOS) measured by STM In Fig. 3, we show the typical LDOS and the QP dispersion. Fig. 3(a) corresponds to the pure dSC at x = 0.14 and T = 0. The LDOS shows d-wave gap features. The two peaks near the gap edges come from the large state density at the points marked as X at (π, 0) and P at the nearby minimum on the dispersion, and their symmetry equivalent points. Note that the distance between neither set of peaks equals to 2∆ where ∆ = 4g ∆ ∆ d is the maximum of the d-wave gap function. One must thus be careful with extracting the SC gap from the peak-to-peak distance even with high STM resolution. The LDOS in the coexistence phase is shown in Fig. 3(c) at x = 0.06 and T = 0. Compared to Fig. 3(a) in the dSC, there is an emergent in-gap resonance peak (marked A) particlehole reflected on both sides of zero-bias. It arises from the resonant scattering of the QP from state k to state k + (π, π) due to period doubling. A similar situation arises in the SF state around a superconducting vortex in the SU(2) formulation of the t-J model [18]. As shown in Fig. 3(d) for the s = + branch, at the resonant energy, the distorted ellipses touch at point A, the low-energy minimum in the dispersion plotted in Fig. 3(c). These resonance peaks, albeit small in weight, are ubiquitous signatures of coexisting order and should be detectable by high resolution STM. Near the gap edges, there are four sets of peaks (two are prominent) due to the doubling of the unit cell. They are labeled by X ± and P ± and marked on the QP dispersion. It is instructive to compare these to the LDOS in the pseudogap SFP at x = 0.06 and T = 0.08J shown in Fig. 3(b). The low energy resonance peaks disappear and the LDOS shifts to the unoccupied side. Interestingly, the minimum of the LDOS in the SFP coincides with the M point next to the in-gap resonance in the coexistence phase. The resonance energy and the spectral shift depend on the band structure and doping and are on the order of 0.043J ∼ 6meV for the parameters used in Fig. 3.
An important consequence of the competing order is the emergence of two energy gaps which can be, however, difficult to observe at low temperatures [19]. In the inset of Fig. 2, the gap to single-particle excitations ∆ 0 obtained from the LDOS at T = 0 is plotted versus doping, which qualitatively agrees with the low temperature ARPES data. It interpolates between a geometric average of the gaps associated with the SF and SC order for x < x c and the gap of the pure dSC for x > x c and closes at x p . The only sign of competing order in ∆ 0 is the mild cusp-like singularity near x c . Signatures of competing order are more apparent in thermodynamic quantities [7,19,20]. In the same figure, the jump ∆γ c in the specific heat coefficient γ(T ) = C p (T )/T at T c and the condensation energy per site ∆E c at T = 0 are shown as a function of doping. ∆γ c rises with decreasing doping on the overdoped side. With the opening of the pseudogap due to SF order, the entropy in the normal state is reduced, ∆γ c turns around at x ′ c and drops continuously to small values in the underdoped regime, in qualitative agreement with specific heat measurements [19]. The condensation energy ∆E c grows almost linearly with doping in the underdoped regime and peaks around x c . The closeness in the free energy between the SFP and dSC, the smallness of the condensation energy, together with a small superfluid density that scales with x promote the appearance of cheap vortices in the under-doped regime as emphasized in the SU (2) theory [17], except that static orbital currents in the present theory live in the bulk as well as inside the vortex.
In summary, we have presented a microscopic large-N theory for the competition between the exchange and the finite-range Coulomb interactions in doped Mott insulators. The t-J-V model shows competing and coexisting order of SF and dSC which can be viewed as a product of SU (2) symmetry breaking by doping through the NN Coulomb interaction. The phase diagram exhibits a QCP separating the coexistence phase from a pure dSC, a finite temperature pseudogap SFP in the underdoped regime, and complete suppression of superconductivity beyond an overdoping. At very low doping, the phase diagram does not match that of the cuprates. The discrepancy is most obvious in the nonzero T c that extends to infinitesimal doping, whereas the SC state in the cuprates disappears at a finite doping below which a spin glass phase prevails. The lack of destruction of the dSC at low doping is because the renormalized NN Coulomb interaction must scale to zero at half-filling due to the constraint forbidding charge fluctuations. We expect the effects of doping induced inhomogeneity [21] due to poor screening of the dopant ionic potential to become very important in this regime [15]. As a result, the local doping concentration (LDC) fluctuates dramatically and its interplay with AF may lead to the destruction of the SC state and the emergence of spin glass and insulating behaviors. Furthermore, the onset of SF order will fluctuate spatially with the LDC, smearing out the sharp finite temperature transition to the pseudogap SFP.