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arXiv:2112.10965 · Physical review. B./Physical review. B

Fermi arcs and pseudogap phase in a minimal microscopic model of d -wave superconductivity

Dheeraj Kumar Singh, Samrat Kadge, Yunkyu Bang, Pinaki Majumdar

Published Feb 2, 2022• 23 citations
arXivHTMLPDF

Abstract

We conclusively show that a pseudogap state can arise at $T>{T}_{c}$, for reasonable pairing interaction strength, from order parameter fluctuations in a two-dimensional minimal model of $d$-wave superconductivity. The occurrence of the pseudogap requires neither strong correlation nor the presence of competing order. We study a model with attractive nearest-neighbor interaction and establish our result using a combination of a cluster-based Monte Carlo method for the order parameter field and a twisted-boundary scheme to compute the momentum-resolved spectral function. Apart from a dip in the density of states that characterizes the pseudogap, the momentum and frequency resolution on our effective lattice size $\ensuremath{\sim}160\ifmmode\times\else\texttimes\fi{}160$ allows two major conclusions: (i) at $T<{T}_{c}$, despite the presence of thermal phase fluctuations, the superconductor has only nodal Fermi points while all non-nodal points on the normal state Fermi surface show a two-peak spectral function with a dip at $\ensuremath{\omega}=0$, and (ii) for $T>{T}_{c}$, the Fermi points develop into arcs, characterized by a single quasiparticle peak, and the arcs connect up to recover the normal state Fermi surface at a temperature ${T}^{*}>{T}_{c}$. We show the variation of ${T}_{c}$ and ${T}^{*}$ with coupling strength and provide detailed spectral results at a coupling where ${T}^{*}\ensuremath{\sim}1.5{T}_{c}$.

DOI
10.1103/physrevb.105.054501

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We conclusively show that a pseudogap state can arise at T>TcT>{T}_{c}T>Tc​, for reasonable pairing interaction strength, from order parameter fluctuations in a two-dimensional minimal model of ddd-wave superconductivity. The occurrence of the pseudogap requires neither strong correlation nor the presence of competing order. We study a model with attractive nearest-neighbor interaction and establish our result using a combination of a cluster-based Monte Carlo method for the order parameter field and a twisted-boundary scheme to compute the momentum-resolved spectral function. Apart from a dip in the density of states that characterizes the pseudogap, the momentum and frequency resolution on our effective lattice size \ensuremath∼160\ifmmode×\else\texttimes\fi160\ensuremath{\sim}160\ifmmode\times\else\texttimes\fi{}160\ensuremath∼160\ifmmode×\else\texttimes\fi160 allows two major conclusions: (i) at T<TcT<{T}_{c}T<Tc​, despite the presence of thermal phase fluctuations, the superconductor has only nodal Fermi points while all non-nodal points on the normal state Fermi surface show a two-peak spectral function with a dip at \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0, and (ii) for T>TcT>{T}_{c}T>Tc​, the Fermi points develop into arcs, characterized by a single quasiparticle peak, and the arcs connect up to recover the normal state Fermi surface at a temperature T∗>Tc{T}^{*}>{T}_{c}T∗>Tc​. We show the variation of Tc{T}_{c}Tc​ and T∗{T}^{*}T∗ with coupling strength and provide detailed spectral results at a coupling where T∗\ensuremath∼1.5Tc{T}^{*}\ensuremath{\sim}1.5{T}_{c}T∗\ensuremath∼1.5Tc​.
Despite the intense theoretical and experimental effort, an understanding of the superconducting pairing mechanism of the high-temperature superconductors leading to an unprecedented high transition temperature Tc{T}_{c}Tc​ is still lacking. An additional puzzle is the unknown connection between the superconducting gap and the so-called pseudogap which is a central property of the most unusual normal state. Angle-resolved photoemission spectroscopy (ARPES) measurements have revealed a gaplike behavior on parts of the Fermi surface, leaving a nongapped segment known as Fermi arc around the diagonal of the Brillouin zone. Two main interpretations of the origin of the pseudogap have been proposed: either the pseudogap is a precursor to superconductivity, or it arises from another order competing with superconductivity. Starting from the t-J model, in this paper we present a microscopic approach to investigate physical properties of the pseudogap phase as well as the superconducting phase in the framework of a renormalization scheme called projector-based renormalization method. This approach is based on a stepwise elimination of high-energy transitions using unitary transformations. We arrive at renormalized ``free'' Hamiltonians for correlated electrons for both phases. Our microscopic approach allows us to explain the experimental findings in the underdoped as well as in the optimal hole doping regime. The ARPES spectral function along the Fermi surface turns out to be in good agreement with experiment: For the pseudogap phase we find well-defined excitation peaks around $\ensuremath{\omega}=0$ near the nodal direction, which become strongly suppressed around the antinodal point. The origin of the pseudogap can be traced back to a suppression of spectral weight from incoherent excitations in a small $\ensuremath{\omega}$ range around the Fermi energy. In the superconducting phase, the order parameter turns out to have ddd-wave symmetry with a coherence length of a few lattice constants. In good agreement with experiments, we find no superconducting solutions for very small hole doping.
We study the electronic structure within a system of phase-decoupled one-dimensional superconductors coexisting with stripe spin and charge-density-wave order. This system has a nodal Fermi surface (Fermi arc) in the form of a hole pocket and an antinodal pseudogap. The spectral function in the antinodes is approximately particle-hole symmetric contrary to the gapped regions just outside the pocket. We find that states at the Fermi energy are extended whereas states near the pseudogap energy have localization lengths as short as the interstripe spacing. We consider pairing which has either local ddd-wave or sss-wave symmetry and find similar results in both cases, consistent with the pseudogap being an effect of local pair correlations. We suggest that this state is a stripe-ordered caricature of the pseudogap phase in underdoped cuprates with coexisting spin-, charge-, and pair-density-wave correlations. Lastly, we also model a superconducting state which (1) evolves smoothly from the pseudogap state, (2) has a signature subgap peak in the density of states, and (3) has the coherent pair density concentrated to the nodal region.
Despite the intense theoretical and experimental effort, an understanding of the superconducting pairing mechanism of the high-temperature superconductors is still lacking. An additional puzzle is the unknown connection between the superconducting gap and the so-called pseudogap which is a central property of the most unusual normal state. Angle-resolved photoemission spectroscopy (ARPES) measurements have revealed a gap-like behavior on parts of the Fermi surface, leaving a non-gapped segment known as Fermi arc around the diagonal of the Brillouin zone. Starting from the ttt-JJJ model, in this paper we present a microscopic approach to investigate physical properties of the pseudogap phase in the framework of a novel renormalization scheme called PRM. This approach is based on a stepwise elimination of high-energy transitions using unitary transformations. We arrive at a renormalized 'free' Hamiltonian for correlated electrons. The ARPES spectral function along the Fermi surface turns out to be in good agreement with experiment: We find well-defined excitation peaks around ω=0ω=0ω=0 near the nodal direction, which become strongly suppressed around the antinodal point. The origin of the pseudogap can be traced back to a suppression of spectral weight from incoherent excitations in a small ωωω-range around the Fermi energy. In a subsequent paper, also the supercunducting phase at moderate hole doping will be discussed within the PRM approach.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature T∗,{T}^{*},T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around T∗{T}^{*}T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
Pseudogap phenomena and the formation of Fermi arcs in underdoped cuprates are numerically studied in the presence of phase fluctuations that are simulated by an XYXYXY model. Most importantly the spectral function for each Monte Carlo sample is calculated directly and efficiently by the Chebyshev polynomials without having to diagonalize the fermion Hamiltonian, which enables us to handle a system large enough to achieve sufficient momentum/energy resolution. We find that the momentum dependence of the energy gap is identical to that of a pure ddd-wave superconductor well below the Kosterlitz-Thouless (KT) transition temperature (TKT)({T}_{\text{KT}})(TKT​) while displays an upturn deviation from $\text{cos}\text{ }{k}_{x}\ensuremath{-}\text{cos}\text{ }{k}_{y}$ with increasing temperature. An abrupt onset of the Fermi arcs is observed above TKT{T}_{\text{KT}}TKT​ and the arc length exhibits a similar temperature dependence to the thermally activated vortex excitations.
I study the effect of Coulomb interaction on superconducting order in a ddd-wave lattice superconductor at T=0T=0T=0 by considering the superconducting saddle point in the two-dimensional $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model with a repulsion UUU. The theory of low-energy superconducting phase fluctuations around this saddle point is derived in terms of the effective hard-core bosons (representing the density of spin-up electrons and the phase of the order parameter), interacting with the fluctuating density of spin-down electrons. Whereas the saddle-point value of the superconducting gap is found to continuously increase towards half filling, the phase stiffness at T=0T=0T=0 has a maximum, and then decreases with further underdoping. Right at half filling the phase stiffness vanishes for large UUU. This argues that the pseudogap phenomenon of the type observed in cuprates is in principle possible without a development of any competing order, purely as a result of growing correlations in the superconducting state. Implications for the finite temperature superconducting transition and the effects of static disorder are discussed qualitatively.