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arXiv:1904.03658 · Chinese Physics Letters

Superconductivity, Pair Density Wave, and Néel Order in Cuprates*

Li-Han Chen, Da Wang, Yi Zhou, Qiang-Hua Wang

Published Jan 1, 2020• 6 citations
arXivHTMLPDF

Abstract

We investigate in underdoped cuprates possible coexistence of the superconducting order at zero momentum and pair density wave (PDW) at momentum Q = (π, π) in the presence of a Néel order. By symmetry, the d-wave uniform singlet pairing dS 0 can coexist with the d-wave triplet PDW dT Q , and the p-wave singlet PDW p Q can coexist with the p-wave uniform triplet pT 0. At half filling, we find that the novel pS Q + pT 0 state is energetically more favorable than the dS 0 + dT Q state. At finite doping, however, the dS 0 + dT Q state is more favorable. In both types of states, the variational triplet parameters dT Q and pT 0 are of secondary significance. Our results point to a fully symmetric Z2 quantum spin liquid with spinon Fermi surface in proximity to the Néel order at zero doping, which may not be adiabatically connected to the d-wave singlet superconductivity at finite doping with intertwining d-wave triplet PDW fluctuations and spin moment fluctuations. The results are obtained by variational quantum Monte Carlo simulations.

DOI
10.1088/0256-307x/37/1/017403

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We investigate in underdoped cuprates possible coexistence of the superconducting order at zero momentum and pair density wave (PDW) at momentum Q = (π, π) in the presence of a Néel order. By symmetry, the d-wave uniform singlet pairing dS 0 can coexist with the d-wave triplet PDW dT Q , and the p-wave singlet PDW p Q can coexist with the p-wave uniform triplet pT 0. At half filling, we find that the novel pS Q + pT 0 state is energetically more favorable than the dS 0 + dT Q state. At finite doping, however, the dS 0 + dT Q state is more favorable. In both types of states, the variational triplet parameters dT Q and pT 0 are of secondary significance. Our results point to a fully symmetric Z2 quantum spin liquid with spinon Fermi surface in proximity to the Néel order at zero doping, which may not be adiabatically connected to the d-wave singlet superconductivity at finite doping with intertwining d-wave triplet PDW fluctuations and spin moment fluctuations. The results are obtained by variational quantum Monte Carlo simulations.
Abstract Over the last two decades, the competing orders in high- T c cuprates have been intensely studied, such as charge density waves (CDW) and pair density waves (PDW), which are thought to play a crucial role in high-temperature superconductivity. Using the t - J model on a square lattice as the simplest model for high- T c cuprates, we employed the fermionic tensor product state (fTPS) method for numerical investigations. Here we show, to the best of our knowledge, several types of PDW states within the low-energy subspace of the t - J model are unveiled thanks to the fermionic cluster update method, alongside the well-known d -wave state. In particular, a kind of C 4 symmetric plaquette PDW state is most likely to survive at low dopings δ < 5%. We believe that the competition among these states in the underdoped region suggests the potential existence of a fluctuating quantum liquid of PDW states, which could serve as a natural starting point for exploring the pseudogap phase and understanding the key mechanisms of d-wave superconductivity. Furthermore, we discuss the potential experimental implications of our discovery.
Abstract Flat-band systems provide an ideal platform for studying strong electron correlations, in which multiple quasi-degenerate charge orders frequently coexist. Motivated by this context, we investigate the pair-density-wave (PDW) superconducting state in a quasi-one-dimensional Creutz lattice featuring two strictly flat bands. Using a renormalized mean-field approach for the strongly correlated t - J model, we obtain self-consistent solutions that demonstrate a PDW phase—characterized by finite-center-of-mass-momentum Cooper pairing—coexisting with charge density wave (CDW) and bond density wave (BDW) orders. Physically, this coexistence phenomenon originates from the perfect band nesting inherent to the flat-band system, which allows for changes in the particle momentum distributions without energy cost. Here, a pure PDW solution without CDW and BDW orders emerges in the underdoped regime when magnetic effects are neglected, while strong Hubbard correlations favor a pure CDW phase. We conclude that a robust PDW phase, accompanied by CDW and BDW orders, occupies the majority of the parameter space. Furthermore, both the PDW order parameter and its critical temperature scale linearly with the superexchange coupling strength—a hallmark of flat-band superconductivity. Overall, our results reveal that a PDW superconducting state can be realized on the Creutz lattice with strong correlations.
We show that the pair-density-wave (PDW) superconducting state emergent in extended Heisenberg-Hubbard models in two-leg ladders is topological in the presence of an Ising spin symmetry and supports a Majorana zero mode (MZM) at an open boundary and at a junction with a uniform d-wave one-dimensional superconductor. Similarly to a conventional finite-momentum paired state, the order parameter of the PDW state is a charge-2e field with finite momentum. However, the order parameter here is a quartic electron operator and conventional mean-field theory cannot be applied to study this state. We use bosonization to show that the 1D PDW state has a MZM at a boundary. This superconducting state is an exotic topological phase supporting Majorana fermions with finite-momentum pairing fields and charge-4e superconductivity.
Pair density waves (PDWs) are superconducting states formed by Cooper pairs of electrons containing a nonzero center-of-mass momentum. They are characterized by a spatially modulated order parameter and may occur in a variety of emerging quantum materials such as cuprates, transition-metal dichalcogenides (TMDs), and Kagome metals. Despite extensive theoretical and numerical studies seeking PDWs in a variety of lattices and interacting settings, there is currently no exact mechanism that spontaneously favors a modulated solution of the superconducting order parameter. Here, we study the problem of two electrons subject to an anisotropic attractive potential. We solve the two-body Schr\"odinger wave equation exactly to determine the pair binding energy as a function of the center-of-mass momentum. We find that a modulated (finite momentum) pair is favored over a homogeneous (zero momentum) solution above a critical, intermediate interaction strength. Hence our exact result justifies previous mean-field approximations that obtain modulated ground states at finite but large interactions. Using this insight from the exact two-body solution, we construct a variational many-body wave function and show that the conclusions of the two-body problem are robust in the many-body limit. Our results thus lay the theoretical and microscopic foundation for the existence of PDWs.
In light of the new experimental and theoretical important developments in high-Tc{T}_{c}Tc​ superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a ddd-wave form factor with the experimentally observed modulation $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the ``hidden'' order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, ddd-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, and ddd-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.
A robust theory of the mechanism of pair density wave (PDW) superconductivity (i.e., where Cooper pairs have nonzero center of mass momentum) remains elusive. Here we explore the triangular lattice $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model, a low-energy effective theory derived from the strong-coupling limit of the Holstein-Hubbard model, by large-scale variational Monte Carlo simulations. When the electron density is sufficiently low, the favored ground state is an s-wave PDW, consistent with results obtained from previous studies in this limit. Additionally, a PDW ground state with nematic d-wave pairing emerges in the intermediate range of electron densities and phonon frequencies. For these s-wave and d-wave PDWs arising in states with spontaneous breaking of time-reversal and inversion symmetries, PDW formation derives from valley-polarization and intrapocket pairing.