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arXiv:1503.00379 · Physical Review B

Renormalization group analysis of the pair-density-wave and charge order within the fermionic hot-spot model for cuprate superconductors

Hermann Freire, Vanuildo S. de Carvalho, C. Pépin

Published Jul 31, 2015• 45 citations
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Abstract

In light of the new experimental and theoretical important developments in high-${T}_{c}$ 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 $d$-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, $d$-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 $d$-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.

DOI
10.1103/physrevb.92.045132

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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±\else\textpm\fiQ0,0)(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)(\ifmmode±\else\textpm\fiQ0​,0) and (0,\ifmmode±\else\textpm\fiQ0)(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})(0,\ifmmode±\else\textpm\fiQ0​) 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±\else\textpm\fiQ0,\ifmmode±\else\textpm\fiQ0)(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})(\ifmmode±\else\textpm\fiQ0​,\ifmmode±\else\textpm\fiQ0​), 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.
It is now well established that superconducting cuprates support a charge-density-wave state in the so-called underdoped region of their phase diagram. We investigate the possibility of charge order in the square-lattice Hubbard model, both alone and in coexistence with ddd-wave superconductivity. The charge order has a period of 4 in one direction, is centered on bonds, and has a ddd form factor. We use the variational cluster approximation, an approach based on a rigorous variational principle that treats short-range correlations exactly, with two clusters of size $2\ifmmode\times\else\texttimes\fi{}6$ that together tile the infinite lattice and provide a nonbiased unit for a period-4 bond-density wave (BDW). We find that the BDW exists in a finite range of hole doping and increases in strength from U=5U=5U=5 to U=8U=8U=8. Its location and intensity depend strongly on the band dispersion. When probed simultaneously with ddd-wave superconductivity, the energy is sometimes lowered by the presence of both phases, depending on the interaction strength. Whenever they coexist, a pair-density wave (a modulation of superconducting pairing with the same period and form factor as the BDW) also exists.
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.
We present a renormalization group (RG) analysis of a fermionic ``hot-spot'' model of interacting electrons on the square lattice. We truncate the Fermi-surface excitations to linearly dispersing quasiparticles in the vicinity of eight hot spots on the Fermi surface, with each hot spot separated from another by the wave vector $(\ensuremath{\pi},\ensuremath{\pi})$. This is motivated by the importance of these Fermi-surface locations to the onset of antiferromagnetic order; however, we allow for all possible quartic interactions between the fermions, and also for all possible ordering instabilities. We compute the RG equations for our model, which depend on whether or not the hot spots are perfectly nested, and relate our results to earlier models. We also compute the RG flow of the relevant order parameters for both Hubbard and J,VJ,VJ,V interactions, and present our results for the dominant instabilities in the nested and non-nested cases. In particular, we find that non-nested hot spots with J,VJ,VJ,V interactions have competing singlet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting and ddd-form factor incommensurate density wave instabilities. We also investigate the enhancement of incommensurate density waves near experimentally observed wave vectors, and find dominant ddd-form factor enhancement for a range of couplings.
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 study the coexistence of pair- (PDW) and charge-density-wave (CDW) states within the single-band $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ and Hubbard models of ddd-wave superconductivity and discuss our results in the context of the experimental observations for the copper-based compounds. In order to take the correlation effects into account with proper precision, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which goes beyond the renormalized mean-field theory in a systematic manner. According to our analysis of the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model, the transition between the pure ddd-wave superconducting phase and the coexistent CDW+PDW phase takes place at $\ensuremath{\delta}\ensuremath{\approx}0.18$ (close to the optimal doping), with the modulated phase located in the underdoped regime. The situation is different for the case of the Hubbard model, where a narrow stability regime of a precursor nematic phase sets in preceding the formation of the modulated CDW+PDW state, with decreasing hole doping. The results conclude our comprehensive discussion of the standard phase diagram for high-TC{T}_{C}TC​ superconducting compounds within the DE-GWF variational approach in the single narrow-band case [see J. Spa\l{}ek et al., Phys. Rev. B 95, 024506 (2017); M. Zegrodnik and Spa\l{}ek, Phys. Rev. B 95, 024507 (2017); Phys. Rev. B 96, 054511 (2017); New J. Phys. 20, 063015 (2018)].
We study the charge-density-wave order and its competition with superconductivity in the one-band Hubbard model for high-Tc{T}_{c}Tc​ superconducting cuprates. We use cluster dynamical mean field theory at T=0T=0T=0. The one-band Hubbard model contains only copper atoms, and a physical charge-density wave with charge excesses located on the oxygen atoms manifests itself as a bond-density wave (BDW) within this context. It arises purely out of local correlation effects and also leads to an ${s}^{\ensuremath{'}}$-wave pair-density wave in the presence of ddd-wave superconductivity. The ddd-wave BDW is suppressed on increasing UUU and is favored on increasing the magnitude of the second-neighbor hopping ${t}^{\ensuremath{'}}$. Further, the ddd-wave BDW order is weakened when in competition with superconductivity, as seen in experiments. Additionally, it behaves quite differently on varying UUU in the superconducting state compared to the normal state.