Charge, bond, and pair density wave orders in a strongly correlated system

Anurag Banerjee, Catherine Pépin, and Amit Ghosal

Phys. Rev. B 105, 134505 – Published 6 April 2022

Abstract

The coexistence of multiple quasidegenerate orders is the hallmark of the strongly correlated materials. Experiments often reveal several spatially modulated orders in the underdoped cuprates. This has come to the forefront with the possible detection of the pair density wave states. However, microscopic calculations often struggle to stabilize such spatially modulating orders as the ground state in the strong correlation limit. This work uses the $t - t^{'} - J$ model with an additional nearest-neighbor repulsion to stabilize spatially oscillating charge, bond, and pairing orders in the underdoped regime. We employ the standard Gutzwiller approach while treating the inhomogeneity for the spatial orders using the self-consistent Hartree-Fock-Bogoliubov methodology. Our calculations reveal that unidirectional bond density states coexisting with charge and pairing modulations can have lower energy than the uniform superconducting state over an extensive doping range. These modulating states vanish monotonically as the modulation wave vector becomes shorter with increased dopings. The finite momentum orders give way to a vestigial nematic phase on increasing doping which only breaks the rotational symmetry of the system. The nematic order vanishes on further increasing doping, and only uniform superconductivity survives. The spatial features of the ground state at each doping reveal multiple wave vectors, which potentially drive the incommensuration of charge orders. Interestingly, the spatially modulating states are absent when the strong correlations criteria are relaxed, suggesting that the removal of double occupancy aids the stabilization of density wave orders.

Received 25 October 2021
Revised 19 February 2022
Accepted 22 March 2022

DOI:https://doi.org/10.1103/PhysRevB.105.134505

Physics Subject Headings (PhySH)

Charge density waves Nematic order Superconductivity d-wave

Strongly correlated systems

Gutzwiller approximation t-J model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Anurag Banerjee ¹, Catherine Pépin¹, and Amit Ghosal²

¹Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France
²Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India

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Vol. 105, Iss. 13 — 1 April 2022

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Images

Figure 1
The figure compares the ground-state energy of the self-consistent solutions with different wave vectors for a few dopings. The thick lines are the guide to the eyes. The dotted lines display the $E_{GS}$ for the uniform and the nematic states. (a) For $δ = 0.075$ there is clearly a minima for $λ = 4$ . As the doping rises to $δ = 0.1$ , the optimal modulation wavelength increases to $λ = 5$ . The wavelength keeps increasing further with reduced doping, becoming $λ = 8$ for $δ = 0.125$ . At $δ = 0.14$ , the nematic state becomes the optimal ground state. The inset of (f) demonstrates the decrease of optimal wave vector with doping.
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Figure 2
The evolution of different order parameters with doping. (a) The evolution of the amplitude of oscillations for the different quantities at the optimal ground state. The blue lines exhibit the oscillation amplitude for the local density at the optimal wave vector. Similarly, the orange and green curves show the evolution of finite-momentum amplitude for extended $s$ -wave and $d$ -wave bond density. Moreover, the same for the superconducting pairing in extended $s$ -wave and $d$ -wave channels is presented in red and violet traces. The lines are the guide to the eyes. All these spatially modulated orders vanish near the $δ \approx 0.14$ . (b) The mean $d$ -wave superconducting pairing in the blue trace compared with the case where no charge or bond density orders are allowed $Δ_{d}^{0}$ in cyan. The average $Δ_{d}$ significantly reduces in the regions where the charge and bond orders are stable. Moreover, we show the renormalized $d$ -wave pairing ${\tilde{Δ}}_{d}$ , which also reduces in the regime where charge and pairing oscillations are dominant, i.e., $δ < 0.16$ . (c) The nematicity in the $χ$ and $Δ$ with red and black traces, respectively. The nematicity vanishes around the doping $δ = 0.16$ .
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Figure 3
Spatial profile for the optimal self-consistent solution at $δ = 0.075$ . Here the periodicity of unidirectional oscillation is $4 a_{0}$ . (a) The bond density $χ_{i j} = \sum_{σ} 〈 c_{i σ}^{†} c_{j σ} 〉$ , where $i, j$ are nearest neighboring sites. (b) The $d$ -wave and extended $s$ -wave components of the $χ$ are extracted in (b). Both clearly show pure oscillations with the periodicity of four lattice spacings. The inset of (c) shows the spatial profile for the local density, which also oscillates. (c) The Fourier transform of these quantities. The dominant wave vector for all the oscillations is at $Q = (1 / 4) 2 π / a_{0}$ as expected. (d) The similar bond plots for local SC pairing $Δ_{i j}$ . (e) The corresponding $d$ -wave and the extended $s$ -wave components of SC pairing. (f) The Fourier transform for the oscillating part of $Δ_{d}$ and $Δ_{s}$ . The pairing modulations also manifest a dominant peak at $Q = (1 / 4) 2 π / a_{0}$ at this doping.
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Figure 4
Spatial profile for the optimal self-consistent solution at $δ = 0.125$ . Here the periodicity of unidirectional oscillation is $8 a_{0}$ . (a) The bond density $χ_{i j}$ , where $i, j$ are nearest-neighboring sites. (b) The $d$ -wave and extended $s$ -wave components of the $χ$ are extracted in (b). Both show some jerky oscillations with the periodicity of eight lattice spacings. The inset of (c) shows the spatial profile for the local density, which oscillates. (c) The Fourier transform of the quantities mentioned above the dominant wave vector for $χ_{d}$ and $χ_{ρ}$ are at $Q = (1 / 8) 2 π / a_{0}$ , whereas the same for $χ_{s}$ is at $Q = (1 / 4) 2 π / a_{0}$ . However, several peaks at the multiples of the ordering wave vector imply that density modulation is not regular over a period at this doping. (d) The similar bond plots for local SC pairing $Δ_{i j}$ . (e) The corresponding $d$ -wave and the extended $s$ -wave components of SC pairing. (f) The Fourier transform for the oscillating part of $Δ_{d}$ and $Δ_{s}$ . The pairing modulations also exhibit a dominant peak at $Q = (1 / 8) 2 π / a_{0}$ at these dopings.
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Figure 5
The spatial dependence for the optimal self-consistent solution at $δ = 0.14$ . Although the optimal self-consistent solution has no modulation in this parameter regime, it breaks the $C_{4}$ -rotational symmetry of the square lattice leading to a charge nematic state. (a) The bond density $χ_{i j}$ , which is different for the x bonds and y bonds. Similarly, the local SC pairing is not purely $d$ wave as shown in (b) as expected in the uniform $t J$ model. Interestingly, there is no spatial variation of these quantities or the local density as presented in (c).
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Figure 6
Schematic symmetry broken phases with doping from our model calculations at zero temperature. The doping regime presented below the broken line is beyond the scope of this study. Below this doping we expect magnetic and other symmetries to be broken. From the low to intermediate dopings, both the translation symmetry, $T$ and the $C_{4}$ rotational symmetry are broken, leading to the observation of the bond, charge, and PDW orders. Beyond a doping level, only $C_{4}$ rotational symmetry is broken, leading to a nematic state. Note that fewer symmetries are broken as the doping is increased. Beyond critical doping, only translational and rotational invariant $d$ -wave SC remains robust. Uniform superconductors only break the particle number conservation symmetry. The uniform $d$ -wave SC remains robust over the whole doping regime studied, only decaying at much higher doping.
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