Hybridization of Higgs modes in a bond-density-wave state in cuprates

Zachary M. Raines, Valentin G. Stanev, and Victor M. Galitski

Phys. Rev. B 92, 184511 – Published 24 November 2015

Abstract

Recently, several groups have reported observations of collective modes of the charge order present in underdoped cuprates. Motivated by these experiments, we study theoretically the oscillations of the order parameters, both in the case of pure charge order and for charge order coexisting with superconductivity. Using a hot-spot approximation we find in the coexistence regime two Higgs modes arising from hybridization of the amplitude oscillations of the different order parameters. One of them has a minimum frequency that is within the single-particle energy gap and which is a nonmonotonic function of temperature. The other—high-frequency—mode is smoothly connected to the Higgs mode in the single-order-parameter region, but quickly becomes overdamped in the case of coexistence. We explore an unusual low-energy damping channel for the collective modes, which relies on the band reconstruction caused by the coexistence of the two orders. For completeness, we also consider the damping of the collective modes originating from the nodal quasiparticles. At the end we discuss some experimental consequences of our results.

Received 24 August 2015
Revised 16 October 2015

DOI:https://doi.org/10.1103/PhysRevB.92.184511

Authors & Affiliations

Zachary M. Raines and Valentin G. Stanev

Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

Victor M. Galitski

Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA and School of Physics, Monash University, Melbourne, Victoria 3800, Australia

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Vol. 92, Iss. 18 — 1 November 2015

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Images

Figure 1
Schematic phase diagram of the hot-spot model [22, 26], which illustrates the transition from superconductivity to charge order, tuned by $V$ (nearest-neighbor Coulomb interaction). In this work we consider the transition from BDW to BDW-superconducting mixed state along the “trajectories” indicated by the dashed lines. Depending on the exact value of $V$ the $T \to 0$ limit of the system could be either in a pure superconducting state (indicated by the blue dashed line) or in a mixed state (green dot-dashed line).
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Figure 2
Mass of the in-gap hybrid Higgs mode $ω_{0} = Re [ω (q \to 0)]$ as obtained from Eq. (11). The frequency is plotted as a function of temperature for two different cases of $ϕ (T \to 0)$ ( $V = 0.2, 0.21$ ) as depicted by the dashed lines in Fig. 1, using the units of Ref. [22]. A soft mixed mode emerges in both cases below the superconducting $T_{c}$ . For reference, twice the single-particle energy gap, which is determined by $2 min (Δ, ϕ)$ , is plotted in the black dashed line. In proximity of a phase transition, the in-gap mode approaches the $2 Δ / 2 ϕ$ Higgs mode of the vanishing order.
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Figure 3
Damping rate $Γ_{0}$ of the in-gap collective mode in the long wavelength limit for two different values of $V$ corresponding to the two trajectories depicted in Fig. 1. Damping is an order of magnitude smaller in the case where $ϕ (T \to 0) \neq 0$ (lower line) and is exponentially suppressed at low temperature due a lack of thermally excited quasiparticles. In both cases, the decay rate is strongly suppressed in the vicinity of the superconducting $T_{c}$ . The transition temperatures are marked for $ϕ (T \to 0) = 0 [ϕ (T \to 0) \neq 0]$ by the dashed (dot-dashed) vertical lines. $T_{c}$ denotes the onset temperature of superconductivity, while $T_{<}$ indicates the boundary between the coexistent and pure superconductivity phases for the case of the blue curve (cf. the upper plot of Fig. 2).
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Figure 4
Slices of the quasiparticle band structure for $k_{y} = 0$ . Dashed lines indicate the particle hole conjugate of the band of the same color. (a) The normal state (gray) and BDW state (green/blue) dispersions. Only transitions between the two solid and dashed bands contribute to $Im Q_{ϕ ϕ}^{R}$ . The onset of superconductivity hybridizes the green and blue bands with their particle-hole conjugates. (b) Bogoliubov band dispersions in the coexistent state. Transitions between all bands may contribute to $Im Q_{ϕ ϕ}^{R}$ leading to damping of the Higgs modes (even for those with mass less than $2 Δ$ ). Processes indicated by horizontal arrows are of particular importance as they occupy a finite phase space for arbitrary frequencies within the gap.
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Figure 5
The evolution of $ω_{0}$ (top panel) and $Γ_{0}$ (bottom panel) of the two Higgs modes with temperature. For each mode the cases of weak and strong damping from the nodal regions are shown $[γ_{Δ} = 0.1 α$ (dashed line) and $γ_{Δ} = 0.6 α$ (solid line), respectively]. $γ_{ϕ}$ , the damping from antinodal region, is the same on both plots.
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Figure 6
The in-gap collective mode mass $ω_{0}$ (relative to the $g_{ep} = 0$ case) as a function of the electron phonon coupling $g_{ep}$ , for $Ω_{Q} = 1 .$ The $g_{ep} = 0$ line corresponds to the behavior shown in Fig. 2.
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