Charge Density Wave Instability Research Articles

Strange metal to insulator transitions in the lowest Landau level

We study the microscopic model of electrons in the partially-filled lowest Landau level interacting via the Coulomb potential by the diagrammatic theory within the GW approximation. In a wide range of filling fractions and temperatures, we find a homogeneous non-Fermi liquid (nFL) state similar to that found in the Sachdev-Ye-Kitaev (SYK) model, with logarithmic corrections to the anomalous dimension. In addition, the phase diagram is qualitatively similar to that of SYK: a first-order transition terminating at a critical end-point separates the nFL phase from a band insulator that corresponds to the fully filled Landau level. This critical point, as well as that of the SYK model—whose critical exponents we determine more precisely—are shown to both belong to the Van der Waals universality class. The possibility of a charge density wave (CDW) instability is also investigated, and we find the homogeneous nFL state to extend down to the ground state for fillings 0.2≲ν≲0.8, while a CDW appears outside this range of fillings at sufficiently low temperatures. Our results suggest that the SYK-like nFL state should be a generic feature of the partially filled lowest Landau level at intermediate temperatures. Published by the American Physical Society 2024

Charge density wave ordering in NdNiO2: effects of multiorbital nonlocal correlations

In this work, we investigate collective electronic fluctuations and, in particular, the possibility of the charge density wave ordering in an infinite-layer NdNiO2. We perform advanced many-body calculations for the ab-initio three-orbital model by taking into account local correlation effects, nonlocal charge and magnetic fluctuations, and the electron-phonon coupling. We find that in the considered material, electronic correlations are strongly orbital- and momentum-dependent. Notably, the charge density wave and magnetic instabilities originate from distinct orbitals. In particular, we show that the correlation effects lead to the momentum-dependent hybridization between different orbitals, resulting in the splitting and shifting of the flat part of the Ni-dz2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${d}_{{z}^{2}}$$\\end{document} band. This strong renormalization of the electronic spectral function drives the charge density wave instability that is related to the intraband Ni-dz2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${d}_{{z}^{2}}$$\\end{document} correlations. Instead, the magnetic instability stems from the Ni-dx2−y2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${d}_{{x}^{2}-{y}^{2}}$$\\end{document} orbital, which remains half-filled through the redistribution of the electronic density between different bands even upon hole doping. Consequently, the strength of the magnetic fluctuations remains nearly unchanged for the considered doping levels. We argue that this renormalization is not inherent to the stoichiometric case but can be induced by hole doping.

Coherent phonon and unconventional carriers in the magnetic kagome metal Fe3Sn2

Temperature- and fluence-dependent carrier dynamics of the magnetic kagome metal Fe3Sn2 were studied using the ultrafast optical pump-probe technique. Two carrier relaxation processes and a laser-induced coherent optical phonon were observed. We ascribe the shorter relaxation (~1 ps) to hot electrons transferring their energy to the crystal lattice via electron–phonon scattering. The second relaxation (~30 ps), on the other hand, cannot be explained as a conventional process, and we attributed it to the unconventional (localized) carriers in the material. The observed coherent oscillation is assigned to be a totally symmetric A1g optical phonon dominated by Sn displacements out of the kagome planes and possesses a prominently large amplitude, on the order of 10−3, comparable to the maximum of the reflectivity change (ΔR/R). This amplitude is similar to what has been observed for coherent phonons in charge-density-wave (CDW) systems, although no signs of such instability were hitherto reported in Fe3Sn2. Our results suggest an unexpected connection between Fe3Sn2 and kagome metals with CDW instabilities and a strong interplay between phonon and electron dynamics in this compound.

Atomistic Origin of Diverse Charge Density Wave States in CsV_{3}Sb_{5}.

Kagome metals AV_{3}Sb_{5} (A=K, Rb, or Cs) exhibit intriguing charge density wave (CDW) instabilities, which interplay with superconductivity and band topology. However, despite firm observations, the atomistic origins of the CDW phases, as well as hidden instabilities, remain elusive. Here, we adopt our newly developed symmetry-adapted cluster expansion method to construct a first-principles-based effective Hamiltonian of CsV_{3}Sb_{5}, which not only reproduces the established inverse star of David (ISD) phase, but also predict a series of D_{3h}-n states under mild tensile strains. With such atomistic Hamiltonians, the microscopic origins of different CDW states are revealed as the competition of the second-nearest neighbor V-V pairs versus the first-nearest neighbor V-V and V-Sb couplings. Interestingly, the effective Hamiltonians also reveal the existence of ionic Dzyaloshinskii-Moriya interaction in the high-symmetry phase of CsV_{3}Sb_{5} and drives the formation of noncollinear CDW patterns. Our work thus not only deepens the understanding of the CDW formation in AV_{3}Sb_{5}, but also demonstrates that the effective Hamiltonian is a suitable approach for investigating CDW mechanisms, which can be extended to various CDW systems.

Controlling the Charge Density Wave Transition in Single-Layer TiTe2xSe2(1-x) Alloys by Band Gap Engineering.

Closing the band gap of a semiconductor into a semimetallic state gives a powerful potential route to tune the electronic energy gains that drive collective phases like charge density waves (CDWs) and excitonic insulator states. We explore this approach for the controversial CDW material monolayer (ML) TiSe2 by engineering its narrow band gap to the semimetallic limit of ML-TiTe2. Using molecular beam epitaxy, we demonstrate the growth of ML-TiTe2xSe2(1-x) alloys across the entire compositional range and unveil how the (2 × 2) CDW instability evolves through the normal state semiconductor-semimetal transition via in situ angle-resolved photoemission spectroscopy. Through model electronic structure calculations, we identify how this tunes the relative strength of excitonic and Peierls-like coupling, demonstrating band gap engineering as a powerful method for controlling the microscopic mechanisms underpinning the formation of collective states in two-dimensional materials.

Competing charge-density wave instabilities in the kagome metal ScV6Sn6

Owing to its unique geometry, the kagome lattice hosts various many-body quantum states including frustrated magnetism, superconductivity, and charge-density waves (CDWs). In this work, using inelastic X-ray scattering, we discover a dynamic short-range sqrt{3}times sqrt{3}times 2 CDW that is dominant in the kagome metal ScV6Sn6 above TCDW ≈ 91 K, competing with the sqrt{3}times sqrt{3}times 3 CDW that orders below TCDW. The competing CDW instabilities lead to an unusual CDW formation process, with the most pronounced phonon softening and the static CDW occurring at different wavevectors. First-principles calculations indicate that the sqrt{3}times sqrt{3}times 2 CDW is energetically favored, while a wavevector-dependent electron-phonon coupling (EPC) promotes the sqrt{3}times sqrt{3}times 3 CDW as the ground state, and leads to enhanced electron scattering above TCDW. These findings underscore EPC-driven correlated many-body physics in ScV6Sn6 and motivate studies of emergent quantum phases in the strong EPC regime.

Charge stripes in the graphene-based materials

We present an analytical model of the charge density wave instability in graphene sheets within the intercalated graphite CaC6 compound. The instability yields the experimentally observed uniaxial charge stripes of periodically modulated electron density, coupled to the softest phonon mode of the superlattice consisting of the Ca atoms intercalated between graphene planes. The Fermi surface of the chemically doped graphene undergoes the novel type of instability driven by the mechanism that gains the condensation energy of the stripe state by the topological reconstruction of the Fermi surface. This mechanism appears to be entirely different from the one based on the Fermi surface nesting, which has been considered a paradigm in the present literature concerning the onset of charge density waves.

Doping evolution of superconductivity, charge order, and band topology in hole-doped topological kagome superconductors Cs(V1−xTix)3Sb5

The newly discovered kagome superconductors $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ ($A=\mathrm{K}$, Rb, Cs) exhibit superconductivity, charge order, and band topology simultaneously. To explore the intricate interplay between the superconducting and charge-density-wave (CDW) orders, we investigate the doping evolution of underlying electronic structures in doped topological kagome superconductors $\mathrm{Cs}{({\mathrm{V}}_{1\ensuremath{-}x}{\mathrm{Ti}}_{x})}_{3}{\mathrm{Sb}}_{5}$ where the Ti dopant introduces hole-like charge carriers. Despite the absence of the CDW phase transition in doped compounds even for the lowest doping level of $x=0.047$, the superconductivity survives in all doped samples with enhanced critical temperatures. The high-resolution angle-resolved photoemission spectroscopy (ARPES) measurements reveal that the Ti dopant in the kagome plane lowers the chemical potential, pushing the van Hove singularity (VHS) at $M$ point above the Fermi level. First-principle simulations corroborate the doping evolution of the band structure observed in ARPES, and affirm that the CDW instability does not occur once the VHS moves above the Fermi level, explaining the absence of the CDW ordering in our doped samples $\mathrm{Cs}{({\mathrm{V}}_{1\ensuremath{-}x}{\mathrm{Ti}}_{x})}_{3}{\mathrm{Sb}}_{5}$. Our results also demonstrate a competition between the CDW and superconducting orders in the kagome-metal superconductor ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$, although the superconductivity is likely inconsequential of the CDW order.

Impact of Sb degrees of freedom on the charge density wave phase diagram of the kagome metal CsV3Sb5

Elucidating the microscopic mechanisms responsible for the charge density wave (CDW) instability of the AV$_3$Sb$_5$ (A=Cs, K, Rb) family of kagome metals is critical for understanding their unique properties, including superconductivity. In these compounds, distinct CDW phases with wave-vectors at the $M$ and $L$ points are energetically favorable, opening the possibility of tuning the type of CDW order by appropriate external parameters. Here, we shed light on the CDW landscape of CsV$_3$Sb$_5$ via a combination of first-principles calculations and phenomenology, which consists of extracting the coefficients of the CDW Landau free-energy expansion from density functional theory. We find that while the main structural distortions of the kagome lattice in the staggered tri-hexagonal CDW phase are along the nearest-neighbor V-V bonds, distortions associated with the Sb ions play a defining role in the energy gain in this and all other CDW states. Moreover, the coupling between ionic displacements from different unit cells is small, thus explaining the existence of multiple CDW instabilities with different modulations along the c-axis. We also investigate how pressure and temperature impact the CDW phase of CsV$_3$Sb$_5$. Increasing pressure does not change the staggered tri-hexagonal CDW ground state, even though the $M$-point CDW instability disappears before the $L$-point one, a behavior that we attribute to the large nonlinear coupling between the order parameters. Upon changing the temperature, we find a narrow regime in which another transition can take place, toward a tri-hexagonal Star-of-David CDW phase. We discuss the implications of our results by comparing them with experiments on this compound.

Electrical and thermal transport properties of the kagome metals ATi3Bi5(A=Rb,Cs)

We report electrical and thermal transport properties of single-crystalline kagome metals $A{\text{Ti}}_{3}{\text{Bi}}_{5}\phantom{\rule{4pt}{0ex}}(A=\text{Rb},\phantom{\rule{4pt}{0ex}}\mathrm{Cs})$. Different from the structrually similar kagome superconductors $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$, no charge density wave instabilities are found in $A{\mathrm{Ti}}_{3}{\mathrm{Bi}}_{5}$. At low temperatures below 5 K, signatures of superconductivity appear in $A{\mathrm{Ti}}_{3}{\mathrm{Bi}}_{5}$ as seen in magnetization measurements. However, bulk superconductivity is not evidenced by specific heat results. Similar to $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}, A{\mathrm{Ti}}_{3}{\mathrm{Bi}}_{5}$ show a nonlinear magnetic field dependence of the Hall effect below about 70 K, pointing to a multiband nature. Unlike $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ in which phonons and electron-phonon coupling play important roles in thermal transport, the thermal conductivity in $A{\mathrm{Ti}}_{3}{\mathrm{Bi}}_{5}$ is dominated by electronic contributions. Moreover, our calculated electronic structures of ${\mathrm{ATi}}_{3}{\mathrm{Bi}}_{5}$ suggest that van Hove singularities are sitting well above the Fermi energy. Compared with $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$, the absence of charge orders in $A{\mathrm{Ti}}_{3}{\mathrm{Bi}}_{5}$ is closely associated with minor contributions from electron-phonon coupling and/or van Hove singularities.

Possible enhancement of the superconducting due to sharp Kohn-like soft phonon anomalies

Phonon softening is a ubiquitous phenomenon in condensed matter systems which is often associated with charge density wave (CDW) instabilities and anharmonicity. The interplay between phonon softening, CDW and superconductivity is a topic of intense debate. In this work, the effects of anomalous soft phonon instabilities on superconductivity are studied based on a recently developed theoretical framework that accounts for phonon damping and softening within the Migdal–Eliashberg theory. Model calculations show that the phonon softening in the form of a sharp dip in the phonon dispersion relation, either acoustic or optical (including the case of Kohn-type anomalies typically associated with CDW), can cause a manifold increase of the electron–phonon coupling constant λ. This, under certain conditions, which are consistent with the concept of optimal frequency introduced by Bergmann and Rainer, can produce a large increase of the superconducting transition temperature . In summary, our results suggest the possibility of reaching high-temperature superconductivity by exploiting soft phonon anomalies restricted in momentum space.

Interplay of charge ordering and superconductivity in two-dimensional 2H group V transition-metal dichalcogenides

Layered $2H\text{\ensuremath{-}}T{X}_{2}$ with $T=\mathrm{Nb}$, Ta and $X=\mathrm{S}$, Se exhibit rich dimensionality effects on charge density wave (CDW) order and superconductivity, but comprehensive understanding of these correlated quantum phases in the two-dimensional limit of $2H\text{\ensuremath{-}}T{X}_{2}$ is still lacking, hindering their practical applications. Here, I calculate from first principles the phonon linewidth and bare susceptibility to study the origin of CDW formation in ${\mathrm{NbSe}}_{2}$, ${\mathrm{TaS}}_{2}$, ${\mathrm{TaSe}}_{2}$, and ${\mathrm{NbS}}_{2}$ monolayers, analyze their relative CDW strength, and evaluate electron-phonon superconductivity within the fully anisotropic Migdal-Eliashberg theory. A peaked linewidth of the longitudinal acoustic branch and no Fermi-surface nesting around ${\mathbf{q}}_{\mathrm{CDW}}$ indicate CDW instability driven by the wave-vector-dependent electron-phonon coupling. The $3\ifmmode\times\else\texttimes\fi{}3$ CDW ground state favors a distinct hollow-centered clustering of transition-metal atoms, with larger distortion amplitude in ${\mathrm{NbSe}}_{2}$ and ${\mathrm{TaSe}}_{2}$ than in ${\mathrm{TaS}}_{2}$ and ${\mathrm{NbS}}_{2}$. Superconducting results of the monolayer CDW phase prove that strong spin-orbit coupling effects are dominant in determining the critical temperature ${T}_{\mathrm{c}}$ of each system via modifying both the strength and anisotropic extent of electron-phonon interactions. I also recapitulate measured opposite thickness dependencies of ${T}_{\mathrm{c}}$ in $\mathrm{Nb}{X}_{2}$ and $\mathrm{Ta}{X}_{2}$, and show that whether superconductivity is weakened or enhanced in monolayer limit relies on specific evolution of CDW-modulated Fermi-level density of states and electron-phonon matrix elements under reduced dimensionality. This work lays the foundation for applications of $2H$ group V transition-metal dichalcogenides in versatile nanoscale quantum devices.

Coexistence of superconductivity and charge density wave instability in A15-Nb3Sn.

A15-type compound Nb3Sn has attracted much attention due to its relatively high critical temperature and critical field of superconductivity, making it a leading material for superconducting applications. In this study, we investigate the structural instability and superconductivity of Nb3Sn under hydrostatic pressure using first-principles calculations. We determine the electronic properties, phonon dispersion, electron-phonon coupling and the superconducting gap for Nb3Sn at pressures ranging from ambient to 9 GPa. Our results show that a significant electron density is present near the Fermi level due to the van Hove singularity, indicating the strong electron-phonon coupling. The phonon dispersion of Nb3Sn exhibits Kohn anomalies at three different wave vectors at a lower temperature. Moreover, above a pressure of 6 GPa, the charge density wave (CDW) instability disappeared, suggesting that pressure inhibits the CDW phase. The superconducting temperature is predicted to be TC = 18.62 K under ambient conditions, which is well consistent with the experimental results. We find that both the CDW and superconducting orders respond to pressure, with their transition temperatures decreasing as the pressure increases below 6 GPa. Above 6 GPa, the superconducting transition temperature increases slowly with pressure. Our results suggest that the instability in Nb3Sn is driven by the softening of the phonon modes due to the CDW caused by strong electron-phonon coupling. Therefore, the CDW phase and superconducting phase of Nb3Sn coexist at low pressure.

Effect of chromium doping on superconductivity and charge density wave order in the kagome metal Cs(V1−xCrx)3Sb5

The recently discovered kagome metal $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ ($A=\mathrm{K}$, Rb, Cs) with intertwined superconductivity and charge density wave (CDW) order has attracted considerable interest. Here we investigate the evolution of CDW and superconductivity in $\mathrm{Cs}{({\mathrm{V}}_{1\ensuremath{-}x}{\mathrm{Cr}}_{x})}_{3}{\mathrm{Sb}}_{5}$, where chromium substitution for vanadium introduces extra electron carriers. Our experiments reveal that the CDW order is gradually suppressed under chromium doping, which can be understood from the band structure modifications induced by electron doping. This highlights the importance of the van Hove singularities near the $M$ and $L$ points for CDW instability. The anomalous Hall effect that is coupled with CDW in ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$ only appears in lightly doped $\mathrm{Cs}{({\mathrm{V}}_{1\ensuremath{-}x}{\mathrm{Cr}}_{x})}_{3}{\mathrm{Sb}}_{5}$ and vanishes at $x\ensuremath{\ge}0.06$ when the CDW order remains relatively strong. Unlike hole- and isovalent-doped $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$, where superconductivity is enhanced upon suppression of CDW, superconductivity is rapidly suppressed under chromium doping, pointing to a conventional $s$-wave pairing.

Suppressed charge-density-wave, robust ferromagnetism and Lifshitz transition in Sm2Ru3Ge5 crystal under high pressure

The recently discovered tetragonal Sm2Ru3Ge5 provides an ideal avenue to study the interplay between the charge-density-wave (CDW) transition and ferromagnetic (FM) order which appear at TCDW = 175 K and Tm = 8 K, respectively. By performing high-pressure structure and electrical transport measurements on the Sc2Fe3Si5-type Sm2Ru3Ge5 (space group: P4/mnc), it is found that the TCDW is gradually suppressed to low temperature while the Tm changes negligibly with the increase of pressure, likely suggesting a weak link between the two states. Besides, a likely rare example of temperature-induced Lifshitz transition is also observed around 93 K, which is gradually suppressed to low temperature upon increasing pressure and eventually becomes invisible at ∼ 8 GPa. Hall resistivity measurements and first-principles calculations provide evidence for the Lifshitz transition. The work covers a topic of high interest in complex quantum materials showing the interplay of ferromagnetism, charge density wave and lattice instability.

Observation of charge density wave in layered hexagonal Cu1.89Te single crystal

Abstract We report comprehensive transport, electron microscopy and Raman spectroscopy studies on the transition-metal chalcogenides Cu1.89Te single crystals. The metallic Cu1.89Te displays successive metal-semiconductor transitions at low temperatures and almost ideal linear MR when magnetic field up to 33T. Through the electron diffraction patterns, the stable room temperature phase is identified as a 3×3×2 modulated superstructure based on Nowotny hexagonal structure. The superlattice spots of Transmission electron microscopy and Scanning tunneling microscopy clearly show the structural transitions from the room temperature Commensurate Ⅰ (C-Ⅰ) phase to the low temperature Commensurate Ⅱ (C-Ⅱ) phase. All the results can be understood in term of charge density wave (CDW) instability, yielding intuitive evidences for the CDW formations in Cu1.89Te. The additional Raman modes below room temperature further reveal that the zone-folded phonon modes may play an important role on the CDW transitions. Our research sheds light on the novel electron features of Cu1.89Te at low temperature, and may provide potential applications for future nano-devices.

Observation of Charge Density Wave in Layered Hexagonal Cu1.89Te Single Crystal

We report comprehensive transport, electron microscopy and Raman spectroscopy studies on transition-metal chalcogenides Cu1.89Te single crystals. The metallic Cu1.89Te displays successive metal-semiconductor transitions at low temperatures and almost ideal linear MR when magnetic field up to 33 T. Through the electron diffraction patterns, the stable room-temperature phase is identified as a 3 × 3 × 2 modulated superstructure based on the Nowotny hexagonal structure. The superlattice spots of transmission electron microscopy and scanning tunneling microscopy clearly show the structural transitions from the room-temperature commensurate I phase, named as C-I phase, to the low temperature commensurate II (C-II) phase. All the results can be understood in terms of charge density wave (CDW) instability, yielding intuitive evidences for the CDW formations in Cu1.89Te. The additional Raman modes below room temperature further reveal that the zone-folded phonon modes may play an important role on the CDW transitions. Our research sheds light on the novel electron features of Cu1.89Te at low temperature, and may provide potential applications for future nano-devices.

Phonon mediated superconductivity in field-effect doped molybdenum dichalcogenides

Superconductivity occurs in electrochemically doped molybdenum dichalcogenides samples thicker than four layers. While the critical temperature (T c ) strongly depends on the field effect geometry (single or double gate) and on the sample (MoS2 or MoSe2), T c always saturates at high doping. The pairing mechanism and the complicated dependence of T c on doping, samples and field-effect geometry are currently not understood. Previous theoretical works assumed homogeneous doping of a single layer and attributed the T c saturation to a charge density wave instability, however the calculated values of the electron–phonon coupling in the harmonic approximation were one order of magnitude larger than the experimental estimates based on transport data. Here, by performing fully relativistic first principles calculations accounting for the sample thickness, the field-effect geometry and anharmonicity, we rule out the occurrence of charge density waves in the experimental doping range and demonstrate a suppression of one order of magnitude in the electron–phonon coupling, now in excellent agreement with transport data. By solving the anisotropic Migdal-Eliashberg equations, we explain the behavior of T c in different systems and geometries. As our first principles calculations show an ever increasing T c as a function of doping, we suggest that extrinsic mechanisms may be responsible for the experimentally observed saturating trend.

Interplay between edge states and charge density wave order in the Falicov-Kimball model on a Haldane ribbon

To determine the impact of including edge states on the phase diagram of a spinless Falicov-Kimball model on the Haldane lattice, a study of a corresponding ribbon geometry with zigzag edges is conducted. By varying the ribbon widths, the distinction between the effects connected to the mere presence of the edges and those originating from interference between the edge states is established. The local doping caused by the former is shown to give rise to a topologically trivial bulk insulator with metallic edge states. Additionally, it gives rise to a charge density wave (CDW) phase with mixed character of the subbands in various parts of the phase diagram. The impact of local doping on the CDW instability is also addressed. Two additional gapless phases are found, caused by the edges but with stability regions depending on the width of the ribbon.

Subgap two-particle spectral weight in disordered s -wave superconductors: Insights from mode coupling approach

We study the two-particle spectral functions and collective modes of weakly disordered superconductors using a disordered attractive Hubbard model on a square lattice. We show that the disorder-induced scattering between collective modes leads to a finite subgap spectral weight in the long-wavelength limit. In general, the spectral weight is distributed between the phase and the Higgs channels, but as we move towards half filling the Higgs contribution dominates. The inclusion of the density fluctuations lowers the frequency at which this mode occurs and results in the phase channel gaining a larger contribution to this subgap mode. Near half filling, the proximity of the system to the charge density wave instability leads to strong fluctuations of the effective disorder at the commensurate wave vector ($[\ensuremath{\pi},\ensuremath{\pi}]$). We develop an analytical mode coupling approach where the pure Goldstone mode in the long-wavelength limit couples to the collective mode at $[\ensuremath{\pi},\ensuremath{\pi}]$. This provides insight into the location and distribution of the two-particle spectral weights between the Higgs and the phase channels.