Charge Density Wave Correlations Research Articles

Electron–electron interactions in partially mixed helical states

We theoretically study the effect of electron–electron interactions in one-dimensional partially mixed helical states. These helical states can be realized at the edges of two-dimensional topological insulators with partially broken time-reversal symmetry, resulting in helical gapped states. Using the bosonization method and renormalization group analysis, we identify weak gap, crossover, and strong gap regimes in the phase diagram. We find that strong electron–electron interaction mixes the helicity of the states, leading to the relevant strong gap regime. We investigate the charge and spin density wave correlation functions in different relevancy regimes of the gap mediated by interactions, where in the case of strong repulsive interaction, the spin density wave dominates the charge density wave. Additionally, employing the Memory function technique, we calculate the effect of mixed helicity on the charge transport in a sufficiently long edge. We find a non-uniform temperature dependence for the charge conductivity in both the strong and weak gap regimes with distinct features.

Pair density wave and superconductivity in a kinetically frustrated doped Emery model on a square lattice

The quest to understand the nature of superconductivity in the cuprates has spotlighted the pair density wave (PDW)–a superconducting state characterized by a spatially modulated order parameter. Despite significant advances in understanding PDW properties, conclusively demonstrating its presence in systems pertinent to cuprate superconductors remains elusive. In this study, we present a systematic density-matrix renormalization group study to investigate the Emery model (or the three-band Hubbard model) on two-leg square cylinders with negative electron hopping term tpp between adjacent oxygen sites. Kinetic frustration - introduced by changing the sign of oxygen-oxygen hopping - leads to a much reduced Cu-Cu antiferromagnetic exchange along with an enlarged charge transfer energy that changes the local properties of the model. At light doping levels, our findings reveal a ground state remarkably consistent with a PDW, exhibiting mutually commensurate superconducting (SC), charge, and spin density wave correlations. Intriguingly, the dominant SC pairing is observed between neighboring oxygen sites, diverging from the expected Cu sites in the positive tpp case. When the system incorporates moderate near-neighbor interactions, particularly an attractive Vpp between adjacent oxygen sites, the SC correlations become quasi-long-ranged, accompanied by a pronounced divergence in the PDW susceptibility. When the attractive Vpp increases further, the system gives way to an unconventional d-wave superconductivity.

Charge Correlations in Cuprate Superconductors

High-temperature superconductivity, with transition temperatures up to ≈134 K at ambient pressure, occurs in layered cuprate compounds. The conducting CuO2 planes, which are universally present, are responsible for the superconductivity but also show a disposition to other competing states including spin and charge order. Charge-density-wave (CDW) order appears to be a universal property of cuprate superconductors. It has been studied via a multitude of probes including X-ray and neutron scattering, nuclear magnetic resonance, scanning probe techniques, electronic transport, and quantum oscillations. Here, we review the microscopic properties of the CDW order. We discuss the nature of the ordered state, that is, its symmetry and microscopic structure. Furthermore, we show how the CDW order is related to quenched disorder, host structure, symmetry breaking perturbations, and magnetic fields. We also describe measurements of dynamic collective charge excitations that are closely related to the quasi-static CDW order. Finally, we highlight some of the debated issues in the field, including the origin of the CDW order, the relationship to spin order, and the nature of the spatial CDW correlations.

Enhancement of charge density wave correlations in a Holstein model with an anharmonic phonon potential

Enhancement of charge density wave correlations in a Holstein model with an anharmonic phonon potential

Pair density wave in the doped three-band Hubbard model on two-leg square cylinders

A pair density wave (PDW) is a superconducting (SC) state with a spatially modulated order parameter. Although much is known about the properties of the PDW state, its realization in microscopic models with divergent susceptibility has been challenging. Here we report a density-matrix renormalization group study of a three-band Hubbard model (also known as the Emery model) for cuprates on long two-leg square cylinders. Upon light doping, we find that the ground state of the system is consistent with that of a PDW state with mutually commensurate and power-law SC, charge (CDW), and spin (SDW) density wave correlations. The SC correlations are dominant between neighboring Cu sites with $d$-wave pairing symmetry. Interestingly, we find that the near-neighbor interactions, especially the near-neighbor attractive ${V}_{pd}$ interaction between neighboring Cu and oxygen sites, can notably enhance the SC correlations while simultaneously suppressing the CDW correlations. For a modestly strong attractive ${V}_{pd}$, the SC correlations become quasi-long-ranged with a divergent PDW susceptibility.

Electronic states of domain walls in commensurate charge density wave ground state and mosaic phase in 1T -TaS2

Domain walls (DWs) in the charge-density-wave (CDW) Mott insulator 1T-TaS2 have unique localized states, which play an important role in exploring the electronic properties of the material. However, the electronic states in DWs in 1T-TaS2 have not been clearly understood, mostly due to the complex structures, phases, and interlayer stacking orders in the DW areas. Here, we explored the electronic states of DWs in the large-area CDW phase and mosaic phase of 1T-TaS2 by scanning tunneling spectroscopy. Due to the different densities of DWs, the electronic states of DWs show distinct features in these phases. In the large area CDW phase, both the domain and the DWs (DW1, DW2, DW4) have zero conductance at the Fermi level; while in the mosaic phase, they can be metallic or insulating depending on their environments. In areas with a high density of DWs, some electronic states were observed both on the DWs and within the domains, indicating delocalized states over the whole region. Our work contributes to further understanding of the interplay between CDW and electron correlations in 1T-TaS2.

Gapless spin liquid and pair density wave of the Hubbard model on three-leg triangular cylinders

We study the ground state properties of the Hubbard model on three-leg triangular cylinders using large-scale density-matrix renormalization group simulations. At half-filling, we identify an intermediate gapless spin liquid phase, which has one gapless spin mode and algebraic spin–spin correlations but exponential decay scalar chiral–chiral correlations, between a metallic phase at weak coupling and Mott insulating dimer phase at strong interaction. Upon light doping the gapless spin liquid, the system exhibits power-law charge-density-wave (CDW) correlations but short-range single-particle, spin–spin, and chiral–chiral correlations. Similar to CDW correlations, the superconducting correlations also decay in power-law but oscillate in sign as a function of distance, which is consistent with the striped pair-density wave. When further doping the gapless spin liquid phase or doping the dimer order phase, another phase takes over, which has similar CDW correlations but all other correlations decay exponentially.

Charge density wave and finite-temperature transport in minimally twisted bilayer graphene

We study phenomena driven by electron-electron interactions in the minimally twisted bilayer graphene (mTBLG) with a perpendicular electric field. The low-energy degrees of freedom in mTBLG are governed by a network of one-dimensional domain-wall states, described by two channels of one-dimensional linearly dispersing spin-$1/2$ fermions. We show that the interaction can realize a spin-gapped interchannel charge density wave (CDW) state at low temperatures, forming a ``Coulomb drag'' between the channels and leaving only one charge conducting mode. For sufficiently high temperatures, power-law-in-temperature resistivity emerges from the charge Umklapp scatterings within a domain wall. Remarkably, the presence of the CDW states can strengthen the charge Umklapp scattering and induce a resistivity minimum at an intermediate temperature corresponding to the CDW correlation energy. We further discuss the conditions that resistivity of the network is dominated by the domain walls. In particular, the power-law-in-temperature resistivity results can apply to other systems that manifest topological domain-wall structures.

Atomistic origin of metal versus charge-density-wave phase separation in indium atomic wires on Si(111)

We investigate in atomic scale the electronic phase separation occurring in the well known quasi 1D charge-density wave (CDW) phase of the In atomic wire array on a Si(111) surface. The characteristic atomic scale defects, originated from excess In atoms, are found to be actively involved in the formation of the phase boundary between the metallic and the CDW phases by extensive analysis of scanning tunneling microscopy images at various temperatures. These particular defects flip the phase of the quasi 1D CDW to impose strong local constraints in the CDW correlation. We show that such local constraints and the substantial interwire CDW interaction induce local condensates of CDW and the phase separation between the metallic and the CDW phases. This work unveils the atomistic origin of the electronic phase separation, highlighting the importance of atomic scale structures of defects and their collective interaction in electronically inhomogeneous materials.

Thermopower across the phase diagram of the cuprate La1.6−xNd0.4SrxCuO4 : Signatures of the pseudogap and charge density wave phases

The Seebeck coefficient (thermopower) $S$ of the cuprate superconductor La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ was measured across its doping phase diagram (from $p = 0.12$ to $p = 0.25$), at various temperatures down to $T \simeq 2$ K, in the normal state accessed by suppressing superconductivity with a magnetic field up to $H = 37.5$ T. The magnitude of $S/T$ in the $T=0$ limit is found to suddenly increase, by a factor $\simeq 5$, when the doping is reduced below $p^{\star} = 0.23$, the critical doping for the onset of the pseudogap phase. This confirms that the pseudogap phase causes a large reduction of the carrier density $n$, consistent with a drop from $n = 1 + p$ above $p^{\star}$ to $n = p$ below $p^{\star}$, as previously inferred from measurements of the Hall coefficient, resistivity and thermal conductivity. When the doping is reduced below $p = 0.19$, a qualitative change is observed whereby $S/T$ decreases as $T \to 0$, eventually to reach negative values at $T=0$. In prior work on other cuprates, negative values of $S/T$ at $T \to 0$ were shown to result from a reconstruction of the Fermi surface caused by charge-density-wave (CDW) order. We therefore identify $p_{\rm CDW} \simeq 0.19$ as the critical doping beyond which there is no CDW-induced Fermi surface reconstruction. The fact that $p_{\rm CDW}$ is well separated from $p^{\star}$ reveals that there is a doping range below $p^{\star}$ where the transport signatures of the pseudogap phase are unaffected by CDW correlations, as previously found in YBa$_2$Cu$_3$O$_y$ and La$_{2-x}$Sr$_x$CuO$_4$.

Charge density waves in cuprate superconductors beyond the critical doping

The unconventional normal-state properties of the cuprates are often discussed in terms of emergent electronic order that onsets below a putative critical doping of xc ≈ 0.19. Charge density wave (CDW) correlations represent one such order; however, experimental evidence for such order generally spans a limited range of doping that falls short of the critical value xc, leading to questions regarding its essential relevance. Here, we use X-ray diffraction to demonstrate that CDW correlations in La2−xSrxCuO4 persist up to a doping of at least x = 0.21. The correlations show strong changes through the superconducting transition, but no obvious discontinuity through xc ≈ 0.19, despite changes in Fermi surface topology and electronic transport at this doping. These results demonstrate the interaction between CDWs and superconductivity even in overdoped cuprates and prompt a reconsideration of the role of CDW correlations in the high-temperature cuprate phase diagram.

Charge density wave and superconductivity in the disordered Holstein model

The interplay between electron-electron correlations and disorder has been a central theme of condensed matter physics over the past several decades, with particular interest in the possibility that interactions might cause delocalization of an Anderson insulator into a metallic state, and the disrupting effects of randomness on magnetic order and the Mott phase. Here we extend this physics to explore electron-phonon interactions and show, via exact quantum Monte Carlo simulations, that the suppression of the charge density wave correlations in the half-filled Holstein model by disorder can stabilize a superconducting phase. Our simulations thus capture qualitatively the suppression of charge ordered phases and emergent superconductivity recently seen experimentally.

Emergent charge order from correlated electron-phonon physics in cuprates

Charge-density wave order is now understood to be a widespread feature of underdoped cuprate high-temperature superconductors, although its origins remain unclear. While experiments suggest that the charge-ordering wavevector is determined by Fermi-surface nesting, the relevant sections of the Fermi surface are featureless and provide no clue as to the underlying mechanism. Here, focusing on underdoped YBa2Cu3O6+x, we propose that charge-density waves form from the incipient softening of a bond-buckling phonon. The momentum dependence of its coupling to itinerant electrons favourably selects the wavevector found in experiments. But, it requires quasiparticle renormalization by strong electronic correlations to enable a unique enhancement of the charge susceptibility near the B1g-phonon selected wavevector. The B1g phonon frequency softens by a few percent, and finite-range charge-density wave correlations will form locally, if nucleated by defects or dopant disorder. These results suggest that underdoped cuprates cannot be understood in the context of strong electronic correlations alone.

Possible Phason-Polaron Effect on Purely One-Dimensional Charge Order of Mo6Se6 Nanowires

In one-dimensional (1D) metallic systems, the diverging electron susceptibility and electron-phonon coupling collaboratively drive the electrons into a charge density wave (CDW) state. However, strictly 1D system is unstable against perturbations, whose effect on CDW order requires clarification ideally with altered coupling to surroundings. Here, we fabricate such a system with nanowires of Mo6Se6 bundles, which are either attached to edges of monolayer MoSe2 or isolated freely, by post-annealing the preformed MoSe2. Using scanning tunneling microscopy (STM), we visualized charge modulations and CDW gaps with prominent coherent peaks in the edge-attached nanowires. Astonishingly, the CDW order becomes suppressed in the isolated nanowires, showing CDW correlation gaps without coherent peaks. The contrasting behavior, as revealed with theoretical modeling, is interpreted as the effect of phason-polarons on the 1D CDW state. Our work elucidates a possibly unprecedented many body effect that may be generic to strictly 1D system but undermined in quasi-1D system.

Enhanced Electron-Phonon Coupling for Charge-Density-Wave Formation in La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4+δ}.

Charge density wave (CDW) correlations are prevalent in all copper-oxide superconductors. While CDWs in conventional metals are driven by coupling between lattice vibrations and electrons, the role of the electron-phonon coupling (EPC) in cuprate CDWs is strongly debated. Using Cu L_{3} edge resonant inelastic x-ray scattering, we study the CDW and Cu-O bond-stretching phonons in the stripe-ordered cuprate La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4+δ}. We investigate the interplay between charge order and EPC as a function of doping and temperature and find that the EPC is enhanced in a narrow momentum region around the CDW ordering vector. By detuning the incident photon energy from the absorption resonance, we extract an EPC matrix element at the CDW ordering vector of M≃0.36 eV, which decreases to M≃0.30 eV at high temperature in the absence of the CDW. Our results suggest a feedback mechanism in which the CDW enhances the EPC which, in turn, further stabilizes the CDW.

Ground state phase diagram of the doped Hubbard model on the four-leg cylinder

We study the ground state properties of the Hubbard model on a 4-leg cylinder with doped hole concentration per site $\delta\leq 12.5\%$ using density-matrix renormalization group. By keeping a large number of states for long system sizes, we find that the nature of the ground state is remarkably sensitive to the presence of next-nearest-neighbor hopping $t'$. Without $t'$ the ground state of the system corresponds with the insulating filled stripe phase with long-range charge-density-wave (CDW) order and short-range incommensurate spin correlations appears. However, for a small negative $t'$ a phase characterized by coexisting algebraic d-wave superconducting (SC)- and algebraic CDW correlations. In addition, it shows short range spin- and fermion correlations consistent with a canonical Luther-Emery (LE) liquid, except that the charge- and spin periodicities are consistent with half-filled stripes instead of the $4 k_F$ and $2 k_F$ wavevectors generic for one dimensional chains. For a small positive $t'$ yet another phase takes over showing similar SC and CDW correlations. However, the fermions are now characterized by a (near) infinite correlation length while the gapped spin system is characterized by simple staggered antiferromagnetic correlations. We will show that this is consistent with a LE formed from a weakly coupled (BCS like) d-wave superconductor on the ladder where the interactions have only the effect to stabilize a cuprate style magnetic resonance.

Unusual Dynamic Charge Correlations in Simple-Tetragonal HgBa2CuO4+δ

The charge-density-wave (CDW) instability in the underdoped, pseudogap part of the cuprate phase diagram has been a major recent research focus, yet measurements of dynamic, energy-resolved CDW correlations are still in their infancy. We report a high-resolution resonant inelastic X-ray scattering (RIXS) study of the underdoped cuprate superconductor HgBa$_{2}$CuO$_{4+\delta}$ ($T_c = 70$ K). At $T=250$ K, above the CDW order temperature $T_\mathrm{CDW} \approx 200$ K, we observe significant dynamic CDW correlations at about 40 meV. This energy scale is comparable to both the superconducting gap and the previously reported low-energy pseudogap. At $T = T_c$, a strong elastic CDW peak appears, but the dynamic correlations around 40 meV remain virtually unchanged. In addition, we observe a new feature: dynamic correlations at significantly higher energy, with a characteristic scale of about 160 meV. A similar scale was previously identified in other experiments as a high-energy pseudogap. The existence of three distinct features in the charge response is highly unusual for a CDW system, and suggests that charge order in the cuprates is closely related to the pseudogap phenomenon and more complex than previously thought. We further observe the paramagnon dispersion along [1,0], across the two-dimensional CDW wavevector $\boldsymbol{q}_\mathrm{CDW}$, which is consistent with magnetic excitations measured by inelastic neutron scattering. Unlike for some other cuprates, our results point to the absence of a discernible coupling between CDW and magnetic excitations.

Superconductivity and charge density wave under a time-dependent periodic field in the one-dimensional attractive Hubbard model

We investigate the competition between superconductivity (SC) and charge density wave (CDW) under a time-dependent periodic field in the attractive Hubbard model. By employing the time-dependent exact diagonalization method, we show that the driving frequency and amplitude of the external field can control the enhancement of either the superconducting pair or the CDW correlations, which are degenerate in the ground state of the half-filled attractive Hubbard model in the absence of the field. In the strong-coupling limit of the attractive Hubbard interaction, the controllability is characterized by the anisotropic interaction of the effective model. The anisotropy is induced by the external field and lifts the degeneracy of SC and CDW. We find that the enhancement or suppression of the superconducting pair and CDW correlations in the periodically-driven attractive Hubbard model can be well interpreted by the quench dynamics of the effective model derived in the strong-coupling limit.

Eliashberg theory of phonon-mediated superconductivity — When it is valid and how it breaks down

We analyze the validity of Eliashberg theory of phonon-mediated superconductivity in 2D systems in light of recent extensive Monte-Carlo studies of the Holstein model. Conventional wisdom says that Eliashberg theory is applicable as long as vertex corrections remain small. For small ratio of the phonon energy Ω0 and the Fermi energy EF, this condition is supposed to hold even when the dimensionless electron–phonon coupling λ is larger than one, i.e., in the strong coupling regime. A comparison between various quantities computed in the Migdal approximation and those computed by Quantum Monte Carlo prove that this belief is wrong, and we identify analytically some of the ways in which this breakdown occurs for various “normal state” properties at λ=λcr, where λcr=O(1). The breakdown occurs at temperatures high enough that neither superconducting nor charge–density wave correlations extend over any significant range of distances, so it cannot be associated with the onset of an instability toward any of the relevant ordered ground-states — rather it is associated with the local physics of classical bipolaron formation. Still, we show that certain properties, including the superconducting Tc and the superconducting gap structure below Tc, can be accurately inferred from the strong-coupling limit of Eliashberg theory at λ≤λcr.

Possible glass-like random singlet magnetic state in 1T-TaS2

Two-dimensional layered transition-metal-dichalcogenide compound 1T-TaS2 shows the rare coexistence of charge density wave (CDW) and electron correlation driven Mott transition. In addition, atomic-cluster spins on the triangular lattice of the CDW state of 1T-TaS2 give rise to the possibility of the exotic spin-singlet state in which quantum fluctuations of spins are strong enough to prevent any long range magnetic ordering down to the temperature absolute zero (0 K). We present here the evidences of a glass-like random singlet magnetic state in 1T-TaS2 at low temperatures through a study of temperature and time dependence of magnetization. Comparing the experimental results with a representative canonical spin-glass system Au(1.8%Mn), we show that this glass-like state is distinctly different from the well established canonical spin-glass state.