Charge Density Wave Order Parameter Research Articles

Deconfined criticalities and dualities between chiral spin liquid, topological superconductor and charge density wave Chern insulator

We propose bi-critical and tri-critical theories between chiral spin liquid (CSL), topological superconductor (SC) and charge density wave (CDW) ordered Chern insulator with Chern number C=2C=2 on square, triangular and Kagome lattices. The three CDW order parameters form a manifold of S^2S2 or S^1S1 depending on whether there is easy-plane anisotropy. The skyrmion defect of the CDW order carries physical charge 2e2e and its condensation leads to a topological superconductor. The CDW-SC transitions are in the same universality classes as the celebrated deconfined quantum critical points (DQCP) between Neel order and valence bond solid order on square lattice. Both SC and CDW order can be accessed from the CSL phase through a continuous phase transition. At the CSL-SC transition, there is still CDW order fluctuations although CDW is absent in both sides. We propose three different theories for the CSL-SC transition (and CSL to easy-plane CDW transition): a U(1)U(1) theory with two bosons, a U(1)U(1) theory with two Dirac fermions, and an SU(2)SU(2) theory with two bosons. Our construction offers a derivation of the duality between these three theories as well as a promising physical realization. The SU(2)SU(2) theory offers a unified framework for a series of fixed points with explicit SO(5), O(4)SO(5),O(4) or SO(3)× O(2)SO(3)×O(2) symmetry. There is also a transparent duality transformation mapping SC order to easy-plane CDW order. The CSL-SC-CDW tri-critical points are invariant under this duality mapping and have an enlarged SO(5)SO(5) or O(4)O(4) symmetry. The DQCPs between CDW and SC inherit the enlarged symmetry, emergent anomaly, and self-duality from the tri-critical point. Our analysis unifies the well-studied DQCP between symmetry breaking phases into a larger framework where they are proximate to a topologically ordered phase. Experimentally the theory demonstrates the possibility of a rich phase diagram and criticality through closing the Mott gap of a quantum spin liquid with projective symmetry group.

Exploring interacting topological insulator in the extended Su-Schrieffer-Heeger model

Exploring topological phases in interacting systems is a challenging task. We investigate many-body topological physics of interacting fermions in an extended Su-Schrieffer-Heeger (SSH) model, which extends the two sublattices of SSH model into four sublattices and thus is dubbed SSH4 model, based on the density-matrix renormalization-group numerical method. The interaction-driven phase transition from topological insulator to charge density wave (CDW) phase can be identified by analyzing the variations of entanglement spectrum, entanglement entropies, energy gaps, CDW order parameter, and fidelity. We map the global phase diagram of the many-body ground state, which contains nontrivial topological insulator, trivial insulator and CDW phases, respectively. In contrast to interacting SSH model, in which the phase transitions to the CDW phase are argued to be first-order phase transitions, the phase transitions between the CDW phase and topologically trivial/nontrivial phases are shown to be continuous phase transitions. Finally, we {also} show the phase diagram of interacting spinful SSH4 model, where the attractive (repulsive) on-site spin interaction amplifies (suppresses) the CDW phase. The models analyzed here can be implemented with ultracold atoms on optical superlattices.

Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions

We investigate the charge-density wave (CDW) transition for one-dimensional spinless fermions at half band-filling with nearest-neighbor electron transfer amplitude $t$ and interaction $V$. The model is equivalent to the anisotropic XXZ Heisenberg model for which the Bethe Ansatz provides an exact solution. For $V> V_{\rm c}= 2t$, the CDW order parameter and the single-particle gap are finite but exponentially small, as is characteristic for a Kosterlitz-Thouless transition. It is notoriously difficult to locate such infinite-order phase transitions in the phase diagram using approximate analytical and numerical approaches. Second-order Hartree-Fock theory is qualitatively applicable for all interaction strengths, and predicts the CDW transition to occur at $V_{\rm c,2}^{(2)}\approx 1.5t$. Second-order Hartree Fock theory is almost variational because the density of quasi-particle excitations is small. We apply the density-matrix renormalization group (DMRG) for periodic boundary conditions for system sizes up to 514 sites which permits a reliable extrapolation of all physical quantities to the thermodynamic limit, apart from the critical region. We investigate the ground-state energy, the gap, the order parameter, the momentum distribution, the quasi-particle density, and the density-density correlation function to locate $V_{\rm c}$ from the DMRG data. Tracing the breakdown of the Luttinger liquid and the peak in the quasi-particle density at the band edge permits us to reproduce $V_{\rm c}$ with an accuracy of one percent.

Charge Density Wave Order and Fluctuations above T_{CDW} and below Superconducting T_{c} in the Kagome Metal CsV_{3}Sb_{5}.

The phase diagram of the kagome metal family AV_{3}Sb_{5} (A=K, Rb, Cs) features both superconductivity and charge density wave (CDW) instabilities, which have generated tremendous recent attention. Nonetheless, significant questions remain. In particular, the temperature evolution and demise of the CDW state has not been extensively studied, and little is known about the coexistence of the CDW with superconductivity at low temperatures. We report an x-ray scattering study of CsV_{3}Sb_{5} over a broad range of temperatures from 300 to ∼2 K, below the onset of its superconductivity at T_{c}∼2.9 K. Order parameter measurements of the 2×2×2 CDW structure show an unusual and extended linear temperature dependence onsetting at T^{*}∼160 K, much higher than the susceptibility anomaly associated with CDW order at T_{CDW}=94 K. This implies strong CDW fluctuations exist to ∼1.7×T_{CDW}. The CDW order parameter is observed to be constant from T=16 to 2K, implying that the CDW and superconducting order coexist below T_{c}, and, at ambient pressure, any possible competition between the two order parameters is manifested at temperatures well below T_{c}, if at all. Anomalies in the temperature dependence in the lattice parameters coincide with T_{CDW} for c(T) and with T^{*} for a(T).

Harnessing interpretable and unsupervised machine learning to address big data from modern X-ray diffraction

The information content of crystalline materials becomes astronomical when collective electronic behavior and their fluctuations are taken into account. In the past decade, improvements in source brightness and detector technology at modern X-ray facilities have allowed a dramatically increased fraction of this information to be captured. Now, the primary challenge is to understand and discover scientific principles from big datasets when a comprehensive analysis is beyond human reach. We report the development of an unsupervised machine learning approach, X-ray diffraction (XRD) temperature clustering (X-TEC), that can automatically extract charge density wave order parameters and detect intraunit cell ordering and its fluctuations from a series of high-volume X-ray diffraction measurements taken at multiple temperatures. We benchmark X-TEC with diffraction data on a quasi-skutterudite family of materials, (CaxSr[Formula: see text])3Rh4Sn13, where a quantum critical point is observed as a function of Ca concentration. We apply X-TEC to XRD data on the pyrochlore metal, Cd2Re2O7, to investigate its two much-debated structural phase transitions and uncover the Goldstone mode accompanying them. We demonstrate how unprecedented atomic-scale knowledge can be gained when human researchers connect the X-TEC results to physical principles. Specifically, we extract from the X-TEC-revealed selection rules that the Cd and Re displacements are approximately equal in amplitude but out of phase. This discovery reveals a previously unknown involvement of [Formula: see text] Re, supporting the idea of an electronic origin to the structural order. Our approach can radically transform XRD experiments by allowing in operando data analysis and enabling researchers to refine experiments by discovering interesting regions of phase space on the fly.

Theory of the charge density wave in AV3Sb5 kagome metals

The family of metallic kagome compounds $A$V$_3$Sb$_5$ ($A$=K, Rb, Cs) was recently discovered to exhibit both superconductivity and charge order. The nature of the charge-density wave (CDW) phase is presently unsettled, which complicates the interpretation of the superconducting ground state. In this paper, we use group-theory and density-functional theory (DFT) to derive and solve a phenomenological Landau model for this CDW state. The DFT results reveal three unstable phonon modes with the same in-plane momentum but different out-of-plane momenta, whose frequencies depend strongly on the electronic temperature. This is indicative of an electronically-driven CDW, stabilized by features of the in-plane electronic dispersion. Motivated by the DFT analysis, we construct a Landau free-energy expansion for coupled CDW order parameters with wave-vectors at the $M$ and $L$ points of the hexagonal Brillouin zone. We find an unusual trilinear term coupling these different order parameters, which can promote the simultaneous condensation of both CDWs even if the two modes are not nearly-degenerate. We classify the different types of coupled multi-$\bf{Q}$ CDW orders, focusing on those that break the sixfold rotational symmetry and lead to a unit-cell doubling along all three crystallographic directions, as suggested by experiments. We determine a region in parameter space, characterized by large nonlinear Landau coefficients, where these phases - dubbed staggered tri-hexagonal and staggered Star-of-David - are the leading instabilities of the system. Finally, we discuss the implications of our results for the kagome metals.

Atomic-scale mapping of pressure-induced deformations and phase defects in the charge density wave order parameter

Understanding the intricate interplay between multiple electronic phases in quantum materials such as charge density wave (CDW), superconducting, and metallic phases is a challenging issue. Systematic introduction of pressure is one approach that has been used to probe this interplay. However, the influence of pressure together with the intricate interaction between electronic and lattice degrees of freedom can trigger complex structural evolution and distribution of various electronic phases at the atomic scale, the characterization of which demands high spatial resolution. We investigate the atomic-scale response of the charge density waves and the underlying atomic lattice in $1T\text{\ensuremath{-}}\mathrm{Ta}{\mathrm{S}}_{2}$ after exposure to hydrostatic pressure. High-resolution transmission electron microscopy images show that the CDW order parameter reacts with an elasticlike strain response to pressure-induced stacking faults and dislocations in the lattice. This is characterized by a proliferation of phase defects including CDW dislocations, discommensurations, and domain walls. Our results evidence the importance of pressure-induced lattice deformations and defects in modulating, stabilizing, or destroying electronic phases at the atomic scale.

Quantum criticality of magnetic catalysis in two-dimensional correlated Dirac fermions

We study quantum criticality of the magnetic field induced charge density wave (CDW) order in correlated spinless Dirac fermions on the $\pi$-flux square lattice at zero temperature as a prototypical example of the magnetic catalysis, by using the infinite density matrix renormalization group. It is found that the CDW order parameter $M(B)$ exhibits an anomalous magnetic field $(B)$ scaling behavior characteristic of the $(2+1)$-dimensional chiral Ising universality class near the quantum critical point, which leads to a strong enhancement of $M(B)$ compared with a mean field result. We also establish a global phase diagram in the interaction-magnetic field plane for the fermionic quantum criticality.

Unveiling charge density wave quantum phase transitions by x-ray diffraction

We investigate the thermal-driven charge density wave (CDW) transition of two cubic superconducting intermetallic systems Lu(Pt1-xPdx)2In and (Sr1-xCax)3Ir4Sn13 by means of x-ray diffraction technique. A detailed analysis of the CDW modulation superlattice peaks as function of temperature is performed for both systems as the CDW transition temperature T_CDW is suppressed to zero by an non-thermal control parameter. Our results indicate an interesting crossover of the classical thermal-driven CDW order parameter critical exponent from a three-dimensional universality class to a mean-field tendency, as T_CDW vanishes. Such behavior might be associated with presence of quantum fluctuations which influences the classical second-order phase transition, strongly suggesting the presence of a quantum critical point (QCP) at T_CDW = 0. This also provides experimental evidence that the effective dimensionality exceeds its upper critical dimension due to a quantum phase transition.

Role of a higher-dimensional interaction in stabilizing charge density waves in quasi-one-dimensional NbSe3 revealed by angle-resolved photoemission spectroscopy

We revisit charge density wave (CDW) behavior in the archetypal quasi-one-dimensional (quasi-1D) material NbSe$_3$ by high-resolution angle-resolved photoemission spectroscopy measurements utilizing a microfocused laser with a photon energy of 6.3 eV. We present a detailed view of the electronic structure of this complex multiband system and unambiguously resolve CDW gaps at the Fermi level ($E_F$). By employing a tight-binding model, we argue that these gaps are the result of interband coupling between electronic states that reside predominantly on distinct 1D chains within the material. Two such localized states are found to couple to an electronic state that extends across multiple 1D chains, highlighting the importance of a higher-dimensional interaction in stabilizing the CDW ordering in this material. In addition, the temperature evolution of intrachain gaps caused by the CDW periodicities far below $E_F$ deviate from the behavior expected for a Peierls-type mechanism driven by nesting; the upper and lower bands of the renormalized CDW dispersions maintain a fixed peak-to-peak distance while the gaps are gradually removed at higher temperatures. This points toward a gradual loss of long-range phase coherence as the dominant effect in reducing the CDW order parameter, which may correspond to the loss of coherence between the coupled chains. Furthermore, one of the gaps is observed above the bulk and surface CDW transition temperatures, implying the persistence of short-range incoherent CDW order. The influence of such higher-dimensional interactions likely plays an important role in a range of low-dimensional systems.

Holographic imaging of the complex charge density wave order parameter

Real-space mapping of the complex charge density wave order parameter reveals coexisting unidirectional charge modulations connected by fundamental crystalline symmetry. These quantum phases develop their distinct order parameter landscapes with a rich variety of features such as domain walls, discommensuration and topological defects.

Staggered ordered phases in the three-orbital Hubbard model

We study ordered phases with broken translational symmetry in the half-filled three-orbital Hubbard model with antiferromagnetic Hund coupling by means of dynamical mean-field theory (DMFT) and continuous-time quantum Monte Carlo simulations. The stability regions of the antiferro-orbital (AFO), antiferro-magnetic (AFM), and charge density wave (CDW) states are determined by measuring the corresponding order parameters. We introduce two symmetrically distinct AFO order parameters and show that these are the primary order parameters in the phase diagram. The CDW and AFM states appear simultaneously with these two types of AFO orders in the weak and strong coupling region, respectively. The DMFT phase diagram is consistent with the results obtained by the Hartree approximation and strong-coupling perturbation theory. In the weak coupling regime, a nontrivial exponent $\beta=3/2$ is found for the CDW order parameter, which is related to the coupling between the CDW and AFO orders in the Landau theory characteristic for the three-orbital model. We also demonstrate the existence of a metallic AFO state without any charge disproportions and magnetic orders, which appears only at finite temperatures.

Effect of Short-Range Correlations on Spectral Properties of Doped Mott Insulators

In the framework of cluster perturbation theory for the 2D Hubbard and Hubbard-Holstein models at low hole doping we have studied the effect of local and short-range correlations in strongly correlated systems on the anomalous features in the electronic spectrum by investigating the fine structure of quasiparticle bands. Different anomalous features of spectrum are obtained as the result of intrinsic properties of strongly correlated electron and polaron bands in the presence of short-range correlations. Particularly, features similar to the electron-like Fermi-pockets of cuprates at hole doping $p\sim0.1$ are obtained without ad hoc introducing a charge density wave order parameter within the Hubbard model in a unified manner with other known peculiarities of the pseudogap phase like Fermi-arcs, pockets, waterfalls, and kink-like features. The Fermi surface is mainly formed by dispersive quasiparticle bands with large spectral weight, formed by coherent low-energy exications. Within the Hubbard-Holstein model at moderate phonon frequencies we show that modest values of local electron-phonon interaction are capable of introducing low-energy kink-like features and affecting the Fermi surface by hybridization of the fermionic quasiparticle bands with the Franck-Condon resonances.

Charge Density Wave and Spin Density Wave in Two-Orbital Model for Iron-Based Superconductors

We present a mean-field theoretical model study for the coexistence of the two strongly interacting charge density wave (CDW) and spin density wave (SDW) for iron-based superconductors in the under-doped region before the onset of the superconductivity in the system. The analytic expressions for the temperature dependence of the CDW and SDW order parameters are derived by using Zubarev’s technique of double-time single particle Green’s functions and solved self-consistently. Their interplay is studied by varying both the CDW and SDW coupling constants. Further, the electronic density of states (DOS) for the conduction electrons are studied in the pure CDW and SDW states and coexistence state for the cases, where the CDW transition temperature is greater than the SDW transition temperature and vice versa, which show two gap parameters.

A model study of tunneling conductance spectra of ferromagnetically ordered manganites

We report here the interplay of ferromagnetism (FM) and charge density wave (CDW) in manganese oxide systems through the study of tunneling conductance spectra. The model Hamiltonian consists of strong Heisenberg coupling in core t2g band electrons within mean-field approximation giving rise to ferromagnetism. Ferromagnetism is induced in the itinerant eg electrons due to Kubo-Ohata type double exchange (DE) interaction among the t2g and eg electrons. The charge ordering (CO) present in the eg band giving rise to CDW interaction is considered as the extra-mechanism to explain the colossal magnetoresistance (CMR) property of manganites. The magnetic and CDW order parameters are calculated using Zubarev’s Green’s function technique and solved self-consistently and numerically. The eg electron density of states (DOS) calculated from the imaginary part of the Green’s function explains the experimentally observed tunneling conductance spectra. The DOS graph exhibits a parabolic gap near the Fermi energy as observed in tunneling conductance spectra experiments.

Narrowband noise study of sliding charge density waves in NbSe3 nanoribbons

Transport properties (dc electrical resistivity, threshold electric field, and narrow-band noise) are reported for nanoribbon specimens of NbSe3 with thicknesses as low as 18 nm. As the sample thickness decreases, the resistive anomalies characteristic of the charge density wave (CDW) state are suppressed and the threshold fields for nonlinear CDW conduction apparently diverge. Narrow-band noise measurements allow determination of the concentration of carriers condensed in the CDW state nc, reflective of the CDW order parameter Δ. Although the CDW transition temperatures are relatively independent of sample thickness, in the lower CDW state Δ decreases dramatically with decreasing sample thickness.

A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS2.

Electron–electron and electron–phonon interactions are two major driving forces that stabilize various charge-ordered phases of matter. In layered compound 1T-TaS2, the intricate interplay between the two generates a Mott-insulating ground state with a peculiar charge-density-wave (CDW) order. The delicate balance also makes it possible to use external perturbations to create and manipulate novel phases in this material. Here, we study a mosaic CDW phase induced by voltage pulses, and find that the new phase exhibits electronic structures entirely different from that of the original Mott ground state. The mosaic phase consists of nanometre-sized domains characterized by well-defined phase shifts of the CDW order parameter in the topmost layer, and by altered stacking relative to the layers underneath. We discover that the nature of the new phase is dictated by the stacking order, and our results shed fresh light on the origin of the Mott phase in 1T-TaS2.

Temperature dependence of the penetration depth of a magnetic field in the presence of dispersion of the superconducting and charge density wave order parameters

A formula for computing the temperature dependence of the London penetration depth of a magnetic field in the regime of coexistence of charge density waves and superconductivity has been proposed taking into account the dependence of both order parameters on the wave vector. It has been shown that an anomalously high diamagnetic response of the system and a finite value of the superconducting current persist even at Tc ≤ T ≤ TCDW.

Enhancement of superconductivity at the onset of charge-density-wave order in a metal

We analyze superconductivity in the cuprates near the onset of an incommensurate charge density wave (CDW) order with momentum ${\bf Q} = (Q,0)/(0,Q)$, as observed in the experiments. We first consider a semi-phenomenological charge-fermion model in which hot fermions, separated by ${\bf Q}$, attract each other by exchanging soft CDW fluctuations. We find that in a quantum-critical region near CDW transition, $T_c = A {\bar g}_c$, where ${\bar g}_c$ is charge-fermion coupling and $A$ is the prefactor which we explicitly compute. We then consider the particular microscopic scenario in which CDW order parameter emerges as a composite field made out of primary spin-density-wave fields. We show that charge-fermion coupling ${\bar g}_c$ is of order of spin-fermion coupling ${\bar g}_s$. As the consequence, superconducting $T_c$ is substantially enhanced near the onset of CDW order. Finally we analyze the effect of an external magnetic field $H$. We show that, as $H$ increases, optimal $T_c$ decreases and the superconducting dome becomes progressively more confined to the CDW quantum-critical point. These results are consistent with the experiments.

Interplay between pair- and charge-density-wave orders in underdoped cuprates

We analyze the interplay between charge-density-wave (CDW) and pair-density-wave (PDW) orders within the spin-fermion model for the cuprates. We specifically consider CDW order with transferred momenta $(\pm Q,0)$/$(0,\pm Q)$, and PDW order with total momenta $(0,\pm Q)/(\pm Q,0)$. We show that both emerge in the spin-fermion model near the onset of antiferromagnetism. We further argue that the two orders are nearly degenerate due to an approximate SU(2) particle-hole symmetry of the model. The ${\rm SU}(2)$ symmetry becomes exact if one neglects the curvature of the Fermi surface in hot regions, in which case ${\rm U}(1)$ CDW and PDW order parameters become components of an SO(4)-symmetric PDW/CDW "super-vector". We develop a Ginzburg-Landau theory for PDW/CDW order parameters and find two possible ground states: a "stripe" state, and a "checkerboard" state. We show that the ${\rm SO}(4)$ symmetry between CDW and PDW is broken by two effects. One is the inclusion of Fermi surface curvature, which selects a PDW order immediately below the instability temperature. Another is the overlap between different hot regions, which favors CDW order at low temperatures. For the stripe state, we show that the competition between the two effects gives rise to a first-order transition from PDW to CDW inside the ordered state. We also argue that beyond mean-field theory, the onset temperature for CDW order is additionally enhanced due to feedback from a preemptive breaking of ${\mathbb Z}_2$ time-reversal symmetry. We discuss the ground state properties of a pure PDW state and a pure CDW state, and show that the PDW checkerboard state yields a vortex-anti-vortex lattice. For the checkerboard state, we considered a situation when both CDW and PDW orders are present at low $T$ and show that the presence of both condensates induces a long sought chiral $s+id_{xy}$ superconductivity.