Wave Order Parameter Research Articles

Optically induced umklapp shift currents in striped cuprates

Motivated by recent experiments that observed low-frequency second-order optical responses in doped striped superconductors, here we investigate the nonlinear electrodynamics of systems exhibiting a charge density wave (CDW) order parameter. Due to the Bragg scattering off the CDW order, an incoming spatially homogeneous electric field in addition to zero momentum current generates umklapp currents that are modulated in space at momenta of the reciprocal CDW lattice. In particular, here we predict and microscopically evaluate the umklapp shift current, a finite momentum analog of the regular shift current which represents the second-order optical process that downconverts homogeneous AC electric field into low-frequency, zero momentum current. Specifically, we evaluate real-time response functions within mean-field theory via the Keldysh technique and use the Peierls substitution to compute observables at finite momenta in lattice models. We find that systems with certain lattice symmetries (such as inversion symmetry), where the regular shift current is disallowed, may give rise to the umklapp one. We apply our framework to investigate lattice symmetries in layered materials with helical-like stripes and show that both types of shift currents provide insight into the nature of intertwined phases of matter. Finally, we discuss the relation of our findings to recent experiments in striped superconductors. Published by the American Physical Society 2024

Investigation of the Interplay Between Spin Density Wave and Superconductivity in High‐Temperature Iron‐Based Superconductor Ba1−xRbxFe2As2

The current research work deals with the theoretical investigation of the interplay between spin density wave (SDW) and superconductivity in a two‐band model high‐temperature iron‐based superconductor by using the double‐time temperature‐dependent Green's function formalism. The superconductivity transition temperature () versus the spin density wave order parameter () for the electron and hole intrabands as well as the interband for the iron‐based superconductor has been plotted. With an increase in of the material, the superconductivity transition temperature decreases. In both intra‐ and interband pairings, the temperature at which the spin density wave transition () rises with increasing . The interplay between superconductivity and spin density wave of the electron and hole intraband interactions for . Furthermore, the phase diagram of the spin density wave parameter () versus temperature has been plotted, and the spin density wave parameter () is observed to decrease as the temperature rises and vanishes at the spin density wave transition temperature ().

Doping phase diagram of a Hubbard model for twisted bilayer cuprates

We study the twisted Hubbard model of a cuprate bilayer at a fixed twist angle $\theta=53.13^{\circ}$ using the variational cluster approximation, a method that treats short-range dynamical correlations exactly. At intermediate interlayer tunneling, the phase difference $\phi$ between the $d$-wave order parameters of two layers is $\pi$ in the overdoped regime, while it is zero in the underdoped regime, close to the Mott phase. At strong interlayer tunneling, we observe a clear time-reversal symmetry breaking phase near optimal doping, in which the phase difference $\phi$ changes continuously from 0 to $\pi$. However, this phase has trivial topology. We also apply a cluster extension of dynamical mean field theory to the same problem, but fail to detect a time-reversal breaking phase with that method.

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.

Josephson effects in twisted cuprate bilayers

Twisted bilayers of high-$T_c$ cuprate superconductors have been argued to form topological phases with spontaneously broken time reversal symmetry ${\cal T}$ for certain twist angles. With the goal of helping to identify unambiguous signatures of these topological phases in transport experiments, we theoretically investigate a suite of Josephson phenomena between twisted layers. We find an unusual non-monotonic temperature dependence of the critical current at intermediate twist angles which we attribute to the unconventional sign structure of the $d$-wave order parameter. The onset of the ${\cal T}$-broken phase near $45^\circ$ twist is marked by a crossover from the conventional $2\pi$-periodic Josephson relation $J(\varphi)\simeq J_c\sin{\varphi}$ to a $\pi$-periodic function as the single-pair tunneling becomes dominated by a second order process that involves two Cooper pairs. Despite this fundamental change, the critical current remains a smooth function of the twist angle $\theta$ and temperature $T$ implying that a measurement of $J_c$ alone will not be a litmus test for the ${\cal T}$-broken phase. To obtain clear signatures of the ${\cal T}$-broken phase one must measure $J_c$ in the presence of an applied magnetic field or radio-frequency drive, where the resulting Fraunhofer patterns and Shapiro steps are altered in a characteristic manner. We discuss these results in light of recent experiments on twisted bilayers of the high-$T_c$ cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.

Dissipative dynamics in the free massive boson limit of the sine-Gordon model

We study the dissipative dynamics of one-dimensional fermions, described in terms of the sine-Gordon model in its free massive boson or semi-classical limit, while keeping track of forward scattering processes. The system is prepared in the gapped ground state, and then coupled to environment through local currents within the Lindblad formalism. The heating dynamics of the system is followed using bosonization. The single particle density matrix exhibits correlations between the left and right moving particles. While the density matrix of right movers and left movers is translationally invariant, the left-right sector is not, corresponding to a translational symmetry breaking charge density wave state. Asymptotically, the single particle density matrix decays exponentially with exponent proportional to -\gamma t|x|\Delta^2−γt|x|Δ2 where \gammaγ and \DeltaΔ are the dissipative coupling and the gap, respectively. The charge density wave order parameter decays exponentially in time with an interaction independent decay rate. The second R'enyi entropy grows linearly with time and is essentially insensitive to the presence of the gap.

Stripe order, impurities, and symmetry breaking in a diluted frustrated magnet

We investigate the behavior of the frustrated $J_1$-$J_2$ Ising model on a square lattice under the influence of random dilution and spatial anisotropies. Spinless impurities generate a random-field type disorder for the spin-density wave (stripe) order parameter. These random fields destroy the long-range stripe order in the case of spatially isotropic interactions. Combining symmetry arguments, percolation theory and large-scale Monte Carlo simulations, we demonstrate that arbitrarily weak spatial interaction anisotropies restore the stripe phase. More specifically, the transition temperature $T_c$ into the stripe phase depends on the interaction anisotropy $\Delta J$ via $T_c \sim 1/|\ln (\Delta J)|$ for small $\Delta J$. This logarithmic dependence implies that very weak anisotropies are sufficient to restore the transition temperature to values comparable to that of the undiluted system. We analyze the critical behavior of the emerging transition and find it to belong to the disordered two-dimensional Ising universality class, which features the clean Ising critical exponents and universal logarithmic corrections. We also discuss the generality of our results and their consequences for experiments.

Observations of the effect of strong Pauli paramagnetism on the vortex lattice in superconducting CeCu2Si2

We present the results of a study of the vortex lattice in the heavy fermion superconductor ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$, using small-angle neutron scattering (SANS). In this material at temperatures well below ${T}_{\mathrm{c}}\ensuremath{\sim}0.6$ K, the value of the upper critical field ${B}_{\mathrm{c}2}\ensuremath{\sim}2.2$ T is strongly limited by the Pauli paramagnetism of the heavy fermions. In this temperature region, our SANS data show an increase in the magnetization of the flux line cores with field, followed by a rapid fall near ${B}_{\mathrm{c}2}$. This behavior is the effect of Pauli paramagnetism and we present a theory-based model, which can be used to describe this effect in a range of materials. The pairing in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ appears to arise from the effect of magnetic fluctuations, but the evidence for a $d$-wave order parameter is rather weak. We find that the vortex lattice structure in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ is close to regular hexagonal. There are no phase transitions to square or rhombic structures; such transitions are expected for $d$-wave superconductors and observed in ${\mathrm{CeCoIn}}_{5}$; however, the temperature dependence of the SANS intensity indicates that both large and small gap values are present, most likely due to multiband $s$-wave superconductivity, rather than a nodal gap structure.

Unconventional superconductivity mediated solely by isotropic electron-phonon interaction

Unconventional superconductivity is commonly linked to electronic pairing mechanisms, since it is believed that the conventional electron-phonon interaction (EPI) cannot cause sign-changing superconducting gap symmetries. Here, we show that this common understanding needs to be revised when one considers a more elaborate theory of electron-phonon superconductivity beyond standard approximations. We self-consistently solve the full-bandwidth, anisotropic Eliashberg equations including vertex corrections beyond Migdal's approximation assuming the usual isotropic EPI for cuprate, Fe-based, and heavy-fermion superconductors with nested Fermi surfaces. In the case of the high-${T}_{c}$ cuprates we find a $d$-wave order parameter, as well as a nematic state upon increased doping. For Fe-based superconductors, we obtain ${s}_{\ifmmode\pm\else\textpm\fi{}}$ gap symmetry, while for heavy-fermion ${\mathrm{CeCoIn}}_{5}$ we find unconventional $d$-wave pairing. These results provide a proof of concept that EPI cannot be excluded as a mediator of unconventional and of high-${T}_{c}$ superconductivity.

Magnetic Properties and Superconductivity in the Two-Dimensional Repulsive Hubbard Model

A new method for estimating the parameter ensuring the fulfillment of the Mermin-Wagner theorem in the strong coupling diagram technique (SCDT) for the two-dimensional Hubbard model is suggested. With the precise parameter value, calculated magnetic quantities are in good agreement with the results of numeric and optical-lattice experiments. Obtained spin and charge vertices are used for investigating superconductivity in the $t$-$U$ and $t$-$t'$-$t"$-$U$ Hubbard models in the regime of strong correlations. We found no superconducting transition in the $t$-$U$ model. In the $t$-$t'$-$t"$-$U$ model, the transition occurs for the singlet $d_{x^2-y^2}$ pairing at $T_c\approx0.016t$. The difference between the two models is in the renormalized hopping describing electron motion in SCDT. In the $t$-$U$ model, it vanishes for momenta $(\pi,0)$, $(0,\pi)$ of extrema of the $d$-wave order parameter, while the hopping is finite in the $t$-$t'$-$t"$-$U$ model. In the one-band model, there are optimal values of $t'$ and $t"$ ensuring the highest $T_c$.

Microscopic Ginzburg-Landau theory and singlet ordering inSr2RuO4

The long-standing quest to determine the superconducting order of Sr$_2$RuO$_4$ (SRO) has received renewed attention after recent nuclear magnetic resonance (NMR) Knight shift experiments have cast doubt on the possibility of spin-triplet pairing in the superconducting state. As a putative solution, encompassing a body of experiments conducted over the years, a $d+ig$-wave order parameter caused by an accidental near-degeneracy has been suggested [S. A. Kivelson et al., npj Quantum Materials $\bf{5}$, 43 (2020)]. Here we develop a general Ginzburg--Landau theory for multiband superconductors. We apply the theory to SRO and predict the relative size of the order parameter components. The heat capacity jump expected at the onset of the second order parameter component is found to be above the current threshold deduced by the experimental absence of a second jump. Our results tightly restrict theories of $d+ig$ order, and other candidates caused by a near-degeneracy, in SRO. We discuss possible solutions to the problem.

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.

Spin texture in a bilayer high-temperature cuprate superconductor

We investigate the possibility of spin texture in the bilayer cuprate superconductor $\rm Bi_2Sr_2CaCu_2O_{8+\delta}$ using cluster dynamical mean field theory (CDMFT). The one-band Hubbard model with a small interlayer hopping and a Rashba spin-orbit coupling is used to describe the material. The $d$-wave order parameter is not much affected by the presence of the Rashba coupling, but a small triplet component appears. We find a spin texture circulating in the same direction around $\mathbf{k}=(0,0)$ and $\mathbf{k}=(\pi,\pi)$ and stable against the superconducting phase. The amplitude of the spin structure, however, is strongly affected by the pseudogap phenomenon, more so than the spectral function itself.

Non-equilibrium [formula omitted]-wave pair density wave order parameter in superconducting cuprates

We investigate the nonequilibrium pair density wave order parameter in a simple microscopic model with the ground state d-wave spatially uniform superconductivity. After pushing the system out of equilibrium by a short-time induced order parameter, the system can exhibit robust free nondecaying oscillations of nonuniform superconducting order. In the weak nonequilibrium regime, the frequency of these oscillations is about 2Δ/ℏ and the amplitude can be explained qualitatively by features of equilibrium free energy. In case when the system was taken far from equilibrium, it transits to a non-linear regime where even metastable coexistence of d-wave superconducting and pair density wave gaps are possible.

Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4

Sr$_2$RuO$_4$ has stood as the leading candidate for a spin-triplet superconductor for 26 years. Recent NMR experiments have cast doubt on this candidacy, however, and it is difficult to find a theory of superconductivity that is consistent with all experiments. What is needed are symmetry-based experiments that can rule out broad classes of possible superconducting order parameters. Here we use resonant ultrasound spectroscopy to measure the entire symmetry-resolved elastic tensor of Sr$_2$RuO$_4$ through the superconducting transition. We observe a thermodynamic discontinuity in the shear elastic modulus $c_{66}$, requiring that the superconducting order parameter is two-component. A two-component $p$-wave order parameter, such as $p_x+i p_y$, naturally satisfies this requirement. As this order parameter appears to be precluded by recent NMR experiments, we suggest that two other two-component order parameters, namely $\left\{d_{xz},d_{yz}\right\}$ or $\left\{d_{x^2-y^2},g_{xy(x^2-y^2)}\right\}$, are now the prime candidates for the order parameter of Sr$_2$RuO$_4$.

Theory of Higgs modes in d -wave superconductors

By applying a microscopic gauge-invariant kinetic theory in $d$-wave superconductors, we analytically derive the energy spectra of the breathing Higgs mode and, in particular, rotating Higgs mode that is unique for the $d$-wave order parameter. Analytical investigation on their dynamic properties is also revealed. We show that the breathing Higgs mode is optically visible in the second-order regime. Whereas the rotating Higgs mode is optically inactive, we show that this mode can be detected in the pseudogap phase by magnetic resonance experiment. It is interesting to find that with a longitudinal temperature gradient, the charge-neutral rotating Higgs mode generates a thermal Hall current by magnetic field in the pseudogap phase, providing a unique scheme for its detection.

Superconducting properties of MoTe2 from ab initio anisotropic Migdal-Eliashberg theory

Molybdenum ditelluride (${\mathrm{MoTe}}_{2}$) is attracting considerable interest since it is the archetypal type-II Weyl semimetal and a candidate for topological superconductivity. We investigate the superconducting phase diagram of two ${\mathrm{MoTe}}_{2}$ polymorphs using the ab initio anisotropic Migdal-Eliashberg theory, and we show that the superconducting dome originates from the synergistic contribution of the density of states at the Fermi level and the transverse acoustic Te modes in the $1{T}^{\ensuremath{'}}$ phase. We find that the electron and hole pockets carry trivial $s$-wave order parameters of slightly different magnitude, reminiscent of a two-gap structure as suggested by recent experiments. We suggest that a possible route for enhancing the superconducting critical temperature, and realizing ${s}_{+\ensuremath{-}}$ pairing, in the ${T}_{d}$ phase is to exploit its nontrivial band topology via electron doping.

Two-band superconductivity with unconventional pairing symmetry in HfV2Ga4

In this letter, we have examined the superconducting ground state of the HfV$_2$Ga$_4$ compound using resistivity, magnetization, zero-field (ZF) and transverse-field (TF) muon-spin relaxation and rotation ($\mu$SR) measurements. Resistivity and magnetization unveil the onset of bulk superconductivity with $T_{\bf c}\sim$ 3.9~K, while TF-$\mu$SR measurements show that the temperature dependence of the superfluid density is well described by a nodal two-gap $s$+$d$-wave order parameter model. In addition, ZF muon relaxation rate increases with decreasing temperature below 4.6 K, indicating the presence of weak spin fluctuations. These observations suggest an unconventional multiband nature of the superconductivity possibly arising from the distinct $d$-bands of V and Hf ions with spin fluctuations playing an important role. To better understand these findings, we carry out first-principles electronic-structure calculations, further highlighting that the Fermi surface consists of multiple disconnected sheets with very different orbital weights and spin-orbit coupling, bridging the way for a nodal multiband superconductivity scenario. In this vein, therefore, HfV$_2$Ga$_4$-family stands out as an open avenue to novel unexplored unconventional superconducting compounds, such as ScV$_2$Ga$_4$ and ZrV$_2$Ga$_4$, and other many rare earths based materials.

Gapless Chiral Superconducting (d + id)-Wave Phase in Strongly Correlated Layered Material with a Triangular Lattice

It is shown that interlayer electron tunneling in the quasi-two-dimensional ensemble of Hubbard fermions leads to the realization of the gapless superconducting phase with the chiral (d + id)-wave order parameter symmetry, not for a single value of sodium ion concentration, but in a wide range of concentrations. Precisely this situation corresponds to experimental data on the layered sodium cobaltite intercalated by water (NaxCoO2 ⋅ yH2O). Intra-atomic electron repulsion that determines the strong electron correlation regime leads to the representation of Hubbard fermions, the interaction of which ensures Cooper instability. Intersite intralayer interactions between fermions considerably affect the positions of nodal points of the chiral order parameter and change the critical concentration at which a topological transition occurs in the 2D system of Hubbard fermions.

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.