Rotational Symmetry Breaking Research Articles

Unconventional density waves and superconductivities in Fe-based superconductors and other strongly correlated electron systems

To seek high-$T_c$ pairing mechanism, many scientists have focused on the mysterious spontaneous rotational symmetry breaking above Tc, such as nematic order at $q=0$ and smectic order at $q\ne0$. Such exotic correlation-driven symmetry breaking in metals has become a central issue in condensed matter physics. We demonstrate the emergence of the nematic and smectic orders due to orbital polarization ($n_{xz}\ne n_{yz}$) and the symmetry breaking in the correlated intersite hopping (= bond order $\delta t_{i,j}$) in Fe-based and cuprate superconductors. In addition, we discuss exotic spontaneous loop current orders driven by the pure imaginary $\delta t_{i,j}$. These interesting ``unconventional density-waves'' originate from the quantum interference between different spin fluctuations that is described by the vertex correction (VC) in the field theory. In the next stage, we discuss electron-correlation driven superconductivity due to the fluctuations of unconventional density-waves. For this purpose, we suggest the beyond-Migdal-Eliashberg gap equation by including the VCs into the equation. In Fe-based superconductors, high-$T_c$ $s$-wave superconductivity can be mediated by nematic and smectic fluctuations because the pairing interaction is magnified by the VCs. We also discuss the multipolar fluctuation pairing mechanism in heavy fermion systems, owing to the cooperation between the strong spin-orbit interaction and the strong electron correlation. To summarize, we suggest that the quantum interference mechanism described by the VCs is the key ingredients to explain not only various unconventional density-waves, but also exotic superconducting states in many strongly correlated metals. We finally discuss some interesting future issues with respect to the quantum interference mechanism.

Anomaly of longitudinal spin susceptibility at a superconducting instability inside a magnetic phase

We study the longitudinal spin susceptibility inside a magnetically ordered phase, which exhibits a superconducting instability leading to a coexistence of the two ordered phases. Inside the magnetic phase, the superconducting gap acquires a linear term in a magnetic field applied along the direction of the magnetic moment. We find that such a linear term generates a jump of the longitudinal spin susceptibility when the superconducting instability occurs via a continuous phase transition. This anomaly at the superconducting instability is a thermodynamic signature of the microscopic coexistence of superconductivity and magnetism and can be a general feature associated with the breaking of spin rotational symmetry inside the magnetic phase.

Nematic transition and nanoscale suppression of superconductivity in Fe(Te,Se)

The interplay of different electronic phases underlies the physics of unconventional superconductors. One of the most intriguing examples is a high-temperature superconductor, FeTe1 – xSex (refs. 1–11). This superconductor undergoes both a topological transition3,4, linked to the electronic band inversion, and an electronic nematic phase transition, associated with rotation symmetry breaking, around the same Se composition where the superconducting transition temperature peaks12,13. In this regime, nematic fluctuations and symmetry-breaking strain could be important, but this is yet to be fully explored. Using spectroscopic-imaging scanning tunnelling microscopy, we study the electronic nematic transition in FeTe1 – xSex as a function of composition. Near the critical Se composition, we find electronic nematicity in nanoscale regions. The superconducting coherence peaks are suppressed in areas where static nematic order is the strongest. By analysing atomic displacement in scanning tunnelling microscopy topographs, we find that small anisotropic strain can give rise to these strongly nematic localized regions. Our experiments reveal a tendency of FeTe1 – xSex, near x ≈ 0.45, to form puddles hosting static nematic order, suggestive of nematic fluctuations pinned by structural inhomogeneity, and demonstrate the effect of anisotropic strain on superconductivity in this regime. The interplay of superconductivity and nematicity of electrons remains unclear in a wide range of materials. Now, more evidence emerges that nematic fluctuations can be pinned into a static phase by disorder, which hinders the superconductivity.

Thermodynamic electric quadrupole moments of nematic phases from first-principles calculations

The electronic nematic phase emerging with spontaneous rotation symmetry breaking is a central issue of modern condensed matter physics. In particular, various nematic phases in iron-based superconductors and high-$T_{\rm c}$ cuprate superconductors are extensively studied recently. Electric quadrupole moments (EQMs) are one of the order parameters characterizing these nematic phases in a unified way, and elucidating EQMs is a key to understanding these nematic phases. However, the quantum-mechanical formulation of the EQMs in crystals is a nontrivial issue because the position operators are non-periodic and unbound. Recently, the EQMs have been formulated by local thermodynamics, and such {\it thermodynamic EQMs} may be used to characterize the fourfold rotation symmetry breaking in materials. In this paper, we calculate the thermodynamic EQMs in iron-based superconductors LaFeAsO and FeSe as well as a cuprate superconductor La$_2$CuO$_4$ by a first-principles calculation. We show that owing to the orbital degeneracy the EQMs in iron-based superconductors are mainly determined by the geometric properties of wave functions. This result is in sharp contrast to the cuprate superconductor, in which the EQMs are dominated by distortion of the Fermi surface.

State densities of heavy nuclei in the static-path plus random-phase approximation

Nuclear state densities are important inputs to statistical models of compound-nucleus reactions. State densities are often calculated with self-consistent mean-field approximations that do not include important correlations and have to be augmented with empirical collective enhancement factors. Here, we benchmark the static-path plus random-phase approximation (SPA+RPA) to the state density in a chain of samarium isotopes $^{148-155}$Sm against exact results (up to statistical errors) obtained with the shell model Monte Carlo (SMMC) method. The SPA+RPA method incorporates all static fluctuations beyond the mean field together with small-amplitude quantal fluctuations around each static fluctuation. Using a pairing plus quadrupole interaction, we show that the SPA+RPA state densities agree well with the exact SMMC densities for both the even- and odd-mass isotopes. For the even-mass isotopes, we also compare our results with mean-field state densities calculated with the finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation. We find that the SPA+RPA repairs the deficiencies of the mean-field approximation associated with broken rotational symmetry in deformed nuclei and the violation of particle-number conservation in the pairing condensate. In particular, in deformed nuclei the SPA+RPA reproduces the rotational enhancement of the state density relative to the mean-field state density.

Electric-field-tunable electronic nematic order in twisted double-bilayer graphene

Graphene-based moiré systems have attracted considerable interest in recent years as they display a remarkable variety of correlated phenomena. Besides insulating and superconducting phases in the vicinity of integer fillings of the moiré unit cell, there is growing evidence for electronic nematic order both in twisted bilayer graphene and twisted double-bilayer graphene (tDBG), as signaled by the spontaneous breaking of the threefold rotational symmetry of the moiré superlattices. Here, we combine symmetry-based analysis with a microscopic continuum model to investigate the structure of the nematic phase of tDBG and its experimental manifestations. First, we perform a detailed comparison between the theoretically calculated local density of states and recent scanning tunneling microscopy data (arXiv:2009.11645) to resolve the internal structure of the nematic order parameter in terms of the layer, sublattice, spin, and valley degrees of freedom. We find strong evidence that the dominant contribution to the nematic order parameter comes from states at the moiré scale rather than at the microscopic scale of the individual graphene layers, which demonstrates the key role played by the moiré degrees of freedom and confirms the correlated nature of the nematic phase in tDBG. Secondly, our analysis reveals an unprecedented tunability of the orientation of the nematic director in tDBG by an externally applied electric field, allowing the director to rotate away from high-symmetry crystalline directions. We compute the expected fingerprints of this rotation in both STM and transport experiments, providing feasible ways to probe it. Rooted in the strong sensitivity of the flat bands of tDBG to the displacement field, this effect opens an interesting route to the electrostatic control of electronic nematicity in moiré systems.

Interacting spin- 32 fermions in a Luttinger semimetal: Competing phases and their selection in the global phase diagram

We compute the effects of electronic interactions on gapless spin-3/2 excitations that in a noninteracting system emerge at a bi-quadratic touching of Kramers degenerate valence and conduction bands, known as Luttinger semimetal. This model can describe the low-energy physics of HgTe, gray-Sn, 227 pyrochlore iridates and half-Heuslers. For the sake of concreteness we only consider the short-range components of the Coulomb interaction. By combining mean-field analysis with a renormalization group (RG) calculation, we construct multiple cuts of the global phase diagram of interacting spin-3/2 fermions at zero and finite temperature and chemical doping. Such phase diagrams display a rich confluence of competing orders, among which rotational symmetry breaking nematic insulators and time reversal symmetry breaking magnetic orders are the prominent excitonic phases. We also show that even repulsive interactions can be conducive for both s-wave and d-wave pairings. The reconstructed band structure inside the ordered phases allows us to organize them according to the energy (entropy) gain in the following (reverse) order: s-wave pairing, nematic phases, magnetic orders and d-wave pairings, at zero chemical doping. But, the paired states are always energetically superior over the excitonic ones for finite doping. The phase diagrams obtained from the RG analysis show that an ordered phase with higher energy (entropy) gain is realized at low (high) temperature. In addition, we establish a "selection rule" between the interaction channels and the resulting ordered phases, suggesting that repulsive interactions in the magnetic (nematic) channels are conducive for the nucleation of d-wave (s-wave) pairing. The proposed methodology can shed light on the global phase diagram of other strongly interacting multi-band systems, such as doped Dirac semimetal, topological insulators and the like.

Twisted and nematic

2D Materials Electrons in quantum materials can break rotational symmetry even when the underlying crystal lattice does not. This phenomenon, called nematicity, has been observed in many unconventional superconductors. Cao et al. found that magic-angle twisted bilayer graphene, in which superconductivity was recently discovered, also exhibits nematicity. The breaking of rotational symmetry was observed through transport measurements, which exhibited characteristic anisotropy. Science , this issue p. [264][1] [1]: /lookup/doi/10.1126/science.abc2836

Nematicity and competing orders in superconducting magic-angle graphene.

Strongly interacting electrons in solid-state systems often display multiple broken symmetries in the ground state. The interplay between different order parameters can give rise to a rich phase diagram. We report on the identification of intertwined phases with broken rotational symmetry in magic-angle twisted bilayer graphene (TBG). Using transverse resistance measurements, we find a strongly anisotropic phase located in a "wedge" above the underdoped region of the superconducting dome. Upon its crossing with the superconducting dome, a reduction of the critical temperature is observed. Furthermore, the superconducting state exhibits an anisotropic response to a direction-dependent in-plane magnetic field, revealing nematic ordering across the entire superconducting dome. These results indicate that nematic fluctuations might play an important role in the low-temperature phases of magic-angle TBG.

Topological phases in N -layer ABC graphene/boron nitride moiré superlattices

Rhombohedral $N = 3$ trilayer graphene on hexagonal boron nitride (TLG/BN) hosts gate-tunable, valley-contrasting, nearly flat topological bands that can trigger spontaneous quantum Hall phases under appropriate conditions of the valley and spin polarization. Recent experiments have shown signatures of C = 2 valley Chern bands at 1/4 hole filling, in contrast to the predicted value of C = 3. We discuss the low-energy model for rhombohedral N-layer graphene (N = 1, 2, 3) aligned with hexagonal boron nitride (hBN) subject to off-diagonal moire vector potential terms that can alter the valley Chern numbers. Our analysis suggests that topological phase transitions of the flat bands can be triggered by pseudomagnetic vector field potentials associated to moire strain patterns, and that a nematic order with broken rotational symmetry can lead to valley Chern numbers that are in agreement with recent Hall conductivity observations.

Tendencies of enhanced electronic nematicity in the Hubbard model and a comparison with Raman scattering on high-temperature superconductors

The pseudogap regime of the cuprate high-temperature superconductors is characterized by a variety of competing orders, the nature of which are still widely debated. Recent experiments have provided evidence for electron nematic order, in which the electron fluid breaks rotational symmetry while preserving translational invariance. Raman spectroscopy, with its ability to symmetry resolve low energy excitations, is a unique tool that can be used to assess nematic fluctuations and nematic ordering tendencies. Here, we compare results from determinant quantum Monte Carlo simulations of the Hubbard model to experimental results from Raman spectroscopy in $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$, which show a prominent increase in the $B_{1g}$ response around 10% hole doping as the temperature decreases, indicative of a rise in nematic fluctuations at low energy. Our results support a picture of nematic fluctuations with $B_{1g}$ symmetry occurring in underdoped cuprates, which may arise from melted stripes at elevated temperatures.

Moving away from singly-magic nuclei with Gorkov Green’s function theory

Ab initio calculations of bulk nuclear properties (ground-state energies, root mean square charge radii and charge density distributions) are presented for seven complete isotopic chains around calcium, from argon to chromium. Calculations are performed within the Gorkov self-consistent Green's function approach at second order and make use of two state-of-the-art two- plus three-nucleon Hamiltonians, $NN$+$3N\text{(lnl)}$ and NNLO$_{\text{sat}}$. An overall good agreement with available experimental data is found, in particular for differential energies (charge radii) when the former (latter) interaction is employed. Remarkably, neutron magic numbers $N=28,32,34$ emerge and evolve following experimental trends. In contrast, pairing gaps are systematically underestimated. General features of the isotopic dependence of charge radii are also reproduced, as well as charge density distributions. A deterioration of the theoretical description is observed for certain nuclei and ascribed to the inefficient account of (static) quadrupole correlation in the present many-body truncation scheme. In order to resolve these limitations, we advocate the extension of the formalism towards incorporating breaking of rotational symmetry or, alternatively, performing a stochastic sampling of the self-energy.

Current switching of valley polarization in twisted bilayer graphene

Twisted bilayer graphene (TBG) aligned with hexagonal boron nitride (h-BN) substrate can exhibit an anomalous Hall effect at 3/4 filling due to the spontaneous valley polarization in valley resolved moir\'e bands with opposite Chern number [Science 367, 900 (2020); Science 365, 605 (2019)]. It was observed that a small DC current is able to switch the valley polarization and reverse the sign of the Hall conductance [Science 367, 900 (2020); Science 365, 605 (2019)]. Here, we discuss the mechanism of the current switching of valley polarization near the transition temperature, where bulk dissipative transport dominates. We show that for a sample with rotational symmetry breaking, a DC current may generate an electron density difference between the two valleys (valley density difference). The current induced valley density difference in turn induces a first-order transition in the valley polarization. We emphasize that the intervalley scattering plays a central role since it is the channel for exchanging electrons between the two valleys. We further estimate the valley density difference in the TBG/h-BN system with a microscopic model and find a significant enhancement of the effect in the magic angle regime.

Application of the uniaxial pressure technique in studying superconductors

<p indent=0mm>Because of the rapid development of the uniaxial pressure device in recent years, the uniaxial pressure or strain technique has been increasingly applied to study unconventional superconductors. The physical properties of a sample will change with the uniaxial strain along a particular lattice axis, providing us with important information regarding superconductivity and other characteristics. When compared with hydrostatic pressure measurement, changes in physical properties with the uniaxial pressure are mainly associated with symmetry, making this methodology particularly suitable for studying the symmetry-related properties of a system such as the electronic nematic phase with rotational symmetry breaking. Furthermore, the uniaxial pressure technique can be used to study the superconducting properties. In this review, we will briefly introduce some typical uniaxial pressure and uniaxial strain devices and techniques and their applications on iron-based superconductors as well as other superconductors. The uniaxial pressure technique exhibits some unique advantages with respect to the study of electronic nematicity, superconductivity, and some other competing orders. With further development, the uniaxial pressure technique is expected to play increasingly important roles in the study of superconductors and other strongly correlated systems.

Modified snaking in plane Couette flow with wall-normal suction

Abstract

Self-bound droplet clusters in laser-driven Bose-Einstein condensates

We investigate a two-dimensional Bose-Einstein condensate that is optically driven via a retro-reflecting mirror, forming a single optical feedback loop. This induces a peculiar type of long-range atomic interaction with highly oscillatory behavior, and we show here how the sign of the underlying interaction potential can be controlled by additional optical elements and external fields. This additional tunability enriches the behavior of the system substantially, and gives rise to a surprising range of new ground states of the condensate. In particular, we find the emergence of self-bound crystals of quantum droplets with various lattice structures, from simple and familiar triangular arrays to complex superlattice structures and crystals with entirely broken rotational symmetry. This includes mesoscopic clusters composed of small numbers of quantum droplets as well as extended crystalline structures. Importantly, such ordered states are entirely self-bound and stable without any external in-plane confinement, having no counterpart to other quantum-gas settings with long-range atomic interactions.

Investigation of Superconducting Properties and Possible Nematic Superconductivity in Self‐Doped BiCh2‐Based Superconductor CeOBiS1.7Se0.3

The superconducting properties and possible nematic superconductivity of the self‐doped BiCh2‐based (Ch: S, Se) superconductor CeOBiS1.7Se0.3 are investigated through measurements of the in‐plane anisotropy of magnetoresistance (MR). Single crystals of CeOBiS1.7Se0.3 are grown using a flux method. The single‐crystal structural analysis reveals that the crystal structure at room temperature is tetragonal (P4/nmm). Bulk superconductivity with a transition temperature of 3.3 K is observed through electrical resistivity and magnetization measurements. Investigation of the anisotropy of the upper critical field suggests relatively low anisotropy in the crystal compared to that of other BiCh2‐based superconductors. In the superconducting states of CeOBiS1.7Se0.3, a twofold symmetric in‐plane anisotropy of MR is observed, which indicates the in‐plane rotational symmetry breaking in the superconducting states of CeOBiS1.7Se0.3.

Dynamic electron correlations with charge order wavelength along all directions in the copper oxide plane

In strongly correlated systems the strength of Coulomb interactions between electrons, relative to their kinetic energy, plays a central role in determining their emergent quantum mechanical phases. We perform resonant x-ray scattering on Bi2Sr2CaCu2O8+δ, a prototypical cuprate superconductor, to probe electronic correlations within the CuO2 plane. We discover a dynamic quasi-circular pattern in the x-y scattering plane with a radius that matches the wave vector magnitude of the well-known static charge order. Along with doping- and temperature-dependent measurements, our experiments reveal a picture of charge order competing with superconductivity where short-range domains along x and y can dynamically rotate into any other in-plane direction. This quasi-circular spectrum, a hallmark of Brazovskii-type fluctuations, has immediate consequences to our understanding of rotational and translational symmetry breaking in the cuprates. We discuss how the combination of short- and long-range Coulomb interactions results in an effective non-monotonic potential that may determine the quasi-circular pattern.

Rotational symmetry breaking and partial Majorana corner states in a heterostructure based on high- Tc superconductors

Applying a microscopic model, we study theoretically the quasiparticle excitation of a twodimensional topological insulator (TI) being in proximity to a high-Tc superconductor. In the momentum space, the proximity induced pairing term in the TI layer includes both the singlet channel and triplet channel, leading to the C4 rotational symmetry breaking of the energy bands and the quasiparticle spectra. For a cylinder geometry, the zero energy edge states may appear but they are localized at the upper boundary. For the finite-size system with open boundaries, the zero energy states at the upper boundary disappear and the Majorana bound states emerge at the two lower corners. All of the results can be understood well through exploring the pairing order parameter and the anomalous Green's function.

Emergent chiral symmetry in a three-dimensional interacting Dirac liquid

We compute the effects of strong Hubbardlike local electronic interactions on three-dimensional four-component massless Dirac fermions, which in a noninteracting system possess a microscopic global U(1) ⊗ SU(2) chiral symmetry. A concrete lattice realization of such chiral Dirac excitations is presented, and the role of electron-electron interactions is studied by performing a field theoretic renormalization group (RG) analysis, controlled by a small parameter ϵ with ϵ = d−1, about the lower-critical one spatial dimension. Besides the noninteracting Gaussian fixed point, the system supports four quantum critical and four bicritical points at nonvanishing interaction couplings ∼ ϵ. Even though the chiral symmetry is absent in the interacting model, it gets restored (either partially or fully) at various RG fixed points as emergent phenomena. A representative cut of the global phase diagram displays a confluence of scalar and pseudoscalar excitonic and superconducting (such as the s-wave and p-wave) mass ordered phases, manifesting restoration of (a) chiral U(1) symmetry between two excitonic masses for repulsive interactions and (b) pseudospin SU(2) symmetry between scalar or pseudoscalar excitonic and superconducting masses for attractive interactions. Finally, we perturbatively study the effects of weak rotational symmetry breaking on the stability of various RG fixed points.